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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 183
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 184
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 185
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 186
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 187
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 188
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 189
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 190
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 191
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 192
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 193
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 194
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 195
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 196
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 197
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 198
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 199
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 200
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 201
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 202
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 203
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 204
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 205
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 206
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 207
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 208
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 209
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 210
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 211
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 212
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 213
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 214
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 215
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 216
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 217
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 218
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 219
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 220
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 221
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 222
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 223
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 224
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 225
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 226
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 227
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 228
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 229
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 230
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 231
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 232
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 233
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 234
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 235
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 236
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 237
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 238
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 239
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 240
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 241
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 242
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 243
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 244
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 245
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 246
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 247
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 248
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 249
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 250
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 251
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 252
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 253
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 254
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 255
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 256
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 257
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 258
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 259
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 260
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 261
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 262
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 263
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 264
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 265
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 266
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 267
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 268
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Page 269
Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Suggested Citation:"THE STUDY ON DEATH RATES." National Research Council. 1969. National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others]. Washington, DC: The National Academies Press. doi: 10.17226/19006.
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Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Abstract of Chapter IV-I To assess the over-all comparative safety of the several anesthetics and the magnitude of the effects of hepatic necrosis requires a study of surgical death rates. This study can provide information about the magnitude of the effects of important variables, such as sex, age, physical status, and opera- tion, as well as information about the anesthetics. Having some firm informa- tion reduces the range of hypotheses available for explaining surgical deaths and suggests the sizes and types of studies still needed. The observed death rates supply baselines for use by various medical groups. The study has produced new methods of analysis dealing with the common problem of analyzing multidimensional contingency tables. In a study of rare events, such as surgical deaths or, even rarer, mas- sive hepatic necrosis, it is the number of rare events observed, rather than the number of opportunities for them to occur, that determines the stability of the averages. Consequently, very large numbers of cases are often re- quired for analyzing rare events. As explained in the second section of this chapter, the process of carrying out unusually large studies may itself create biases and inaccuracies that can invalidate the differences observed as rep- resentative of those that are to be detected. And therefore it may frequently happen that small quantitative differences cannot be empirically established. CHAPTER IV-I. INTRODUCTION TO THE STUDY OF DEATH RATES* Byron W. Brown, Jr. University of Minnesota School of Public Health Minneapolis, Minnesota Frederick Mosteller Harvard University Cambridge, Massachusetts Lincoln E. Moses Stanford University Stanford, California W. Morven Gentleman Bell Telephone Laboratories Murray Hill, New Jersey WHAT CAN BE LEARNED? Reports of a few postoperative deaths from hepatic necrosis in patients anesthetized with halothane, together with uneasiness about halog- enated anesthetics, led to the National Halothane Study. This portion of the Study primarily com- pares the relative frequencies of postoperative death from all causes after the administration of the five major anesthetics (or combinations of anesthetics). Using death rate as the index em- phasizes the over-all danger of the surgical pro- cedure, including the anesthetic, rather than just the consequences to the liver. Early in the planning phases, calculations of orders of magnitude clearly indicated that, even if halothane did lead to a higher death rate from hepatic necrosis than other anesthetics did, the fraction of deaths from this source would be numerically very small, compared with the over-all operative mortality of around 2 percent. Of this 2 percent total mortality, the deaths at- tributable to hepatic necrosis could account for but a small part. Indeed, if halothane were only slightly superior to other agents in over-all operative mortality, then that slight superiority could far outweigh any plausible excess of deaths due to massive hepatic necrosis. These remarks •The first section of this chapter was written by B.W. Brown, Jr., and F. Mosteller; the second section was written by L. E. Moses, F. Mosteller, and W.M. Gentleman. 183

are valid whether or not experts can identify those deaths due to massive hepatic necrosis or can establish a causal connection between the use of halothane and those deaths. The issue is mag- nitude, rather than causation. Thus, a prelimi- nary question was: "How do the postoperative death rates after the several anesthetics com- pare, first, for all patients combined, and, sec- ond, for specific classes of patients, such as those defined by type of operation, physical status (anesthetic risk), and age?" The issue discussed briefly above, deaths from massive hepatic necrosis vs. deaths from all sources, exemplifies a common conflict of interest present in medical investigations. By identifying causes of deaths and subsequently adjusting the conditions leading to these causes, we often make scientific and medical progress. Nevertheless, appraising the consequences of a small portion of cases of a complicated medical procedure, in addition to being difficult and tenuous itself, always invites the pitfall of over- looking the total effect of the procedure on the patient population. At a ridiculous extreme, we can consider reducing the anesthetic death rate to zero by eliminating the use of anesthetics. A more realistic example would be the abandon- ment of penicillin because we know that a small proportion of people are allergic to it and may die from it. One important measure of the total effect of a surgical procedure is the total death rate, not just anesthetic death rate or death rate from some specific disorder, such as massive hepatic necrosis. Naturally, a question of great medical im- portance, although not so primary from a public- health point of view, is whether hepatic necrosis is especially associated with halothane or with any other anesthetic. If so, what is the mechanism and how can the hepatic damage be avoided? Fortunately, the National Halothane Study has been able to review both the problem of fatal hepatic necrosis and that of death rates. What has not been investigated by the Study, except through a survey of the literature, is the behavior of the liver and other organs in living patients before surgery and after surgery in which dif- ferent anesthetics were used. Chapter II-1 discusses the pro's and con's of studies of past data in general. Below is a de- scription of what may be learned from a study of death rates in particular. The Study Shows the Postoperative Death Rates That Have Been Associated With Anesthetics In the Recent Past Historical comparison of anesthetics If, as the establishment of a National Halo- thane Study might suggest, death rates among some special classes of patients have been much higher after the administration of one anesthetic, such as halothane, than after the administration of other agents, we might hope to discover that effect in the data. In the absence of such effects, we can be encouraged about the over-all safety of the anesthetic. Effects of other variables on death rate By examining the variation in death rates associated with other variables (e.g., hospital, type of operation, and length of operation), we can put the variation in death rates associated with anesthetics into perspective. Generally speaking, research expenditures directed toward improvement of practices in any field are more rewardingly directed toward reduction of large effects than toward small ones (unless some specific method of dealing with a small effect is in sight). Choice of hypotheses for investigation Above all, in any investigation of this sort, we must not underestimate the value of estab- lishing a few facts with reasonable firmness. Before examining a large, systematically col- lected body of data, an investigator with a fertile mind can think of thousands of hypotheses of varying credibility. Some possible hypotheses for this Study (all false) are: (1) differences in death rates for anesthetics may be accounted for by one or two hospitals; (2) massive hepatic ne- crosis is entirely associated with extracorporeal circulation in cardiac operations; and (3) hospitals differ in surgical death rates only because they perform operations of different seriousness. A large body of data can often help the investigator by disposing of some hundreds of these hypoth- eses, thus eliminating the necessity for whole lines of investigation. Guidance toward study size and type By looking at the sizes and kinds of effects observed in the past, we can get an idea of the size and kind of new study required for the de- tection of effects. What is especially valuable is that some inquiries may be shown to be unfeasi- ble. Let us not overemphasize the adequacy of a historical investigation, as compared with a randomized prospective study. Retrospective studies, such as the present one, inevitably have potential biases that cannot be rooted out, satis- factorily assessed, or even identified. To summarize, we can find out the death rates that have occurred in the past in the hos- pitals in the sample, and hope that the trends are similar to those of the national population of hos- pitals and patients. Inevitably, the retrospective information offers a better foundation for plan- ning and policy than the guesses that must be made in its absence. 184

The Collection and Analysis Of a Large Body Of Data Create a Set Of Baselines With Which Various Groups Can Compare Experiences Hospitals Administrators of hospitals and medical in- vestigators customarily compare the experi- ences of their hospitals with those of others. Only active study by qualified investigators can tell what suggestions the experience of an especially low death rate in one hospital will offer to a hos- pital with a higher rate. Within reason, the com- petitive urge for excellence can be a healthy force for achieving and maintaining high standards. Operations The determination of death rates that follow particular sorts of operations can similarly stimulate inquiry as to why some surgical groups achieve lower death rates for specific operations than others do. Such comparisons cannot be made in a superficial manner; consideration must be given, for example, to the composition of the pa- tient population. Surgical teams may wish to com- pare their experiences on their own substantial series of operations with this larger collection of such series. Investigations That Fall Between Correla- tional Analysis and Experimental Studies Help To Check Over-All Findings and Are Of Value In Themselves Hospitals differ, sometimes radically, in their anesthetic practices. To the extent that they do, the subtle biases arising from an anesthesi- ologist's preferential use of agents in a particular clinical situation may be overridden. For example, one hospital in this Study used halothane in over 40 percent of its operations and cyclopropane in fewer than 0.5 percent. Another hospital used halothane in 6 percent of its operations and cy- clopropane in 46 percent. Swings in usage this large surely arise from hospital or departmental policy, rather than from individual choices based on the conditions of single patients. The patient is considered, and individual choices are made, but the range of choices in the first hospital did not really include cyclopropane and in the second hospital scarcely included halothane. Insofar as swings in usage can be attributed more to policy than to individual choices, the comparisons are freer of internal selective biases than the usual criticisms of retrospective studies would suggest. Unfortunately, we cannot make much direct use of this finding because the direct comparison of hospitals that use anesthetic A and not B with those that use B but not A would have anesthetic effects heavily confounded with hospital differences, especially if relatively few hospitals offer such pure usage and nonusage of anesthetics. Nevertheless, the indirect use of the observation of large swings as based on policy rather than on individual decision strengthens our over-all in- ference considerably. To turn to the other extreme, we can make special studies based on hospitals that use two or more anesthetics extensively, thus maintaining a high level of experience with each, while con- trolling on hospital. The Attempt To Analyze This Large Collection Of Data Has Developed New Methods Of Analysis Not only in medical investigations, but also in social and engineering problems, the analysis of rates and counts associated with many back- ground variables is a recurring and very awkward problem. In this study, we compare the anes- thetics in terms of overt death rates, but the agents have been used on patients with different physical conditions. For a careful comparison, allowance should be made for such differences. And after ad- justment for anesthetic risk, the patient's age must be taken into consideration, and the type of operation, etc. Obviously, the number of such al- lowances the investigator would like to use is not even limited by the variables he has observed. Let us especially note the need to handle var- iables jointly. Variables controlled separately may not tell us the simultaneous effect of two or more variables. In some problems, effects add up simply, but in others, variables may interact so that their joint effect cannot be computed by add- ing up their separate effects. Frequently, in studies of several variables, one table shows the rates associated with dif- ferent ages, and another the rates associated with the sexes, as the tables of Chapter IV-2 do. Such tables adjust for variables singly, but not in combination. Less frequently, rates are com- puted with control on two variables, perhaps for specific age-sex classes. To go further and give rates for classes of patients described by three variables—say, age, sex, and anesthetic—the in- vestigator would often be hard pressed for the cases needed for a reliable rate. If we use eight age classes, two for sex, and five for anesthetics (a deliberate oversimplification), we have 8X2X5 = 80 classes. As further variables are added, more and more classes are created, and the most direct analysis--deaths in class divided by ad- ministrations in class--fails because a class has too few cases to lend stability to the estimate. The resulting frustration is the bane of the man who must work with counts and several variables. It is appropriate to create new methods for handling this nearly universal problem at just this time. High-speed computers and experience with them have now developed to such a stage that we can afford to execute extensive manipulations repeatedly on large bodies of data with many control variables, whereas previously such heavy arithmetic work was impossible. The presence of the large sample from the National Halothane Study has encouraged the investigation and 185

development of flexible methods of adjusting for several background variables. Although this ad- justment problem is not totally solved by the work in this Study, substantial advances have been made and directions for further profitable re- search are clearly marked.. At first, one might suppose that the needs for adjustments would be limited to nonrandomized studies and that a randomized prospective study would have no use for them because it could maintain balance. Perhaps if only a pair of num- bers for the entire study were the assured goal, the randomized prospective study would not have to use adjustment; but adjustment is likely to increase precision, perhaps very substantially— an opportunity not to be missed. Some reasons are suggested below. Pure or complete randomization does not produce either equal or conveniently proportional numbers of patients in each class; attempts at deep poststratification are doomed to failure be- cause for several variables the number of possible strata quickly climbs beyond the thousands. Fur- thermore, randomization within strata during a study is rarely feasible. It is easy to ensure that sex is treated equally by randomly assigning equal numbers of women and men to each treatment. But then age will not be controlled, nor risk, nor kind of operation. To make this even clearer, con- sider some of the control variables used in the National Halothane Study: age, 10 categories; risk, eight categories; sex, two categories; pre- vious operation, three categories; and operation, about 75 categories. These five variables alone produce 10X8X2X3X75 = 36,000 cells. This fail- ure of chance to equate for strata demands and rewards adjustment for the inequalities. Briefly, the "control" that we associate with the idealized physical-science experiment is not available when we have many variables whose values we must accept, as we do with surgical patients, rather than change. Insofar as we want rates for special groups, we need some method of estimation that borrows strength from the general pattern of the variables. Such a method is likely to be similar, at least in spirit, to some of those that were developed and applied in this Study. At some stage in nearly every large-scale, randomized field study (a large, randomized prospective study of postoperative deaths would be no exception), the question arises whether the randomization has been executed according to plan. Inevitably, adjustments are required to see what the effects of the possible failure of the randomization might be. Again, the desired ad- justments would ordinarily be among the sorts that we discuss. To summarize, the sorts of adjustments needed and developed for this retrospective study of death rates will be needed in future large retrospective studies in many fields and may often be useful as well in large, randomized prospective studies. That the data on operations are gathered by means of a sample survey scarely calls for com- ment today. Abstracting every one of 856,000 charts in a short period would have been too ex- pensive for this Study and would almost inevitably have led to work of poor quality or to nonpartic- ipation by the institutions. Consequently, the use of a sample was necessary. What is a bit unusual is that data are available on essentially all the deaths. Usually, when sampling is used, nearly all the information comes from the objects sampled. When the event of interest is rare, the in- formation in the sample is usually more sensibly measured by the number of times the rare event happens than by the sample size. When the event occurs about half the time (say, 0.2 to 0.8), then the sample size is often a good measure because we are usually interested in accuracy as a per- centage of the true value. If, however, an event occurs two times in a million trials, after noting that the event is rare we rightly feel that the rate is poorly determined; the next million might easily give zero or five occurrences--variations of 100 and 150 percent, respectively. But if an event occurs 200,000 times in a million trials, we feel that under similar conditions the next million will produce the same result within a few percent, say, 2 or 3 percent. The strong implication of these remarks is that the information in a study of deaths after surgery is measured by the number of deaths, rather than by the number of operations. For ex- ample, a sample of 10,000 with a death rate of 1 percent is a small sample because it produces about 100 deaths; although replication under exactly equated conditions could readily produce from 80 to 120 deaths, a more realistic variation might run from 60 to 140, for a variation of 40 percent. But 40 percent would probably be about the size of difference that we would hope to detect in planning the study. Working with retrospective data in the Na- tional Halothane Study gives an unusual advantage in that data on the deaths are fairly readily avail- able. The cost of gathering the data on a death is much the same as that for a randomly chosen case. Because most of the accuracy in the study of the rare event, death, is contained in the deaths, we lose only modest information from sampling the 856,000 cases, rather than making a census of them. Thus, the retrospective study has a great saving. Its sample of 38,000 could have been expected to generate perhaps 760 deaths, but in- stead we have available the whole 17,000 deaths. The matter of size is discussed in connection with a possible prospective study in the following section. ON THE BASIC UNRESOLVABILITY OF VERY SMALL DIFFERENCES In a properly randomized clinical trial, many new problems arise because the randomization does not fit the usual medical or institutional 186

practice. Achieving a balance in the treatments presents another problem. Should experts on treatment A handle all patients assigned to A, and experts on treatment B handle those assigned to B, or should the same persons handle both kinds of treatment? Obviously, this choice will vary with the medical problem being studied. If the same persons are to administer both treat- ments, there may be a need for retraining, and in any case a considerable need for cooperation. No one wants a study merely to affirm that lack of practice may give poor practice. We cannot always use experimental techniques that we pre- fer; for instance, in comparing anesthetics, the anesthetist rarely can be blind to his anesthetic (an exception arises when the anesthetics have nearly identical physical and pharmacologic prop- erties, as do halothane and chloroform).All these matters make control more difficult, especially the control intended to ensure that the effect measured is the effect that occurs in the general population. Many studies can produce definitive results even when done on a modest scale. But studies of rare events are invariably irksome, first, because of the large numbers required, and, second, because of the likely existence of sources of bias at least comparable in magnitude with the effects needing to be detected. Although the second of these considerations may be more im- portant, the first is more easily illustrated. As a working number, let us say that in the United States around 10 million operations are performed each year. In our study a 2 percent death rate has been observed; if this rate were applied to all operations, there would be 200,000 deaths. A slight change in the death rate, to 1.9 percent, would reduce the number of deaths by 10,000, a substantial change. What size of inves- tigation would it take to detect reliably a dif- ference in death rate of 0.1 percent? As a prac- tical matter, many experienced workers would say that it could not be done with a study of any size. But, as an idealization to help guide our thinking, we can readily make some calculations to emphasize that large numbers are required. For simplicity, suppose that we intend to compare only two treatments, the anesthetic halo- thane (H) and anesthetic X. Suppose further that we have no variation other than binomial. This means that, as opposed to a real situation, we will underestimate the needed sample size by an unknown amount. Let p be the death rate after halothane and p+ 6 that after anesthetic X. If the death rates differ by 0.1 percent, then 6 would be ±0.001. To be confident of detecting this difference, we want the standard deviation of the observed difference to be small, say, one-sixth as big as 6, because fluctuations of two or three standard deviations in the observed difference can easily occur by chance. When 6 is small compared with p and the sample sizes for H and X are each N, the stand- ard deviation of the difference in observed rates is about V 2p/N. To meet the criterion of six stand- ard deviations, we need i/V2p/N = 6, or N = 72p/6 2. For 6 = 0.001, p =0.02, we find that N = 1,440,000, for a total clinical trial of 2,880,000 based on this oversimplified model. To face the more realistic difficulties requires contemplating samples several times as large as this. Obviously, before embarking on such a large-scale research program, the medical profession wants to be sure that society has a good chance of gaining much; that was the feeling, for example, toward the trial of the Salk poliovirus vaccine, which did require such a large-scale program. In considering the extreme of a difference of 0.001, we should not overlook the drastic reduc- tion in necessary sample size that occurs if we consider instead a difference of 0.005. Here we would need, with the simplified model, about 115,000. Such a study could perhaps be borne. If we required 6 to be a fraction of p, say, 6 = fp, then using the separation of 6 as before yields N = 72/f2p. Consequently, for fixed f as p increases, the number required decreases. This suggests that a study restricted to operations with higher values of p might also require fewer cases and permit concentration on the portion of the surgical population especially likely to pro- duce the deaths. But, to be realistic, we must be talking about big numbers if we are to consider the matter at all. Let us return to the second point concerning biases. First, there is a difficulty in principle. The training, cooperation, and communication needed to carry out a huge study might very well change the small differences that we are trying to measure, so that the real differences in the un- trained noncooperating institutions would not be assessed by the study. Next, just as the evidential value of a retrospective study is sensitive to the confounding of therapeutic acumen with choice of anesthetic, so is a prospective study very sensi- tive to slight biases of follow-up, coding decision, and so on, when directed at establishing the pres- ence or absence of minute differences. Small studies can sometimes be handled by a staff that has a pre-existing background of effective com- munication, and so high standards of accuracy and uniformity can be maintained. But if a study of very great size is to be undertaken, new per- sonnel must be trained, and the facility with which errors can be prevented, or identified and cor- rected, diminishes. Consequently, when one ex- pands a sample size to reduce sampling error, he sets in motion at the same time processes that will increase nonsampling errors. (These consid- erations underlie the experience of the Census Bureau that a sample survey conducted by per- manent staff is often more accurate than a com- plete enumeration for which staff must be built.) We therefore conclude that very small quan- titative differences cannot be empirically estab- lished. 187

Abstract of Chapter IV-2 The main purpose of the death-rate study was to find out about death rates associated with different anesthetic agents. An early task in approach- ing this problem was to reduce the several hundreds of different combina- tions of anesthetics that were administered in surgery to a small number of categories of anesthetic practice. We divided them into five groups: halothane, nitrous oxide-barbiturate, cyclopropane, ether, and "Other." The first four represented anesthetics and combinations of anesthetics that involved the named agent but not any of the other three. The category "Other" included all combinations of anesthetic agents that involved either none of the fore- going or two or more of them. The different anesthetics were used with widely varying frequency from institution to institution. Halothane was the most frequently used and ether the least. Other variables associated with the surgical procedure have a much greater influence on survival rate than the choice of anesthetic. Such vari- ables measured in this study include the age of the patient, the type of op- eration, the length of the operation, the sex of the patient, his physical status (anesthetic risk), whether the patient had had an operation in the same or the previous month, the year, and the institution. Inevitably, some anesthetics were used more than others in particular operations, for old patients, for patients with poor physical status, etc. If these imbalances had not been corrected, they would have introduced spurious differences among the anesthetics. A way of adjusting for such interfering variables is to compute a standardized death rate. In this chapter, each in- terfering variable is used as a basis for computing standardized death rates. We are thus able to look at the comparisons among anesthetics purged, at least partially, of the effects of the interfering variables taken one at a time. The effect of this standardization is to enhance some differences and to re- duce others. This is one indication for the need to construct standardized rates that take into account several variables at one time. The results reported in this chapter are usually exhibited for three large subportions of the data: the high-death-rate group, which involves four dif- ferent operations, about 62,000 surgical procedures, and 6300 deaths; the low-death-rate group, which involves seven very common operations, about 370,000 surgical procedures, and 800 deaths; and the middle-death-rate group, which involves the remaining 62 operations, 426,000 surgical pro- cedures, and 9600 deaths. The study was not designed to show death-rate differences among in- stitutions, but dramatic differences occurred. The institutional death rates ranged from 0.0027 to 0.0641, numbers that stand in a ratio of about 1:25. Standardization on age, operation, physical status, and a composite variable called "shock-likelihood index" reduced the apparent institutional variation, but by no means eliminated it. Standardization on other interfering variables had little effect. In summary, this chapter introduces the data of the death-rate study, exhibits the importance of several interfering variables, and gives prelimi- nary notions of anesthetic differences after allowance for the effects of single interfering variables. The chapter also identifies two problems that call for further investigation: (1) There is a need to compare the anesthetics after multiple ad- justment for all important interfering variables. (2) A more careful appraisal of differences among institutional death rates is called for. 188

CHAPTER IV-2. COMPARISON OF CRUDE AND STANDARDIZED ANESTHETIC DEATH RATES Lincoln E. Moses Stanford University Stanford, California CLASSIFICATION OF ANESTHETIC PRACTICES AND OPERATIONS The comparison of anesthetic agents is made somewhat complex by the fact that it is common to use several agents in the course of a single operation. In this Study, a very large number of combinations of anesthetics appeared, which was not surprising. We have defined five common anesthetic practices. The detailed definitions ap- pear in Appendix 1, but the five anesthetic prac- tices can be thought of, roughly, as: (1) halothane: combinations of agents including halothane but excluding cyclopropane and ether; (2) nitrous oxide-barbiturate: combina- tions of agents including nitrous oxide and thiopental but excluding halo- thane, cyclopropane, and ether; (3) cyclopropane: combinations of agents including cyclopropane but excluding halothane and ether; (4) ether: combinations of agents in- cluding ether but excluding halo- thane and cyclopropane; and (5) Other: all other combinations of agents, many of which include at least two of halothane, cyclopropane, and ether; all practices using meth- oxyflurane or Fluor omar; and all practices that include nitrous oxide but exclude Pentothal. Later we investigate the differences in in- terpretation that arise in comparing agents if the categories overlap so that any patient who re- ceived halothane is charged to halothane (even if he also received ether or cyclopropane) and simultaneously charged to any other major agent that he received. Every operation was assigned a two-digit operation code. When the crude death rates for the different operations were examined, it was found that four had more than 1000 deaths each, and these have been treated separately as the high-death-rate operations. It was also found that seven had more than 30,000 estimated ex- posed each, and these have been separated out as the low-death-rate operations. The purpose of these separations was to avoid swamping the remainder of the data; the operations in those two groups were not the ones with the four highest or the seven lowest crude death rates. The high-death-rate operations are: (1) large bowel operations, (2) exploratory laparotomy, lysis of adhesions, control of hemorrhage, and drainage of abscess, (3) craniotomy, and (4) heart operations with pump. The low-death-rate op- erations are: (1) mouth and dental operations, (2) eye operations, (3) dilatation and curettage, etc., (4) hysterectomy, (5) herniorrhaphy, (6) cystoscopy, and (7) plastic surgery. The middle-death-rate operations involved all other specific procedures (excluding mammoplasty, for which there were no deaths and an estimated 530 operations). The estimated numbers of procedures ("es- timated exposed," abbreviated "EE") and the estimated death rates for these three groups of operations are displayed in Table 1. A very large share of the procedures and a very small share of the deaths are associated with the seven low- death-rate operations. The over-all death rate in the middle-death-rate operations is about 10 times as great as that in the low-death-rate cate- gory, and nearly 60 percent of all the deaths are in the middle-death-rate group. The high-death-rate group includes only about 7 percent of the ex- posures to anesthesia, but nearly 40 percent of the deaths, and the over-all death rate is nearly 50 times as great in this group as in the low- death-rate group. Comparisons of the agents with each other differ from group to group. For ex- ample, ether appears best in the low- and high- death-rate groups but not in the middle-death- rate group, Other appears to be worst in the high- and low-death-rate groups but not in the middle-death-rate group, etc. Before attaching any significant weight at all to the agent comparisons in Table 1, it is im- portant to recognize that many aspects of sur- gical procedure affect the outcome, e.g., the age of the patient, his physical condition, the opera- tion performed on him, and its duration. If we were satisfied that all these variables were bal- anced equally among the five anesthetic practices, then we might take the death rates in Table 1 at face value; in fact, study of the data shows that these other variables are not equally balanced among the anesthetic agents. We must therefore examine the effects of such imbalances. AGENT CONTRASTS ALLOWING FOR ONE VARIABLE AT A TIME Among the variables observed in this Study that exert an important effect on mortality are: whether there has recently been a previous 189

TABLE 1.—AGENT COMPARISONS FOR LOW-, MIDDLE-, AND HIGH-DEATH-RATE OPERATIONS H N-B C E 0 All Estimated exposed (in thousands) Low-death-rate operations 86.6 105.1 67.3 50.0 57.3 366.3 Middle-death-rate operations 145.7 100.8 68.1 44.2 67.2 426.0 High- death- rate operations 21.8 11.6 11.7 7.6 9.1 61.7 All operations 254.9 218.2 147.4 102.0 134.0 856.5 Number of deaths Low-death-rate operations 234 209 175 63 163 844 Middle-death-rate operations 2559 1753 2397 836 2070 9615 High-death-rate operations 2064 1323 1270 492 1205 6354 All operations 4863 3292 3845 1396 3444 16840 Death rate Low-death-rate operations 0.00270 0.00199 0.00260 0.00126 0.00285 0.00230 Middle-death-rate operations 0.0176 0.0174 0.0352 0.0189 0.0308 0.0226 High-death-rate operations 0.0948 0.1143 0.1086 0.0647 0.1327 0.1030 All operations 0.0191 0.0151 0.0261 0.0137 0.0257 0.0197 operation, the anesthetic risk (physical status) as judged at entrance to surgery, the duration of the operation, the patient's age, and the patient's sex. The other variables investigated—the insti- tution, the year, and the type of operation—may also prove important. All these variables are de- fined in Appendix 2. Tables EE-1 through EE-26 (at the end of this chapter) display for the different anesthetic practices* how the cases were distributed on the variables. They are followed by Tables DR-1 through DR-26, which display the death rates ("DR") by anesthetic practice and the variables. The death rates exhibited after Table 1, generally throughout Part IV, are calculated as the quotient "deaths divided by deaths plus esti- mated exposed"; this is sometimes written as "D/D+EE." This estimate is somewhat unconven- tional, and a few words of explanation are in order here (a fuller discussion appears as Appendix 3). In a cell with few cases, it can happen (and does) that some deaths are observed although the ran- dom sample provided no cases for the cell; in that case the plausible "ordinary" estimate, D/EE, would have a zero denominator. The es- timator D/D+EE yields the value 1.0 in that case. Of course, invariably the estimate we have used is smaller than the "ordinary" estimator. But if EE is large and the death rate is small, there is little difference between the values of the two •Tables EE-1 through EE-26 and DR-1 through DR-26are early computer printouts; the columns for nitrous oxide- barbiturate are labeled "NP" instead of the later abbrevi- ation, "N-B." estimates. In cells with both high death rates and large EE, it may be worthwhile to reconvert the estimates to D/EE form. For comparisons of death rates, however, this will rarely be useful. Interpretation of these tables and their use is illustrated by the following discussion of Tables EE-5 and DR-5. - Table DR-5 displays the death rates for each of the anesthetic practices, by category of phys- ical status. In studying this table, one sees that the death rate depends strongly on the physical status, ranging from about one-fourth of 1 per- cent for physical status 1 to over 30 percent for physical status 7 (moribund). The row labeled "over-all" shows the death rates for each of the agents. The next row, labeled "standardized death rate," can be explained by reference to Table EE-5. This table shows for each of the common practices in the middle-death-rate op- erations the percentage of patients in each phys- ical status category. In comparing the cyclopro- pane column with the "all" column, one is struck by the fact that cyclopropane has a dispropor- tionately large share of patients in status cate- gories 5, 6, and 7. These are very high death- rate categories. It is clear that the death rate for cyclopropane will tend to be high because it is used so often in patients in these physical status groups. The standardized death rate ad- justs for this kind of imbalance. It is determined by taking the cyclopropane death rates for the various physical status categories and calculat- ing what the over-all death rate for cyclopropane would have been if these rates had been applied to a population of the kind shown in the "all" 190

column of Table EE-5. The resulting number (Table DR-5) is 0.0286, quite different from the raw value of 0.0340. The standardized rates for the other agents are similarly computed.* In comparing the standardized rates with the over-all rates, it is seen that standardization in correcting for imbalance between agents on phys- ical status has made halothane, nitrous oxide - barbiturate, and ether look somewhat less favor- able and both cyclopropane and Other somewhat more favorable than they did before standardiza- tion. The next row of Table DR-5, "standard (ratio)," is calculated by dividing all the entries in the row above it by 0.0221, which is the crude over-all death rate, or equivalently, the stand- ardized rate calculated by applying average death rates within risk categories to the over-all population. The figures in this row enable one to see conveniently whether an agent has a higher or lower death rate than "average." An alternative to (direct) standardization, just described, is the computation of an "in- directly standardized mortality ratio." The ob- jective is, again, to compare agents in a way that is corrected for disparities among the popu- lations to which the agents were applied, but the method is different in detail. The death rates for "standard population" (in this case, the "all agents" population) are applied to the population exposed to, say, cyclopropane; this leads to a putative number of deaths, i.e., the number that "would have" occurred in the cyclopropane- treated patients if the general death rates had applied (this is a sort of par value). Then the ratio of the actual number of deaths to that par is calculated; it would exceed unity if there were excessive deaths with cyclopropane. This indirect method can still be used where some operations are not done at all with, say, cyclopropane, and the direct method cannot. (For example, if no hysterectomies were done with ether, the direct method would fail for the lack of an observed ether death rate, but the indirect method would work, both par and observed deaths being zero.) This illustrates the advantage that can accrue with the indirect method. Generally, in our study the two approaches lead to conclusions that agree closely. The bottom row of Table DR-5 shows this indirectly standardized mortality ratio. Tables EE-1 through EE-26 all show per- centage distributions of the patients assigned to the anesthetic practices with respect to the in- terfering variables (risk, etc.). The percentages in each column therefore add (approximately) to 100. The bottom row of each table gives the total numbers of estimated exposures in the anesthetic practices expressed in thousands. «Detailed formulas used in these calculations appear in Appendix 4. The weights used are based on D+EE, rather than EE, because the death rates have denominators of the D+EE type, rather than the EE type. The Tables of Estimated Exposures Previous Operation This variable has three classes. The great bulk of the procedures were performed on pa- tients who had had no previous surgery in either the month of the surgery relevant to this Study or the month before. Only 7.5 percent of the cases in the low-death-rate group, 10.3 percent in the middle-death-rate group, and 14.3 percent in the high-death-rate group had had a previous operation. Cases in which there had been a previous operation were classified according to whether halothane was used then or not. A strik- ing but not surprising feature is that the fraction of patients who received halothane for previous operations was far greater in the halothane pop- ulation than in any of the others. Physical Status In the low-death-rate operations there was a marked tendency for cyclopropane to be used relatively more often in emergency operations (physical status 5, 6, and 7) than any of the other anesthetic agents, and for ether to be used far less in emergency operations. The findings were similar, although not quite so marked, in the middle- and high-death-rate groups. Length From low- through middle- to high-death- rate operations, there was a striking increase in the fractions of longer procedures. In the low- death-rate group, nitrous oxide-barbiturate was used more than the other agents. In the middle-death-rate operations, ether had more than its share of long procedures; and in the high-death-rate operations, both cyclopropane and ether had less than their shares of long pro- cedures. Age In the low-death-rate group, both ether and Other were used very heavily for children and ether was used more heavily than the other agents in the 10-19 age group. In the middle- death-rate operations, ether and Other were again used relatively more for children. In the high-death-rate group, Other had a dispropor- tionately high share of administrations to chil- dren. Sex In the low-death-rate operations, cyclopro- pane and nitrous oxide-barbiturate were used in females in far greater than ordinary proportions. This is probably a reflection of the heavy use of both in dilatation and curettage and of cyclopro- pane in hysterectomy. In the middle-death-rate 191

operations, there was some tendency for cyclo- propane and nitrous oxide-barbiturate to be used more in females. But there were no clear sex effects at all for the high-death-rate operations. Year There was a marked increase in the use of halothane over the period 1959-1962, compen- sated primarily by a decrease in the use of ni- trous oxide-barbiturate and ether, although cyclo- propane also showed a downward trend over that period. Institution The most notable feature of the hospital dis- tribution of the different anesthetic agents is the diversity of usage patterns. The number of hospi- tals responsible for as few as 1 percent of all the operations that involved halothane (in the whole Study) is less than the number responsible for so few cases involving any of the other agents. At the other extreme, over half the hospitals were each responsible for fewer than 1 percent of all the operations that involved ether. Indeed, three institutions (1, 8, and 9) together accounted for 49.5 percent of the low-death-rate operations that involved ether; and institutions 8 and 9 to- gether accounted for 53.4 percent and 50.0 per- cent of the ether operations in the middle- and high-death-rate groups, respectively. To study usage patterns further by institu- tion, it is convenient to assemble all the opera- tions. This is done in Table EE-22. This table has been constructed so that it shows what per- centage of each hospital's operations involved each anesthetic practice. Thus, each row of this table adds to 100 percent, whereas in the other EE tables the columns add to 100 percent. This table shows the great diversity in the popularity of the agents. Halothane was the most commonly used agent in the entire Study, and it was also the most com- monly used in 17 individual institutions. Only in institution 20 was it the least commonly used agent. The over-all use of halothane was about 30 percent, but its use in individual hospitals ranged from about 6 percent to nearly 63 percent. Nitrous oxide-barbiturate, the second most com- monly used agent in the entire Study, was used more commonly than all the other agents in six institutions, and was the least used agent in one institution; its percentage of use ranged from 1.9 to 73.3. Cyclopropane, ranking third in over-all use, presented a very heterogeneous use pattern from institution to institution, from 0.2 to 48 per- cent. In seven institutions it was the most com- monly used agent, and in eight the least commonly used. Ether had the lowest over-all use, 11.9 per- cent. In 22 institutions it was less used than all the other agents, but in two it was the most com- monly used; its percentage of use ranged from 0 to 38.5. The use of Other ranged from 2.1 to 41.4 percent; it was the most commonly used in two institutions and the least commonly used in three. Operation The operation tables are the only ones in which the labels of the rows are different for the low-, middle-, and high-death-rate groups. The seven low-death-rate operations show quite dif- ferent patterns as to the agents used. Ether and Other are prominent for the mouth operations (code 1). Nitrous oxide-barbiturate was very commonly given for dilatation and curettage (code 60) and for cystoscopy (code 73). Cyclopropane was very commonly used for dilatation and cu- rettage (code 60) and for hysterectomy (code 65). The names (or codes) of the middle-death-rate operations are not shown; the 25 rows correspond to collections of operations that were grouped in such a way that the death rate steadily increases with the group number. For these operations halo- thane was more commonly used than any other agent, and ether least. Halothane was also the most commonly used and ether the least com- monly used in the high-death-rate operations. In fact, halothane was used in almost twice as many procedures as its nearest competitor. Ether was never used in operation 33 (heart with pump), and cyclopropane was almost never used in oper- ation 12 (craniotomy) and only rarely in opera- tion 33. The Tables of Death Rates Tables DR-1 through DR-26 display death rates (estimated as D/D+EE) for the various an- esthetic agents and the interfering variables. The crude over-all death rate for each agent is shown, together with the death rate standardized on the all-agents population for the variable under con- sideration. The standardized death rate is also exhibited in the ratio form, and the indirectly standardized mortality ratio is given for each agent. Previous Operation Death rates were much higher if there had been a previous operation than if there had not; that was true in the low-, middle-, and high- death-rate groups. In the low- and middle-death- rate groups little difference in death rate was associated with the distinction between previous operations with and without halothane. In those groups, standardizing with respect to previous operation has little effect, leading to rates hardly different from the crude rates. In the high- death-rate group, the death rates were lower if halothane had been used in a previous operation (than if it had not) for two of the agents (halothane and Other), and higher for the other three agents. Standardization has a small effect, and does not alter the rank order of the death rates. 192

Physical Status Physical status has a very strong association with death rate. In the low-death-rate group, the rate changes from 0.0004 to 0.0565 in the non- emergency operations; the effect of standardizing with respect to physical status is to cause ether to look less advantageous and Other to look even worse. In the middle-death-rate group, a simi- larly strong gradient is seen, and the effect of standardization is to reduce the apparent infe- riority of cyclopropane, as well as that of Other. In the high-death-rate group, there is again a marked gradient, and the effect of standardization is to reduce greatly the apparent superiority of ether and to reverse the standings of cyclopro- pane and halothane, the latter appearing worse. Length Length is another variable that has a pro- found association with death rate. In the low- death-rate group, the shortest operations had a death rate of less than 0.1 percent, and the longest operations more than 2 percent. The only important contrast among the death rates stand- ardized with respect to length is the markedly lower rate associated with ether. In the middle- death-rate group, the gradient with length ranges from 0.6 percent to almost 11 percent; after standardization, ether, halothane, and ni- trous oxide-barbiturate look about equally good. In the high-death-rate group, the regularity of the gradient with length is not so clear. Short operations (up to 28 min) had the lowest death rate, but there is no strong or clear gradient between 0.5 and 4 hr; when operations were longer than that, the death rate rose with length. The effect of standardization in the high-death- rate operations is primarily to make cyclopropane and Other look worse. Age For all three groups of operations, the death rate was higher for the patients 0-9 years old than for young adults, and increased steadily with age beginning at 10 years. In the low-death- rate group, standardization with respect to age raises the death rates for cyclopropane and Other rather substantially; in the middle-death-rate group, standardization has little effect; and in the high-death-rate group, standardization hardly changes the rates and does not change their rank order, but there is a modest increase for nitrous oxide-barbiturate. Sex In all three groups, the death rates for males were higher than for females: in the low-death- rate group, 0.003 and 0.002 respectively; in the middle-death-rate group, 0.027 and 0.017; and in the high-death-rate group, 0.104 and 0.083. Standardization with respect to sex results in little modification of death rates, except that it substantially increases the rate for cyclopropane in the low-death-rate group. Year There was no important regular variation in death rate with year, and standardization for this variable has little effect. Institution Death rates for institutions exhibit large and important variability. In the low-death-rate group, three institutions (1, 21, and 28) had death rates under 0.1 percent, and three institutions (24, 25, and 30) had death rates over 1 percent. The range of death rates over the 34 institutions for the low-death-rate operations was from 0.0005 to 0.0141. In the middle-death-rate group, one institution (28) had a death rate under 0.0050, and three (2, 7, and 25) had death rates over 0.0500. In the high-death-rate group, three insti- tutions (14, 21, and 28) had death rates under 0.0600, and seven (2, 5, 7, 24, 25, 29, and 31) had death rates over 0.1300; the range in the high- death-rate group was from 0.024 to 0.160. The over-all crude death rates for all groups ranged from 0.0027 to 0.0641, with six institutions (1, 3, 14, 17, 23, and 28) having death rates under 0.0100, and three institutions (2, 24, and 25) having death rates over 0.0600. Comparison among the crude death rates by anesthetic agent in the various institutions pre- sents a confused picture. For example, although cyclopropane had the highest crude death rate, in nine institutions the cyclopropane rate was below the over-all rate. Other had the second highest crude death rate, but in eight institutions its death rate was lower than the over-all death rate. Ether had the lowest over-all crude death rate, but in nine institutions the ether death rate was higher than the over-all death rate—and in two it was not used at all. The effect of standardizing the agent death rates with respect to institution is generally to raise the rates; this happens for each agent in each group, except halothane in the low-death- rate group. This phenomenon is presumably a result of there being an average tendency for high death rates to accompany an anesthetic agent used only infrequently in a hospital; when such rates are applied to the (larger) general-usage fraction, a large contribution to the standardized death rate results. Similarly, a much-used agent in a particular hospital may have a lower death rate than usual, but if the low rate is applied to the (smaller) general-usage fraction, this will also tend to make the standardized death rate larger than the observed one. The indirectly standardized mortality ratio seems a better index for standardization with respect to institution. (Indeed, institution is one of the very few vari- ables for which there is much difference between the two approaches.) In the low-death-rate group, standardization (indirect) does not greatly change 193

the comparison among agents from that indicated by the crude rates. In the middle-death-rate group, the positions of cyclopropane and Other are reversed by standardization (indirect), and the apparent virtue of ether is much reduced at the same time. In the high-death-rate group, stand- ardization (indirect) alters the interpretation but little; nitrous oxide-barbiturate, which has a slightly higher crude rate than cyclopropane, now seems slightly superior to cyclopropane. Operation In the low-death-rate group, there was sub- stantial variability in the death rates by opera- tion, from 0.00066 to 0.00689. Ether appears to be markedly the best of the agents in this group, although standardization with respect to operation somewhat reduces its apparent superiority and at the same time causes halothane to look better than before and cyclopropane and Other to look worse. In the middle-death-rate group, the gra- dient of over-all death rate with operation sub- group number is obvious. The effect of standard- ization in this group is not very large, although it does alter the relative positions of halothane, nitrous oxide-barbiturate, and ether. In the high- death-rate group, there was great variability in the death rates for the four operations (ranging from 0.0565 to 0.1601), and the effects of stand- ardization are large. In terms of indirect stand- ardized mortality ratios, ether and halothane appear to be best and about equally good. Tables EE-26 and DR-26 provide a more de- tailed breakdown of the data by operation. These tables show the operations by operation code. The table of estimated exposures shows some interesting patterns. First, halothane was the least used for certain "pet" operations. For each of nitrous oxide-barbiturate, cyclopropane, and ether, there are two operations that together ac- counted for some 30 percent of the administra- tions: for nitrous oxide-barbiturate, operation 60 (20.5 percent) and operation 73 (10.8 percent); for cyclopropane, operation 60 (21.5 percent) and operation 65 (8.8 percent); and for ether, opera- tion 1 (24 percent) and operation 65 (6.4 percent). This usage concentration was less marked for Other; the two commonest operations together account for only 22.1 percent of the administra- tions: operation 1 (16.7 percent) and operation 55 (5.4 percent). But with halothane, the two com- monest operations together accounted for less than 14 percent: operation 1 (6.5 percent) and operation 60 (6.9 percent). Second, inspection of the number of operation codes for which the in- dividual agents were very rarely used shows that halothane was the most widely used. Any opera- tion listed in the table of estimated exposures for which an agent shows 0.0, 0.1, or 0.2 is one for which the agent was used in less than one-fourth of 1 percent of all administrations of that agent. The frequencies of such "avoided" operations are (not counting "unknown" operations): halo- thane, 17; nitrous oxide-barbiturate, 19; cyclo- propane, 27; ether, 21; and Other, 20. In fact, five of the operations are "rare," in that they show 0.0 for all five agents. If these are omitted from the count, the figures become: halothane, 12; nitrous oxide-barbiturate, 14; cyclopropane, 22; ether, 16; and Other, 15. In the sense of having the fewest "avoided" operations, halothane is the most widely used in this body of data. (This is not because it was used most often of the agents; the discussion is in terms of the percentage of the times the agent was used, and each agent was on an equal footing: its own 100 percent was dis- posed onto 76 + "unknown" operations.) Table DR-26 shows the death rates for the different operations, by anesthetic agent. (Here we see some instances of deaths occurring where estimated exposed is zero: operation 43 for nitrous oxide-barbiturate and operation 33 for ether both had death rates of 1.00000.) The death rates ranged from 0.00000 for operation 31 (an ill-determined rate) to 0.16012 for operation 33. Several operations had death rates under 0.001, and several over 0.120. The data are complex; standardization with respect to operation re- moves much of the variation among agent death rates, and changes the rank order of death rates greatly, chiefly as a result of ether's having the smallest crude rate and the largest standardized rate. In terms of the indirect mortality ratio (probably a more appropriate index here), there is much less change from the pattern of the crude death rates. Tables 2, 3, and 4 summarize the stand- ardized death rates and direct mortality ratios for the low-, middle-, and high-death-rate groups exhibited in Tables DR-1 through DR-26. In Table 2, we see that Other has the highest or next highest standardized death rate among the agents in every row; that ether and nitrous oxide- barbiturate always have the two lowest stand- ardized death rates, in an order depending on what has been standardized for; and that halothane and cyclopropane are always in third, fourth, or fifth positions, again depending on what has been standardized for. Furthermore, there is good quantitative agreement between corresponding standard mortality ratios and indirectly stand- ardized mortality ratios, with one notable excep- tion: standardizing for institution produces very different ratios for cyclopropane (direct, 2.151; indirect, 0.981) and for Other (direct, 1.779; in- direct, 1.324). In the middle-death-rate group (Table 3), there is much more uniformity. Cyclopropane always has the highest standardized rate, Other always the next highest, and halothane and nitrous oxide-barbiturate ordinarily the two lowest posi- tions, with ether third, except when the standardi- zation is with respect to length or operation. As was the case with the low-death-rate operations, there is good quantitative agreement between the direct and indirect mortality ratios, except for standardization with respect to institution, for 194

TABLE 2.— AGENT-STANDARDIZED DEATH RATES ON -ALL AGENTS" POPULATION (Low-Death-Rate Operations) Stand- ardizing Standardized death rate Direct mortality ratio Indirect mortality ratio variable H N-B C | E | 0 H | N-B C E 0 H N-B | C E 0 Previous 0.00258 0.00201 0.00263 0.00123 0.00293 1.124 0.875 1.146 0.536 1.274 1.088 0.885 1.142 0.572 1.260 Physical status 0.00264 0.00192 0.00235 0.00210 0.00349 1.149 0.835 1.024 0.912 1.517 1.081 0.836 0.968 0.733 1.470 Length 0.00247 0.00235 0.00275 0.00114 O.J0268 1.076 1.022 1.194 0.495 1.164 1.078 1.036 1.156 0.488 1 125 Age 0.00264 0.00175 0.00333 0.00192 0.00381 1.147 0.761 1.450 0.834 1.656 1.109 0.715 1.276 0.715 1.415 Sex 0.00265 0.00209 0.00300 0.00132 0.00279 1.151 0.908 1.306 0.576 1.212 1.145 0.888 1.200 0.521 1.1S6 Year 0.00290 0.00191 0.00260 0.00129 0.00284 1.261 0.832 1.130 0.560 1.236 1.181 0.859 1.127 0.547 1.234 Institution 0.00269 0.00233 0.00494 0.00137 O.0O409 1.168 1.015 2.151 0.596 1.779 1.096 0.907 0.981 0.639 1.324 Operation 0.00228 0.00200 0.00306 O.0OU7 0.00329 0.994 0.870 1.331 0.639 1.433 1.014 0.881 1.214 0.583 1.316 TABLE 3.--AGEWT-STANDARDIZED DEATH RATES ON "ALL AGENTS" POPULATION (Middle-Death-Rate Operations) Stand- ardizing variable Standardized death rate Direct mortality ratio Indirect mortality ratio H N-B C E 0 H N-B C E 0 H N-B 1 C E 0 Previous 0.31750 0.01745 0.03483 0.01849 0.02898 0.793 0.791 1.578 0.838 1.313 0.787 0.779 1.581 0.846 1.284 Physical status 0.01943 0.01941 0.02855 0.02119 0.02532 0.880 0.879 1.293 0.960 1.147 0.871 0.871 1.223 0.919 1.154 Length 0.01736 0.01809 0.03551 0.01648 0.03069 0.787 0.820 1.609 0.747 1.390 0.779 0.802 1.599 0.729 1.382 Age 0.01831 0.01701 0.03405 0.01838 0.02820 0.830 0.771 1.543 0.833 1.278 0.820 0.766 1.535 0.826 1.266 Sex 0.01714 0.01727 0.03513 0.01835 0.02955 0.776 0.783 1.592 0.832 1.339 0.772 0.779 1.600 0.826 1.345 Year 0.01724 0.01717 0.03419 0.01898 0.02988 0.781 0.778 1.549 0.860 1.354 0.790 0.761 1.548 0.830 1.365 Institution 0.01879 0.01790 0.04441 0.02994 0.03488 0.851 0.811 2.012 1.356 1.580 0.796 0.757 1.317 1.05'. 1.405 Operation 0.01857 0.02080 0.03004 0.01819 0.02793 0.841 0.942 1.361 J.824 1.265 0.828 0.906 1.320 0.741 1.257 TABLE 4.—AGENT-STANDARDIZED DEATH RATES ON "ALL AGENTS" POPULATION (High-Death-Rate Operations) Stand- Standardized death rate Direct mortality ratio Indirect mortality ratio variable H N-B C E 0 H N-B C E 0 H N-B C E 0 Previous 0.08763 0.10401 0.10168 0.06091 0.11661 0.939 1.114 1.089 0.653 1.249 0.929 1.114 1.052 0.616 1.282 Physical status 0.09467 0.09832 0.09254 0.07787 0.10494 1.014 1.053 0.991 0.834 1.124 0.988 1.072 0.966 0.777 1.112 Length 0.08237 0.09440 0.11011 0.07978 0.12364 0.882 1.011 1.179 0.855 1.324 0.876 ' 1.011 1.172 0.689 1.351 Age 0.08907 0.10887 0.09704 0.06010 0.11906 0.954 1.166 1.039 0.644 1.275 0.949 1.140 1.028 0.614 1.228 Sex 0.08641 0.10279 0.09804 0.06096 0.11693 0.926 1.101 1.050 0.653 1.253 O.WO 1.114 1.056 0.630 1.285 Year 0.09121 0.10541 0.09822 0.06103 0.11758 0.977 1.L29 1.052 0.654 1.259 0.955 1.-074 1.058 0.610 1.264 Institution 0.08845 0.11781 0.14317 0.13107 0.13645 0.946 1.261 1.532 1.403 1.460 0.956 0.989 1.011 0.716 1.318 Operation 0.07763 O.OJ251 0.11184 0.20216 0.12370 0.832 0.991 1.198 2.166 1.325 0.849 1.036 1.245 0.745 1.249 which the direct ratios are much higher than the indirect for both cyclopropane and ether. In the high-death-rate group (Table 4), there is also considerable uniformity, best seen in the indirect ratios, where, for each standardization, ether has the lowest ratio and Other the highest. For all standardizations except with respect to physical status, halothane is second; and nitrous oxide-barbiturate and cyclopropane are usually third and fourth. All these statements hold true for both standardized death rates and direct mortality ratios, except for standardization with respect to institution and operation, where ether has high direct mortality ratios and the lowest indirect ratios, and other inversions occur. INVESTIGATION AND DISCUSSION OF INSTITUTIONAL VARIATION IN DEATH RATES One of the interfering variables that has been very influential in affecting agent compari- sons is the institution. Review of Tables 2, 3, and 4 shows that standardization for institution sometimes leads to results that are at variance with those obtained by standardization for other variables. Standardization for institution could be expected to have peculiar effects, inasmuch as there are marked variations in death rates from hospital to hospital, marked differences in the degrees of use of the anesthetic agents, and, presumably, rather sharply different kinds of 195

TABLE 5.—STANDARDIZED MORTALITY RATIOS FOR 34 INSTITUTIONS (All Operations) Institution D/D+EE Ind. S.M.R. Previous opera- tion Physical status Sex Year Opera- tion Age Shock likeli- hood 1 0.00732 0.374 0.404 0.524 0.386 0.373 0.716 0.443 0.523 2 0.06049 3.274 3.404 1.062 3.417 3.306 2.577 2.629 1.975 3 0.00803 0.411 0.440 1.059 0.411 0.407 0.688 0.420 0.842 4 0.02394 1.247 1.224 0.773 1.294 1.240 1.149 1.307 1.120 5 0.03512 1.851 1.835 1.411 1.862 1.851 1.620 1.856 1.236 6 0.01085 0.558 0.562 0.602 0.597 0.558 0.974 0.595 0.724 7 0.04532 2.414 2.398 1.498 2.414 2.458 2.309 1.956 1.762 8 0.O1456 0.751 0.743 0.926 0.724 0.754 0.577 0.645 0.757 9 0.01730 0.895 0.849 0.924 0.877 0.892 0.897 0.912 0.963 10 0.02661 1.390 1.358 1.122 1.361 1.384 1.115 1.231 1.476 11 0.02070 1.075 1.008 0.659 1.046 1.072 0.887 1.309 0.992 12 0.01966 1.019 1.023 1.709 1.055 1.013 0.976 1.000 1.083 13 0.03447 1.815 1.883 4.307 1.370 1.857 0.979 1.670 1.367 14 0.00518 0.264 0.280 0.274 0.266 0.263 0.472 0.321 0.274 15 0.01899 0.984 1.005 1.118 1.026 0.982 1.223 0.992 0.941 16 0.03128 1.642 1.641 1.408 1.610 1.631 1.086 1.728 1.148 17 0.00933 0.478 0.421 0.505 0.436 0.485 0.557 0.830 0.549 18 0.03544 1.868 1.935 0.953 1.895 1.870 1.109 1.611 1.145 19 0.02583 1.348 1.355 1.471 1.372 1.352 1.121 1.275 1.418 20 0.01507 0.778 0.783 0.838 0.810 0.779 0.973 0.741 0.754 21 0.01235 0.636 0.664 0.961 0.640 0.636 0.627 0.593 0.627 22 0.01932 1.002 0.999 0.916 1.042 0.996 1.301 1.061 1.198 23 0.00842 0.431 0.472 0.720 0.447 0.434 0.651 0.542 0.615 24 0.06340 3.442 3.270 2.122 3.334 3.461 1.477 3.067 1.899 25 0.06405 3.480 3.280 1.817 3.374 3.503 2.413 1.968 2.268 26 0.01030 0.529 0.533 0.782 0.543 0.525 1.036 0.571 0.839 27 0.01074 0.552 0.563 1.018 0.561 0.555 0.780 0.623 0.723 28 0.00268 0.136 0.148 0.348 0.136 0.137 0.287 0.243 0.203 29 0.04185 2.221 2.064 1.324 2.141 2.243 1.121 2.435 1.257 30 0.04816 2.573 2.178 0.880 2.433 2.589 0.935 3.521 1.291 31 0.01421 0.733 0.747 1.662 0.730 0.731 1.261 0.773 1.363 32 0.01746 0.903 0.929 1.086 0.909 0.897 0.845 0.891 0.861 33 0.02476 1.291 1.203 0.731 1.232 1.299 1.015 1.193 1.045 34 0.01561 0.806 0.827 1.346 0.797 0.808 1.122 0.836 1.030 TOTAL 0.01928 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Maximum ratio among institu- tions 25.4 22.9 15.6 25.1 25.5 9.0 12.6 11.1 patient populations. Further investigation of in- stitutional differences should be illuminating. In Table DR-22, it was seen that death rates by in- stitution (for all operations) ranged from 0.0027 to 0.0641. This approximately 25-fold variation is, in itself, important and provocative. This section is addressed to studying institutional variation in death rate more carefully. (It is studied further in the last section of Chapter IV-6.) Generally, the institutions differ from one another as to age distribution of patients, kinds of operations performed, etc. Thus, it is in- evitable that some of the institutional variation in death rate is properly ascribed to differences in such interfering variables. A way of adjusting for these interfering variables is to compute for each institution an indirectly standardized mor- tality ratio for any particular interfering vari- able, e.g., age, sex, and operation. This has been done, and the results are summarized in Table 5. The extreme left-hand column identifies the institutions by their code numbers. The next column gives the over-all death rate (all opera- tions) for each institution. The next column, headed "Ind. S.M.R.," gives the indirectly stand- ardized mortality ratio for each institution. If such a ratio is less than unity, the institution's death rate was less than the "par" value com- puted for its population using the entire Study's death rates for the standardizing variable; if it is higher than unity, the institution's death rate was higher. Each of the next seven columns shows the mortality ratio for the death rates of each institution standardized with respect to a single variable. If the entire difference in institutional death rates were ascribable to differences in age distribution of patients, this would cause all the mortality ratios in the column headed "Age" to be 1.000. Variation among the ratios in that column reflects institutional variation not as- sociated with age distribution. Other columns are analogously interpreted. 196

Study of Table 5 shows that standardization with respect to physical status, age, operation, and shock-likelihood index (SLI)* each substan- tially reduces the institutional variation, but standardization with respect to previous opera- tion, sex, and year each has very little effect. A rough-and-ready index of variation among the mortality ratios is the ratio of the highest num- ber in the column to the lowest. These figures are displayed in the bottom row of the table. Standardization for operation has the greatest effect of all; the smallest standardized mortality ratio is 0.287 (institution 28) and the largest is 2.577 (institution 2). This is a 9-fold range, which, although smaller than the original 25-fold range, is still large. Our conclusion from Table 5 is that some, but by no means all, of the in- stitutional variation in death rates is ascribable to the interfering variables age, operation, and physical status. The part that is not stands as an unsolved problem. ADJUSTMENT FOR MORE THAN ONE VARIABLE AT A TIME Two features of the preceding presentation de- serve comment. First, the entire body of data has been summarized into three groups—the low-, middle-, and high-death-rate operations—rather than fractionated into many small, independently meaningful subclasses. Second, contrasts among the agents have been examined after adjustment for each of several interfering variables, but these adjustments have all been made for one variable at a time. Such an analysis fails to reveal many as- pects of the data. It does not answer such ques- tions as: "How do the agents compare for people 30 to 50 years old who undergo nonemergency cholecystectomies?" Many questions of this kind could be posed, and the experience of this Study can give reasonable answers to some. This ap- proach to analysis is elaborated in considerable detail in Chapter IV-3. Over-all summaries of the kind presented above occasion different an- swers, depending on which interfering variable is adjusted for. This variety in over-all answers is troublesome and invites a search for methods to adjust for all variables simultaneously. Comparison of death rates by anesthetic agent after adjusting for all the identified inter- fering variables, simultaneously, poses diffi- culties. In principle, it would appear that the complete pigeonhole array of all possible com- binations among the variables could be set forth, and standardized death rates for each agent calculated using the agent death rates and the population belonging to each of the many pigeonholes. Unfortunately, this procedure is not practical if 10 ages (or even seven), 25 op- erations (or even five), five lengths of opera- tion, seven levels of physical status (or even four), two sexes, and two classes for previous operation are used. It is not practical because there are too many ill-defined agent death rates and cell frequencies involved in a table with so many pigeonholes, compared with the number of observations. There are several basic ways of trying to take account simultaneously of all these inter- fering variables. One way is to condense the classifications greatly so that the over-all set of pigeonholes has a relatively small number of cells; for example, if the six variables above were all dichotomized, there would be only 64 pigeonholes, a manageably small number. A second way is to fit a model (e.g., regression or a parameterized multinomial) that involves all the interfering variables and the agents; then the estimates of the parameters for agent effects could be studied. A third way is to fit a model using the interfering variables (but ignoring agent) to establish strata graduated with regard to mortality; for example, the top stratum might involve a few extremely serious operations and very old, high-risk patients in certain less- serious operations, etc., and the lowest stratum might involve all subjects on some very trivial operation, and young, low-risk subjects with short operations or certain not-quite-trivial operations, etc.; once the strata have been es- tablished, the standardized death rates for the various agents can be calculated and compared. Most of the approaches just sketched have been applied in Part IV in one or more ways. In Chapter IV-3, the analysis is addressed pri- marily to making adjustments for several vari- ables simultaneously so that death rates can be compared at the level of single cells. In addition, the fitting process used for that purpose does enable construction of strata, homogeneous as to fitted death rate, and thus agents can be com- pared after standardization with respect to these strata. This is done in a part of Chapter IV-5. In Chapter IV-6, the anesthetic agents are compared in terms of standardized death rates and indirect mortality ratios, with standardization based on strata that were constructed by coalescing cells of the various cross-classifications in order of observed (rather than fitted) death rates. Chapter IV-5 compares the agents in a similar way; the only difference is that the strata were constructed by coalescing cells of various cross-classifications in order of estimated death rates as fitted by a multiple-regression approach. The methods of Chapter IV-4 may be thought of as partaking something of both cell aggregation and regression methods. These three chapters (IV-4, -5, and -6) all construct "strata" with a view to dividing the data into subgroups of relatively homogeneous death rates, which enable agents to be compared *SLI is a composite variable determined by operation and physical status; a detailed definition of it is given in Chapter IV-7. 334-553 O-69—14 197

(by standardization), with the influence of all other relevant variables largely controlled through the stratification. This discussion has used the word "stratifi- cation" in a not very usual sense, which deserves remark. Ordinarily, strata consist of identifiable sub-parts of a population that can be identified without reference to the value of the random quantity being studied. For example, in studying income, we can use occupations as strata and we know a person's stratum before we find out his income. In the present context, we have set up the "strata" in terms of what the death rate turned out to be. This is where the difference lies. In all these stratification processes,* we have ignored the anesthetic agents, so that the stratification schemes have been "blind to the *An exceptional instance arises in Chapter IV-5. agents." All these approaches, where the "strata" depend on the observed death rates, present theoretical problems, whose full understanding still awaits research not done. Where only a single variable has been ad- justed for, the categories of that single variable have defined "strata" in the classical sense of the word. The only instance in which two or more variables have been simultaneously used to yield "strata"—in the classical sense—for standardization of agent death rates is in Chapter IV-7, where the composite variable, SLI, yields the strata, and where that composite variable is an a priori combination of operation and physical status. It should be added that the composite variable, SLI, has also been treated in the sequel as one of several variables by such methods as smear and sweep, cell aggregation, and regression. 198

TABLE EE-1 PREVIOUS OPERATION vs AGENT - - ESTIMATED EXPOSED (LOW) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN .0 .0 .0 .0 .0 .0 NO PREVIOUS OP 89.8 93.3 92.9 94.0 93.2 92.5 NO HALOTHANE 3.9 5.5 5.9 5.4 4.8 5.1 HAD HALCTHANE 6.3 1.2 1.2 .6 2.1 2.4 TOTAL EE 86.6 105.1 67.3 50.0 57.3 366.3 (IN THOUSANDS) TABLE EE-2 PREVIOUS OPERATION vs AGENT - - ESTIMATED EXPOSED (MID) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E O ALL UNKNOWN .0 .0 .0 .0 .0 .0 NO PREVIOUS OP 90.3 90.7 90.9 90.7 84.6 89.6 NO HALOTHANE 4.9 8.1 7.9 8.3 12.1 7.6 HAD HALOTHANE 4.8 1.2 1.3 1.0 3.2 2.7 TOTAL EE 145.7 100.8 68.1 44.2 67.2 426.0 (IN THOUSANDS) TABLE EE-3 PREVIOUS OPERATION vs AGENT - - ESTIMATED EXPOSED (HIGH) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E O ALL UNKNOWN .0 .0 .0 .0 .0 .0 NO PREVIOUS OP 85.9 87.2 86.2 83.3 84.6 85.7 NO HALOTHANE 6.6 11.5 12.8 15.4 10.7 10.4 HAD HALOTHANE 7.4 1.3 .9 1.3 4.7 3.9 TOTAL EE 21.8 11.6 11.7 7.6 9.1 61.7 (IN THOUSANDS) 199

3.7 3.3 2.6 2.3 2.7 3.0 .4 .2 .4 .0 .1 .2 3.0 3.3 8.2 .5 2.2 3.6 .8 .6 1.5 .2 .6 .3 .0 .0 .0 .0 .0 .0 TABLE EE-4 PHYSICAL STATUS vs AGENT - - ESTIMATED EXPOSED (LOW) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 14.2 15.8 15.8 15.3 15.2 15.3 RISK 1 58.9 57.9 56.9 69.1 66.1 6C.7 RISK 2 19.0 18.9 14.5 12.5 13.1 16.3 RISK 3 RISK 4 RISK 5 RISK 6 RISK 7 TOTAL EE 86.6 105.1 67.3 50.0 57.3 366.3 (IN THOUSANDS) TABLE EE-5 PHYSICAL STATUS vs AGENT - - ESTIMATED EXPOSED (MID) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 16.2 16.3 14.9 16.3 17.7 16.2 RISK 1 -45.0 46.1 40.2 45.8 37.9 43.5 RISK 2 25.2 24.7 20.7 24.4 26.0 24.4 RISK 3 6.1 6.8 5.6 8.2 9.1 6.9 RISK 4 RISK 5 RISK 6 RISK 7 TOTAL EE 145.7 IOC.8 68.1 44.2 67.2 426.0 (IN THOUSANDS) TABLE EE-6 PHYSICAL STATUS vs AGENT - - ESTIMATED EXPOSED (HIGH) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 16.0 12.6 15.1 10.6 20.5 15.2 RISK 1 RISK 2 RISK 3 RISK 4 RISK 5 RISK 6 RISK 7 TOTAL EE 21.8 11.6 11.7 7.6 9.1 61.7 (IN THOUSANDS) .8 .6 1.0 .5 1.2 .8 4.5 3.3 11.0 2.9 4.4 5.1 1.8 1.9 5.6 1.6 3.0 2.6 .3 .3 l.l .4 .6 .5 21.7 17.2 18.0 25.9 15.5 19.8 31.8 34.1 29.1 39.8 28.9 32.3 17.4 19.4 11.9 12.5 15.4 15.8 3.6 4.C 2.8 1.9 1.9 3.1 5.2 4.1 9.0 3.1 4.5 5.3 3.5 7.6 11.5 5.9 10.3 7.1 .8 1.0 2.4 .3 2.9 1.4 200

.3 .3 .1 .0 .2 .2 16.8 31.8 26.8 9.4 12.8 21.3 33.3 37.7 32.0 37.5 38.5 35.7 31.6 2C.4 22.1 34.8 29.8 26.8 15.8 8.6 17.5 15.9 16.4 14.2 2.1 1.0 1.3 2.2 1.9 1.6 .2 .2 .0 .2 .3 .2 TABLE EE-7 LENGTH vs AGENT - - ESTIMATED EXPOSED (LOW) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 1-28 MIN 29-58 59-118 119-238 239-388 369 UP TOTAL EE 86.6 105.1 67.3 50.0 57.3 366.3 (IN THOUSANDS) TABLE EE-8 LENGTH vs AGENT - - ESTIMATED EXPOSED (MID) PERCENT CISTRIBUTICN WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 1-28 MIN 29-58 59-118 119-238 239-388 369 UP TOTAL EE 145.7 IOC.8 68.1 44.2 67.2 426.0 (IN THOUSANDS) TABLE EE-9 LENGTH vs AGENT -- ESTIMATED EXPOSED (HIGH) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 1-28 MIN 29-58 59-118 119-238 239-388 369 UP TOTAL EE 21.8 11.6 11.7 7.6 9.1 61.7 (IN THOUSANDS) .5 .0 .0 .4 1.2 .4 5.6 8.0 3.9 2.7 13.8 6.9 17.5 22.3 18.3 11.9 15.0 17.8 32.7 29.5 37.5 30.3 29.5 31.9 33.7 28.5 30.8 41.2 28.8 32.0 8.6 1C. 3 8.1 11.4 9.9 9.4 1.4 1.3 1.4 2.3 1.9 1.6 .3 .1 .1 .0 .3 .2 .9 .3 .6 .0 1.4 .7 2.1 5.3 6.0 4.5 3.5 3.9 19.0 15.8 31.2 30.1 19.1 22.1 37.4 33.1 44.5 43.5 46.0 39.9 28.6 3C.3 15.6 20.0 25.2 24.9 11.7 15.0 2.1 2.0 4.6 8.2 201

TABLE EE-10 AGE vs AGENT - - ESTIMATED EXPOSED (LOW) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN G-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90 UP TOTAL EE (IN THOUSANDS) .0 .0 .2 .0 .1 .0 20.7 2.6 12.1 55.4 52.7 23.7 9.7 7.8 5.7 11.3 7.3 8.3 12.4 17.4 18.6 5.3 6.4 13.1 16.0 21.9 21.4 8.5 8.7 16.5 15.1 18.9 21.2 7.1 9.7 15.4 11.7 14.3 11.2 5.5 6.6 1C. 8 10. I 11.1 6.7 4.4 4.6 8.1 3.2 4.6 2.4 1.9 3.0 3.3 .7 1.3 .3 .5 .7 .8 .2 .0 .0 .0 .0 .0 86.6 105.1 67.3 50.0 57.3 366.3 UNKNOWN 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90 UP TOTAL EE (IN THOUSANDS) TABLE EE-11 AGE vs AGENT - - ESTIMATED EXPOSED (MID) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE 145.7 NP IOC.8 68.1 44.2 67.2 ALL .0 .0 .1 .0 .1 .0 10.5 2.9 6.7 15.6 18.9 9.9 10.9 9.4 8.1 8.0 8.5 9.4 9.9 12.5 13.6 9.3 8.1 1C. 8 13.9 14.6 16.1 11.9 11.9 13.9 17.2 19.3 16.9 17.2 14.4 17.2 16.5 17.0 14.7 15.8 15.7 16.2 14.2 14.6 14.8 14.5 13.3 14.3 5.8 7.5 6.7 6.4 7.2 6.6 1.1 2.1 2.1 l.l 1.8 1.6 .0 .0 .2 .0 .1 .0 202

TABLE EE-12 AGE vs AGENT - - ESTIMATED EXPOSED (HIGH) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN .0 .0 .0 .0 .3 .0 0-9 12.9 6.0 8.4 10.0 26.2 12.3 10-19 20-29 30-39 4C-49 bC-59 60-69 70-79 80-89 90 UP TOTAL EE 21.8 11.6 11.7 7.6 9.1 61.7 (IN THOUSANDS) 10.6 1C. 4 4.4 5.5 6.3 8.1 8.3 7.4 10.0 5.1 3.6 7.4 12.0 8.9 10.6 9.9 7.6 1C. 2 18.1 17.3 12.1 10.0 8.6 14.4 15.3 17.9 18.7 22.9 14.5 17.3 13.4 21.5 16.2 18.6 16.6 16.6 7.9 9.4 14.2 14.7 12.7 1C. 9 1.4 1.2 5.3 3.3 3.3 2.6 .0 .0 .0 .0 .2 .0 203

TABLE EE-13 SEX vs AGENT -- ESTIMATED EXPOSED (LOW) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN KALE FEMALE • 0 8 2 .0 28.6 71.4 .0 .9 .1 • 0 2 8 .0 .0 45.1 35.2 54.9 64.8 40. 59. 20 79 47. 52. TOTAL EE (IN THOUSANDS) 86. 6 105.1 67 .3 50. 0 57.3 366.3 TABLE EE-14 SEX vs AGENT - - ESTIMATED EXPOSED (MID) PERCENT CISTR I8UT ION WITHI N COMMON PRACTICE H NP C E 0 ALL UNKNOWN WALE Ft-MALE 0 0 0 .o 46.9 53.1 .0 .7 .3 • 0 1 9 .0 .0 52.3 49.2 47.6 5C.8 51. 49. 43 56 52. 47. TOTAL EE (IN THOUSANDS) 145. 7 IOC. 8 68 .1 44. 2 67.2 426.0 TABLE EE-15 SEX vs AGENT -- ESTIMATED EXPOSED (HIGH) PERCENT LISTR IBUT ION WITHI N COMMON PRACTICE H KP C E 0 ALL UNKNOWN MALE FfcMALE • 0 2 8 .0 47.5 52.5 .0 .3 .7 0 6 .0 .0 49.8 48.6 50.2 51.4 49. 48 51 47. 52. 50. 4 TOTAL EE 21.8 11.6 11.7 7.6 9.1 61.7 (IN THOUSANDS) 204

l.l .2 .4 .3 .4 .5 7.9 31.4 26.9 32.6 23.2 23.9 17.8 26.7 27.0 29.4 25.0 24.7 28.1 24.9 23.8 20.5 24.4 24.7 45.1 16.9 21.9 17.3 27.0 26.1 TABLE EE-16 YEAR vs AGENT - - ESTIMATED EXPOSED (LOW) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 1959 I960 1961 1962 TOTAL CE 86.6 10*).1 67.3 50.0 57.3 366.3 (IN THOUSANDS) TABLE EE-17 YEAR vs AGENT - - ESTIMATED EXPOSED (MID) PERCENT LISTRIBUTICN WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 1959 1960 1961 1962 40.7 15.3 18.4 13.5 25.1 25.8 TOTAL EE 145.7 ICC.8 63.1 44.2 67.2 426.0 (IN THOUSANDS) TABLE EE-18 YEAR vs AGENT -- ESTIMATED EXPOSED (HIGH) PERCENT DISTRIBUTION WITHIN COfMCN PRACTICE H NP C E 0 ALL UNKNOWN 1959 196C 1961 1962 TOTAL FE 21.8 11.6 11.7 7.6 9.1 61.7 {IN THOUSANDS) 2.1 .7 1.0 1.0 1.5 1.4 9.1 34.8 30.3 35.1 23.6 23.6 19.6 28.3 28.2 31.5 25.1 25.1 28.5 2C.9 22.1 18.9 24.8 24.1 3.7 .2 2.2 .6 .9 2.0 7.9 32.8 25.4 39.5 21.8 21.8 IR.5 32.6 24.1 26.5 28.3 24.6 33.7 22.4 28.6 21.6 21.2 27.3 36.2 12.0 19.7 11.7 27.8 24.3 205

TABLE EE-19 INSTITUTION vs AGENT - - ESTIMATED EXPOSED (LOW) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 1 2 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3C 31 32 33 34 TOTAL EE 86.6 105.1 67.3 50.0 57.3 366.3 (IN THOUSANDS) .0 .0 .0 .0 .0 .0 6.8 6.2 .0 10.7 l.l 5.6 1.6 .8 .9 .5 .7 1.0 1.6 6.1 3.2 .9 4.6 3.5 4.5 4.5 1.0 2.8 1.4 3.1 3.0 1.7 10.3 .9 2.7 3.6 4.0 6.4 .6 l.l 5.4 3.9 .5 .3 3.7 .9 1.6 1.3 10.6 9.3 .9 14.8 4.7 8.1 1.8 7.9 1.5 24.0 2.6 6.7 6.4 1.8 2.5 3.8 1.4 3.2 3.5 2.1 1.3 5.6 2.4 2.8 3.0 1.5 10.5 .1 .8 3.2 .5 .2 .4 .0 .4 .3 2.7 4.4 5.2 .0 9.6 4.4 2.4 3.7 6.6 3.7 4.8 4.1 .7 2.3 .8 .2 2.8 1.5 1.6 1.1 .5 5.5 2.4 1.9 .3 .6 .8 .8 .2 .5 2.6 1.5 2.5 .9 .8 1.7 1.4 2.6 16.8 6.6 8.5 6.4 10.2 5.6 9.9 2.2 13.5 8.2 5.9 2.1 5.9 1.6 3.0 3.8 1.0 1.7 3.7 .0 .3 1.4 .5 2.9 .1 .0 .4 1.0 1.4 .1 1.3 .2 1.1 .8 8.3 7.4 l.l 3.0 3.9 5.3 3.0 2.2 2.3 2.4 9.7 3.6 1.8 1.6 .0 1.1 .1 l.1 .8 .C .3 .0 .3 .3 .3 .3 .1 .0 .6 .3 2.2 2.2 .8 2.6 2.0 2.0 3.6 .7 1.5 .0 2.4 1.7 1.2 6.2 .9 2.7 3.0 3.1 .2 .0 1.9 .1 l.l .6 206

TABLE EE-20 INSTITUTION vs AGENT - - ESTIMATED EXPOSED (MID) PtRCtNT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 TOTAL EE 145.6 IOC.8 68.1 44.2 67.2 425.9 (IN THOUSANDS) .0 .0 .0 .0 .0 .0 6.7 6.4 .0 .7 2.0 4.2 1.7 .9 1.4 .6 .9 1.2 .7 7.2 .9 .1 2.9 2.6 4.8 2.9 3.5 1.4 3.5 3.6 2.7 3.1 11.4 .7 2.3 3.9 3.3 5.0 .7 2.3 5.8 3.6 1.2 1.0 2.4 .4 1.7 1.3 15.6 5.5 .8 35.1 11.5 12.2 2.9 1C.1 3.1 18.3 3.3 6.3 4.5 3.2 3.8 2.1 1.0 3.3 3.8 1.9 1.9 6.2 3.3 3.2 3.3 1.4 5.8 .4 1.8 2.7 1.1 .3 1.7 .3 2.7 1.2 1.8 1.7 4.8 .2 3.3 2.4 1.1 4.3 6.8 3.0 2.9 3.2 .6 4.6 .8 .2 3.9 2.1 1.1 1.3 .5 10.2 5.6 2.7 1.1 1.2 2.3 2.5 .7 1.4 3.0 4.2 1.3 .1 1.2 2.4 1.0 2.2 13.5 6.5 7.4 4.9 7.8 7.3 7.3 1.0 7.6 6.9 5.7 1.4 5.2 l.l 3.1 3.7 .8 1.8 1.3 .0 .4 1.0 1.1 6.1 .7 .3 .4 2.0 1.2 .5 1.5 .3 1.8 l.l 6.8 4.9 1.5 .7 1.9 4.1 4.0 2.0 3.9 1.2 2.4 2.9 .9 1.2 .0 .7 .0 .7 1.0 .1 .2 .0 .7 .5 .9 .5 .3 .0 .5 .5 1.4 1.0 2.4 .2 .3 1.2 2.9 1.1 1.4 .3 2.4 1.9 2.7 3.6 1.9 2.1 4.6 3.1 .7 .1 5.0 .6 5.9 2.1 207

TABLE EE-21 INSTITUTION vs AGENT - - ESTIMATED EXPOSED (HIGH) PERCENT UISTRIBUTICN WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 1 2 3 6 7 8 9 10 11 12 13 14 Ib 16 L7 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 TOTAL EE 21.7 11.6 11.7 7.6 9.1 61.6 (IN THOUSANDS) .0 .0 .0 .0 •0 .0 4.2 2.0 .0 .9 .7 2.1 1.4 .5 2.2 .5 .6 1.2 .8 3.7 .3 .0 1.3 1.2 2.1 1.5 4.2 8.7 5.9 3.8 4.0 3.5 9.1 1.8 2.4 4.4 1.2 .9 .0 3.4 2.3 1.3 2.0 .8 1.4 1.2 3.0 1.7 20. 2 6.4 2.6 34.4 16.4 15.5 6.7 5.9 1.6 15.6 4.0 fc.3 6.3 b.3 8.3 3.5 2.1 5.5 3.0 1.8 4.6 4.6 5.2 3.6 3.3 .4 7.2 .0 1.4 2.8 1.0 .3 1.6 .2 2.8 1.2 .2 .0 3.3 .0 3.2 1.2 1.2 1.8 4.2 1.4 4.0 2.3 .9 9.3 2.3 .2 1.9 2.8 1.9 .6 .1 3.4 3.3 1.7 1.5 2.0 2.4 2.0 .6 1.7 2.7 5.6 2.1 .0 2.4 2.8 1.4 l.C 10.1 10.9 9.1 5.3 10.2 15.3 8.8 1.5 5.0 9.1 3.1 2.C 2.7 .7 2.9 2.5 .4 1.2 .5 .0 .4 .5 2.0 17.5 2.7 .3 .8 4.7 1.0 .7 2.3 .6 2.8 1.4 1.5 2.5 1.1 .0 .7 1.3 3.1 .2 4.1 .0 2.7 2.3 .4 .4 .0 .3 .1 .3 1.3 .3 .9 .0 1.3 .9 3.2 1.7 .6 .0 .4 1.6 .6 .0 2.0 .0 .1 .6 3.9 .4 1.5 .0 1.7 2.0 2.8 4.3 4.0 4.0 7.0 4.0 .0 .2 1.2 .0 1.2 .5 208

TABLE EE-22 PERCENTAGE USAGE OF COMMON PRACTICES FOR EACH OF 34 INSTITUTIONS INSTI- COMMON PRACTICES EE FOR TUTION H NP C E 0 INSTITUTION 1 41.7 38.6 .2 14.4 5.1 39888 2 43.9 19.4 19.7 6.0 11.1 9536 3 10.6 57.0 11.1 2.0 19.2 24826 4 38.9 26.9 12.1 9.2 12.8 29227 5 22.8 16.2 48.0 2.7 10.2 32884 6 27.9 39.4 3.0 6.0 23.8 30259 7 23.6 12.4 37.6 6.2 20.2 11417 8 39.8 17.6 1.6 28.0 13.1 91224 9 13.1 35.0 6.0 38.5 7.3 55452 10 46.3 19.5 17.8 10.8 5.7 29227 11 35.4 16.4 10.3 22.6 15.4 26252 12 32.6 12.0 47.3 .9 7.2 25134 13 32.3 8.5 23.0 2.6 33.7 6889 14 19.2 23.8 26.4 .5 30.1 27083 15 13.2 27.8 31.6 10.8 16.6 30271 16 11.3 51.1 8.7 1.4 27.5 16072 17 17.4 12.9 3.6 38.3 27.7 19759 18 24.0 23.0 26.9 18.9 7.2 8899 19 39.1 34.8 15.5 2.8 7.9 18325 20 6.2 10.7 45.8 14.7 22.5 47508 21 34.4 23.1 19.5 2.5 20.5 65149 22 45.4 12.1 25.1 4.2 13.2 31158 23 21.7 38.4 34.8 0 5.1 9774 24 16.2 73.3 5.6 1.2 3.7 15289 25 37.5 8.4 25.7 3.6 24.8 8445 26 46.2 34.4 5.0 4.9 9.5 37658 27 33.2 16.1 17.3 6.3 27.1 27257 28 43.1 41.8 .7 12.4 2.1 7061 29 62.7 5.7 11.1 1.2 19.3 3916 30 52.3 22.7 7.6 0 17.4 4406 31 32.8 26.5 19.0 10.8 10.8 12488 32 53.3 11.8 13.9 1.1 19.9 15530 33 21.2 39.9 9.0 9.7 20.2 26822 34 11.3 1.9 42.7 2.8 41.4 11350 EE AS PERCENT OF TOTAL 4.7 1.1 2.9 3.4 3.8 3.5 1.3 10.7 6.5 3.4 3.1 2.9 .8 3.2 3.5 1.9 2.3 1.0 2.1 5.5 7.6 3.6 1.1 1.8 1.0 4.4 3.2 .8 .5 .5 1.5 1.8 3.1 1.3 TOTAL 29.8 25.5 17.2 11.9 15.6 856435 209

TABLE EE-23 OPERATION vs AGENT - - ESTIMATED EXPOSED (LOW) PERCENT CISTRIBUTION WITHIN COHMON PRACTICE NP ALL UNKNOWN 1 3 55 60 65 73 90 TOTAL EE (IN THOUSANDS) .0 .0 .0 .0 .0 .0 18.6 11. 4 9.0 48.0 38.7 21.9 13.0 6.7 2.8 8.4 10.8 6.4 8.4 3.5 10.6 10.8 12.4 8.4 19.7 41.5 47.0 4.1 9.4 27.2 9.5 6.3 19.3 12.8 11.9 11.2 14.7 21.9 2.9 4.0 7.2 12.0 16.1 8.8 8.5 12.0 9.5 11.0 86.6 105.1 67.3 50.0 57.3 366.3 TABLE EE-24 OPERATION vs AGENT - - ESTIMATED EXPOSED (MID) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H ALL UNKNOWN 1 2 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 TUTAL EE 'IN THOUSANDS) .0 .0 .0 .0 .0 .0 32.3 36.1 28.2 33.4 24.8 31.5 12.0 9.9 14.3 8. 1 14.1 11.8 11.9 12.0 8.8 9.2 7.1 1C. 4 4.8 5.0 3.8 3.7 7.3 5.0 3.6 2.8 6.6 4.1 5.4 4.3 2.4 4.7 1.4 2.6 3.7 3.0 3.6 4.0 1.6 1.9 7.1 3.8 1.2 1.2 1.5 2.1 1.5 1.4 3.8 2.4 3.3 3.2 2.2 3.1 1.9 1.9 1.0 1.6 1.6 1.7 2.3 1.8 2.5 2.2- 1.7 2.1 1.4 1.3 1.5 2.1 1.8 1.5 1.4 l.fi 1.8 1.0 1.4 1.5 1.2 1.3 3.1 2.8 1.7 1.8 .4 .3 .7 l.l 1.1 .6 3.0 3.1 .6 .8 2.0 2.2 3.0 2.2 1.6 2.6 1.9 2.4 2.1 1.9 7.1 8.1 4.0 3.8 2.2 2.C 1.5 2.7 2.3 2.1 2.2 1.9 1.4 1.0 2.2 1.9 1.5 1.1 2.3 3.0 1.9 1.7 .4 .5 1.6 .8 ,7 .7 .3 .2 .2 .2 .4 .3 .8 .6 2.8 1.7 1.5 1.3 .2 .1 .9 .1 .3 .3 145.6 ICC.8 68.1 44.2 67.2 426.0 210

TABLE EE-25 OPERATION vs AGENT ~ ESTIMATED EXPOSED (HIGH) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL UNKNOWN 12 33 44 48 TOTAL EE 21.8 11.6 11.7 7.6 9.1 61.7 (IN THOUSANDS) .0 .0 .0 .0 .0 .0 43.4 32.4 1.5 26.1 15.8 27.2 17.6 19.8 7.1 .0 19.1 14.1 19.2 22.6 46.6 37.2 36.3 29.8 19.8 25.3 44.8 36.7 28.9 29.0 211

UNKNOWN 1 ? 3 <t 5 (» 7 a 0 10 12 13 Ib 16 17 20 21 22 23 25 26 27 28 30 31 32 3* 14 36 39 40 41 42 43 44 45 46 47 4!i ', 0 -JL 52 5^ 55 58 59 TABLE EE-26 OPERATION vs AGENT - - ESTIMATED EXPOSED (ALL) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL .1 .0 .0 .0 .1 .0 6.5 5.6 4.1 24.0 16.7 9.6 .9 .9 .0 .9 .6 .7 4.6 3.3 1.3 4.2 4.7 3.6 1.1 1.8 .2 1.2 .8 1.1 .6 .4 .0 .3 .3 .4 .9 1.0 .2 .5 .2 .7 .3 .3 .2 .5 .2 .3 .1 .0 .0 .3 .1 .1 .2 .1 .2 .0 .1 .2 1.9 1.1 1.2 1.7 1.9 1.5 3.8 1.8 .1 2.0 1.1 ?.o .5 .3 .0 .2 .4 .3 .3 .0 .0 .0 1.5 .4 1.6 1.4 .0 .3 2.8 1.3 .0 .1 • 0 .1 .1 .0 1.3 l.l .3 .9 2.7 1.2 .8 .8 .0 .3 .9 .6 .2 .0 .1 .0 .2 .1 .3 .3 .0 .3 .3 .2 .0 .0 .0 .2 .2 .1 .5 .2 .2 .3 .3 .3 1.8 1.0 .7 l.l .9 1.2 .2 .1 .2 .2 .1 .2 3.2 3.7 3.0 1.0 1.1 2.7 .0 .0 .0 .0 .0 .0 1.0 .8 .6 .8 .4 .8 1.6 1.1 .6 .0 1.3 1.0 1.3 .9 .6 .4 1.1 .9 .6 .5 .2 .8 .5 .5 .1 .0 .2 .1 .1 .1 1.6 1.1 2.9 1.7 1.1 1.6 .8 .6 1.0 .7 .6 .7 .7 .6 1.4 1.3 .9 .9 .0 .0 .0 .0 .0 .0 1.7 1.2 3.7 2.8 2.5 2.2 1.2 .9 3.3 3.6 2.0 1.9 .1 .0 .3 .0 .1 .1 .5 .3 1.3 .8 .8 .6 1.3 1.4 3.6 2.8 2.0 2.1 1.9 1.1 3.5 1.3 1.4 1.8 .3 .1 .3 .5 .5 .3 .1 .0 .4 .0 .2 .1 .9 1.1 .8 .6 .9 .9 3.0 1.7 4.9 5.4 5.4 3.6 .0 .0 .0 .0 .0 .0 .9 .5 1.0 1.3 .9 .9 .4 .4 .3 .3 .4 .4 .3 .3 .7 .3 .4 .4 212

TABLE EE-26 (Cont'd) OPERATION vs AGENT - - ESTIMATED EXPOSED (ALL) PERCENT DISTRIBUTION WITHIN COMMON PRACTICE H NP C E 0 ALL 60 62 63 65 66 67 68 70 71 72 73 75 76 7f 80 81 «4 85 86 87 88 69 90 92 93 95 9H 99 TOTAL hC (IN THOUSANDS) 6.9 2C.5 21.5 2.1 4.1 11.9 .0 .3 .9 .0 .3 .3 .0 .0 .3 .0 .0 .0 3.3 3.1 8.8 6.4 5.1 4.9 .9 1.0 3.2 1.4 1.0 1.4 .0 .0 .0 .0 .0 .0 .3 .4 .8 .5 .4 .4 1.3 2.2 .6 1.1 1.8 1.5 1.1 1.1 1.4 1.9 1.0 1.2 .2 .2 .0 .0 .1 .1 5.2 10.8 1.3 2.0 3.1 5.2 1.5 .8 1.4 1.2 .9 1.2 .6 .5 .7 .6 .8 .6 .6 .5 .5 .8 .6 .6 1.8 1.4 .6 .3 .6 l.l 2.9 1.6 1.5 1.4 2.1 2.0 3.8 2.5 1.0 1.8 1.9 2.4 1.9 1.0 .8 .7 1.1 1.2 1.8 L.4 .3 .4 1.0 1.1 .4 .3 .4 .2 .2 .3 .8 .9 .8 .5 .7 .8 .8 .6 1.0 2.0 .8 1.0 5.7 4.3 3.9 6.0 4.1 4.8 2.5 2.1 1.7 1.0 4.2 2.3 .7 .5 .5 .8 .8 .6 1.3 1.0 1.4 .5 .7 l.l .8 .4 .0 .2 .2 .4 .0 .0 .0 .0 .0 .0 245.9 2U.O 147.2 100.0 132.6 83H.7 334-553 O-69—15 213

TABLE DR-1 PREVIOUS OPERATION vs AGENT - - DEATH RATES (LOW) UNKNOWN ,NU PREVIOUS UP NO HALJTHANE HAD HALOTHANE LVLR-ALL STANDARDIZED LEATH RAIE STANDARD (RATIO) INDIRECT M.R. H NP 0 ALL 0/0 0/0 0/0 0/0 0/0 0/0 00210 .00162 .00211 .00117 .03194 .00181 00913 .00742 .00873 .00296 .01340 .00325 00713 .00559 .00991 .00000 .01830 .00841 .00269 .00198 .00259 .00126 .03284 .00230 .00258 .00201 .00263 .00123 .00293 .00230 1.12421 .87474 1.14594 .53562 1.27422 1.08807 .88490 1.14219 .57205 1.25955 TABLE DR-2 PREVIOUS OPERATION vs AGENT - - DEATH RATES (MID) UNKNOWN NO PREVIOUS UP NO HALUTHANE HAD HALOTHANE DVfcR-ALL STANDARDIZED UEATH RATE STANDARD (RATIU) INDIRECT h.R. H NP U ALL 0/0 0/0 0/0 0/0 0/0 0/0 01518 .01519 .02862 .01671 .02697 .01931 03784 .03416 .08156 .03780 .04448 .04606 03455 .04289 .10270 .02123 .04972 .04314 .01727 .01709 .03401 .01854 .02988 .02207 ,01750 .01745 .03483 .01849 .02898 .02207 79275 .79055 1.57806 .83768 1.31295 .78731 .77943 1.58125 .84619 1.28431 214

TABLE DR-3 PREVIOUS OPERATION vs AGENT - - DEATH RATES (HIGH) UNKNOWN NO PREVIOUS OP NU HALOTHANE liAD HALOTHANE UVFR-ALL STANDARDIZED DEATH RAIE STANDARD (RATIO) INDIRECT i-i.R. H NP 0 ,08763 .10401 .93880 1.11427 1.08928 .65251 1.24920 .92943 1.11439 1.05155 .61601 1.28239 ALL 0/0 0/0 0/0 0/0 0/0 0/0 .08201 .09486 .08784 .05854 .10831 .08678 .12606 .14971 .14676 .07058 .18151 .13541 .10249 .17416 .27333 .08491 .11618 .11834 .08661 .10261 .09798 .06076 .11715 .09335 10168 .06091 .11661 .09335 TABLE DR-4 PHYSICAL STATUS vs AGENT - - DEATH RATES (LOW) H NP C E UVER-ALL STANDARDIZED DEATH RATE STANDARD (RATIO) INDIRECT .00269 .00198 .00259 .00126 .00284 ALL UNKNOWN .00365 .00150 .00356 .00104 .00321 .00257 !U SK 1 .00045 .00038 .00034 .00026 .00034 .00036 R I SK 2 .00308 .00331 .00316 .00272 .00320 .00315 RISK 3 .01968 .01/88 .02128 .01548 .02179 .01924 RISK 4 .04192 .03158 .03543 .08000 .23171 .05650 RISK 5 .00116 .00115 .00126 .00778 .00623 .00183 RISK 6 .03360 .02508 .02647 .04348 .06571 .03351 RISK 7 .53333 .09/22 .16923 .08000 .54167 .20398 .00230 .00264 .00192 .00235 .00210 .00349 .00230 1.14933 .83521 1.02436 .91190 1.51715 1.08144 .83644 .96755 .73254 1.47004 215

UNKNOWN KISK KISK KI SK KISK RISK :cISK RISK TABLE DR-5 PHYSICAL STATUS vs AGENT - - DEATH RATES (MID) H NP C E UVER-ALL STANDARDIZED DEATH RATE STANDARD (RATIU) IMDIRECT M.R. 0 .01727 .0170" .03401 .01855 .02988 .01941 .02855 .02119 .02532 ,88013 .87930 1.29332 .96003 1.14707 ,87097 .87120 1.22261 .91890 1.15360 ALL 01778 .01507 .02927 .01896 .02328 .01991 00192 .00172 .00335 .00256 .00356 .00238 01517 .01600 .02555 .01476 .02116 .01775 06917 .06922 .10561 .05521 .06916 .07241 15543 .15395 .18664 .22426 .18445 .17272 01123 .01231 .01358 .02075 .02326 .01444 10898 .12158 .12488 .14422 .14384 .12531 P5232 .23720 .33753 .28507 .38659 .31475 .02207 .02207 UNKNOWN KISK 1 KISK RISK KISK RISK KISK KISK 2 3 V b 6 7 TABLE DR-6 PHYSICAL STATUS vs AGENT - - DEATH RATES (HIGH) H NP C E 0 uVtR-ALL STANDARDIZED I'EATH RATE STANDARD (RATIU) INDIRECT ,08663 10263 ALL 11369 .08682 .08147 .06148 .09957 .09642 C2454 .02452 .01634 .01355 .03358 .02243 05471 .06893 .04941 .03812 .05859 .05470 12471 .16599 .18187 .11453 .15306 .14605 15892 .23140 .18582 .17919 .35531 .20587 04665 .06759 .05725 .04898 .07606 .05699 P2143 .12996 .17615 .16003 .17741 .17439 44164 .43689 .36689 .62121 .43348 .42410 .09800 .06080 .11720 .09337 .09467 .09632 .09254 .07787 .10494 .09337 1.01393 1.05300 .99112 .83396 1.12387 .98819 1.07250 .96583 .77747 1.11162 216

UVbR-ALL TABLE DR-7 LENGTH vs AGENT - - DEATH RATES (LOW) H NP ALL UNKNOWN .01333 .00735 .00000 .00000 .02913 .01144 1-28 MIN .00069 .00075 .00078 .00043 .03164 .00081 29-58 .00153 ,.00164 .00130 .00021 .00145 .00132 59-118 .00309 .00306 .00448 .00121 .03292 .00293 119-238 .00501 .00438 .00413 .00365 .03517 .00452 239-388 .01053 .00639 .01772 .00360 .01327 .01009 J89 UP .02286 .02299 .02083 .03846 .01005 .02077 .00269 .00196 .00259 .00126 .00284 .00230 STANDARDISED DEATH RATE STANDARD (RATIO) .00247 .00235 .00275 .00114 .03268 .00230 1.07589 1.02224 1.19433 .49523 1.15420 INDIRECT M.R. 1.07798 1.03ti35 1.15552 .48846 1.12542 TABLE DR-8 LENGTH vs AGENT - - DEATH RATES (MID) H NP C E 0 .01727 .01709 .03401 .01855 .02988 ALL UNKNOWN * 01606 • 05714 • 26190 • 02128 • 02331 .02722 1-28 MIN • 00390 • 00481 • 01157 • 00673 • 03748 .00609 29-58 • 00586 • 00b&7 • 01341 • 00607 • 01515 .00831 59-118 • 01077 • 01215 • 02421 • 01254 • 02252 .01551 114-238 • 02137 • 02309 » 04307 • 01875 • 03923 .02734 239-388 • 041.30 • 03333 • 07729 • 03788 • 05380 .04697 3b9 UP • 09914 . 1 Id83 • 13291 • 07024 • 11555 .10701 .02207 STANDARDIZED DEATH RATE STANDARD (RATIO) .01736 .01809 .03551 .01648 .03069 .78658 .81973 1.60864 .74681 1.39039 ,02207 INDIRECT M.R. .77941 .80157 1.59893 .728t)4 1.38169 217

LiVtR-ALL TABLE DR-9 LENGTH vs AGENT - - DEATH RATES (HIGH) H IMP C ,08662 10262 .09799 .06078 .11717 ALL UNKNOWN • 09859 • 13333 • 07692 1. 00000 • 10000 .10769 1-28 MIN • 03590 • 07692 • 09589 1. 00000 • 05263 .05669 29-58 • 08317 • 03785 • 08786 • 04972 • 16931 .08142 59-118 • 05023 • 08897 • 10067 • 05455 • 12720 .08029 119-238 . 07209 • 07614 • 08536 • 05708 • 09442 .07742 239-388 • 10603 • 10439 • 10394 • 06064 • 12052 .10532 .ib9 UP • 14126 • 18362 • 26866 • 21990 • 23636 . 17 i73 .09336 STANDARDIZED DEATH RATE STANDARD (RATIO) .08237 .09440 .11011 .07978 .12364 .09336 .88232 1.01108 1.17938 .85455 1.32432 INDIRECT M.R. .87587 1.01109 1.17219 .68932 1.35117 OVER-ALL TABLE DR-10 AGE vs AGENT - - DEATH RATES (LOW) H rtP .00269 .00198 .00259 .00126 .00284 ALL UNKNOWN • 01316 0/0 . 00000 0/0 • 00000 .00385 0-9 • 00178 • 00110 • 00208 • 00069 • 00093 .00114 10-19 • 00107 • 00073 • 00103 • 00000 • 00096 .00076 20-29 • 00139 • 00071 • 00080 • 00150 • 03271 .00108 30-39 • 00123 • 00035 • 00111 • 00118 • 03160 .00089 40-49 • 00176 • 00100 • 00133 • 00226 • 00215 .00145 50-59 • 00314 • 00245 • 00290 • 00324 • 03887 .00339 60-69 * 00487 • 00411 • 00748 * 00407 • 01048 .00541 7J-79 * 01341 • 00856 • 01846 • 00730 • 01329 .01165 80-89 • 03448 • 01797 • 08190 • 00741 • 03623 .02818 90 UP • 01351 • 13462 • lllll 0/0 • 03030 .05000 ,00230 STANDARDIZED DEATH RATE STANDARD (RATIU) .00264 .00175 .00333 .00192 .00381 1.14677 .76067 1.45030 .83438 1.65565 .00230 INDIRECT M.R. 1.10858 .71451 1.27633 .71484 1.41535 218

UNKNOWN 0-9 10-19 20-29 3C-39 40-49 50-59 60-69 70-79 80-89 90 UP OVER-ALL TABLE DR-11 AGE vs AGENT - - DEATH RATES (MID) H NP C E 0 ALL 01613 .12500 .00000 .00000 .01408 .01230 02050 .01427 .03966 .02054 .02426 .02332 00334 .00366 .00752 .00282 .00802 .00462 00586 .00458 .00613 .00265 .01146 .00594 00716 .00596 .01023 .00474 .00894 .00746 01087 .01015 .02038 .00601 .01820 .01266 01769 .01828 .03892 .01820 .03022 .02296 02688 .02773 .05434 .03092 .04271 .03451 05135 .04938 .10367 .05972 .07734 .06495 10732 .07429 .13260 .14017 .15274 .11416 26804 .41429 .18354 .07500 .33945 .26160 .01727 ,01709 .03401 .01855 .02988 02207 STANDARDIZED DEATH RATE STANDARD (RATIO) .01831 .01701 .03405 .01838 .02820 .02207 ,82961 77076 1.54251 .83255 1.27759 INDIRECT M.R. .82036 76624 1.53528 .82643 1.26605 UNKNOWN C-9 10-19 i-C-29 30-39 40-49 50-59 60-69 70-79 80-89 90 UP UVER-ALL TABLE DR-12 AGE vs AGENT - - DEATH RATES (HIGH) H NP C U ALL .00000 0/0 .00000 0/0 .00000 .oocoo .08309 .14020 .14645 .08785 .09681 .10186 .05979 .07875 .05839 .02093 .09479 .06529 .06298 .10685 .03239 .04000 .11260 .06601 .08104 .13260 .04637 .02461 .09766 .07891 .08579 .11077 .08382 .06863 .13925 .09473 .11013 .10814 .08256 .04334 .12608 .09587 .09693 .06572 .13462 .07218 .11183 .09576 .08555 .11093 .12808 .07749 .13479 .10775 .13928 .19101 .13250 .11744 .20833 .15216 .41667 1.00000 .57895 1.00000 .33333 .50000 .08663 10262 .09802 .06079 ,11718 .09337 STANDARDIZED DEATH RATE STANDARD (RATIU) .08907 .10887 .09704 .06010 .11906 .09337 .95397 1.16603 1.03924 .64369 1.27517 INDIRECT M.R. .94874 1.14026 1.02764 .61434 1.22769 219

UNKNOWN KALE FtMALE UVER-ALL TABLE DR-13 SEX vs AGENT - - DEATH RATES (LOW) H NP C ALL .04000 0/0 .00000 .00000 .00000 .02083 .00313 .00308 .00487 .00097 .03312 .00292 .00237 .00154 .00199 .00151 .00260 .00196 .00269 .00198 .00259 .00126 .00284 .00230 STANDARDIZED DEATH RATE STANDARD (RATIO) .00265 .00209 .00300 .00132 .00279 .00?30 1.15133 .90845 1.30611 .57551 1.21165 INDIRECT M.R. 1.14473 .88841 1.19965 .52053 1.18607 UNKNOWN KALE htMALt OVER-ALL TABLE DR-14 SEX vs AGENT - - DEATH RATES (MID) H NP C ALL .00000 0/0 0/0 0/0 .03571 .03030 .02091 .02120 .04620 .02176 .03478 .02708 .01345 .01343 .02432 .01501 .02444 .01717 .01727 .01709 .03401 .01854 .02988 .02207 STANDARDIZED DEATH RATE STANDARD (RATIO) ,01714 .01727 .03513 .01835 .02955 .02207 77642 .78259 1.59187 .83150 1.33883 INDIRECT N.R. .77213 .77857 1.59996 .82606 1.34500 220

UNKNOWN MALE FLMALE UVhR-ALL TABLE DR-15 SEX vs AGENT - - DEATH RATES (HIGH) H NP C E 0/0 0/0 0/0 0/0 .10001 .11U12 .11022 .06900 .07325 .09570 .08625 .05318 0 ALL 3/0 0/0 12650 .10431 13767 .08274 .08661 .10261 .09798 .06077 .11715 .09335 STANDARDIZED DEATH KATE STANDARD (RATIU) .H8641 .10279 .09804 .06096 .11693 .09335 .92571 1.10118 1.05026 .65306 1.25265 INDIRECT M.R. .91958 1.11364 1.05587 .63010 1.23482 UNKNOWN 1959 1960 1961 1962 UVtR-ALL TABLE DR-16 YEAR vs AGENT - - DEATH RATES (LOW) H NP 0 ALL 00000 .00000 .00000 .00000 .03000 .00000 00378 .00230 .00204 .00110 .03263 .00219 00233 .00221 .00329 .00116 .03279 .00237 00320 .00164 .00224 .00176 .03371 .00250 00240 .00157 .00284 .00116 .03232 .00219 ,00269 .00198 .00259 .00126 .03284 .00230 STANDARDIZED DEATH RATE STANDARD (RATIO) .00290 .00191 .00260 .00129 .03284 .00230 1.26136 .83176 1.13028 .55971 1.23621 INDIRECT M.R. 1.18112 .85875 1.12689 .54700 1.23449 221

UVER-ALL TABLE DR-17 YEAR vs AGENT - - DEATH RATES (MID) H NP 0 ,01727 ALL UNKNOWN .00000 .00000 .00000 .00000 .00000 .00000 1959 .01681 .01721 .03085 .01924 .03196 .02265 1960 .01666 .01577 .03565 .01596 .03153 .02216 1961 .01963 .01835 .03652 .02063 .02999 .02365 1962 .01688 .01830 .03542 .02112 .02783 .02113 ,01709 .03401 .01854 .02988 .02207 STANDARDIZED DEATH RATE STANDARD (RATIO) INDIRECT M.R. ,01724 .01717 .03419 .01898 .02988 .02207 ,78121 .77808 1.54914 .85985 1.35381 78995 .76117 1.54840 .82974 1.36460 UVi-R-ALL TABLE DR-18 YEAR vs AGENT - - DEATH RATES (HIGH) H NP C E ,08662 10261 ,09798 .06079 .11716 ALL UNKNOWN .00000 .00000 .00000 .00000 .00000 .00000 1959 .10131 .10342 .11324 .05768 .12919 .09950 1960 .10303 .09227 .11293 .06704 .12004 .10059 1961 .08112 .09871 .07147 .05677 .13567 .08627 1962 .08799 .13583 .10667 .06716 .09310 .09520 ,09336 STANDARDIZED DEATH RATE STANDARD (RATIU) ,09121 .10541 .09822 .06103 .11758 .09336 .97706 1.12907 1.05213 .65370 1.25946 INDIRECT M.R. .95480 1.07415 1.05757 .60992 1.26433 222

UNKNOWN 1 2 3 4 5 6 7 d 9 1G 11 12 13 14 15 16 17 16 19 20 21 22 13 24 25 26 27 28 29 30 31 32 33 3* uVER-ALL TABLE DR-19 INSTITUTION vs AGENT - - DEATH RATES (LOW) H NP ALL 0/0 0/0 0/0 0/0 0/0 0/0 .00119 .00035 .01515 .00000 .03482 .00068 .00910 .01044 .01085 .00000 .01571 .00980 .00000 .00124 .00047 .00234 .00114 .00100 .00155 .00084 .00436 .00215 .00738 .00191 .00424 .00762 .00302 .00216 .00717 .00433 .00029 .00178 .00686 .00557 .00032 .00141 .00837 .00940 .00281 .00221 .00987 .00516 .00142 .00051 .00168 .00149 .03186 .00118 .00187 .00204 .00495 .00117 .OD341 .00180 .00357 .00484 .00885 .00156 .03384 .00422 .00330 .00322 .00796 .00287 .03581 .00390 .00651 .00381 .00071 .00000 .01499 .00297 .00924 .00000 .00830 .00000 .00000 .00513 .00211 .00065 .00172 .00000 .00090 .00119 .00237 .00285 .00113 .00000 .03254 .00187 .00462 .00162 .00177 .00000 .00251 .00222 .00142 .00000 .00290 .00109 .03293 .00142 .01261 .00000 .00576 .00246 .OW24 .00676 .00539 .00391 .00419 .00000 .00895 .00459 .00169 .00221 .00053 .00211 .03227 .00137 .00136 .00051 .00090 .00184 .03039 .OOC86 .00271 .00277 .00226 .00000 .03115 .00224 .00000 .00169 .00121 0/0 .03000 .00114 .01656 .00690 .08333 .00000 .04206 .01165 .01144 .01266 .01764 .00000 .01713 .01410 .00140 .OOlfl .00538 .00000 .03221 .00154 .00307 .00127 .0050H .00000 .03072 .00173 .00063 .00058 .00000 .00000 .00000 .00051 .00742 .02632 .00565 .03704 .01031 .00901 .01379 .01038 .02062 0/0 .03804 .01144 .00211 .00214 .00938 .00000 .03351 .00250 .00255 .00143 .00499 .00000 .03148 .00257 .00382 .00336 .00158 .00291 .03176 .00301 .00575 .00000 .00076 .00000 .01061 .00399 .00270 ,00198 .00259 .00126 .03284 .00230 STANDARDIZED DEATH RATE STANDARD (RATIO) .00269 .00233 .00494 .00137 .03409 1.16799 1.01519 2.15054 .59642 1.77922 ,00230 INDIRECT M.R. 1.09606 .90638 .98075 .63944 1.32422 223

TABLE DR-20 INSTITUTION VB AGENT - - DEATH RATES (MID) H NP ALL UNKNOWN 1 2 3 4 b 6 7 H 9 10 11 12 13 14 1>» 16 17 18 19 20 21 22 23 it* 25 26 27 26 29 30 31 3? 33 UVER-ALL 0/0 0/0 0/0 0/0 0/0 0/0 01247 .00460 .13514 .00915 .01765 .01022 06195 .06720 .11907 .05479 .09841 .07812 01194 .OOH43 .02483 .00000 .01549 .01089 00398 .02198 .06240 .06056 .04888 .02862 04534 .05396 .02745 .09668 .06346 .04151 01185 .01300 .02954 .03494 .01113 .01417 04189 .03216 .03736 .17561 .08657 .05263 01301 .01058 .04044 .01166 .00869 .01199 02527 .01488 .03089 .01605 .02481 .01698 02019 .02966 .06577 .03163 .07588 .03460 01053 .01343 .02846 .02631 .04236 .02115 01990 .00919 .02692 .01765 .05513 .02485 0207? .01813 .05281 .01613 .02392 .02918 00631 .00283 .01235 .03030 .01026 .00879 02260 .01513 .02695 .02861 .04760 .02587 03186 .02657 .07083 .03093 .03106 .03244 01517 .00297 .02550 .00243 .01482 .00^05 02349 .02617 .05105 .00534 .06098 .03120 02185 .00964 .05945 .02941 .04321 .02198 01870 .01431 .01802 .02146 .02993 .02104 01128 .01405 .02186 .02772 .01927 .01546 01811 .03094 .02472 .01266 .04186 .02379 00762 .01292 .01029 0/0 .02062 .01141 04970 .02629 .19089 .14815 .13261 .04H81 04701 .07227 .09498 .12422 .11029 .08029 00828 .Ol-i69 .05138 .01534 .01800 .01376 00552 .00853 .02194 .01119 .01585 .01111 00506 .00169 .00000 .00662 .03226 .00440 02033 .04459 .08284 .00000 .02876 .02797 03630 .03592 .05978 0/0 .03073 .03717 01431 .01331 .02939 .01449 .05069 .02C69 01393 .00647 .03529 .00000 .02228 .01702 02674 .01512 .05460 .04016 .03065 .02833 00824 .02222 .01567 .01852 .01527 .01477 .01727 .01709 .03401 .01855 .02989 02208 STANDARDIZED DEATH RATE STANDARD (RATIO) ,01879 .01790 .04441 .02994 .03488 ,85119 .81091 2.01162 1.35627 1.58020 .02208 INDIRECT M.R. .79612 .75740 1.31702 1.05388 1.40541 224

UNKNOWN 1 I 6 1 b 9 1C 11 12 13 16 I/ IB 19 20 21 22 23 24 25 26 2T 28 29 30 31 32 33 34 UVFR-ALL TABLE DR-21 INSTITUTION vs AGENT - - DEATH RATES (HIGH) H NP C E DEATH RATE STANDARD (RATIO) INDIRECT f.R. .08663 .10273 .09808 .06084 .11734 ALL 0/0 0/0 0/0 0/0 0/0 0/0 06076 .08800 1.00000 .02985 .14474 .06960 14169 .21795 .12759 .11628 .28378 .15493 06186 .07792 .09091 0/0 .10370 .07904 07186 .14070 .11252 .06901 .11221 .09466 13074 .15528 .08927 .13924 .26441 .13213 11150 .19048 1.00000 .05904 .06849 .10231 I 1725 .09091 .14595 .18919 .21221 .16013 07544 .06289 .08589 .03867 .08528 .06660 09193 .Hu39 .21212 .06969 .10099 .09595 050HO .06748 .09311 .02909 .10599 .06785 07950 .14516 .06713 .09254 .08721 .08690 08021 .21154 .06480 1.00000 .22360 .09072 13333 .16667 .14414 .00000 .06545 .11250 lllll 1.00000 .02284 1.00000 .03987 .04^44 08156 .19767 .10364 .17600 .09476 .11H19 14103 .09572 .12378 .00000 .12935 .10843 05760 .05797 .36000 .01504 .08669 .06267 08989 .15985 .11285 .01911 .13846 .10549 16597 .04889 .11957 0/0 .15326 .11664 04268 .02479 .05600 .06877 .11087 .07141 04586 .04169 .06210 .04202 .13800 .05572 10106 .15217 .11364 .03704 .13043 .11483 09000 .08387 .10294 0/0 .00000 .07989 11359 .13939 .26037 .21875 .37398 .16015 10843 .11957 .14148 .06383 .17105 .13659 09065 .12689 .13514 1.00000 .23810 .12921 07798 .29412 .08539 0/0 .07308 .08441 02439 .04167 0/0 .00000 .00000 .02367 15497 .15909 .16529 1.00000 .11364 .15000 09834 .11312 .16250 0/0 .11364 .10638 15432 .31250 .09843 0/0 .21429 .13t'04 07476 .10526 .12981 0/0 .10920 .08884 13734 .11011 .05859 .05937 .08696 .09608 04762 .09091 .12270 0/0 .12500 .11677 .09345 .08845 .11781 .14317 .13107 .13645 .09345 .94644 1.26061 1.53207 1.40255 1.46008 .95621 .98H78 1.01135 .71600 1.31755 225

UNKNOWN 1 5 6 / li 9 10 11 12 13 14 li 16 17 18 19 20 21 22 23 26 27 ?8 29 30 31 32 33 34 UVtR-ALL TABLE DR-22 INSTITUTION vs AGENT - - DEATH RATES (ALL) H NP C E 0 ALL 0/0 0/0 0/0 0/0 D/0 0/0 01124 .00356 .06731 .00037 .01846 .00732 05125 .04845 .08622 .03723 .09218 .06049 00938 .00743 .00719 .00205 .01016 .00803 00914 .01232 .05912 .03374 .04997 .02394 04244 .04766 .02144 .05771 .05530 .03512 01056 .00857 .02508 .02995 .00826 .01085 05331 .03091 .02232 .07542 .07581 .04532 01814 .00712 .03474 .01159 .01745 .01456 03466 .01300 .03499 .01089 .02469 .01730 01659 .02598 .05357 .01319 .0*718 .02661 01359 .01627 .03004 .02019 .03586 .02070 02129 .00936 .01434 .02954 .05992 .01966 03095 .02017 .05826 .01111 .02643 .03448 00535 .00171 .00778 .04000 .03488 .00518 01576 .01510 .01950 .01831 .02742 .01899 03556 .02989 .05469 .01299 .02539 .03128 01488 .00312 .02624 .00237 .01599 .00933 03299 .03529 .04930 .00592 .06569 .03544 03087 .01238 .03307 .00391 .05201 .02584 01436 .00801 .01113 .01836 .02433 .01507 01096 .01212 .01438 .01194 .01307 .01236 01688 .02382 .01732 .00612 .03135 .01933 00842 .01055 .00555 0/0 .01188 .00842 05591 .04388 .21009 .13043 .17277 .06340 03830 .06553 .07109 .07034 .09229 .06406 00713 .01000 .04012 .00542 .01326 .01030 01073 .00679 .02337 .00349 .00658 .01074 00328 .00169 .00000 .00228 .01342 .00268 03500 .06250 .07463 .04000 .03822 .04184 05341 .04498 .07182 0/0 .02538 .04817 01419 .00691 .03218 .00074 .01312 .01421 01628 .00761 .03048 .00000 .01808 .01746 03567 .01292 .04208 .02361 .02897 .02476 00854 .02283 .01525 .01558 .01756 .01561 .01872 ,01486 .02543 .01350 .02506 .01928 ^TANDAkDIZtf) UEATH RAFfc STANDARD (RATIO) .01986 .01524 .03458 .01949 .02988 .01928 1.02997 .79004 1.79313 1.01086 1.54955 INDIRECT h.R. .96998 .75605 1.19044 .79763 1.38191 226

UVtR-ALL TABLE DR-23 OPERATION vs AGENT - - DEATH RATES (LOW) H NP U .00269 ,00196 ALL UNKNOWN 0/0 0/0 0/0 0/0 3/0 0/0 i • 00099 • 00067 • 00166 • 00029 • 03054 .00066 3 • 00097 • OOC99 • 00215 • 00000 • 03000 .00072 55 • 00327 • 00355 9 00571 • 00260 • 03364 .00384 60 • 00088 • 00092 • 00082 • 00097 • 03130 .00090 6"i • 00171 • 00226 • 00262 • 00328 • 03408 .00272 73 • 00368 • 00342 • 00458 • 00249 • 00700 .00384 90 • 00760 * 00508 • 00882 • 00234 • 01104 .00689 ,00259 .00126 .00284 .00230 STANDARDIZED DEATH RATE STANDARD (RATIU) ,00228 .00200 .00306 .00147 .03329 .00230 ,99395 .87012 1.33105 .63878 1.43286 INDIRECT 1.01449 .88122 1.21383 .58257 1.31601 227

UNKNOWN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 16 17 18 19 20 21 22 23 <.4 25 UVER-ALL TABLE DR-24 OPERATION vs AGENT - - DEATH RATES (MID) H NP C E 0 ALL 0/0 0/0 0/0 0/0 0/0 0/0 001 78 .00219 .00311 .00250 .00317 .00233 00372 .00488 .00746 .00583 .00701 .00545 00719 .00615 .00829 .00538 .01099 .00730 00957 .00970 .01685 .01033 .00945 .01052 00841 .01035 .01181 .01139 .01860 .01192 01781 .01280 .02152 .01135 .02579 .01726 01996 .02006 .03526 .01675 .01959 .02075 02089 .02088 .03507 .01178 .04175 .02552 02325 .02952 .03139 .01205 .03959 .02653 02488 .02617 .05063 .04313 .03053 .03046 02514 .03636 .04815 .03153 .04136 .03469 03733 .03214 .05652 .02784 .03221 .03709 01889 .01965 .08252 .04651 .07436 .04209 04028 .03304 .05047 .03615 .05137 .04291 04018 .09122 .04839 .02800 .03624 .04393 03499 .04531 .07565 .05181 .07232 .04607 03566 .05193 .06452 .03422 .07207 .04688 04191 .05586 .06254 .02644 .05333 .04841 08156 .06462 .11581 .05063 .10599 .08225 07644 .07207 .13180 .10526 .11945 .09214 06762 .09983 .13604 .05659 .13600 .09776 10511 .08347 .10280 .08719 .10657 .09887 09627 .06989 .17910 .04651 .16573 .12033 10378 .13146 .14020 .09953 .11292 .12107 11340 .14694 .13656 .21519 .13693 .13747 .01727 .01709 .03401 ,01855 .02988 .02207 STANDARDIZED DEATH RA1E STANDARD (RATIU) ,01857 .02000 .03004 .01819 .02793 .02207 ,84126 .94218 1.36097 .82400 1.25533 INDIRECT M.R. ,82815 .90605 1.32020 .74133 1.25693 228

TABLE DR-25 OPERATION vs AGENT -- DEATH RATES (HIGH) H NP C E UNKNOWN 12 33 OVER-ALL bTANDARDIZED DEATH RATE STANDARD (RATIU) 0 ALL 0/0 0/0 0/0 0/0 3/0 0/0 .09162 .09742 .12371 .06538 .18462 .09902 .14068 .19416 .16867 1.00000 .14342 .16012 .06212 .07901 .11504 .06548 .10240 .08852 .04601 .04663 .06579 .04679 .07578 .05653 .086t)2 .10262 .09798 .06077 .11715 .09335 .07763 .09251 .11184 .20216 .12370 .09335 83155 ,99096 1.19798 2.16552 1.32504 INDIRECT M.R. .84865 1.03609 1.24483 .74537 1.24911 334-553 O-69-16 229

TABLE DR-26 OPERATION vs AGENT - - DEATH RATES (ALL) H NP C E 0 ALL UNKNOWN 1 2 3 4 5 o f U 9 10 12 13 15 16 17 20 21 22 23 25 28 30 31 32 33 34 36 39 40 41 42 43 46 47 48 50 51 52 54 55 56 57 58 59 60 62 02273 .05882 .03846 .11364 .03109 .03818 00099 .00067 .00166 .00029 .00054 .00066 00277 .00164 .00990 .00215 .03131 .00224 00097 .00099 .00215 .00000 .00000 .00072 00110 .00131 .00000 .00168 .00284 .00143 02520 .03317 .03390 .05788 .02180 .03047 00174 .00246 .01111 .00187 .03403 .00274 00375 .00000 .01351 .00435 .03386 .00422 00366 .00495 .00000 .00000 .00000 .00181 01972 .01320 .00743 .04255 .01282 .01560 00188 .00392 .00346 .00422 .03446 .00324 09162 .09742 .12371 .06538 .18462 .09902 06551 .08537 .01786 .C3382 .13201 .07419 00727 .02338 .00000 .00000 .01329 .01209 02068 .02319 .06494 .01873 .01987 .02133 18009 .05882 .08333 .03906 .04908 .08629 01012 .01137 .04183 .00887 .03888 .01138 00797 .00355 .02985 .00909 .03632 .00679 09627 .06989 .17910 .04651 .15573 .12033 01662 .00722 .00000 .00000 .01053 .01017 01778 .01914 .05797 .03030 .03846 .02854 02864 .04833 .06250 .02147 .04076 .03608 03566 .05193 .06452 .03422 .07207 .04688 03604 .03860 .03245 .03590 .08054 .04037 00101 .00076 .00114 .00289 .00068 .00101 00000 .00000 .00000 .00000 .03000 .00000 00319 .00226 .00000 .00242 .03733 .00276 14068 .19416 .16867 1.00000 .14342 .16012 07644 .07207 .13180 .10526 .11945 .09214 09299 .04944 .16787 .05495 .11303 .08611 01393 .02069 .03143 .02830 .03175 .02437 0072? .00876 .01212 .01106 .02633 .01155 02288 .03156 .04595 .03628 .04162 .03409 04028 .03304 .05047 .03615 .05137 .04291 07407 1.00000 .11111 .00000 .24000 .08920 ^6212 .07901 .11504 .06548 .13240 .08852 04191 .05586 .06254 .02644 .05333 .04841 04098 .08287 .09158 .18421 .12183 .08358 10378 .13146 .14020 .09953 .11292 .12107 04601 .04663 .06579 .04679 .07578 .05653 00324 .00297 .00601 .01027 .03908 .00543 04018 .09122 .04839 .02800 .03624 .04393 11340 .14894 .13656 .21519 .13693 .13747 00172 .00215 .00337 .00524 .03336 .00263 00327 .00355 .00571 .00260 .03364 .00384 00971 .00000 .12000 .00000 .02000 .0 it/62 06846 .09949 .13727 .05077 .14030 .09788 02601 .02162 .05354 .03409 .03373 .0310* 10511 .08347 .10280 .08719 .13657 .09887 00088 .00092 .00082 .00097 .03130 .00090 00800 .00168 .00512 .00000 .03257 .00402 230

TABLE DR-26 (Cont'd) OPERATION vs AGENT - - DEATH RATES (ALL) H NP C E ALL 63 66 66 67 68 70 71 72 73 75 76 77 BO 81 84 85 86 87 88 89 90 92 93 95 98 99 OVER-ALL 00000 .00000 .00518 .00000 .03000 .00249 00171 .00226 .00262 .00328 .03408 .00272 00563 .00457 .00667 .00488 .01159 .00641 04902 .16667 .08108 .27778 .07843 .09541 00250 .00524 .00177 .00000 .03361 .00268 01832 .01287 .01895 .01156 .02604 .01722 00144 .00451 .00720 .00053 .03152 .00315 00000 .00211 .00000 .00000 .03000 .00082 00368 .00342 .00458 .00249 .00700 .00384 02257 .02410 .03305 .01088 .04043 .02599 01799 .01170 .03304 .01585 .02041 .01988 02268 .02091 .03693 .00966 .04405 .02580 00669 .00869 .01278 .01176 .01826 .00917 00126 .00118 .00181 .00143 .03179 .00142 00268 .00381 .00470 .00708 .00619 .00396 00377 .00691 .00650 .00280 .03392 .00470 03499 .04531 .07565 .05181 .07232 .04607 00398 .00543 .00781 .00000 .03441 .00503 01889 .01965 .08252 .04651 .07436 .04209 00000 .00118 .00198 .00049 .03177 .00097 00760 .00508 .00882 .00234 .01104 .00689 00398 .00545 .01018 .00400 .03742 .00606 03768 .03039 .06737 .02584 .02582 .03627 00761 .00849 .01093 .01887 .01114 .00958 02451 .04135 .00685 .01860 .03529 .02813 0/0 0/0 0/0 0/0 3/0 0/0 ,01912 .01507 .02545 .01369 .02523 ,01954 STANDARDIZED DEATH RATE STANDARD (RATIO) .01649 .01901 .02597 .02636 .02549 .01954 .R4361 .97284 1.32859 1.34854 1.33450 INDIRECT M.R. .84071 .95^*20 1.29104 .73697 1.25339 231

APPENDIX 1 TO CHAPTER IV-2 DETAILED DEFINITIONS OF ANESTHETIC PRACTICES William H. Forrest, Jr. Stanford University School of Medicine Palo Alto, California Halothane: All operations with halothane that do not also use any of: ether, cyclopropane, methoxyflurane (code 2), Fluoromar (code 19), and miscellaneous (code 21). Nitrous oxide-barbiturate: all operations with nitrous oxide-barbiturate that do not use any of: halothane, cyclopropane, ether, methoxyflurane, Fluoromar, and miscellaneous (code 21). Cyclopropane: all operations with cyclopropane that do not use any of: halothane, ether, methoxyflurane, Fluoromar, and miscellaneous (code 21). Ether: all operations with ether that do not use: halothane, cyclopropane, methoxyflurane, Fluoromar, or miscellaneous (code 21). "Other": all operations not included above, i.e.: all operations using miscellaneous (code 21), all operations using methoxyflurane or Fluoromar, all operations using two or more of halothane, cyclopropane, or ether, and all operations using combinations or single agents that do not in- clude halothane, nitrous oxide-barbiturate, cyclopropane, or ether. 232

APPENDIX 2 TO CHAPTER IV-2 DEFINITIONS OF VARIABLES Lincoln E. Moses Stanford University Stanford, California Previous Operations: If the patient had had no operation (preceding the one that brought him into the Study) during the same or previous calendar month, this variable was scored "no previous op." If he had had one or more in that period in which halothane was used, the variable was scored "had halothane." If he had had one or more in that period, but none involved halothane, the variable was scored "no halothane." Physical Status (Anesthetic Risk): Both names appear in the text and have the same meaning. The variable has eight classes: 0 - unknown; 1 - no complicating systemic disturbance, nonemergency; 2 - moderate complicating systemic dis- turbance, nonemergency; 3 - severe complicating systemic dis- turbance; 4 - extreme complicating systemic dis- turbance; 5 - no complicating or moderate com- plicating systemic disturbance, emer- gency; 6 - severe or extreme complicating systemic disturbance, emergency; and 7 - moribund. Length: This variable has seven categories: Unknown, 1-28 min, 29 - 58 min, 59 - 118 min, 119 - 238 min, 239 - 388 min, and over 388 min. Age: There are 11 categories: "unknown" and then by decades, less than 10 years old, 10 and over and less than 20, etc. The highest category is "over 90." Sex: Male and female. Year: The calendar years 1959, 1960, 1961, and 1962. Institution: Thirty-four institutions have been arbitrarily assigned the numbers 1 through 34. Operation: The classification of operations is based on the standard two-digit operation codes. Of the 100 operation codes (00 to 99), 12 did not appear in any institution (codes 11, 14, 19, 24, 29, 37, 61, 64, 82, 83, 96, and 97); 10 pre- sented exactly one random case all together (codes 18, 35, 38, 49, 53, 69, 74, 78, 79, and 94); and one (91) had two random cases. All 23 of these operation codes were dropped. Operation code 31 (mammoplasty), with no deaths and 530 estimated exposed, was inadvertently omitted. This left 76 operation codes. Some tables pre- sent these 76 operation codes condensed to 25 groups of codes formed by combining the 76 in order of death rate. Most commonly in the Study, the 76 are grouped as high (codes 12, 33, 44, and 48), low (codes 1, 3, 55, 60, 65, 73, and 90), and middle (the remaining 65 codes). 233

APPENDIX 3 TO CHAPTER IV-2 ESTIMATION OF DEATH RATES Frederick Mosteller Harvard University Cambridge, Massachusetts Death rates are ordinarily estimated from counted data by dividing the number of deaths by the number exposed. In the National Halothane Study, the sample is used to estimate the num- ber exposed during the period, but we have a count of all the deaths. The rate of sampling of operations, when averaged over hospitals, comes to about one case in 25. Consequently, in a por- tion of the population where few cases occur, sampling variation can produce cells with sev- eral observed deaths, but too few random cases to account for them; i.e., a cell can produce more deaths than the number of estimated ex- posed. Because we have occasion to look at many breakdowns of the data, with many cells, this situation can happen noticeably often, even though relatively infrequently. When this occurs, the death rate given by deaths/(estimated exposed) either exceeds unity (a bizarre state of affairs), or is undefined be- cause the denominator is zero. Although reports of estimated death rates in excess of unity are not necessarily wrong or bad, carrying with them as they do their own urgent warning of unreliability, they can lead to misunderstanding, and they may not give as good an estimate of p, the death rate, as could be pro- vided. By and large, in this report we have used the death-rate index, number of deaths D /,\ number of deaths + estimated exposed D + EE ' ' instead of the more usual D/EE. (Here D = num- ber of deaths and EE = estimated exposed.) Esti- mate (1) has the advantage that if an estimate is made at all, as it would be unless the denomina- tor were zero, the rate cannot exceed 1.00. Secondly, unless the death rates are very high or the cells very large, a commonly used meas- ure of excellence, the mean square error, of this estimate of the true death rate, p, is smaller than that of the usual estimate. Most of our data concern death rates well under 0.5, and when the cells have large counts misunderstanding is not so likely. Once one considers using estimate (1), the question arises whether augmenting EE in the denominator by a larger multiple of the number of deaths would give even more favorable re- sults. We considered estimates of the form and computed the mean square error (average value of [f(c) - p]2) using the binomial distribu- tion with number of exposed N = 25, 50, 100, 200, and 400 for various values of c and p. Estimate (2) was replaced by 1.00 when the numerator ex- ceeded the denominator. Note that a random sampling rate of one in 25 corresponding to that used for hospitals in this Study was used here, so that, for example, when N = 25, the number observed in the sample has a probability of 0.37 of being zero. When both EE and D were zero, we made no contribution to the mean square error, on the grounds that one estimator was as bad as another in these straits. For given values of p, N, and c, we computed the percentage reduction produced by estimate (2) compared with the mean square error given by that expression when c = 0. Then, for each p, c pair, we averaged the percent reduction over the five chosen values of N. These calculations led to the summary in Table 1. TABLE 1.--SUMMARY OF AVERAGE PERCENTAGE REDUCTION IN MEAN SQUARE ERROR OF D/(oD + EE) FOR VARIOUS VALUES OF C, WITH THE AVERAGE OVER N = 25, 50, 100, 200, and 400.* p - 0.02 0.05 0.1 0.3 0.5 0.7 f(c) = D 0.0 0.2 -1 -3 -7 -17 -21 -2 0.4 -2 -6 -14 -36 -31 3 0.6 -3 -9 -18 -37 -33 22 0.8 -4 -11 -22 -40 -29 50 1.0 -5 -13 -25 -41 -21 82 1.2 -27 -34 -44 -54 -25 106 1.4 -40 -46 -56 -60 -25 137 1.6 -49 -55 -64 -62 -12 169 1.8 -55 -61 -69 -62 -2 201 2.0 -60 -66 -73 -60 a 231 2.5 -67 -73 -79 -53 33 299 3.0 -72 -78 -81 -43 57 356 3.5 -75 -81 -82 -34 78 404 4.0 -78 -83 -81 -25 96 445 4.5 -80 -84 -80 -17 112 479 5.0 -81 -85 -73 -9 126 502 *A positive percentage means an increase; a nega tive percentage, a. decrease. cD + EE (2) The basic calculations leading to the table were made by William S. Mosteller. 234

The table shows that the mean square error for c = 1 is generally less than that for c = 0, ex- cept for high values of p. Indeed, from more detailed tables not presented here, up to p =0.5, nearly all c's up to 1.00 produce reductions in mean square error for all N's up to 400. Thus, on the basis of mean square error, the use of c = 1 is preferable to c = 0. Just how large c might best be is a matter of judgment, but the table suggests that if a single value of c must be chosen, c might well have been larger than 1, perhaps 1.5 or even 2.0, for the death rates of the size we ordinarily discuss in connection with surgery. The table then offers a further reason for adding at least D to EE for the denominator of the estimate. It gives a better estimate of the death rate, and it keeps the estimated value -within its natural bounds. 235

APPENDIX 4 TO CHAPTER IV-2 FORMULAS FOR DIRECT AND INDIRECT STANDARDIZATION USED IN CHAPTER IV-2 Lincoln E. Moses Stanford University Stanford, California Let the strata be indexed by s = 1,..., S. Let the anesthetic agents be indexed by a= 1,..., A. In the sth stratum we have: dsa = deaths in stratum s where agent a was used, EE8a= estimated exposed for agent a in stratum s, nsa = dsa + EEBa, A ds+ = Zi dsa deaths (all agents) in stra- a = l turn s, psa = estimated death rate in sa stratum s for agent a. Then the directly standardized death rate (in terms of the system of strata 1,..., S) for agent a is given in the following expression: D* - P = 8+ s = sa (Note: Where some psa was undefined, it was replaced in the expression for p* by dsa PS. = "8+ = E naa deaths-plus-EE's (all a = l agents) in stratum s, S n++ = 2 ns+ total of all deaths-plus- s = l EE's, S L S=l deaths (all strata) for agent a , and And the indirectly standardized mortality ratio for agent a is given in the following expression: Vn d« / B&^ L~l na 236

Abstract of Chapter IV-3 Over-all adjusted summary rates can improve the comparison of death rates associated with different anesthetic agents by taking into account the effects of some of the underlying variables, but they do not indicate whether there are important reversals for a particular subsection of the population. The smoothed contingency-table analysis provides estimated rates that are more stable than the observed rates for each cell of a multidimensional contingency table. These rates can then be examined for trends and rever- sals within the population. Smoothed rates have been computed for each of five anesthetic groups for various strata of four population segments. The segments were formed by splitting the operations into three groups according to their crude death rate: low, middle, and high. Cholecystectomies were treated separately, as well as being included in the middle-death-rate segment. The strata were defined by risk and age, and, where feasible, by sex and operation. For each segment, the smoothed rates within strata did not vary greatly between agents. The reversals of ordering by agent between strata seemed to be random, rather than systematic. For the low-death-rate operations and for the cholecystectomies, the highest rates most often occurred with the anesthetic group Other; and for the middle-death-rate operations, with cyclopropane and Other. The lowest rates for the low-death-rate operations most often occurred with ether and nitrous oxide-barbiturate; for the middle-death-rate operations, with ether, halo- thane, and nitrous oxide-barbiturate; and for the cholecystectomies, with halothane and nitrous oxide-barbiturate. Thus, for these three segments of data, nitrous oxide-barbiturate and halothane rates are usually among the lowest three, and cyclopropane rates are among the highest three. Ether is associated with the lowest rates for the low-death-rate and middle-death- rate operations, but with the second-highest rates for cholecystectomies. This may be a reflection of its uneven pattern of usage from one institution to another. For the high-death-rate operations, this unevenness persists: ether is associated with the lowest rates for exploratory laparotomy (operation 44) and craniotomy (operation 12), is rarely used for heart and great vessel operations (operation 33), and seems to be associated with higher rates than halothane or nitrous oxide-barbiturate for large bowel operations (opera- tion 48). For the high-death-rate operations, the great increases in death rate for older persons and persons rated as poor risks are more striking than any differences between anesthetics. Rates for these operations, based on break- downs that do not include anesthetic as a variable, point up these trends and provide baselines against which the effect of new techniques may be measured. 237

CHAPTER IV-3. SMOOTHED CONTINGENCY-TABLE ANALYSIS Yvonne M. M. Bishop Harvard University Cambridge, Massachusetts Frederick Mosteller Harvard University Cambridge, Massachusetts PURPOSE Part IV of this report answers the compre- hensive question: "How does the postoperative death rate after administration of halothane compare with the rates after other commonly used anesthetics?" The question is difficult to answer, because no single rate is associated with each anesthetic--instead, there is a range of rates. Even for the same operation, the risk of death for an individual patient depends far more on the patient's sex, physical status, and age, to mention but a few of the factors involved, than it does on the particular anesthetic used. Indeed, the physician thinks of his patient as a unique case. In other sections of this report, effort has nevertheless been directed toward providing a set of over-all rates that are com- parable for each anesthetic. These have been computed by a variety of methods, but all meth- ods have achieved comparability by the device of standardization, either direct or indirect. No standardization would be necessary if there were no differences in the types of pa- tients for whom each anesthetic was used, i.e., if the population did not differ between anesthetics. Even in the presence of differing population dis- tributions, the answer to the primary question would not be difficult to obtain if, for every type of patient, the observed crude rates for each anesthetic always showed the same ordering by anesthetic. Standardization is a means of obtain- ing comparable over-all rates when the crude rates may not be comparable. They may not be comparable because: (1) the rates for a single anesthetic differ between segments of the population, (2) the relative frequency of use of anesthet- ics differs between segments of the pop- ulation, and (3) there are reversals in the crude rates for different anesthetics between seg- ments of the population. Over-all adjusted rates take into account the first two sources of noncomparability, and en- able the primary public-health question to be answered. But the anesthesiologist about to ad- minister an anesthetic is not as interested in an over-all rate based on the average for some hypothetical population as he is in the anesthetic likely to be best for this particular patient. He already knows the age, sex, operation, and physical status of the patient, and he may wish to use that knowledge in making his decision. Rates that are specific for each type of patient may help him. The two questions are of course closely related. The first concern in answering the comprehensive question must be whether a higher death rate is in general associated with a particular anesthetic; but, as soon as we suspect that the relative rates for different anesthetics may be reversed for some segments of the popu- lation, then we must examine the data further. We try to determine whether the observed re- versals could be merely random fluctuations or show clearly defined patterns. If patterns occur, then we may be able to define the segments of the population for whom a particular anesthetic is associated with a higher rate. For this pur- pose, also, we need to look at rates for each seg- ment of the population. The smoothed contingency-table analysis presented in this chapter is an attempt to provide such rates for each segment of the population; it therefore represents a long step, although not the whole trip, toward the objective appraisal of a unique case. The many strata provided repre- sent single individuals better than usual, but, of course, the physician must take the further steps of recognizing and allowing for the variables not considered in the statistical tables. If we introduce simultaneously several of the variables that describe attributes of the pa- tient and that are known to affect death rates, we find that the population has been split into a great many segments, or "cells." Breaking up our sample observations into the correspond- ing number of dimensions and then looking at the observed death rate in each cell is unsatis- factory because the number of observations in each cell becomes small and so the crude rate is not very reliable. The smoothed contingency- table analysis "smooths" the data in the cells by replacing the original counts with fitted counts based on the one-, two-, and sometimes three- dimensional margins of the original table. The analysis gains strength by paying great attention to these margins of low dimension where the number of cases in a cell is usually fairly sub- stantial. This method makes relatively few as- sumptions and, as described here, leans very hard on the original variables and the original distribution of the cases into their cells. The smoothed rates are more reliable than the crude rates. In the process of computing them, care was taken to ensure that they did not depart systematically from the observed 238

rates and, hence, nullify the examination of trends or patterns to determine whether a par- ticular subsection of the population had an unusually high rate with any one anesthetic. By providing both crude (observed) death rates and fitted death rates in the cells, we enable the reader to see the extent of the smoothing and to make alternative analyses if he prefers. For the high-death-rate operations we have also provided a smoothed analysis that does not include anesthetic as a variable, because there may be some interest in the death rates as- sociated with patients in various categories in these high-death-rate operations. This sort of use for the smoothed contingency-table analysis may turn out to be valuable in a variety of prob- lems. The contingency-table method is especially suited for the study of single cells and their comparison with one another. The method can, however, also be used as a means of defining the segments of the population that are subject to equal rates, and that can therefore be amal- gamated for the purposes of deriving over-all adjusted rates. This is done in Chapter IV-5. The reader who wishes to see results at once without a background of the method should turn to the second half of this chapter, beginning with the discussion of low-death-rate opera- tions in particular. It should be kept in mind that death rates are computed as deaths/(deaths + estimated exposed), abbreviated as "D/D+EE." GENERAL METHOD In the contingency-table analysis, we ex- amine separately the information on deaths and the information on operations performed (the random sample). Each of these two sets of data is laid out as a multidimensional contingency table. The number of dimensions of the table equals the number of variables being considered, such as age, sex, and anesthetic risk (physical status). Hereafter, "anesthetic risk" is abbre- viated as "risk." Each variable has several categories and these jointly define the cells in the table. In a three-dimensional table with four categories of age, two of sex, and four of risk, the total number of cells is 4 X 2 X4 = 32. As we have mentioned earlier, when the num- ber of variables increases, the number of cells becomes very large, and for a fixed total number of observations, the number of estimated exposed per cell (or deaths per cell) must become small. Indeed, contingency tables based on several var- iables can easily have many empty cells. Estimates based on small counts in indi- vidual cells have large sampling errors. Some- times for a large contingency table, the method of analysis is merely to look at each two-way summary table (two-dimensional margin) to see whether the two variables are "correlated." One searches for a two-factor effect involving the two variables. In the three-dimensional case of age, sex, and risk, mentioned above, such a two-way table, or "face," might display, for example, risk against sex. The presence of a two-factor effect would imply that one sex had more high-risk operations than the other sex. The weakness of this approach has often been demonstrated: if, say, older persons have more high-risk opera- tions than younger persons, an apparent two- factor effect between risk and sex might be due entirely to different age distributions for the two sexes, and would then disappear if the risk X sex table for each age were examined separately; on the other hand, a two-factor effect between risk and sex that appeared when each age was examined separately might disappear if all ages were combined and only the face table examined. Thus, in the presence of three variables, we need to examine two-factor effects for each pair of variables at every level of the remaining variable. To illustrate, if age is variable 1, sex is variable 2, and risk is variable 3, the 1-2 table* for each level of 3 must be examined, and similarly the 1-3 table for each level of 2 and the 2-3 table for each level of 1. As an example, we give in Table 1 the four 2-3 tables (sex-risk), one for each level of variable 1 (age), for the deaths from the middle-death-rate operations. Also given in the table is the ratio of fe- males to males. If there were no 2- 3 effect (sex- risk effect), then for any particular age this ratio would be the same for each risk level. In this example, there is a marked 2-3 effect: the ratio for age level 2 ranges between 0.68 and 0.99. Similarly, if there were no 1-2 effect (age-sex effect), then for any particular risk level the ratio would be the same for each age. We could examine these data for a 1-3 effect (age-risk ef- fect) by computing similar ratios for successive age levels instead of successive sex levels, and again we would find that the effect was present. Having established that all three two-factor effects are present, we can proceed to look for a three-factor effect. The absence of a three- factor effect implies that each two-factor effect is the same for each level of the third variable. In this example, it is most convenient to look at the 2-3 effect (sex-risk) and see whether it changes with age. To do this, we need to examine whether the ratio of the sex ratios for any two risk levels is the same for each age level. We can look, for instance, at the ratio of sex ratios for risk levels U and C. We see that this ratio is 0.78/0.65 for age level 1 and 0.99/0.68 for age level 2, and we conclude that the data show a marked three-factor effect. If we had found that all three two-factor effects were present, but that each two-factor effect was of about the same magnitude for each level of the remaining variable, then we could have said that there was little evidence of a •The abbreviation "1-2 table" denotes the two-way con- tingency table obtained when variable 1 Is crossed with variable 2. 239

TABLE 1.—THE AGE X SEX X RISK DISTRIBUTION OF DEATHS FOR MIDDLE-DEATH-RATE OPERATIONS Age level 1: 0-9 years + unknown 2: 10-49 years 3: 50-69 years 4: 70 years and older Risk level U: unknown A: anesthetic risks 1, 2, and 5 B: anesthetic risks 3, 4, and 6 C: anesthetic risk 7 Age level 1 Age level 2 Age level 3 Age level 4 Risk level Female Male Female Male Female Male Female Male D 69 89 119 120 178 368 183 278 A 88 118 241 283 385 797 291 436 B 239 307 414 446 612 1131 596 867 C 41 63 90 132 105 215 133 181 SEX-RATIOS Risk level Age level 1 Age level 2 Age level 3 Age level 4 U 0.78 0.99 0.48 0.66 A 0.75 0.85 0.48 0.67 B 0.78 0.93 0.54 0.69 C 0.65 0.68 0.49 0.73 three-factor effect. In the absence of a three- factor effect, more stable estimates for every cell in the three-dimensional matrix could have been obtained by taking the two-factor effects as exactly the same for each level of the remaining variable. This exemplifies the source of strength mentioned at the beginning of this chapter. This procedure can be extended to more than three variables: with five variables, for example, we can look at the five one-factor effects of each variable, the 10 two-factor ef- fects, the 10 three-factor effects, the five four- factor effects, and the one five-factor effect. After a preliminary examination of the data, a mathematical model would be chosen that in- cluded only the effects that seemed to be large. To be effective in adding strength, such a model would include a selection of only the lower- order effects. Then estimates for every cell would be obtained in accordance with the chosen model. The observed values would be compared with the fitted values in each cell of the body of the table; if the differences were not large, then the model would be accepted as giving more stable cell estimates. Accepting the model for this purpose does not mean that we believe that all other effects are zero, but only that they are small enough that the cell estimates are im- proved by neglecting those effects. The situation is quite similar to the common one in regres- sion, in which, to get improved precision, a straight-line or a quadratic function is delib- erately fitted for estimation purposes, even when it is known that the true curve is more complex. This, in very general terms, describes the approach adopted in the contingency- table analy- ses. Stable cell estimates have been obtained separately for the deaths and for the estimated exposed by fitting the same model to these two bodies of data, and then rates have been com- puted from these smoothed estimates. MODELS AND NOTATION The mathematical model of the contingency table in common use, and that used here, is the proportional one described by Birch (1). In a two- way table of sex X age, the absence of a two- factor effect means that the proportion of males is the same in every age category, and, of course, that the sexes have identical age dis- tributions. To discuss a three-variable model we need some notation. Let the count in the cell at the ith level of variable 1, the jth level of variable 2, and the kth level of variable 3 be denoted as x,|k. The + subscript indicates that a variable has been summed over. Thus, if variable 3 has K categories, jj2 1JK = 240

and, if N is the total number of observations in the table, x+J+ and x++lcare defined similarly. We have used the notation m^ to represent the fitted count in each cell. The "fitted" or "smoothed" values are obtained by fitting a theoretical model to the data. Thus, the fitted miIk are maximum likelihood estimates of the theoretical values postulated by the model. We have also used m1]k to denote the theoretical values, in the belief that this will be less con- fusing to the reader than introducing yet another notation. Suppose that, on examining the data of Table 1, we had found no large two- factor or three- factor effects. Then we could have replaced the observed cell estimates, xjjk , with fitted cell estimates, m^ , which exhibited no two-factor or three-factor effects at all. This is equivalent to saying that each m^ is proportional to the product of its three marginal proportions: mjjk = N N N k=1,2 ..... K. (1) For example, x1++/N could be the proportion of 50-year-olds, x+j+/N the proportion of males, and x++ic/N the proportion of low-risk patients. When we examined the data to see whether two-factor effects existed, we looked at ratios; then, to see whether three-factor effects existed, we looked at ratios of ratios. If we had first taken the logarithm of every cell entry, we could have looked instead at logarithmic sums and dif- ferences. The smoothed estimates can also be expressed additively in the logarithmic scale. If we take logarithms of both sides of Eq. 1, we have: logml]k = logxi++ +logx+j + + log x++k- 2 log N. (2) An attraction of the additive form is that it brings with it the usual analysis of variance notions of grand mean, row effect, column ef- fect, layer effect, and interactions of various orders. We can achieve precisely these ideas in the logarithmic model by converting to Birch's notation, shortly to be discussed. In this notation, if we do not distinguish between the theoretical values and our estimates of them, we can write Eq. 2 as: u+u1i + u2 + u3k. (3) Here we only sketch the ideas of the u's; a fuller development is given in Appendix 1. In this nota- tion, u is the grand mean of the logarithm of the theoretical counts in all LJK cells; u = 2 logmfn./IJK, l.j,k all nijjk > 0. Similarly, (u + u1(l)) is the mean of the logarithm of the counts in all JK cells at level i of variable 1, ( 2 log mni/JK); J,k ' J thus, uj(1) is the deviation of this mean from u, the grand mean. For variables 2 and 3, u2( and U3(k> are similarly defined. Because the values of u, are deviation scores, 'l(i) u2(J),andu 3(k) I i 2(j, 3(k) = 0. The advantage of changing to this notation is that it can be extended to include multifactor ef- fects by adding further terms, which are again deviation scores. In Eq. 2 we were able to adjust for the fact that log xi++, log x+.+, and log x++k were not deviation scores by subtracting 2 log N. We cannot do this so simply in models that in- clude all the two-factor effects. For example, in the three-dimensional case, if all three two- factor effects were present, but the three-factor effect were so small that it could be neglected, the model could be written: log mjjk = u u u 3(k) a2(j) U13(lk) + U23(jk) The first three subscripted u-terms are defined as deviations from u. The three additional terms in brackets represent the three two-factor ef- fects; u12(ij) represents the two-factor effect (joint effect) of variables 1 and 2 and can assume a different value for each combination of cate- gories i and j of these variables. Thus, if varia- ble 1 has two categories and variable 2 has three categories, the values of u12 can be repre- sented by a 2 X 3 matrix. An example is given in Appendix 1. In this 2X3 matrix of values of u12, every row and every column totals zero, because these two-dimensional terms are again deviations from the linear terms. The number of degrees of freedom associated with the matrix is (2-1) X (3-1) = 2. To obtain these estimates, we would use the three observed two-dimensional faces with entries x^+,x+jk,and x1+k. The fitted m^ would be obtained by an iterative process and would add to the same totals, that is: mjj+ = xjj+ foralli,j, m+jk = x+jk for a11 J'k' mi+k= x1+k for all i,k. Thus, although it is necessary to change the notation to express the model, the fitted values 241

are computed entirely from the summary x's. That is true for all models of this type that in- volve multifactor effects of lower order than the dimensionality of the table. The computations are carried out entirely from the summary edges, two-dimensional faces, or, when needed, three- dimensional faces that are involved in the lower- order effects and without further reference to the individual cells, once the summary statistics are obtained by addition. Table 2 shows the summary two-dimensional faces for the age-sex-risk data on deaths pre- TABLE 2.--THE FACES FOR THE AGE X SEX X RISK DISTRIBUTION OF DEATHS FOR THE MIDDLE-DEATH- RATE OPERATIONS Age level 1: 0-9 years + unknown 2: 10-49 years 3: 50-69 years 4: 70 years and older Risk level U: unknown A: anesthetic risks 1, 2, and 5 B: anesthetic risks 3, 4, and 6 C: anesthetic risk 7 The 1-2 table (age X sex): Age level Female Male 1 2 3 437 864 1280 1203 577 981 2511 1762 The 1-3 table (age X risk): Age level Risk level U A B C 1 158 206 546 104 2 239 524 860 222 3 546 1182 1743 320 4 461 727 1463 314 The 2-3 table (sex X risk): Sex Risk level U A B C Female 549 1005 1861 369 Male 855 1634 2751 591 seated in Table 1. These are the summary faces that would be used to fit the model given in Eq. 4. As we observed from our inspection of Table 1, we would not expect the model given in Eq. 4 to fit these data well, because the data show a large three-factor effect. When fitting a five-variable model to the middle-death-rate data, we in- cluded all 10 three-factor effects, and the entire 32 cells given in Table 1 constituted one of the summary three-dimensional faces used. With other sections of the data, we have found that it was sufficient to include only two-factor ef- fects. When the effects fitted are of low order, the summary faces have fewer cells and the entries in each summary cell are relatively stable. In one analysis for the cholecystectomies, for example, six variables were considered: three operation codes, two periods, three risk categories, three age groups, two sexes, and five anesthetic agents, for a total of 540 cells. When we scattered 672 deaths into 540 cells, many cells remained empty. The model chosen involved eight of the possible 15 two-factor ef- fects and none of the three-factor, four-factor, five-factor, or six-factor effects. The eight rele- vant summary faces provided all the necessary information for fitting the model. (The one-factor effects have always been included, but these one- dimensional totals can be derived from the face totals.) Thus, the largest face had 3X5 =15 cells, and the distribution of the 672 deaths put a fairly large count into every cell of every face, so estimates could be obtained for all 540 ele- mentary cells. Appendix 1 gives details of the method of selecting a model, the method of computing estimates from the model, and the criteria for deciding whether the model is satisfac- tory. SEGMENTS OF DATA ANALYZED Three death-rate groups divided the opera- tions into three segments: (1) the seven low-death-rate operations, (2) the four high-death-rate operations, and (3) the remaining (middle-death-rate) opera- tions (see Chapter IV-2 for description). For our contingency-table method, we analyzed these three death-rate groups separa- tely. In addition, we made a separate analysis of cholecystectomies, which were represented by three codes of operations in the middle-death- rate segment: code 40: cholecystectomy alone, code 41: cholecystectomy and/or bile duct, and code 42: cholecystectomy and/or bile duct and other major proce- dures. 242

Cholecystectomies were chosen for special consideration because they form a fairly stand- ard group of operations that are performed fre- quently in most institutions. Furthermore, we believed that they could be expected to show the deleterious effects of halothane if such effects existed. Risk 7 cases and deaths were not used in the cholecystectomy analysis, because they were moribund cases and it would be reasonable to analyze them separately; however, the number of cases (six observed cases, which means about 150 estimated exposed and 52 deaths) was so small that a satisfactory separate analysis can- not be made. The numbers of deaths, randoms (i.e., cases in the random sample), and estimated exposed for each segment are shown in Table 3. This table also gives the number of categories for each variable considered in each segment, the resulting total number of cells, and, where pertinent, the number of the table in which the death rates are presented. The definition of categories varied slightly in the different seg- ments, as discussed below and summarized in Table 5. The models fitted are given in Table 4. SELECTION OF VARIABLES We selected five variables as offering satis- factory data, being important, and being feasible for control: agent, sex, risk, age, and operation. Length of operation would have given an analysis of some value, but circumstances prevented its use in the smoothed contingency-table analysis. Other methods do use length for some analyses. Because the sample size was not large enough, it was not feasible to consider more than six variables at once, and for some segments of the data even six represented too fine a breakdown. Although other methods of analysis showed important differences between institutions (see Chapter IV-6), we could not include institution as a variable in the contingency-table method. Treating each institution separately makes too many cells with too few operations per cell; at- tempts to put institutions into homogeneous classes failed. We also analyzed the middle-risk operations, omitting the effect of sex differences. In analyz- ing the low-risk operations, we pooled the seven operation codes. CATEGORIES PER VARIABLE Whenever the variable "agent" is used, it has five categories, corresponding to the five common-practice groups. The principal constitu- ents were: agent 1: halothane, agent 2: nitrous oxide-barbiturate (N-B), agent 3: cyclopropane, agent 4: ether, and agent 5: Other and combinations of above. As explained in Appendix 1 to Chapter IV-2, these principal constituents are seldom, if ever, used alone. We will, however, in our discussion use them as the names for our five categories and, for the sake of brevity, speak of category 5 as "Other." The dichotomous variable "sex" has females in the first category, males in the second. This variable is used for all segments except models B for middle and high death rate. In many but not all institutions, it is cus- tomary to rate the patient, before operation, on a seven-point scale of operative risk, called "anesthetic risk" or "physical status." Because this information was not available for all institu- tions, there is also a large category of "unknown" risk. For high- and middle-death-rate opera- tions, the risk classifications have been grouped according to their associated death rates. Thus, low-risk emergencies have been grouped with low-risk nonemergencies, leading to the follow- ing categories: category U: unknown; category A: risks 1, 2, and 5; category B: risks 3, 4, and 6; and category C: risk 7. For cholecystectomies, category C was omitted. For the low-death-rate operations, cate- gory C was omitted and the emergency opera- tions were separated from the nonemergency, so that the categories were: category U: unknown, category Al: risks 1 and 2, category Bl: risks 3 and 4, category A2: risk 5, and category B2: risk 6. The age categories were: ages 0-9 years plus a very few un- known ages, ages 10 - 49 years, ages 50 - 69 years, and ages 70 years and older. For the cholecystectomies, the count in the first of these categories was small and so was com- bined with that in the next category. The variable "operation" was omitted from the low-death-rate analysis. For cholecystec- tomies, this variable had three categories; codes 40, 41, and 42. The high-death-rate segment had four categories: codes 48, 44, 12, and 33, in 243

TABLE 3.—BREAKDOWN OF DATA INTO SEGMENTS AND CELLS Death-rate segment Deaths Observed cases Estimated exposed Categories per variable Total no. of Table of rates Agent Sex Risk Age Operation cells Low* 803 15,577 366, 189 5 2 5 4 -- 200 6 Middle, model A 9615 19,490 426, 033 5 2 * 4 5 800 (not given) Middle, model B 9615 19,490 426, 033 5 — 4 4 5 400 16 Cholecystectomy* 672 1,220 27, 620 5 2 3 3 3 270 21 High, model A 6354 2,875 61, 720 « 2 4 4 4 128 24 High, model B 6354 2,875 61, 720 5 ~ 4 4 4 320 26 *Risk 7 cases are omitted from the low-death-rate and Cholecystectomy segments. TABLE 4.—MODELS FITTED Death-rate segment No. of variables 1 -dimensional margins 2 -dimensional faces 3 -dimensional faces Low 4 all 4 all 6 none Middle, model A 5 all 5 all 10 all 10 Middle, model B 4 all 4 all 6 all 4 Cholecystectomy 5 all 5 agent X sex none High, model A High, model B* 4 4 all 4 all 4 agent x risk agent x age agent X operation risk x age risk X operation age X operation all 6 all 6 none *Special device used of introducing constants into all cells, fitting, then removing constants. 244

TABLE'5.— SUMMARY OF VARIABLES USED AND CATEGORIES FOR EACH VARIABLE FOR EACH SEGMENT OF DATA Standard categories Agent Sex Risk Age 1 = halothane Female U = unknown 0 - 9 + unknown 2 = nitrous oxide-barbiturate Male Al = risks 1+2 10-49 3 = cyclopropane Bl = risks 3+4' 50-69 4 = ether A2 = risk 5 70 + 5 = Other B2 = risk 6 C = risk 7 Categories used Death-rate segment Agent Sex Risk Age Operation Low standard standard standard except standard 7 operation C omitted codes combined Middle, model A standard standard U = unknown standard 5 categories A = A1 + A2 of increasing B = Bl + B2 death rate C = risk 7 Middle, model B standard omitted U = unknown standard 5 categories A = A1 + A2 of increasing B = Bl + B2 death rate C =risk 7 Cholecystectomy standard standard U= unknown 0 - 49 + unknown code 40 A = A1 + A2 50-69 code 41 B = Bl + B2 70 + code 42 C omitted High, model A omitted standard U= unknown standard code 48 A=A1 + A2 code 44 B = Bl + B2 code 12 C = risk 7 code 33 High, model B standard omitted U= unknown standard code 48 A = Al + A2 code 44 B = Bl + B2 code 12 C =risk 7 code 33 order of increasing death rate. For the middle- death-rate operations, we condensed the 25- category code used in other analyses to five categories by adding successive groups of five. We derived this 25-category code by arranging all operations in order of increasing crude death rate and then combining consecutive operations so that each group had approximately the same number of deaths. Table 5 summarizes these definitions of variables and categories. READING AND CONSIDERING CELLS OF TABLES In every cell of the tables of death rates for the smoothed contingency-table analysis, we present: (1) observed death rate, (2) fitted number of deaths, (3) fitted number of randoms (EE), and (4) fitted death rate. Observed death rates are computed from deaths/ (deaths + EE), and the fitted death rates from fitted deaths/(fitted deaths + fitted EE). In inter- preting these data, a number of cautions are in order. The method of smoothed contingency tables improves the stability of estimated death rates, but here it cannot do quite as much as it would in some studies because of the sampling design. Because hospitals contributed equal numbers of randomly chosen cases, rather than equal pro- portions of their own cases, each case in the sample must be weighted by a factor depending on the total number of operations performed in the hospital to give the estimated exposed. This factor varies between 5 and 75. It was necessary to do this weighting before fitting the model. Thus, although the face or cube summary cells 334-553 O-69—17 245

from which the model estimates were derived seemed to be larger than those for the deaths, they were subject to greater sampling variability. A cell entry of 200 estimated exposed, for in- stance, could represent on the average only eight sample cases, whereas a cell of the same size for deaths would represent 200 independent pieces of information. This means that any measure of deviation between observed and fitted cell values will be large for the estimated ex- posed (see Appendix 1). From the point of view of interpretation of the results, it means that, even when the model has been fitted, rates based on fewer than 100 estimated exposed (about four randoms) are subject to such large errors that they are of little value taken individually. The method ensures that the total number 'of entries in each cell of each two-way or three- way summary table used is unaltered; if the model predicts too many observations in one cell, it necessarily predicts too few in another for the margins used. The fitted cell entries depend entirely on the summary tables. Occa- sionally, even with condensation of categories and inclusion of only the most important varia- bles, the cells of the summary tables are small or empty; this is particularly true of the high- death-rate operations. When the cell estimates are small, they are obtained from small cells in the summary tables. Small cells are not as reliable as large cells; consequently, the esti- mates derived from them are not as reliable. This is particularly true of the estimated ex- posed. Thus, when the estimated deaths and esti- mated exposed are combined to give rates in each cell, the rates will not be reliable if the number of exposed is small. This will show up in the tables of rates in the form of large dif- ferences between neighboring cells. LOW-DEATH-RATE OPERATIONS The variables selected were sex, age (four categories), and risk (five categories), giving 40 cells for each of five agents. Two models were fitted, one using all two-dimensional faces and the other all three-dimensional faces. The former was judged to give an adequate fit and did not introduce empty cells, as did the latter; the following discussion is therefore based on the fit using two-dimensional faces. Table 6 shows both the observed and the fitted death rates in detail. But much can be learned from the shorter summary tables (Tables 7 through 15). For this segment of the data, risk categories Al (elective, anesthetic risks 1 and 2), A2 (emergency, anesthetic risk 5), Bl (elective, anesthetic risks 3 and 4), and B2 (emergency, anesthetic risk 6) are used. Comparison of Halothane (Agent 1) with Nitrous Oxide-Barbiturate (Agent 2} The fitted rates of Table 7 for halothane and for nitrous oxide-barbiturate come from cells of Table 6 in which at least one of the two agents has at least 2500 estimated exposed. The num- bers in parentheses are the numbers of esti- mated exposed in hundreds. Nitrous oxide- barbiturate appears more popular than halothane, but both were used extensively. Observe that, although nitrous oxide-barbiturate generally had the lower death rate, the values are rather close together, except for risk U. Of the 40 pairs of cells shown in the com- plete table, halothane had the higher observed (unsmoothed) death rate in 22 pairs and nitrous oxide-barbiturate in 14 pairs, and there were four ties at zero deaths. Here, for assessing consistency, observed rates seem more appro- priate, because smoothed rates may offer mis- leading consistency. Risks B1, A2, and B2 have fewer cases, and they need a separate summary. Table 8 gives rates added over age and sex. These will be the same for fitted and observed, because the risk X agent table was one of the configurations fitted. It shows that the rates are close together for the two agents. A difference of 1 percent, as for risk B2, between the death rates for halothane and nitrous oxide-barbiturate matters, if it is real, but its sampling error is large. As a grand summary, experience with nitrous oxide-barbiturate has produced a slightly lower rate than halothane. Comparison of Halothane (Agent 1) with Cyclopropane (Agent 3) The fitted rates of Table 9 for halothane and cyclopropane come from cells of Table 6 in which at least one of the two agents has at least 2500 estimated exposed. Table 9 shows the two agents to be very closely matched, with dif- ferences of no more than a few hundredths of 1 percent. Cyclopropane looks slightly better over-all. Table 10 compares the halothane and cyclo- propane rates for risks B1, A2, and B2. Again, the rates for the two agents are close to one another. Except for the death-rate excess of 0.8 percent for halothane in risk B2 of Table 10, Tables 9 and 10 leave the two agents closely matched. Comparison of Halothane (Agent 1) with Ether (Agent 4) The fitted rates of Table 11 for halothane and ether come from cells of Table 6 in which at least one of the two agents has at least 2500 246

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TABLE 7.—FITTED DEATH RATES IN LARGE CELLS FOR LOW-DEATH-RATE OPERATIONS, HALOTHANE COMPARED WITH NITROUS OXIDE-BARBITURATE TABLE 9.--FITTED DEATH RATES IN LARGE CELLS FOR LOW-DEATH-RATE OPERATIONS, HALOTHANE COMPARED WITH CYCLOPROPANE Risk level: U = 11 . 12 = unknown risks 1 + 2 risk 5 Female Male Risk Age H N-B H N-B Risk level: U=unknown Al = risks 1 + 2 A2 = risk 5 U 10-49 0.0018 (41)» 0.0005 (82) 0.0021 (18) 0.0006 (25) U 50-69 0.0040 (13) 0.0016 (26) Al 0-9 0.0012 (56) 0.0009 (10) 0.0004 (85) 0.0004 (11) Al 10-49 0.3006 (256) 0.0004 (418) 0.0005 (107) 0.0003 (121) Al 50-69 0.0020 (78) 0.0020 (127) 0.0016 (70) 0.0018 (80) A2 10-49 0.0002 (17) 0.0001 (27) •Numbers in parentheses are estimated exposed in hundreds. TABLE 8.--SUMMARY OF FITTED DEATH RATES FOR LOW- DEATH-RATE OPERATIONS OVER ALL AGES FOR RISK CATEGORIES Bl, A2, and B2, HALOTHANE COMPARED WITH NITROUS OXIDE-BARBITURATE Risk level: Bl= risks 3+4 A2= risk 5 B2= risk 6 Risk H N-B Bl 0.0221 (36)» 0.0011 (26) 0.0349 (7) 0.0190 (36) 0.0012 (34) 0.0258 (6) A2 B2 "Numbers in parentheses are estimated exposed in hundreds. estimated exposed. The differences are small, with ether showing lower rates for the risk U and young risk Al patients. Table 12 compares the agents for risks B1, A2, and B2; the two agents are extremely closely matched. Elsewhere, we explain that our results for ether are more un- certain than for the other anesthetics because ether is little used by some institutions. Comparison of Halothane (Agent 1) with Other (Agent 5) The fitted rates of Table 13 for halothane and Other come from cells of Table 6 in which at least one of the two agents has at least 2500 estimated exposed. It does not seem worthwhile to produce the death-rate comparison for risks B1, A2, and B2, because the numbers of esti- mated exposed are small. Although closely matched, Other had a lower rate for the young, halothane a slightly lower rate for the older patients. Discussion Inspection of the five agents for each risk- age-sex category shows that the estimated Female Male Risk Ace H C H C U 10-49 0.0018 (41)» 0.0014 (56) Al 0-9 0.0012 (56) 0.0009 (36) 0.0004 (85) 0.0004 (24) Al 10-49 0.0006 (256) 0.0004 (270) 0.0005 (107) 0.0004 '49) Al 50-69 0.0020 (78) 0.0018 (65) 0.0016 (70) 0.0020 (26) A2 10-49 0.0002 (17) 0.0002 (44) "Numbers in parentheses are estimated exposed in hundreds. TABLE 10.—SUMMARY OF FITTED DEATH RATES FOR LOW- DEATH-RATE OPERATIONS OVER ALL AGES FOR RISK CATEGORIES Bl, A2, and B2, HALOTHANE COMPARED WITH CYCLOPROPANE Risk level: Bl = risks 3+4 A2 = risk 5 B2=risk 6 Risk Bl A2 52 0.0221 (36)« 0.0011 (26) 0.0349 (7) 0.0235 (20) 0.0013 (55) 0.0270 (10) "Numbers in parentheses are estimated exposed in hundreds. halothane death rates exceed those for all the other four agents in only five instances. These were all cases of young females of different risks, as shown in Table 14. Cells Bl and B2 may be disregarded, because they are based on such small numbers; but risks U and Al should be considered more seriously. Inspection of Table 6 for nitrous oxide- barbiturate and cyclopropane rates would sup- port the suspicion that halothane might have disadvantages for females, particularly young ones. If this were true and halothane rates for females were outstandingly high, then when we compare the halothane rate with that of any other agent, the ratio of the two rates for females will be larger than the corresponding ratio for males, unless the other agent also produces exception- ally high rates for females. We do not find this to be true when halothane rates are compared with ether rates; here the difference in ratios between the sexes is reversed. The comparison with ether shows that, although ether rates are generally lower than halothane rates, for the risk A1, age 10-49 females listed above, the two rates are very close, the ether rate being 0.00062, compared with the halothane rate of 0.00063 (see Table 6). Inspection of the other two cells under suspicion shows that the cyclopropane and 250

TABLE ll.— FITTED DEATH RATES IN LARGE CELLS FOR LOT-DEATH-RATE OPERATIONS, HALOTHANE COMPARED WITH ETHER TABLE 13.—FITTED DEATH RATES IN LARGE CELLS FOR LOW-DEATH-RATE OPERATIONS, HALOTHANE COMPARED WITH OTHER Risk level: U = unknown Al = risks 1 + 2 Female Male Risk Age H B H E U 0-9 0.0017 (20) 0.0004 (29) U 10-49 0.0018 (41)» 0.0010 (15) Al 0-9 0.0012 (56) 0.0006 (93) 0.0004 (85) 0.0002 (131) Al 10-49 0.0006 (256) 0.0006 (98) 0.0005 (107) 0.0004 (37) Al 50-69 0.0020 (78) 0.0025 (22) 0.0016 (70) 0.0016 (18) "Numbers in parentheses are estimated exposed In hundreds. Risk level: U =unknown Al = risks 1 + 2 Female Male Risk Age H 0 H 0 U 0-9 0.0017 (20) 0.0009 (30) U 10-49 0.0018 (41)« 0.0022 (17) Al 0-9 0.0012 (56) 0.0004 (104) 0.0004 (85) 0.0002 (136) Al 10-49 0.0006 (256) 0.0006 (109) 0.0005 (107) 0.0005 (39) Al 50-69 0.0020 (78) 0.0033 (29) 0.0016 (70) 0.0031 (22) •Numbers In parentheses are estimated exposed In hundreds. TABLE 12.—SUSMARY OF FITTED DEATH RATES FOR LOW-DEATH-RATE OPERATIONS OVER ALL AGES FOR RISK CATEGORIES Bl, A2, and B2, HALOTHANE COMPARED WITH ETHER Risk level: Bl = rie A2 = rle B2 = ri£ ks 3 + 4 k 5 k 6 Risk H E Bl 0.0221 (36)» 0.0172 (12) A2 0.0011 (26) 0.0078 (3) B2 0.0349 (7) 0.0445 (11) "Numbers in parentheses are estimated exposed in hundreds. TABLE 14.—CELLS IN WHICH HALOTHANE FITTED DEATH RATES FOR LOW-DEATH-RATE OPERATIONS ARE HIGHER THAN THOSE OF THE OTHER FOUR AGENTS Risk level: 0 =unknown Al = risks 1 + 2 Bl = risks 3+4 B2 = risk 6 Risk Age Sex Halothane rate Eetimi (in ited exposed hundreds ) U 0-9 f 0.00319 12 Al 0-9 t 0.00116 56 U. 10-49 t 0.00063 256 XL 0-9 t 0.03962 1 B2 0-9 t 0.06336 <1 halothane rates are also very close together. They are: Risk U Al -Age_ 0-9 0-9 Rate EE Rate EE 0.00293 0.00090 1016 3580 0.00319 0.00116 1241 5568 Thus, in the risk U cell, one more cyclopropane death would make the ratio of the cyclopropane rate to the halothane rate larger than unity, and the same is true for the risk Al cell. On the basis of indicating a sex difference, and in terms of the actual magnitude of the rates in- volved, these three cells do not seem to indi- cate excessive deaths related to halothane. A further comparison supports this conclusion. If halothane caused excessive deaths in age group 0-9, we would expect the rate ratios to be smaller for this age group compared with other age groups, for every agent comparison. They are not. When halothane is compared with other agents on a cell-by-cell basis, no consistent evidence of excessive deaths from halothane arises in any sector of the population, nor is there any tendency for relatively high halothane rates to be associated with any of the variables considered. Essentially, we have found that the magni- tude of differences in rates between the several agents and halothane is neither large nor very consistent. The impression from Tables 7, 9, and 11 is that for low-death-rate operations halothane did not perform quite as well as nitrous oxide-barbiturate, cyclopropane, or ether. To check this impression, we ranked all the agents except Other from 1 (lowest crude death rate) to 4 (highest crude death rate) for each of the eight (sex, age) groupings within each risk. Then we summed the ranks within each risk. The resulting totals are shown in Table 15. All told, then, one gets the distinct impres- sion that for the low-death-rate operations, although the differences are small and incon- sistent, the order of increasing death rate is ether, nitrous oxide-barbiturate, halothane, and cyclopropane, and that the first two are es- pecially close. Nevertheless, the rates are all small and nearly comparable. It must be empha- sized here, as everywhere else in this report, that the rates were obtained from existing pat- terns of use, and not from the results of a prospective randomized study. And, of course, no account could be taken of why one or another anesthetic may have been especially indicated in an individual operation. 251

TABLE 15.--SUMMARY OF RANKINGS OF OBSERVED DEATH RATES FOR THE LOW-DEATH-RATE OPERATIONS* Risk level: U = unknown Al = risks 1 + 2 Bl= risks 3 + 4 A2= risk 5 Risk 1 B2 = risk 6 H N-B C E Totals 0 25 13.5 26.5 15 80 Al 22 18 23 17 80 Bl 20 19 24.5 16.5 80 A2 21 **19.5 21.5 **18 80 B2 19 **21 26.5 **13.5 80 Totals 107 91 122 80 400 *A score of 1 was assigned to the agent with the lowest crude death rate for each sex-age grouping within each risk, 2 for the next lowest, etc. **Low scores for nitrous oxide-barbiturate and ether for risks A2 and B2 may be due largely to the lack of use, rather than to good performance. MIDDLE-DEATH-RATE OPERATIONS General Usage For the middle-death-rate operations, it was necessary to fit three-dimensional faces. Two models are fitted on differing variables. Model A included five variables: operation (five cate- gories), risk (four categories), sex (two cate- gories), age (four categories), and agent (5 cate- gories). This gave a total of 800 cells, and spread the data too sparsely, so model B was fitted, in which sex was not considered, reducing the num- ber of cells to 400. We first present results from model A and then some from model B. The basic table for model A is not presented, but the detailed data of model B are given in Table 16. All rates presented are deaths/(deaths + EE). Because young females were the most suspicious-looking group in the low-death-rate operations, as far as a possibility of excess deaths from halothane was concerned, they were studied again for the middle-death-rate opera- tions. We looked for a tendency, in the compari- son of halothane with other agents on a cell-by- cell basis, for halothane to have higher death rates when two cells for young females were compared than when the corresponding cells for males were compared. For the youngest age group this occurred only for operation categories 3, 4, and 5 for nitrous oxide-barbiturate; for cyclopropane it almost always occurred; and for ether it was consistently true. For the next age group, the consistency was less: for nitrous oxide-barbiturate it was again not found for op- eration categories 1 and 2 (except for risk U); for cyclopropane there was no systematic pat- tern; and for ether the relationship existed for risk U but not for risks A and B. Again, the fact that risk U (unknown) does not reflect the average pattern of the other risks is surprising. We did not find halothane to be especially disadvanta- geous for young females in middle-death-rate operation categories 1 and 2, which should most nearly resemble the low-death-rate operations. Because the suspicion about females did not seem justified, we examined the cells to see where halothane rates were higher than those of the other four agents. This involved 120 com- parisons; risk C (moribund), for which the rates were erratic because of few cases, was ex- cluded. If each of the five agents had about the same death rates, random distribution would predict that halothane would get the worst rate in 24 instances. In fact, halothane has the worst fitted rate in only the 11 instances shown in Table 17. All these death rates are based on small numbers of estimated exposed; in all ex- cept the first one listed, the number of estimated exposed was less than 150. Because the cells listed are spread among different sexes, ages, and operation groups, the table provides no evidence of any real tendency of halothane rates to be excessive, and suggests chance variation. To confirm this conclusion, the same pro- cedure of selecting cells where the halothane death rate was higher than for the other four agents was carried out on model B, which omitted the sex variable. The five cells that were se- lected from the 60 possibilities, again excluding risk C (moribund), are shown in Table 18. Be- cause age 0-9, operation category 5, and all three risks showed up in model A for both sexes, that group still appears here, but the two largest cells listed do not show the same combinations of age, risk, and operation categories as in model A, so this supports the previous conclusion of chance variation. Rates in Large Cells Although the full table of rates (Table 16) for model B is large for casual examination, much of its message is contained in the larger cells whose rates are especially stable. Such data are given in Table 19. Cyclopropane has a somewhat higher death rate than the other four agents, Other is second highest, halothane and nitrous oxide-barbiturate are nearly equal, and ether may have a slightly lower rate. Ranking Analysis of Death Rates for the Four Anesthetics As in the analysis of the low-death-rate operations, another sort of summary statistic can be had by ranking halothane, nitrous oxide- barbiturate, cyclopropane, and ether in ascend- ing order of crude death rates within a cell 252

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TABLE 17.—CELLS IN WHICH HALOTHANE DEATH RATES ARE HIGHER THAN THOSE OF THE OTHER FOUR AGENTS, MIDDLE-DEATH-RATE OPERATIONS, MODEL A,* RISK C EXCLUDED Operation category Risk Age Sex Rate Estimated exposed 2 U 10-49 m 0.0108 714 3 B 0-9 f 0.1334 14 3 B 10-49 f 0.1208 68 3 B 70 + t 0.1826 107 5 U 0-9 m 0.2329 20 5 0 0-9 f 0.2066 21 5 A 0-9 m 0.1484 41 5 A 0-9 f 0.0944 63 5 B 0-9 f 0.4112 19 5 B 0-9 m 0.2852 30 5 B 10-49 f 0.1215 142 TABLE 18.--CELLS IN WHICH HALOTHANE HAD THE HIGHEST DEATH RATE IN THE MODEL B ANALYSIS OF MIDDLE-DEATH-RATE OPERATIONS, RISK C EXCLUDED Estimated Operation category Risk Age Rate exposed halothane (hundreds ) 2 U 10-49 0.012 13 4 B 70 + 0.182 56 5 U 0-9 0.220 «0 5 A 0-9 0.116 1 5 B 0-9 0.340 -0 *Basic table for model A is not presented. *42 estimated exposed. **48 estimated exposed. TABLE 19.--FITTED DEATH RATES FOR CELLS IN WHICH AT LEAST ONE AGENT HAS AT LEAST 2500 ESTIMATED ; EXPOSED, MIDDLE-DEATH-RATE OPERATIONS, MODEL B Operation category 1 Risk Age H N-B C E 0 U U A A A A 10-49 50-69 0-9 10-49 50-69 70 + 0.0018 (75)* 0.0047 (44) 0.0015 (78) 0.0008 (466) 0.0040 (182) 0.0155 (27) 0.0017 (66) 0.0042 (32) 0.0024 (15) 0.0008 (351) 0.0033 (131) 0.0142 (24) 0.0028 (39) 0.0079 (13) 0.0017 (20) 0.0014 (227) 0.0046 (72) 0.0252 (10) 0.0019 (20) 0.0058 (13) 0.0013 (36) 0.0007 (115) 0.0027 (43) 0.0194 (6) 0.0017 (33) 0.0051 (20) 0.0020 (53) 0.0013 (161) 0.0054 (70) 0.0228 (11) Operation category 2 Risk Age H N-B C E 0 A 10-49 50-69 0.0043 (52) 0.0115 (51) 0.0062 (35) 0.0128 (43) 0.0047 (17) 0.0170 (18) 0.0022 (14) 0.0125 (14) 0.0085 (21) 0.0179 (23) A Operation category 3 Risk Age H N-B C E 0 A A 10-49 50-69 0.0061 (26) 0.0194 (30) 0.0056 (21) 0.0212 (17) 0.0093 (14) 0.0260 (19) 0.0024 (12) 0.0241 (12) 0.0086 (12) 0.0259 (13) Operation category 4 Risk Age H N-B C E 0 A 10-49 50-69 0.0103 (58) 0.0299 (35) 0.0109 (36) 0.0304 (24) 0.0132 (23) 0.0373 (19) 0.0036 (22) 0.0223 (W) 0.0166 (20) 0.0366 (16) A "Numbers in parentheses are estimated exposed in hundi (fixed risk, age, and operation category). This ranking was done and then summed over ages to give the summaries in Table 20 (low totals imply low death rates). As before, we did not include agent 5 (Other) in the comparison. The summed rankings are fairly consistent from one risk level to another within an operation category, and from one operation category to another. Recall 258

TABLE 20.—AGENT RANKS TOTALED OVER AGES FOR EACH RISK CATEGORY WITHIN OPERATION CATEGORY, MIDDLE- DEATH-RATE OPERATIONS, MODEL B Operation category 1 Risk H N-B C E Total U 9 6 13 12 40 A 10 7 14 9 40 B 8 10 15 7 40 Total 27 23 42 28 120 Operation category 2 U 12 9 14 5 40 A 6 11 15 8 40 B 9 8 11 12 40 Total 27 28 40 25 120 Operation category 3 0 7 5 10 8 *30 A 5 8 11 6 *30 B 7 4 9 10 *30 Total 19 17 30 24 90 Operation category 4 Operation category 5 U 4 9 10 7 *30 A 11 8 14 7 40 B 8 9 10 3 *30 Total 23 26 34 17 100 Operation category Grand summary H N-B C E Total 1 27 23 42 28 120 2 27 28 40 25 120 3 19 17 30 24 90 4 23 32 42 23 120 5 23 26 34 17 100 Total 119 126 188 117 550 U 7 12 13 8 40 A 8 10 14 8 40 B 8 10 15 7 40 Total 23 32 42 23 120 "Subtotals come to 40 unless some age group had few cases. that crude rates are preferred to fitted rates to get a notion of the consistency of the ordering in the face of the variability of the data. The implication of these tables is that for the middle-death-rate operations, cyclopropane's death rates are consistently higher than those of the other three agents. The others, in turn, are roughly equal as measured by ranks. Discussion There were differences between ages in the relative death rates of the agents and in the usage patterns. No one agent always had higher rates than the others, but for the first three risk categories, cyclopropane death rates were higher than those of halothane, nitrous oxide- barbiturate, and ether. Cyclopropane's rate was much the same as that for agent 5 (Other). As always, we must be wary of concluding that the same results would hold in an experiment, and -we must not attribute the death rates themselves to the anesthetic. The difference between death rates for anesthetics is as close as we can hope to come to assigning cause in such an investiga- tion. CHOLECYSTECTOMIES Table 21 displays smoothed death rates by agent, sex, risk, and age for each of three op- eration codes: code 40: cholecystectomy alone, code 41: cholecystectomy and/or bile duct, and code 42: cholecystectomy and/or bile duct, and other major proce- dures. In fitting these rates, we used the two-factor effects for all pairs of the five variables except sex X operation code, sex X risk, and sex X age. Inasmuch as about 60 percent of patients having cholecystectomies fall into categories risk A (1+2+5), and age 0-49 or 50-69, a general idea of the data can be had from a shorter table, Table 22. Tables 21 and 22 show that death rates for males were about twice those for females for corresponding operation, agent, risk, and age group. When anesthetics are compared, halothane looks relatively better for the older of the two age groups. Considering the sizes of the samples, a visual analysis of Table 22 leaves little basis 259

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TABLE 22.—FITTED DEATH RATES FOR CHOLECYSTECTOMIES RISK A, AGE 0-49, AGE 50-69 H N-B C E 0 Operation code 40: Cholecystectomy alone Age 0-49: Age 50-69: Female 0.0005 (8)* Male 0.0010 (4) Female 0.0028 (10) Male 0.0054 (5) 0.0007 (7) 0.OOU (2) 0.0034 (5) 0.0067 (2) 0.0004 (10) 0.0007 (5) 0.0035 (9) 0.0071 (5) 0.0004 (4) 0.0010 (2) 0.0040 (4) 0.0112 (2) 0.0018 (3) 0.0043 (1) 0.0080 (3) 0.0196 (1) Operation code 41 Cholecystectomy and/or bile duct Age 0-49: Age 50-69 Female 0.0021 (3) Male 0.0041 (2) Female 0.0101 (4) Male 0.0193 (2) 0.0031 (3) 0.0061 (1) 0.0139 (3) 0.0267 (1) 0.0018 (3) 0.0036 (1) 0.0159 (3) 0.0317 (1) 0.0014 (1) 0.0039 (1) O.0142 (1) 0.0384 (1) 0.0034 (1) 0.0085 (1) 0.0144 (1) 0.0350 (1) Operation code 42: Cholecystectomy and/or bile duct, and other major procedures Age 0-49: Age 50-69: Female 0.0064 (2) Male 0.0122 (1) Female 0.0211 (4) Male 0.0400 (2) 0.0058 (3) 0.0111 (1) 0.0180 (3) 0.0343 (1) 0.0034 (3) 0.0069 (2) 0.0213 (4) 0.0424 (2) 0.0025 (2) 0.0070 (1) 0.0177 (3) 0.0475 (1) 0.0073 (2) 0.0179 (1) 0.0215 (2) 0.0516 (1) *Numbers in parentheses are estimated exposed in hundreds. TABLE 23.—SUMS OF RANKS OF CRUDE DEATH RATES FOR CHOLECYSTECTOMIES BY AGENT FOR CATEGORIES HAVING AT LEAST 300 ESTIMATED EXPOSED Operation code H N-B C E Total 40 18.5 21.5 28 22 90 (9 rows) 41 1 4 U 13 19 60 (6 rows) 42 21 19.5 23 26.5 90 (9 rows) Totals 53.5 55 64 67.5 240 for choosing among the four agents, but if one had to choose, halothane looks best except for opera- tion code 42, where, of course, the cases are few and the estimates unreliable. The agent Other had a systematically high set of death rates. We want to get a general notion of the rank- ing of halothane, nitrous oxide-barbiturate, cyclopropane, and ether. For each operation code-sex-risk-age category of Table 21 that contains at least 300 estimated exposed, spread among all agents, we ranked the crude death rates from least to greatest. These ranks were then summed for each operation to give the totals shown in Table 23. On this measure, halothane and nitrous oxide-barbiturate perform about equally well and slightly better than cyclopropane and ether. HIGH-DEATH-RATE OPERATIONS We wish primarily to supply some general information gained in the Study, rather than to compare anesthetic agents, when we give smoothed death rates for the four high-death- rate operations: code 48: large bowel, code 44: exploratory laparotomy, etc., code 12: craniotomy, and code 33: heart and great vessel with pump. In deciding whether to recommend an operative procedure at a given time, a physician or sur- geon might wish to consult a death-rate table in which he can allow for the patient's anesthetic risk (physical status), sex, and age. In doing this, he will probably also have to consider at least whether the hospital or surgeon has a higher or a lower death rate than the average of the hospitals in this Study and whether the technology associated with these operation codes has improved much since the years on which these data are based (1958-1962). The latter point also suggests that these data offer some- thing in the way of a baseline for these high- death-rate operations. Table 24 gives death rates for the four high- death-rate operation codes. About 60 percent of the 60,000 cases in Table 24 are contained in the eight lines reproduced in Table 25. Recall that, because we use deaths/(deaths + EE) as the in- dex of death rate, the rate r tends in the high- death-rate operations to be somewhat lower than the correct rate. To convert to the usual D/EE, one uses r/(l-r); for example, if our tabled rate is 0.2, then the usual rate is 0.2/(1-0.2) = 0.25. Table 26 gives the fitted death rates for the high-death-rate operation codes with the agent included in the fitting. The data were somewhat 263

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TABLE 25.-- FITTED DEATH RATES IN LARGE CELLS FOR HIGH-DEATH-RATE OPERATIONS, ICDEL A Riek level: A ' risks 1 + 2 + 5 B - risk! 3+4 + 6 Operation code 48 44 12 33 A t 0-9 0.027 (3)» 0.041 (3) 0.037 (8) 0 .11/4 <*) A m 0-9 0.021 (4) 0.042 (3) 0.040 (9) c .109 (5) A A f 10-49 10-49 0.01/4 0.012 (17) (17) 0.017 0.020 (24) (21) 0.040 0.048 (29) (28) 0 0 .092 .097 (8) (9) A f 50-69 '0.024 (27) 0.036 (21) 0.061 (17) c .168 (1) A 2j 50-69 0.027 (25) 0.055 (17) 0.095 (15) 0 .222 (1) B f 10-49 0.087 (3) 0.092 w 0.135 C 0 .168 (11) B 10-49 0.075 (3) 0.104 (7) 0.158 (8) 0 .175 "Number* In parentheses are estimated erposed in hundreds. patchy, so we had to adopt an additional smooth- ing device. We added 4 to the number of deaths in every cell and 40 to the estimated exposed in every cell before fitting the model. These num- bers were removed after fitting and so occa- sionally produced negative estimated cell values. Where negative values occur in the table, the fitted rate appears as 9.99999, a signal that no useful estimate appeared. The largest cells from Table 26 have been abstracted to give Table 27. The most striking feature of Table 27 is the very great increase in death rate for older persons and persons rated as being poorer risks. These four operations were separated from the main body of data because they contained so many deaths, but the operations themselves are not closely related to each other. It is therefore desirable to consider agent differences sepa- rately for each operation. For operation 48 (large bowel), for all but the youngest age group, halothane and nitrous oxide-barbiturate, on the whole, had the lowest rates, and Other had the highest rates. For operation 44 (exploratory laparotomy), ether seems to have had the lowest rates, fol- lowed by halothane and nitrous oxide-barbiturate. Recall that our ether rates are poorly deter- mined. Cyclopropane appears to have had the highest rates. Cyclopropane was rarely used for opera- tion 12 (craniotomy). For the four large cells with ages 10 and over, ether had the lowest rates. The other three agent categories did not show consistent differences. For operation 33 (heart and great vessel with pump), in the three cells listed nitrous oxide-barbiturate had the highest rates; halo- thane, cyclopropane, and Other had similar rates; and ether was seldom used. Table 28 gives the rankings of anesthetic agents (excluding Other) by observed death rates in the high-death-rate operations for rows of Table 26 having at least 1000 estimated ex- posed. Rank 1 was assigned to the agent with lowest death rate, rank 2 to the next lowest, and so on. The ranks were then summed. We note that for operation 48 (large bowel), halothane has the lowest score of the four agents. For operation 44 (exploratory laparotomy), halothane is tied with ether for the lowest. Ether had the lowest score for operation 12 (craniotomy); cyclopro- pane had the lowest score for operation 33 (heart with pump), although there were few cases. 266

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TABLE 27.—FITTED DEATH RATES PER 100 IN LARGE CELLS FOR HIGH-DEATH-RATE OPERATIONS, MODEL B Risk level: U = unknown A = risks 1+2+5 B = risks 3 + 4 + 6 Operation code 48: Large bowel Risk Age H N-B C E 0 U A A A 50-69 10-49 50-69 70 + 50-69 70 + 2.9 (4)« 1.3 (8) 2.4 (4) 1.2 (6) 2.1 (9) 5.2 (3) 9.8 (2) 10.5 (2) 8.3 (2) 1.5 (11) 3.1 (13) 3.3 (9) 12.8 (5) 19.5 (4) 3.9 (1) 0.9 (6) 2.0 (9) 5.3 (4) U.8 (1) 15.5 (2) 3.8 (1) 0.8 (5) 3.6 (7) 6.9 (3) 16.4 (2) 16.6 (3) B 2.2 (13) 4.9 (5) 8.0 (4) 12.4 (3) B Operation code 44: Exploratory laparotomy, etc. U U A A A B B B 10-49 50-69 10-49 50-69 70 + 10-49 50-69 70 + 5.1 (3) 9.0 (3) 1.3 (13) 3.5 (8) 3.3 (3) 9.4 (3) 16.1 (3) 12.5 (3) 7.9 (1) 9.3 (2) 1.8 (6) 3.3 (6) 7.8 (2) 9.7 (2) 13.2 (3) 14.4 (3) 3.0 (5) 11.6 (3) 2.4 (14) 6.4 (10) 8.4 (3) 10.9 (6) 21.4 (6) 39.5 (2) 2.8 (1) 4.7 (1) 5.9 (2) 2.1 (6) 5.4 (6) 9.9 (2) 11.0 (2) 14.9 (4) 16.2 (4) Operation code 12: Craniotomy 6.1 (l) 1.8 (6) 3.2 (9) 4.5 (2) 9.5 (2) 16.5 (2) 12.2 (3) Age H N-B C £ 0 A A A 0-9 10-49 50-69 10-49 50-69 3.6 (11) 3.9 (37) 9.5 (14) 13.0 (10) 24.3 (5) 6.0 (2) 8.4 (1) 8.3 (0)»* 5.4 (0)»*i 4.9 (2) 3.4 (5) 4.4 (6) 8.0 (2) 13.8 (1) 3.6 (2) 9.5 (3) 11.6 (2) 3.4 (1) 28.5 (1) B B Operation code 33: Heart vessel with pump 5.6 (11) 6.3 (9) 14.1 (4) 14.7 (5) and great Age H N-B C E 0 A B B 10-49 7.7 (7) 12.2 (6) 20.0 (3) 24.5 (2) 12.7 (12)20.8 (9) 9.7 (1) 21.1 (2) 12.0 (2) 9.1 (2) 19.2 (3) 20.2 (2) 0-9 10-49 •Numbers in parentheses are estimated exposed in hundreds. »«41 estimated exposed. ***35 estimated exposed. TABLE 28.—RANKINGS OF ANESTHETICS BY OBSERVED DEATH RATES IN THE HIGH-DEATH-RATE OPERATIONS FOR ROWS OF TABLE 26 HAVING AT LEAST 1000 ESTI- MATED EXPOSED Physical „ Anesthetic status Age H N-B C E Operation 48: Large bowel Unknown 50-69 1 2 4 3 1, 2, 5 10-49 2 3 4 1 1, 2, 5 50-69 3 1 4 2 1, 2, 5 70+ 3 2 1 4 3, 4, 6 50-69 1 3 2 4 3, 4, 6 70+ 1 2 3 4 Totals 11 13 18 18 60 Operation 44: Exploratory laparotomy Unknown 10-49 2 3 1 4 Unknown 50-69 3 2 4 1 1, 2, 5 10-49 1 2 4 3 1, 2, 5 50-69 3 2 4 1 1, 2, 5 70+ 1 3 4 2 3, 4, 6 10-49 2 4 3 1 3, 4, 6 50-69 2 1 3 4 3, 4, 6 70+ 3 2 4 1 Totals 17 19 27 17 80 Operation 12: Craniotomy 1, 2, 5 0-9 1 3 (*) 2 1, 2, 5 10-49 2 3 («) 1 1, 2, 5 50-69 3 2 (*) 1 3, 4, 6 10-49 2 3 (*) 1 3, 4, 6 50-69 3 2 (») 1 Totals 11 13 (») 6 30 Operation 33: Heart and great vessel i fith pump 3, 4, 6 10-49 2 3 1 ii*) 3, 4, 6 0-9 2 3 1 («) 3, 4, 6 10-49 1 3 2 («) Totals 5 9 4*« (*) 18 "Too seldom used to be ranked. **Based on small numbers of cases. CONCLUSIONS In a complicated system requiring many variables for its description, one cannot expect a simple conclusion. Insofar as we can sum- marize, halothane showed up satisfactorily, cyclopropane and Other sometimes did not, and nitrous oxide-barbiturate and ether often had the lowest death rates. For the low-death-rate operations, although the differences in rates were small, the order of increasing death rates seemed to be ether, nitrous oxide-barbiturate, halothane, cyclopro- pane. For the middle-death-rate operations, halothane, ether, and nitrous oxide-barbiturate were associated with similar death rates, but cyclopropane and Other with higher rates. For the cholecystectomies, although the differences in death rates between agents were not large, halothane had the lowest death rates for both "cholecystectomy alone" and "chole- cystectomy and/or bile duct," but matters were not so clear for "cholecystectomy and/or bile duct, and other major procedures," where cases were fewer and the control poorer because of the great variety of "other major procedures." The results were a bit of a surprise because halothane is sometimes thought to be an un- desirable anesthetic for use in cholecystecto- mies. Among the high-death-rate operations, for large bowel, halothane and nitrous oxide- barbiturate had the lowest rates; for explora- tory laparotomy, ether, then halothane, and nitrous oxide-barbiturate had the lower rates, and cyclopropane rates were high; for craniot- omy, cyclopropane was rarely used, ether seems to have had the lowest rates (although not much used), and the other three agents did not 271

show consistent differences; and for heart and great vessel with pump, ether was rarely used and cyclopropane only occasionally, nitrous oxide-barbiturate had the highest rates, and halothane rates were similar to those for cyclo- propane and Other. As a final warning: (1) Death rates asso- ciated with ether were poorly determined be- cause some institutions rarely used it. (2) The rates reported here are related to the choice of common practices in the institutions, and therefore may be subject to various sorts of biases that would be controlled only in a pro- spective randomized study. (3) The death rates are thought of as caused not by the anesthetics, but by the whole procedure, including the pa- tient's disease; however, differences in death rates are an indication of deaths caused by anesthetic differences (plus some uncontrolled variables). (4) Observed differential death rates are not necessarily an automatic basis for changing medical practices, especially inasmuch as the training and skills of the practitioner must be considered, as well as the properties of the individual patient. REFERENCE 1. Birch, M. W. Maximum likelihood in three- way contingency tables. J. Roy. Statist. Soc., Ser. B, 25:220-233, 1963. 272

APPENDIX TO CHAPTER IV-3 CALCULATING SMOOTHED CONTINGENCY TABLES Yvonne M. M. Bishop Harvard University Cambridge, Massachusetts This appendix describes briefly the theory behind the use of a linear model in the logarithmic scale for smoothing contingency tables. Several papers in the cur- rent literature deal with this approach, one of the most recent being that of Birch. His notation of "u-terms" has been adopted here. This appendix also describes the method used to compute the estimated cell values for the particular model selected, the method used to choose the model, and the criteria used to assess whether a particular model was satisfactory. THE LOGARITHMIC MODEL Birch (1) has described the usual linear model in the logarithmic scale for a three-dimensional contingency table. For ease of writing, the following description is also limited to three dimensions. For smoothing contingency tables, we have found that using at least four dimensions leads to more satisfactory fitting in the data on anesthetics. The method used here can be extended to the number of di- mensions that seems to be appropriate for the size of the particular collection of data. For a given linear model in the logarithmic scale, the same maximum likeli- hood estimates for the cell entries arise under a variety of sampling conditions. Nevertheless, for purposes of discussion we need to derive the distribution appro- priate for our data. Let the subscript i refer to one of the I categories of variable 1, subscript j to one of the J categories of variable 2, and subscript k to one of the K categories of variable 3. Let x^ denote the number of observations (that is, the count) in cell (i,j,k) and N = I x^. In the present data only N, the total number of observations in the sample, is fixed. It is therefore convenient to think of the observations x^k as following the multinomial distribution. Cochran (4) reports that Fisher first gave a simple deriva- tion of the multinomial under these conditions. If the observed x^ are regarded as following independent Poisson distributions with means yljk, their joint fre- quency function is: xjjk ~vjjk y^k e (i) The total N = I xljk also follows the Poisson distribution with mean Y= x yljk i,j,k i,j,k J" The frequency function of N is therefore YNe-Y N: (2) The conditional distribution of the x^, given that their total is N, is obtained by dividing expression 1 by expression 2. We then have the multinomial expressed in terms of probabilities, p xjjk where pjjk= yjjk /Y and 2 n = 1. l»J|K 273

Let us now define mjjk as the expected cell value that would be obtained under the multinomial of expression 3. For fixed N we have m^ = Np^, and so can sub- stitute for Pjjk in expression 3. If we do this and take the logarithm (to the base e) of the likelihood, we have: r n Nxijkt1 - . J . |-i,j,k J •,*i* log - xi loe N (4) The first two terms of this expression are unchanged, whatever the values of The last term, or "kernel," will change with mi.k, and that is the part of the ex- pression that is important in our development. Having defined the distribution, we may now proceed to express the m^ in terms of a log -linear model. In Chapter IV-3 we introduced the u-term notation and described how individual u-terms can be regarded as representing simple or multiple -factor effects of the variables. We also showed that all subscripted u-terms are deviation scores, and hence sum to zero when added over one of the subscript variables. For the sake of completeness, we will recapitulate here. Let us first write out the full three-dimensional model: Iogmjjk=u+u1(l) + u2(j) + u3(k) + U12(jj) + U23(jk) + U13(lk) + U123(ljk) - (5) The definitions of the u-terms are then as follows: over-all mean: u= '' UK main effect of variable 1 for category i: - 2 log mjjk ><» J,k JK - u ; two-factor effect between variables 1 and 2 for all cells at level i of variable 1, level j of variable 2: v log nii > 1. / » U12(u) = I Kiik - No + U2(j> + u); and similarly for the remaining terms. The consequence of the additive model is that, for the one-factor terms, 1 j k for the typical two-factor term involving variables 1 and 2, and similarly for the other two-factor terms, u.,... . and u,.,... . We also have for .i,. , i JVIK; ,£ the three-factor term: ^ U123(ljk) = ^ U123(ljk) = * U123(jjk) 274

The complete model of expression 5 does not impose any further restrictions on the mjjk. If we substitute this linear expression for log m^ in the log- likelihood of expression 4, we have: log Nj , - N log N + x 2 xjjk[u+ U1(l) + U2(J) + U3(k)+ U12(iJ) + U23(Jk) + U13(llc) i,j,k + U123(ljk)]- (6) Because the first two terms are independent of the u-parameters, they may be ignored. The remaining terms of expression 6 may be rewritten: Nu + l xj++ (u1(l)) + s x+j+(u2(j) ) + 2 x++k(u3(k) ) 1 j k + 2 x1+ + 2 xjjk (u123(jjk) ). (7) Uk If, instead of a model with all possible effects, we define a model with, for instance, no three-factor effect, this would be equivalent to postulating that U123(ljk) = 0 for all i,j,k, and the term 2 xjjk (u123(ljk) ) would disappear from i,j,k expression 7. Then the sufficient statistics for the remaining seven u-terms would be N, x1+4, x+j+, x++k, xy+, x1+k, and x+jk. These can all be derived from the three marginal face totals with typical elements x^, xI+k, and x+Jk. In other words, the kernel of the distribution no longer contains terms as detailed as xjjk . It is this reduction of dimensionality that enables us to obtain more stable cell estimates with models that have only low-order u-terms. To fit such models to our observed xjjk, we need some theoretical results. We will denote maximum likelihood estimates of the parameters by a prime; thus, u' is the maximum likeli- hood estimate of u and m jjk is the maximum likelihood estimate of m^ . This distinction between theoretical values and maximum likelihood estimates, or "fitted" values, has not been made in Chapter IV-3. The necessary theoretical results are given by Birch, who shows that for any reduced model: (1) the marginal totals are maximum likelihood estimates of their expectations, for example, x1++ = m'1++, which estimates nn++; (2) there is only one set of u's, and hence m's, for which the likelihood func- tion is a maximum; and (3) the maximum likelihood estimates of the m's are determined uniquely by the appropriate marginal totals and the marginal totals are preserved. In order to prove (2), Birch assumes that no cell is empty. This assumption is not met in our body of data; but consideration of the degrees of freedom in- volved leads to the conclusion that, if the appropriate marginal totals are not zero, they still provide a unique solution, regardless of whether the elementary cells are zero. Occasionally we have used a model when a marginal total was itself zero. In such a model, because our method preserves totals, no estimates different from zero can be derived for the expected cell entries for any of the empty cells whose sum is involved in the particular marginal entry. In effect, then, the number of independent equations to be solved simultaneously is reduced by one, and a unique solution is obtained for the remaining cells. It should be noted that: (1) other methods can be developed that do not have the empty-cell limitation, and (2) when data come from several sources (institu- tions in this Study), the multinomial model may not apply and so the sufficiency cannot be counted on. Nevertheless, the idea of fitting in this manner can still be used without change. To obtain the fitted m'^ in every cell for the model that involves the mean u, the three one-factor u's, and the three two-factor u's, the three configurations of two-dimensional face totals of x's are sufficient. The maximum likelihood esti- mates of the u-terms or of the m's are ordinarily derived iteratively. The need 275

for iteration occurs for this particular model only in three dimensions, but it occurs for most models of interest when the number of dimensions exceeds three. The method of iteration used in this study (to be described in the next section) produces the fitted m'^ without directly computing the u'-terms. Thus, although the u'-terms are a useful device from a theoretical point of view, and can readily be obtained from the m^, they are not needed in the calculations. They can be useful for purposes of interpretation, in that they give an analysis-of-variance summary of the data. Nevertheless, for illustrative purposes we have computed the u'-terms for the low-death-rate operations (Table 1). This was a four-dimensional set of data and we fitted the same model to both deaths and estimated exposed. The model contained the mean u, the four main-effect terms, and the six two-factor terms. All higher-order terms were equated to zero. Suppose we wish to find log m'^ (i = j = k = t = 1), the logarithm of the ex- pected value of the first cell, the cell that is at level 1 of each of the four variables. We look at the u'-terms for the estimated exposed, and take the sum of the mean and the main-effect u-terms. These are: u/ + u d) '2(1) + U 3(1) 4(1)' with values 5.5135 + 0.5078 + 0.2532 - 0.0538 + 1.1324 = 7.3531. To these must be added the sum of the two-factor terms: '12(1,1) 13(1,1) '14(1,1) U 34(l,l) with values -0.1442-0.0501 -0.1716 -0.2536 -0.0960 + 0.4858 = -0.2297. TABLE 1.—AN EXAMPLE OF u'-TERMS FOR LOW-DEATH-RATE OPERATIONS FITTED TO TW0-DIMENSIONAL FACES (AGBfT, SEX, ACE, AND RISK) [Cne-factor terms are vectors and have as many parameters as the variable has categories. Two-factor terms are two-dimensional matrices; the number of the column corresponds with the number of the category of the first variable, and the number of the row with the number of the category of the second variable.] u'-TEHU3 FOR DEATHS u' -TERMS FOR ESTIMATED LX;I.:KD Main u -term: 0 7313 Main u'-term: 5.5135 One-factor u'-terms: One-factor u'-terms , Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 ui 0.5078 0.1967 0.1927 -0.8230 -0.0742 l 0.4281 -0.2119 0.3047 -0.8157 0.2947 r Female Male u2 0.2532 -0.2532 "a 0.0784 -0.0784 ui -O.0538 1.1728 -0.0472 -1.0718 "3 Age 0-9 -0.7419 Age 10-49 0.0676 Age 50-69 0.4689 Age 70+ u» 1.1324 2.7105 -0.1651 -1.5200 -2.1577 0.2054 , Risk U Risk Al Risk Bl Risk A2 Risk B2 Two-factor u'-terms * 0.2321 0.8032 0.8609 -1.6676 -0.2287 "12 0.14*2 -0.1442 0.0391 0.2694 -0.0993 -0.0651 Two-factor u' -terms: -0.0391 -0.2694 0.0993 0.0651 u'i2 -0.1086 0.1086 0.0080 -0.0080 0.1478 -0.1478 0.0518 -0.0518 -0.0989 0.0989 -0.0501 0.0026 -1.7774 0.4797 -0.2755 0.2701 1.1278 -0.3013 0.9752 -O.4512 f 0.2037 -1.5580 -0.0929 1.0350 0.4121 0.1333 0.6152 0.1706 -0.4659 -0.4533 Ui3 0.0111 0.2595 -0.0125 0.0041 -0.2622 -0.0859 0.6825 -0.1652 -0.3606 -0.0708 -0.1846 0.4876 -0.0690 -0.2659 0.0319 -0.0302 0.8109 0.1743 -0.7732 -0.1818 ult -0.1109 0.0847 0.0794 0.1184 -0.1716 0.1574 -0.2448 -0.3780 -0.1191 uii 0.2799 -0.1391 0.4586 0.1855 -0.3009 0.1594 -0.3991 -0.3129 0.2236 -0.2442 0.2494 0.0225 0.2844 -0.0560 -0.4454 -0.0767 0.0290 -0.0825 0.2036 -0.3074 -0.4257 -0.3942 0.6685 0.3962 0.5151 0.3922 -0.0075 -0.0001 0.1364 0.1817 -0.6519 0.4448 -0.9698 -0.3155 -0.0184 0.1083 0.0539 -0.2041 0.3291 -O.3121 0.1333 -0.2536 0.2536 r 0.0714 -0.0714 0.3925 -0.3925 u23 0.3367 -0.3367 0.0090 -0.0090 -0.1024 0.1024 -0.1480 0.1480 -0.3057 0.3057 "2* 0.0346 0.2931 0.0998 0.004S -0.4324 -0.0346 -0.2931 -0.0998 -0.0048 0.4324 -0.0960 u2' -0.0672 -O.0420 0.2192 -0.0139 0.0960 0.0672 0.0420 -0.2192 0.0139 , 0.1692 -0.0654 -0.1915 0.0877 0.4858 -0.2337 0.0832 -0.3353 a* -0.0246 0.0087 0.2058 -0.1898 0.3198 -0.0798 0.1986 -0.4385 -0.0440 -0.0076 0.1096 -0.0580 -0.7014 -0.9278 0.5327 1.0965 -0.3479 -0.2301 0.3070 0.2711 0.6194 0.9647 -0.5173 -1.0667 0.2474 0.2945 -0.4309 -0.1110 -0.7235 0.2766 -0.2971 0.7440 276

And the log mykl = 7.3531 - 0.2297 = 7.1234 (i = j = k = 1 = 1). The antilogarithm, 1241 (recall that base e was used), gives the fitted number of estimated exposed in this cell. This number is found in Table 6 of Chapter IV-3 on the line labeled "Fit EE" for agent 1, female, age 0-9, risk U. If we were to compute the death rate in the customary way, as deaths/estimated exposed, the logarithm of the expected rate in a particular cell would be the dif- ference between the logarithm of the expected value for deaths and the logarithm of the expected value for estimated exposed. Thus, the logarithm of the rate may be expressed as a linear combination of w-terms; dropping suffixes in parentheses, this is: w + wj + w2 + w3 + w4 + w12 + w13 + WM + w23 + w24 + w34, where w= (u' for deaths) - (u' for estimated exposed), wl(D = (ui<1) for deatns) - (ui(l) for estimated exposed), and similarly for all w-terms. The w-terms have the same additive properties as the u-terms. For the first variable, agent, the main effect w-term has parameter values as follows: w1(1) = 0.4281 - 0.5078 = -0.0797, w1(2) = -0.2119 - 0.1967 = -0.4086, W1(3) = 0.3047 - 0.1927 = 0.1120, wl(4) = -0.8157 +0.8230 = 0.0073, and w,,o> = 0.2947-0.0742= 0.3689. 1(5) This orders the different agents in terms of their main effect. If the data could be fitted by a model that did not contain any w-terms involving variable 1, except the main effect Wj, then the ordering of the wj terms would correspond with the order- ing of the over-all adjusted rates, however they were computed. In the present instance, the ordering of the wj terms is the same as the ordering of the adjusted rates given in Chapter IV-6, except for the interchange of agents 1 and 4. Our main purpose, however, is to obtain stable cell estimates, not over-all relationships. If the method of fitting a log-linear model is to be effective in producing more stable cell estimates than would be obtained by using only observed cell values, then most of the variability between cells in the raw data should be described by the u-terms of lower order. In the three-dimensional case, for instance, it is theoretically possible, but unlikely, that the three-factor term u123 is large and the two-factor terms u12, u23, and u13 are small or nonexistent. Birch has postu- lated a "hierarchy" principle, and strictly speaking the methods used here are applicable only when such a principle holds. (Other methods could be developed.) The principle, as followed here, implies that, if a lower-order effect is zero, then all higher-order effects involving the same combination of variables are also zero. In the three-dimensional case, we can think of the three-factor effect u123 as measuring differences in the two-factor effect u12 for each level of variable 3. The hierarchy principle states that, if u12 = 0 (that is, u^.^ = 0 for all values of i and j), then u123 must also be zero. The argument applies naturally to the other two two-factor effects as well. Thus, if a two-factor effect is on the average negligible, the principle implies that it is negligible for every level of the third variable. When more than three dimensions are involved, the argument is extended to multifactor effects. In the four-dimensional case, if u123 = 0, then u12,4 = 0. If u12 = 0, the four-dimensional model is reduced from a full complement of 16 u-terms down to 12 u-terms, because u123 , u124 , and u1234 are also zero. It may be shown that for s dimensions the total number of u-terms in the full model is 2s. This is reduced by one-fourth to 3X2S"2 if a two-factor effect is zero. One 334-553 0-69-19 277

effect of this principle is to reduce greatly the possible number of models that could be fitted to a given set of data. This principle does more. It defines groups of models for which the m-values may be obtained without iteration. These are referred to as "multiplicative" models. A special set of these models is called "simple multiplicative." These by definition have only one two-factor effect (and higher-order related terms) of zero; this definition holds for any number of dimensions. Because the m'-values are easily obtained for these models, they can be used in a first examination of the data to determine which of the iterative models is the most feasible to fit. This examination is described later (under "Selection of Model"). II we keep to the hierarchy principle, there is only one model that requires iterative fitting for three-dimensional data. This is the model that uses seven parameters, the only zero term being u12j . As soon as any other parameters are zero, we have a multiplicative model. Let us consider the simple multiplicative model in three dimensions with u13 = 0 for all i, k. This model may be written: log mjjk = u + u1(l) + u2(j) + u3(fc) + u12(lj) + u23(jk) . (8a) It follows that mjj+ = 2 exp[u+ u1(l) + u2(j) + u3(k) + u12(jj) + u23(jk)] k = exp[u + u1(l) + u2(j) + u12(jj)] X exp[u3(k) + u23(jk) ], (8b) and similarly: m+jk = exp[u + u2(j) + u3(k) + u23(jk) ] 2 exp[u1(l) + u12(jj)]. (8c) We also have : exp[u +u2(j) ] S S exp[u1(l) + u3(k) + uj2(lj) + u23(Jk)], (8d) I K. which may be written: expfu + u2(j) ][S exp(u1(l) + u12(lJ) )][S exp(u3(k) + u23(jk) )]. (Be) 1 1C From expressions 8a, 8b, 8c, and 8e, we find that (8f) and so m'1jk — . (9) x+J+ These values of mjjk are the expected, or fitted, cell values for this model. Thus, when there is no two-factor effect between variables 1 and 3, to compute the m^ we need the two two-dimensional faces with typical elements xjj+ and x+jk. These are obtained by adding over the categories of variable 3 and over the categories of variable 1, respectively. We also need the one-dimensional margin with typical element x+j+, obtained by adding over the categories of both the mutually inde- pendent variables 1 and 3. This margin could also be obtained from either of the faces. We will use the word "configuration" and capital X's to refer to any sub- matrix obtained by adding over the categories of one or more variables of the original matrix. In this language, for this model, the two configurations Xy+ and X+jic are sufficient, because we can derive from them the configuration X+j+. 278

If variable 2, say, has only one category, so that the suffix j may be dropped, then rn'ik may be estimated by or the product of the marginal totals divided by the grand total. Then expression 9 reduces to the familiar relationship commonly used in a two-way contingency table to obtain the expected value in each cell under the hypothesis of independence of row and column effects. Thus, in a two-way contingency table the test for inde- pendence is equivalent to setting up a linear model in the log-scale with three parameters, namely, the mean u and the two relevant one -dimensional u-terms, and comparing the results with the observed values. Expression 9 may be inter- preted as an extension of this formula to the case of J two-way tables where variable 2 has J categories. In other words, the three-dimensional matrix may be regarded as J layers of two-way tables of size IXK, involving variables 1 and 3, the mutually independent variables. FITTING THE MODEL BY THE ITERATIVE METHOD Once a model has been selected, the fitted values in every cell are derived from the summary configurations. For illustrative purposes, discussion here centers again on a three-dimensional array of data for which the sufficient con- figurations are the three sets of two-dimensional face totals having typical elements xjj+> x1+k, and x+jk. The dimensions of these faces are IXJ, IXK, and JXK. We wish to obtain the fitted value m'jjk for every cell. We will first describe the proc- ess and then give an example. Step 1: Obtaining configurations. The data xjjk are read into the computer and the three configurations of totals with typical cells, xjj+ , xi, k, and x jl; obtained. These totals are unaltered throughout the computation. When referring to the entire configurations we will use Xjj+, Xi+k, and X+jk. Step 2: Starting values. A constant is assigned to every cell of the mjjk matrix. In practice we used the value 1, thus mfa = 1, for all i, j, k. Every cell of this matrix is changed at every subsequent step. The superscript refers to the stage of iteration and indicates the current value of m'^. Step 3: Beginning the iteration. The m' -matrix is fitted to the first configura- tion, Xjj + . The procedure is multiplicative, thus for all i, j, and k. Step 4: Continuation of the iteration. The m'-matrix is fitted to the second configuration, X1+k, thus for all i, j, and k. Step 5 and subsequent steps: The m'-matrix is similarly fitted to the third configuration, X+jk. Then the cycle using the configuration of Step 3 is repeated. At the end of each cycle the amount of change that the last step has introduced is assessed. If the last step did not change any cell value by as much as 0.1, the pro- cedure is terminated; otherwise, another cycle is performed. Suppose termination occurs after the rth cycle. Then the values of m^ ' are our approximation to the fitted cell values for this model, mjjk . It can be shown that this process will always terminate (3). Discussion of speed of convergence must, of course, involve the criterion used for termination. Initially other criteria were tried. If the criteria were too strict, or involved more com- putation than comparing each cell difference with a constant value, the amount of machine-time required increased greatly. Inspection of intermediate values showed 279

that the criterion of a difference less than 0.1 for every cell was satisfactory. The intermediate values inspected were the sums of the absolute differences between the cells of the observed configurations being used and the corresponding values for the current m's. This was done at every step. The number of cycles involved in fitting various models is given in the final section of this appendix. The program used was written in general terms so that any data matrix could be fitted to any selection of configurations of three dimensions or fewer. Provision was made for a data matrix of 3000 cells and as many as nine variables. Ten configurations could be incorporated, provided that none contained more than 100 cells. The program would also obtain the estimates for any multiplicative models in one cycle. Example: Suppose we have a 2X2X2 matrix and a sample of size 1214 distributed in the eight cells as follows: Cell x Cell X 1,1,1 213 1,1,2 527 2,1,1 27 2,1,2 198 1,2,1 60 1,2,2 48 2,2,1 22 2,2,2 119 We postulate u123 = 0 and proceed to fit the configurations First we obtain these configurations: X1+k, and X+jk. XiJ- M+k i=l i=2 i=l i=2 j=l J=2 J=l 740 225 k=l 273 49 k=l 240 82 J=2 108 141 k=2 575 317 k=2 725 167 We initially put m\jk' = 1 for all i, j, and k, and then proceed to adjust the values of m' by fitting each configuration in turn. The values after the first three fits and the final values after five cycles are given in Table 2. (The subscripts i, j, and k TABLE 2.—EXAMPLE OF ITERATIVE FITTING Cell .<"> .«*) J2> n,(3> m1 F.T.* X2 1,1,1 1.0 370.0 238.23 213.06 218.90 -0.384 0.159 2,1,1 1.0 112.5 30.12 26.94 21.10 1.246 1.650 1,2,1 1.0 54.0 34.77 53.15 54.08 0.814 0.648 2,2,1 1.0 70.5 18.88 28.85 27.92 -1.128 1.255 1,1,2 1.0 370.0 501.77 522.19 521.07 0.270 0.068 2,1,2 1.0 112.5 194.88 202.81 203.93 -0.400 0.172 1,2,2 1.0 54.0 73.23 62.60 53.96 -0.797 0.658 2,2,2 1.0 70.5 122.12 104.40 113.04 0.575 0.314 Total 1214.OO 1214.00 1214.00 1214.00 4.832 4.924 *Freeman-Tukey deviate; these are squared before they are added. 280

have been dropped from all column headings in the tables.) We also give two meas- ures of goodness-of-fit between the myk and the xijk. These are the Freeman- Tukey deviate (defined below) and the usual squared deviate, x2 =(x-m')2/m'. Turning to computational details to obtain the m,,k, which are fitted to X1]+, we add the mj,k over the categories of variable 3, for example, •ffi.-S-SCT•. Then we adjust each of the terms forming this sum by multiplying by the ratio xll+ /mn+» so we have m(D _(1) = 1X140 = 370 mill=mil2 1X 2 370- To obtain the mjjk', which are fitted to X1+k, we add the mik over the categories of variable 2, for example, mii!+ m12i = miVi =370+ 54 = 424' and we get the ratio x1+1/mj+'= 273/424. Then we have ii>\ fi\ xi+] 97-5 ™W U^_iil. = 370X4£1 = 238.231 "'in ~ "'m" (l) and w» V*/ — ~« \ / V / 1 \ — K.A \S *^ ' " — 1A HRc\ 121 ^21 ' ' ~ 494 ~~ "4.769. ml + l We proceed to fit X+jk and then repeat the cycle until successive values for the same cell do not differ by more than 0.1. This example took five cycles, so we have taken the m\jk values to be our estimates of mijk. If we look at the sum of the squared Freeman-Tukey deviates and at the summed x2 deviates, we have the values of 4.832 and 4.924, respectively. Both measures are large for a x2 variable having one degree of freedom, but inspection of the pattern for individual cells shows that there is not any one cell where the fitted values depart extremely from the observed values. Usually, we only have values of xjjk and mjjk, and if they agree closely we ac- cept the m'jjk as giving more stable cell estimates. In the present example, the data are a random sample taken from a distribution for which the ml]k were known. We show the construction of these theoretical m^ in Table 3. We set up a model with u123 = 0 but all other u-terms present, choosing the mean u = 4.5 so that the total sample size exceeded 1000. We summed the seven u-terms for each cell and took antilogarithms to give us the m1jk. In Table 4 we compare the sample observations, xjjk, and the fitted cell estimates, mjjk, with the theoretical values. We find the x2 value for the sample observations is 8.939 with seven degrees of freedom, but for the fitted estimates it is reduced to 4.081 with six degrees of freedom. Thus, in this example, our fitted values agree more closely with the theoretical values than with the observed values. 281

TABLE 3.—CONSTRUCTION OF MODEL WITH NO THREE-FACTOR INTERACTION Cell ui ua U3 Uj 2 U13 U2 , Log m for u=4..' m 1,1,1 0.4 0.5 -0.6 0. 4 0.3 -0 .2 5.3 200.337 2,1,1 -0.4 0.5 -0.6 -0. 4 -0.3 -0 .2 3.1 22.198 1,2,1 0.4 -0.5 -0.6 -0 4 0.3 0 .2 3.9 49.402 2,2,1 -0.4 -0.5 -0.6 0. 4 -0.3 0 .2 3.3 27.113 1,1,2 0.4 0.5 0.6 0. 4 -0.3 0 .2 6.3 544.572 2,1,2 -0.4 0.5 0.6 -0. 4 0.3 0 .2 5.3 200.337 1,2,2 0.4 -0.5 0.6 -0. 4 -0.3 -0 .2 4.1 60.340 2,2,2 -0.4 -0.5 0.6 0. 4 0.3 -0 .2 4.7 109.947 TABLE 4. --COMPARISON OF SAMPLE VALUES AND FITTED VALUES WITH THEORETICAL VALUES Cell m X (x-m)2 m m' (m' -m)2 m 1,1,1 200.34 213 0.800 218.90 1.719 2,1,1 22.20 27 1.038 21.10 0.055 1,2,1 49.40 60 2.275 54.08 0.443 2,2,1 27.11 22 0.963 27.92 0.024 1,1,2 544.57 527 0.567 521.07 1.014 2,1,2 200.34 198 0.027 203.93 0.064 1,2,2 60.34 48 2.524 53.96 0.675 2,2,2 109.95 119 0.745 113.04 0.087 Total 1214.25 8.939 4.081 SELECTION OF MODEL The selection of variables to be included and the numbers of categories per variable was largely an arbitrary process, based on discussion with anesthesiol- ogists and inspection of the data. In general, the number of categories was kept as small as possible. Models were attempted that included differing numbers of varia- bles. In the following discussion, the number of variables is s. If s was as small as 4, then it was sometimes found necessary to fit configurations of dimension as great as s-1. This could introduce too much random fluctuation of the raw data into the fitted values. If s was too large, then the total number of cells became large and the individual cell entries became very small, and each computed m'depended on so many parameters that its error of estimation might be large. We used simple multiplicative models to decide which u-terms seemed to be largest once the appropriate dimensions for a particular set of data had been fixed. We have derived the simple three-dimensional expression for m^ when a single two-factor term is zero in expression 9. An expression of similar form exists for models of any dimension. Suppose, for example, we have five-dimensional data (the highest dimension we have used in this report). If uj2 = 0 (and from the 282

hierarchy principle this implies that u123 , U124, u12s, "1234. U1235 » U1245» and U12345 are also zero), then it may be shown that m I2345(jjklm) = (m+2345(jklm) ) (ml+345(llclm) ) m++345(klm) Consequently, the two configurations X+2345 and X1+345 are sufficient. They are of dimension 4, and from them the third three-dimensional configuration that is needed, X++345 , may be obtained. In general, for s dimensions two configurations of dimension (s-1) are needed, and from them a third configuration of dimension (s-2) can be derived. Initially, the models obtained by removing in turn each two-factor term and its higher-order relatives were investigated. For each model, simple multiplica- tive values for m'were computed for every cell. Then a measure of divergence be- tween the observed x-value and the fitted m '-value was obtained for each cell. These measures were then summed to give a measure of deviation, which was compared with the appropriate degrees of freedom. We chose to look at simple multiplicative models first, to find out which effects were large. By considering the example of expression 10 as consisting of many "layers" of a two-way table of variable 1 crossed with variable 2, it can be seen that each layer has (1-1) (J-1) degrees of freedom. Thus, if C is the total number of cells in the matrix, we have C/IJ layers. It may be shown that each layer is independent. Birch (2) has shown this for three dimensions. The independence means that we can add the degrees of freedom for each layer. Thus, the degrees of freedom for this simple multiplicative model are C(I-1)(J-1)/IJ, provided that there are no empty cells in the (s-1)-dimensional matrices. The measure of deviation obtained when u12 and its relatives were equated to zero was compared with this number of degrees of freedom. This fitting of a simple multiplicative model was repeated for each two- factor u-term, and the discrepancy between the measure of deviation and the degrees of freedom was obtained for each. When the measure of deviation was small relative to the degrees of freedom, the two-factor term would ordinarily not be included in the iterative model to be fitted later. Because interest was focused primarily on the effect of agent in this Study, the two-factor effects involving agent were always included, regardless of the relative size of the measure of deviation. All other two-factor terms that were associated with relatively large measures of deviation were included. For each set of data, all the possible "l simple multiplicative models of this type were fitted. If several of these models showed small measures of deviation relative to the degrees of freedom, the two-factor terms that were equated to zero in these models were considered to be unimportant. The next procedure was to fit a model that included only the important two-factor terms, in addition to the s one-factor terms and the mean u-term. Usually, this model required iterative fitting. When it had been fitted, a measure of deviation and degrees of freedom were again computed. On the basis of these, a decision was made either to use this model for final presentation or to try fitting a different model. If none of the simple multiplicative models gave relatively small measures of deviation, the first iterative model attempted included all the two-factor effects. If this did not fit the data well, we proceeded to models that included three -factor effects. Effects of higher order than three were never used in this study. When it was necessary to include three-factor effects, they were all included. It is pos- sible that prolonged search would have detected other models that fitted the data as well as those we used, or even better. We were concerned not with finding the optimum model, but with finding a model that fitted the data adequately and pro- vided more stable cell estimates than the original observations themselves. In doing this, it is well if the number of cells in the configuration to be fitted is kept as small as possible. Throughout this section the term "measures of deviation" has been used as a means of assessing how well the model fitted the data. The reason for this circum- locution is that at various times we used three such measures. In some of the preliminary investigations, X2 was used, although it was realized that it might be a 283

poor choice for small cells. When the final models were to be fitted, John Tukey suggested a convenient measure and it was used throughout this study. This meas- ure, referred to as the "Freeman-Tukey deviate," (5) is defined as follows. If x is the observed count in a cell and m' the fitted value, the deviate for this cell is: d = \/x + vx+l - \/4m' + 1. The transformation to /x +\/x + 1 is known to aid stabilization of variance. The asymptotic value of the mean of these transformed variates is v 4m' + 1 for large m'. The deviate is approximately distributed according to the standard normal. The sum of squares of these deviates may be regarded as a X2 measure with the appropriate degrees of freedom for the data on deaths. It is more difficult to assess how well the model fits the data on estimated ex- posed. To get an estimate of the numbers exposed, each observation in the random sample of operations was weighted by the sampling ratio for the particular institu- tion from which the observation was drawn. This weighting had to be done before the model was fitted. When measures of deviation were computed in the manner just described for the death data, the estimates were always much larger than the related degrees of freedom, as would be expected, inasmuch as the weights were much larger than unity. The measures could be reduced to values more nearly resembling the usual x2 by dividing by the average weighting factor. We have done this in Table 5 and find that, although the measures are much closer, they are still large, relative to the measures obtained by fitting deaths. Greater reduction is achieved if, instead of using the arithmetic average of the weighting factors^ we divide by the square root of the average squared weighting factor. We could obtain further refinement by taking into account differences in total numbers of observa- tions between institutions. To justify any of these average factors, rather broad assumptions about dif- ferences in distributions between institutions are needed. Instead of attempting to determine the most desirable factor, we checked, by using a different approach, that the large measures of deviation observed were indeed due to the weighting. For some sets of data, we fitted the same model to the cases before they were weighted. We found that the measures of deviation then obtained were comparable with those obtained for the deaths. Because there is no reason to suppose that the weighted data would exhibit multiple-factor effects not shown by the unweighted data, we were satisfied that our measures of deviation were not unduly large. Apart from providing an over-all measure, the computation of deviates served a better purpose. In every instance in which a model was fitted, the Freeman- Tukey deviates were printed out for every cell. These were then inspected to ensure that their distribution was compatible with what would be expected from random fluctuations. DETAILS OF MODELS USED IN THIS STUDY This section discusses some of the models fitted in this study in some detail. The discussion covers models selected for display in this report, that is, the models that we believed gave the most satisfactory fit for each particular set of data, and covers some models which we fitted but have not displayed. Details of the models discussed are given in Table 5. For each set of data, the description of the model is followed by values of 2d2 for deaths and estimated ex- posed. As explained in the preceding section, the measure of deviation is large for the estimated exposed because every observation was weighted. We have divided the measure by the average weighting factor and this number appears directly below the uncorrected value in brackets. This average weighting factor is an underestimate of the additional amount of variability introduced, and so the corrected value is still large but much closer to the value obtained for the deaths. The next column, labeled "d.f.," shows the number of degrees of freedom with which 2d2 should be compared. The last column gives the number of cycles to convergence. All models required iterative fitting. The expression 2d2 refers to the sum of squares of the Freeman-Tukey deviate. The degrees of freedom were computed by counting the number of non- empty cells in the fitted model and subtracting from it the number of independent parameters fitted. Thus, because of zeros, even though the same model is pre- sented for both deaths and estimated exposed, the number of degrees of freedom 284

TABLE 5.--MEASURES OF DEVIATIONS FOR SELECTED MODELS Data No. set vari °f Model ables Zd2 d.f. Cycles Low 4 All 6 D 144 136 3 2-dimensional faces EE 10422 136 8 (443) 4 All 4 D 51 33 7 3-dimensional faces EE 1578 44 8 (67) Mid 5 All 10 D 482 447 3 3-dimensional faces EE 15540 418 5 (711) 4 All 4 D 167 141 3 3-dimensional faces EE 4767 130 7 (218) High 4 All 6 D 562 243 8 2-dimensional faces EE 7846 213 10 (including agent but (365) not sex) 4 All 6 D 96 81 5 2-dimensional faces EE 2867 74 6 (without agent) (133) Chole 6 A selection of 8 D 384 483 2 from 15 possible EE 16800 483 4 2-dimensional faces (743) 5 The corresponding 7 D 182 218 2 2-dimensional faces EE 9012 218 4 (10 possible) (398) may differ slightly. We use the abbreviations "Low," "Mid," and "High" for the low-death-rate, middle-death-rate, and high-death-rate sets of data, and "Chole" for cholecystectomies. Details of the variables involved and numbers of cells are given in Tables 3 and 5 of Chapter IV-3. We turn now to Table 5 of this appendix. The first models listed in Table 5 were fitted to the low-death-rate operations. For the low-death-rate operations, one advantage of fitting to two-dimensional faces, compared with three-dimensional configurations, is seen from the relative magnitude of the deviations: for the deaths, the former gives a X2 value of 144 with 136 degrees of freedom, whereas the latter gives 51 with 33 degrees of free- dom. Another reason for preferring the former model is that fitting to three- dimensional configurations involves 14 empty cells among the fitted configurations for the estimated exposed, whereas the two-dimensional model involves none. For the middle-death-rate operations, which are listed next, there seems to be little basis for choosing between four and five variables if the goodness-of-fit is considered. Use of four variables yields fewer empty cells. For the high-death-rate operations, it is difficult to fit a model to a multi- dimensional contingency table that includes both agent and operation as variables. A reason is that, for one of the four operation codes, the sample of operations did not include any observations where ether was used, and for another operation code, hardly any where cyclopropane was used. Yet it would be unsatisfactory to amalgamate the four operation codes, because their death rates vary considerably. To overcome the empty-cell difficulties and permit agent-specific estimates, an additional device was introduced. Constants were added to every cell before fitting and subsequently removed from the expected values. For the estimated ex- posed, the constant chosen was 40, approximately equivalent to two more observa- tions per cell before weighting. For the deaths, the constant was 4; 4/40 = 10 per- cent, approximately equal to the over-all death rate for these operations. Although 285

the 10 percent ratio seems reasonable, the choice of 4 and 40 is somewhat arbi- trary and more work is needed to determine either the wise selection of con- stants or alternative methods. These ideas are suggested by Bayesian estimation in multinomial problems. The deviations for this model with added constants are not given in Table 5. The model given shows why the device was necessary; the fit obtained from two-dimensional faces is poor and the number of cycles to con- vergence is greater than for any of the other models fitted. In addition to this model, which includes the variables agent, operation, risk, and age, we also fitted another model to the high-death-rate operations where the variable agent was omitted but the variable sex was introduced. We provided a fitting exclusive of agent because physicians may be interested in the death rates for particular categories of patient, regardless of the anesthetic used. We see that for this model the irregularity of the data is not as great and a satisfactory fit was obtained from the two-dimensional faces. The cholecystectomies were originally fitted to six variables. The sixth variable, "period," was a dichotomy in which each category represented 2 years of observations. With this large number of variables, it was enough to include a selec- tion of the faces. In the second model listed, the variable period was removed, re- ducing the number of variables to five. When the same combinations of the re- maining variables were fitted, the sums of deviations were relatively closer to the value expected for the degrees of freedom, but still smaller than this value. This is the model presented in Chapter IV-3. To summarize, Table 5 shows for the low-death-rate data two models of dif- fering dimensions fitted to the same set of data and the reasons for preferring one over the other. For the middle-death-rate data, we show the same dimension of model fitted to differing numbers of variables, and there is not much difference between their goodness-of-fit. For the high-death-rate data, we show two exam- ples where the same dimension of model was fitted to the same number of varia- bles but a different selection; here one model is satisfactory, but the other is a very poor fit. For the cholecystectomy data, we show two examples where a good fit was obtained from a selection among the possible two-dimensional faces, in- stead of using the total number available. REFERENCES 1. Birch, M. W. Maximum likelihood in three-way contingency tables. J. Roy. Statist. Soc., Ser. B, 25:220-233, 1963. 2. Birch, M. W. The detection of partial association, I: The 2x2 case. J. Roy. Statist. Soc., Ser. B, 26:313-324, 1964. 3. Brown, D. T. A note on approximations to discrete probability distributions. Information and Control 2:386-392, 1959. 4. Cochran, W. G. The X2 test of goodness of fit. Ann. Math. Statist. 23:315-345, 1952. 5. Freeman, M. F., and J. W. Tukey. Transformation related to the angular and the square root. Ann. Math. Statist. 21:607-611, 1950. 286

Abstract of Chapter IV-4 Smear-and-sweep is a procedure for aggregating sparse multidimen- sional contingency tables by repeatedly spreading out the data into a two-way classification (of which one variable, except at the first stage, is the con- glomerate variable formed at the preceding stage), and then pooling the re- sulting cells according to their ordering on "death rate" to produce cate- gories of a new conglomerate variable ready for the next stage. The final conglomerate variable defines a small number of categories that are more- or-less homogeneous with respect to death rate and that allow, in part, for the effects of all the variables and their interactions; such categories form a basis for calculating comparative standardized death rates. A process like this is subject to questions regarding many of the choices involved, e.g., the order in which the variables are introduced and the method of estimating within-cell death rates to be used in pooling. In applying smear-and-sweep to the middle-death-rate operations in the National Halo- thane Study, many choices were tried; this chapter presents and compares the results. To eliminate ratio estimator bias and obtain confidence limits for the interagent death-rate differences, the standardized death rates are jackknifed over institutions. The resulting tables of pseudo-values of death rate by agent, population, and institution are carefully examined by standard (e.g., ANOVA) and nonstandard (e.g., robust location estimation) techniques. The conclusions reached correspond to those found by other approaches: halothane and nitrous oxide-barbiturate are assigned death rates of about 2 percent; cyclopropane and Other appear measurably higher, with death rates of about 21A percent; the difference in death rate between these groups is statistically significant (at 5 percent) when examined in the most sensitive way; the death rate for ether looks low but is too ill-determined to yield definite comparisons. CHAPTER IV-4. THE SMEAR-AND-SWEEP ANALYSIS W. Morven Gentleman Bell Telephone Laboratories Murray Hill, New Jersey John P. Gilbert* Harvard Computing Center Cambridge, Massachusetts John W. Tukey** Princeton University, Princeton, New Jersey and Bell Telephone Laboratories, Murray Hill, New Jersey WHAT SMEAR-AND-SWEEP IS preted individually. One way to handle this situa- tion is to limit ourselves to questions that may The large number of variables available in be answered in terms of summary statistics, the National Halothane Study produces such a such as standardized death rates (by contrast, multiplicity of cells in the complete cross- for example, with procedures designed to fit, classification that, as has been noted, the data and directly interpret, detailed models). Our are too sparse for cell death rates to be inter- problem is then to compute versions of these •Part of this work was done while the author was L. L. Thurstone Distinguished Fellow, University of North Carolina; the fellowship was supported in part by Public Health Service research grant M-10006 (National institute of Mental Health). ••Prepared in part in connection with research at Princeton University sponsored by the Army Research Office (Durham). 287

summary statistics that will have most of the ad- vantages of control on all the known and available interfering variables. We must state the problem this way for three reasons: (1) data are so dilute that direct control on all the available variables is almost as impractical as direct interpretation of the cross-classified data; (2) variables that are not available (that may not even have been thought of) may have such important influences that their control would be most desirable; and (3) the available control variables are given in broad groups and involve classification error and at- tenuation. The second and third considerations are not peculiar to this Study; in fact, with re- spect to them, the data in this Study are more adequate than usual. Smear-and-sweep is a form of analysis that attempts to pool the cells of the cross-classifi- cation into categories in which the death rates (averaged over agents) are reasonably similar, and then calculates standardized death rates based on those categories. In all such methods, category formation is a qualitative analogue of index formation, and standardization provides control in a way analogous to the common use of stratified sampling. Smear-and-sweep differs from the other methods of this class in its ability to bring in many more control variables than can be used simultaneously in methods that do not assume a specific functional form. It does this by working with two variables at a time, one being (except at the initial step) the categorization obtained in the previous step. Categories must be formed in a way that adequately avoids the (direct) effects of the particular anesthetic agent used, because those effects will almost certainly not be bal- anced across the cells defined by the control variables. To ensure this, the selection of cate- gories must be based on all the relevant data in such a way that the name of the anesthetic used has no direct effect. On the null hypothesis, anesthetic has no effect; hence, it could have no effect in the category formation, and finding a significant effect in the death-rate analysis gives solid evidence against the null hypothesis— indeed, it gives solid evidence as to the direction of the major differences. (The estimation of the amount of the differences, which is not our con- cern here, raises some difficulties, which are commented on in Appendix 3.) The advantage of this approach is that it readily permits us to handle complicated bodies of data that would otherwise require nonlinear models or high-order interactions. In use, smear-and-sweep requires many choices, not only as to the variables included and the order in which they are considered, but as to general frameworks and the many details in- volved in providing cell "death rates" and using them to determine how the cells are to be com- bined into categories. Moreover, because the grouping must be on the basis of the observed data, the category into which a particular cell is pooled is influenced by sampling fluctuations, through their effect on cell "death rates." We might observe here that an analysis of the data as a stratified sample can be carried out directly, provided that the categories to be used can be determined a priori. The study of shock, presented in Chapter IV-7, is such an analysis. The distinction between that approach and smear-and-sweep is that the categories used in the latter are selected to try to make the death rates within each category about the same, rather than to give the categories any intrinsic medical meaning. Death rates can be estimated only after the data have been observed; hence, these categories must be formed a posteriori. To avoid spreading the data too thinly, we proceed sequentially, "smearing" the data out onto the cross-classification of two variables, then "sweeping" them up into categories of ap- proximately equal numbers of deaths. These categories then form the levels of a composite variable. We then take the third variable on which we wish to control, and smear the data out onto the cross-classification of this third vari- able with the composite variable just formed. This procedure is continued, smearing and sweeping repeatedly until we have introduced all the variables on which (or on suitable combina- tions of which) we are to control. Once the final set of categories has been chosen, we may compute standardized death rates over them, corresponding to various popu- lations. These standardized rates are weighted averages of the death rates associated with a particular agent in each category, the weights being the fractions of the given population in that category. The standardized rates may be thought of (subject to issues to be discussed later) as the death rates one might expect if the agent in ques- tion were applied to the given population. We choose this direct standardization because we feel that it is easier to interpret than indirect standardization. If the categories are not too small, there is essentially no difference between direct and indirect standardization. We want an over-all comparison of the agents and a com- parison for some special populations. We want an error term, so that we can test whether dif- ferences between the standardized death rates for the various agents are real or might rea- sonably be attributed to random error. The relevant error term would appear to be the com- parison across hospitals, inasmuch as we would like the conclusions of this analysis to apply to hospitals other than those in the National Halo- thane Study. Hospital main effects (differential death rates not accounted for by standardization) must be removed, however, because they are known to exist and to be large enough to swamp the effects we are looking for. 288

We may obtain such an error term and pro- vide what may be thought of as an adjusted esti- mate for each hospital by a procedure known as "jackknifing," which is discussed later. To summarize, smear-and-sweep analysis is a technique for grouping data into categories (sequentially introducing each interfering vari- able), then estimating various standardized death rates over these categories and jackknifing the estimates to obtain "pseudo-values" of the death rates for each hospital, so that we may compare the anesthetics. HOW TO USE SMEAR-AND-SWEEP This section describes the implementation of the general technique presented in the last sec- tion. Several parameters will appear; we make no attempt to specify them at this point, but they will be considered again, after we have looked at some results of applying the technique to the data of the National Halothane Study. The smear-and-sweep analysis has three distinct parts: forming the categories, estimating standardized death rates, and comparing the standardized rates. As outlined above, the conglomeration proc- ess that produces the categories is carried out sequentially. The parameters include the order in which the interfering variables are introduced and the number of the available variables to be used. Anesthetic agent, as noted above, is to be completely ignored in this step. Once we have chosen two variables, and smeared the data into the cells of that two-way cross-classification, we must estimate the av- erage death rate in each cell. We are tempted to use only the data in an individual cell and thus make the pooling wholly data-directed, i.e., in- dependent of any assumed regularity. However, because of the small numbers of observations that may appear in the cells, it is better to bor- row strength from a smoothed estimate by using an estimator of the form D+Mc . u+Mp' P = EE+M where D and EE are, respectively, the numbers of deaths and estimated exposed in the cell; p' is a smoothed estimate of the death rate in the cell, in this instance ob- tained from an additive model fitted by least squares to a table whose entries are a suitable function of the crude death rates (D/EE); and M is a constant that may be interpreted as the number of EE that we feel the estimate p' is worth, in terms of the smallness of its variability. The value of M and the function of crude death rate to be used in obtaining p' are operating parameters, which must be selected. Having estimated the average death rate in each cell, we must pool the cells into categories. This is done by listing the cells in order of in- creasing death rate, and dividing the list into segments containing approximately equal num- bers of deaths; each segment becomes a cate- gory. Other rules for sweeping are possible, such as a modified ordering, or division of the list into segments containing equal numbers of estimated exposed or equal variances of esti- mated death rates. Even with the method used, we must choose the size of the categories. In practice, this size was fixed in terms of a mini- mum number of deaths per category; other choices are possible. For any set of choices of the parameters, the categories may now be determined. The next step is to calculate the standardized death rates. As mentioned earlier, the standardized death rate is the sum over all categories, of the prod- ucts of death rate by population proportion. The death rate associated with an agent in a particu- lar category is simply estimated as the ratio of the number of patients receiving the agent who died to the estimated number exposed to the agent (or 1, if that ratio exceeds 1). The fraction in each category of the population of patients exposed to a given agent is estimated by the corresponding fraction observed, based on the total estimated number exposed to the agent. We compute the standardized death rate for each agent, first with respect to the over-all population, and second with respect to the char- acteristic population to which each of the other agents is normally applied. These standardized death rates are now jackknifed. This is a technique due to Quenouille and Tukey for estimating variance and removing any 1/N bias, such as is present in ratio esti- mators like death rates (1,3-5). (Because the hos- pitals, which we use as blocks, are not of equal size, the leading term in the bias is not com- pletely eliminated, but may be shown to be less than 2 percent of its previous size; see Appendix 1.) Briefly, jackknifing is a technique for use with nonlinear statistical estimators, whereby the data are divided into B blocks, and then the B quantities (called "pseudo-values"), B X /estimate computed-! \ from all the data J - (B-l) /-estimate computed from all the-) \data except that in the ith block/ are formed. The mean of the pseudo-values is a new estimator, unbiased at order 1/N; and 1/B times the variance of the pseudo-values is an estimate of the variance of this new estimator. The pseudo-values themselves may be regarded 289

for some purposes, including the application of certain significance tests, as approximately in- dependent replications of the original estimator. Thus, it is these jackknifed standardized death rates that we shall see in the next section. Before we look at them, however, we should give some thought to how the comparisons should be carried out. The most naive approach is to observe that for each agent-population combination, we have 34 pseudo-values; and hence we can estimate a mean and standard deviation. These enable us to compare the means. This approach is fine for a start, in that it gives a crude interval, but if we attempt to ask too delicate a question, we find that it is inadequate. It fails in two ways: (1) no account is taken of the effects of differences be- tween hospitals, which are large enough to swamp the differences that we are looking for; and (2) means and variances are always badly affected by outliers or long-tailed distributions, and the death-rate data are very long-tailed. The most obvious way to cure the first fail- ing is to use analysis of variance on the table of pseudo-values, removing the hospital main ef- fects. This approach is still sensitive to the out- lier problem, however, and in addition we have another problem. As we will see, it seems un- reasonable to assume homogeneity of variance: death rates associated with ether are not as well determined as the others. If we remove outliers and repeat the analysis with and without ether rates, the comparison of the resulting analyses of variance will be instructive. Another way to cure the failings of the naive technique is to use a nonparametric procedure. Ranova, or analysis of variance of ranks, re- moves hospital effects by doing the ranking with- in hospital, and ranks are insensitive to numeric values of outliers. Unfortunately, this technique lacks power and has the disadvantage of not giv- ing estimated death rates. The final method we use is simultaneous comparison between all pairs of the agents. This analysis, like the ranova, is carried out only on the 34 X 5 table of pseudo-values of over-all death rate. We regard its rows as 34 observa- tions of a five-dimensional vector, and estimate a mean vector and covariance matrix. This permits us to make simultaneous comparisons of all agent-agent differences without assuming any special form for the correlation between the observations, or any specific ratios between variances, while canceling out hospital main effects. If, instead of a mean vector and covari- ance matrix, we use a robust multivariate loca- tion procedure, obtained by least pth devia- tions (2), this approach is not sensitive to outliers. One last tool that we will mention is a cri- terion for comparing choices of parameters. We take logarithms of each entry in the 5X5 table of pseudo-value means for anesthetic agent vs. population. A simple additive model with agent and population main effects is fitted, and the residual sum of squares is then used as our cri- terion. Because we would like statements about populations and agents to be independent, small values of the criterion are desirable. Why do we choose this particular criterion? Other than the interaction term interpretation just mentioned, we may view the residual sum of squares as a measure of error in approximating the 5X5 table that is a function of 10 vectors by a simpler 5X5 table that is merely a function of nine scalars. The preliminary logarithmic transformation seems sensible, both intuitively and from inspec- tion of the data. Moreover, for many of the para- meters of the smear-and-sweep that can be se- lected by ad hoc arguments, this criterion yields the same choice. These may not seem very tell- ing arguments, but fortunately the final results are so similar that the choice of the parameter- determining criterion is not critical. RESULTS Various combinations of parameters were tried on the middle-death-rate operations and are available for comparison. They are listed in Table 1. Broadly, the results are the same for all these cases. For example, consider the death rates by agent, standardized to the over-all popu- lation, as given in Table 2. We see that, regardless of the parameters chosen in the smear-and-sweep, the standardized death rate associated with halothane is similar to that for nitrous oxide-barbiturate, at about 2.0 percent. The death rate associated with Other is noticeably higher, at about 2.4 percent, and that with cyclopropane higher yet, at about 2.5 per- cent. The analyses for ether, however, behave erratically, the standardized death rate appear- ing to change wildly with the parameters. This may be traced directly to the infrequent use of ether in operations with higher death rates. Hence, there are exceedingly few cases in the random sample, and it makes a large difference which category they fall into. We might wish to inquire whether the differ- ences observed in Table 2 are statistically sig- nificant. Our first approach is to consider the variance of the pseudo-values corresponding to each entry in the table; from these we can calcu- late a standard error for each. These standard errors are given in Table 3. The remarkable agreement in the columns of Table 3 again indicates that the choice of the various parameters in smear-and-sweep analysis is not as important as might have been feared. Notice also the large difference between the standard error of the ether death rate and any of the others. This is not surprising: ether is used much less than the other agents, and so we might have expected its death rate to be poorly deter- mined. Although these standard errors are useful in that they show the degree of precision of our 290

TABLE 1.—ANALYSES PERFORMED Analysis Variables* (in order) Transfor- mation** U*** No. of categories Criterion I oui SQRT 5000 24 0.0164 II QU SORT 5000 12 0.0214 III out SORT 500 25 0.0943 I» QUA. in 5000 24 0.0508 V dtt SORT 5000 24 0.0138 VI QRA SQRT 500 24 0.0392 VII ORA SORT 50 24 0.0365 VIII UD SQRT 500 25 0.0132 IX ORAL SQRT 500 24 0.0755 X OARL Sya 500 24 0.0257 XI LQAR SORT 500 24 0.0405 zn LORA SORT 500 34 0.0249 *0 = operation, A = age, R = risk (physical status), and L = length (duration) of anesthesia. 0 and L themselves have been grouped to reduce the number of levels. **S9tT - square root; LN - natural logarithm. ***A constant that weights the death rate observed in the cell. TABLE 2.—OVER-ALL DEATH RATES FOR MIDDLE-DEATH-RATE OPERATIONS ESTIMATED BY VARIOUS ANALYSES Analysis Agent H N-B C E 0 I 0.0193 0 .0202 0.0253 0.0179 0.0241 II 0.0186 0 .0201 0.0259 0.0165 0.0255 III 0.0191 0 .0203 0.0250 0.0247 0.0249 IT 0.0209 0 .0199 0.0255 0.0200 0.0250 V 0.0203 0 .0205 0.0255 0.0183 0.0245 VI 0.0208 0 .0212 0.0245 0.0209 0.0241 VII 0.0214 0 .0213 0.0264 0.0163 0.0243 VIII 0.0192 0 .0199 0.0254 0.0181 0.0241 IX 0.0215 0 .0207 0.0253 0.0195 0.0241 z 0.0199 0 .0212 0.0254 0.0186 0.0269 n 0.0195 0 .0219 0.0250 0.0214 0.0238 XII 0.0197 0 .0210 0.0254 0.0190 0.0240 range 0.0029 0 .0020 0.0019 0.0084 0.0031 TABLE 3.— STANDARD ERRORS OF ENTRIES IN TABLE 2 Agent Analysis H N-B 0 E | 0 I 0.0028 0 .0018 0.0016 0.0051 0.0024 II 0.0035 0 .C016 0.0015 0.0052 0.0024 III 0.0027 0 .0016 0.0016 0.0053 0.0018 IV 0.0021 0 .0016 0.0017 0.0039 0.0022 V 0.0029 0 .0016 0.0016 0.0038 0.0022 VI 0.0017 0 .0019 0.0016 0.0030 0.0022 VII 0.0018 0 .0013 0.0017 0.0049 0.0021 VIII 0.0027 0 .0018 0.0017 0.0048 0.0020 n 0.0019 0 .0016 0.0016 0.0060 0.0019 X 0.0021 0 .0018 0.0014 0.0043 0.0019 XI 0.0025 0 .0018 0.0014 0.0044 0.0020 XII 0.0025 0 .0014 0.0015 0.0040 0.0018 death-rate estimates, they have two serious de- fects: they do not take account of hospital main effects, and they are sensitive to outliers. The former is not completely undesirable, for it en- ables us to consider the question: "To what ex- tent can we detect differences between the agents if we ignore differences between hospitals?" That question concerns not only the error variance, but also the relative magnitudes of the hospital and agent main effects. We are led to the answer that, ignoring the effect of outliers, we are not able to detect any differences. As we shall see, this is due both to outliers and to hospital effects. We may go on to ask the question: "Taking into account the hospital main effects, can we detect differences between agents?" We naturally are tempted to apply ANOVA to the 34 X 5 table of pseudo-values of over-all death rates for hospital vs. agent. We will illustrate, using only analysis VIII (in which we standardized on age, risk, and operation in the order ARO, fitted p' using a square root transform with M = 500, and used 25 categories); results for other analyses are similar. The table of pseudo-values for this case ap- pears as Appendix 4. Negative death rates, such as appear there, should not disturb us. These are pseudo-values, and if small ratios are to be un- biased, occasional negative numbers must be ex- pected. Proceeding in a straightforward manner to compute an analysis of variance, we produce Table 4. Because the F ratio for agent effects is less than its critical value, we might be tempted to conclude that there was no significant agent ef- fect. Before jumping to such a conclusion, how- ever, we should observe the lack of homogeneity in the variances, and also observe that over two- thirds of the error sum of squares comes from four residuals defined by hospital-agent pairs (9,E), (ll.E), (8,H), and (20,E), three of which involve ether. We feel compelled to repeat the analysis, leaving out ether; doing this gives Table 5. We see that both the effects are now signifi- cant, although the residual (8,H) still contributes 40 percent of the error sum of squares. Trim- ming this outlier, if appropriate, would increase both F ratios. Another way to carry out these tests is by ranova. Here we rank the death rates for each hospital, then sum the ranks for each agent. The statistic where W = A2 ., 2R? - 3n(k+l), nk(k+l) i k = number of items being ranked, n = number of times we rank, and Rj = sum of ranks for ith item, is approximately x k-r In our case the sums of the ranks are: iL N-B 81 83 —C 121 E 109 O 116 for a W of 16.56 on four d.f., which is significant at the 0.5 percent point. The last way we will consider the table of death rates cross-classified by agent and hos- pital is to make multiple comparisons by estab- lishing simultaneous confidence intervals on the difference in death rates between each pair of agents. From the five-dimensional mean and variance matrix, we get direct estimates of these pairwise differences and their variances; addi- tive hospital effects do not enter, nor variances 291

for other agents. However, with only 33 degrees of freedom, the analysis is very susceptible to outliers, so that when we first attempt it, we obtain Table 6, in which none of the differences is significant because all the intervals include zero. However, if we decide to use the more ro- bust estimators obtained by minimizing pth devi- ations for p <2, we obtain Table 7 for p = 1.5 and Table 8 for p = 1.25. The asterisks indicate the significant differences H - C, H - O, and N-B - C. Notice that the robust estimators both gave about the same intervals, and that the intervals are only about half as long as those of the least squares method, indicating how badly the outliers had increased the variance estimates appropriate to simple arithmetic means. TABLE 4.--TWO-WAY ANALYSIS OF VARIANCE TABLE 7.—DEATH-RATE ESTIMATES BASED ON LEAST 1.5 POWER DEVIATIONS Sum of squares Mean square 5? d.f. F critical value Hospitals 0.01553 33 0.000471 2.06 1.55 Agents 0.00138 4 0.000346 1.50 2.44 Error 0.03026 132 0.000229 TABLE 5.—TWO-WAY ANALYSIS OF VARIANCE WITHOUT ETHER Sum of squares d.f. Mean square F 5% critical value Hospitals 0.008997 33 0.000273 2.56 1.57 Agents 0.000941 3 0.000314 2.94 2.70 Error 0.01058 99 0.000107 TABLE 6.— DEATH-RATE ESTIMATES BASED ON LEAST SQUARED DEVIATIONS H N-B C E 0 0.0192 0.0199 0.0254 0.0181 0.0241 5,i simultaneous intervals for differences H - N-B H - C H - E H - 0 N-B - C N-B - E N-B - 0 C - E C - 0 E - 0 (-0.0105, 0.0090) (-0.0149, 0.0026) (-0.0142, 0.0164) (-0.0112, 0.0014) (-0.0121, 0.0012) (-0.0146, 0.0182) (-0.0112, 0.0029) (-0.0066, 0.0210) (-0.0047, 0.0072) (-0.0196, 0.0076) H N-B 0 0.0201 0.0195 0.0256 0.0217 0.0247 5% simultaneous intervals for differences H - N-B H - C H - E H - 0 N-B - C N-B - E N-B - 0 C - E C - 0 E - 0 (-0.0057, 0.0068) (-0.0106, -0.0003)» (-0.0095, 0.0063) (-0.0082, -0.0010)» (-0.0118, -0.0003)» (-0.0113, 0.0069) (-0.0108, 0.0005) (-0.0042, 0.0119) (-0.0038, 0.0055) (-0.0107, 0.0047) TABLE 8.—DEATH-RATE ESTIMATES BASED ON LEAST 1.25 POWER DEVIATIONS N-B 0.0202 0.0195 0.0255 0.0227 0.0248 5% simultaneous intervals for differences -H - N-B H - C H - E H - 0 N-B - C N-B - E N-B - 0 C - E C - 0 E - 0 (-0.0050, 0.0064) (-0.0098, -0.0009)* (-0.0083, 0.0031) (-0.0074, -0.0017)* (-0.0117, -0.0005)* (-0.0100, 0.0034) (-0.0108, 0.0002) (-0.0033, 0.0089) (-0.0036, 0.0051) (-0.0078, 0.0038) We must note that the effects of combining these robust estimation procedures with jack- knifing have not been explicitly studied. The con- nection of jackknif ing with t- and F-tests is par - ticularly close, in that jackknifing is intended to reduce bias in terms of averages and to provide mean squares whose average values are almost correct. For such tests, the greatest difficulty, which is usually not too serious, arises from loss of degrees of freedom and is confined to 292

situations that are usually easily identifiable. According to our present understanding of jack- knifing and robust estimation, we expect the largest part of these good features to continue in the present case, but we must admit that this view rests on understanding and judgment, rather than on trial or theory. We turn now to a consideration of the 5X5 table of death rates (means of pseudo-values) for anesthetic agents vs. populations, Table 9. We will again look only at analysis vm. Corresponding to this is a table of the stand- ard deviations of these numbers (Table 10), as computed from the pseudo-values. (The pseudo- values themselves appear in Appendix 5.) One instructive table (Table 11) is computed "x by using the fact that, if y = In x, then iry «—, x and estimating the standard error of the logar- ithms of the death rates in Table 9. We see that all these standard errors are more-or-less the same, except that those involving either the ether population or ether as an anesthetic are notice- ably larger. I f we neglect this, we can pool the standard errors of Table 11 to give a combined mean square of 0.0491 on 825 d.f. (This 825 should not be interpreted literally, inasmuch as the ele- ments in the 5x5 tables are not independent, but we may assume that the correct figure is still very large.) Fitting an additive model to the lo- garithms of the death rates gives a mean square of 0.2344 on four d.f. for population differences and a mean square of 0.1003 on four d.f. for agent differences. We can therefore form Table 12, the analysis-of-variance-like table. Here, too, we could have omitted ether, getting instead Table 13. The last two tables must be taken with a grain of salt and not construed as a proper test of significance. But they are instructive in in- dicating that the kinds of patients given the va- rious anesthetics are distinctive in terms of age, risk, and operation, even after the pooling in- volved in forming smear-and-sweep categories. We also see again here that, when we are not deceived by the excessive variance of ether, nor too seriously bothered by outliers, we can show clearly that the differences between agents are significant. It is interesting to look at the estimates of logarithmic main effects. They are (to base e): H N-B E agent -0.1043 -0.0577 0.1759 -0.1392 0.1253 popu- lation -0.1767 -0.1970 0.2985 -0.0755 0.1507 We have referred several times to analysis VHI. We chose this analysis for two reasons: (1) the choices of the parameters in this analysis are the ones we prefer, as discussed immedi- ately below; and (2) it has the smallest value of the criterion introduced at the end of the last TABLE 9.--STANDARDIZED DEATH RATES FOR FIVE AGENTS ON FIVE POPULATIONS (from analysis VIII) Population Agent H N-B C E 0 H 0.0172 0 .0169 0.0219 0.0154 0.0205 N-B 0.0164 0 .0173 0.0214 0.0148 0.0203 C 0.0261 0 .0274 0.0346 0.0267 0.0327 E 0.0178 0 .0190 0.0240 0.0182 0.0226 0 0.0221 0 .0240 0.0301 0.0219 0.0296 TABLE 10.—STANDARD ERRORS OF ENTRIES IN TABLE 9 Population Agent H N-B C E 0 H 0.0020 0.0024 0.0026 0 .0045 0.0026 N-B 0.0029 0.0022 0.0028 0 .0051 0.0031 C 0.0061 0.0043 0.0046 0 .0079 0.0056 E 0.0058 0.0054 0.0050 0 .0044 0.0061 0 0.0066 0.C037 0.0037 0 .0082 0.0045 TABLE 11.— APPROXIMATE STANDARD ERRORS OF LOGARITHMS OF DEATH RATES IN TABLE 9 Agent Population H N-B C E 0 H 0.114 0.141 0.118 0.295 0.125 N-B 0.178 0.126 0.129 0.348 0.155 C 0.236 0.157 0.132 0.296 0.172 E 0.327 0.285 0.208 0.242 0.269 0 0.300 0.155 0.124 0.374 0.152 section, as we can see in Table 1. We mention in passing that the values of this criterion are ex- tremely small if we interpret them as interac- tion sums of squares and compare them with the mean squared error of Table 12. It is better to think of the criterion as the error in an approxi- mation—an approximation that we see is exceed- ingly good. FURTHER CONSIDERATION OF THE PARAMETERS In considering the question of what para- meter values are best, the simplest problem is 334-553 O-69—20 293

TABLE 12.—ANALYSIS OF LOGARITHMS OF DEATH RATES Mean square d.f. Population 0.2344 4 4.78 (very significant) Agent 0.1003 4 2.05 Error 0.0491 825 TABLE 13.—ANALYSIS OF LOGARITHMS OF DEATH RATES (OMITTING ETHER) Mean square d.f. Population 0.2404 3 8.9 (very significant) Agent 0.0760 3 2.8 (significant) Error 0.0270 528 that of the choice of M. We may approximate the problem by considering the estimator D+Mp' kR+M where D is binomial, mean Np and variance Np(l-p), R is Poisson, mean N/k and variance N/k, p' has mean p and variance a2, D, R, and p' are independent, and k is the inverse of the sampling fraction. Then by the theory of propagation of error we find that to leading order the mean squared error of this estimator as an estimate of p is Np(l+kp-p) + M 2 2 (N+M)2 which is minimized when M =P(1+kP-P) . a^ In this study, k may be taken as 25 and we may obtain estimates of a2 by looking at the residuals when p' is obtained by fitting. Because we used a square root transformation, a 2 a 4p X (average squared residual). For the sake of ro- bustness, we have chosen to estimate the average squared residual as the square of the 70th per- centile of the squared residuals. For the three permutations of age, risk, and operation we get Table 14. We see that for the average death rate in the Study, 2 percent, M = 500 is a fairly rea- sonable choice, but for a higher death rate, such as 20 percent, a much larger M, such as 2000, seems better. Because the cells that we are most concerned about are those with few data, which typically are those with high death rates, it seems that M = 5000 might have been a better choice than the 500 we used in most of our analyses. As a further comment on this, in Table 15 we repeat from Table 1 the values of our cri- terion for analyses V, VI, and VII, which differ only in the choice of M, and similarly for anal- yses I and III. The next question is what transformation one should use to fit for p'. For this we consider analyses IV and V, which differ only in that IV used the logarithmic transformation and V used the square root. The criterion values were 0.0508 and 0.0138, respectively, indicating that the square root was better. But separate evi- dence may be obtained by reordering the resid- ual tables so that the rows and columns are in order of increasing main effects. For analysis IV, using the logarithm, we get Table 16. For analysis V, using the square root, we get Table 17. The pattern of minus signs from upper left to lower right in Table 16 is an indication that the curvature in the transformation was too strong. (If it had been too weak, the diagonal would have had a band of plus signs, with minus signs in the upper right and lower left corners, as the less marked pattern in Table 17 tends to show.) These statements can be made more quantitative by counting the excess of plus signs in the indicated corners of Tables 16 and 17, subtracting the resulting excesses in the upper right and lower left from those in the upper left and lower right. Carrying out the calculation gives -52 for Table 16 but +26 for Table 17, in- dicating that the best transformation is between these two, closer to the square root (perhaps close to the cube root). TABLE 14.—BEST VALUES OF M ACCORDING TO SIMPLE MODEL Variables smeared on Estimated a1 Optimal M P = 0-2 | p = 0.02 A and R 0.000275 p 21,000 5,400 AR and 0 0.00364 p 1,590 406 0 and A 0.00110 p 5,250 1,35O OA and R 0.0061 p 950 240 0 and R 0.00347 p 1,670 430 OR and A 0.00252 p 2,300 590 TABLE 15.— EFFECT OF M ON THE CRITERION Analysis M Criterion V 5000 0.0138 VI 500 0.0392 VII 50 0.0365 I 5000 0.0164 III 500 0.0943 294

TABLE 16.--SIGNS OF RESIDUALS OF FIT TO LN CRUDE OPERA- TION BY RISK DEATH RATES TABLE 17.—SIGNS OF RESIDUALS OF FIT TO SQRT CRUDE OPERA- TION BY RISK DEATH RATES To consider what order the variables should be used in, and the related question of how many variables to use, we could argue by analogy with stepwise regression that the most important va- riables should be used first, and that in general we should use as few variables as possible, using a variable only if it substantially affects the re- sults. The analogy seems inappropriate to smear-and-sweep, however, because this is an ordering and partition problem, which must be rather more like ordering punchcards, where any preliminary passes on irrelevant variables do not detract from the final results, but final irrelevant passes can be quite harmful. (The serious harmfulness of irrelevant passes in sorting is clearly a false analogy, but mild harmfulness from using an irrelevant variable in the last smear-and-sweep is quite possible.) If we consider analyses III, VI, and VnI, which differ only in order, and analyses IX, X, XI, and XII, which also differ only in order, we have Table 18. (Because the order of the first two variables is irrelevant, we have given both orders.) Because we have reason to believe that operation is the most important variable, the comparison of the first three looks right. On the other hand, because we believe that length of anesthesia (L) is less important than at least two of the other three variables, we would expect that using it first would be preferable to using it last, 295

TABLE 18.—EFFECT OF ORDER ON THE CRITERION Variables used Analysis Criterion III AOR=OAR 0.0943 VI ORA=ROA 0.0392 VIII ARO=RAO 0.0132 IX ORAL=ROAL 0.0755 X AORL=OARL 0.0257 XI LOAR=OLAR 0.0405 XII LORA=OLRA 0.0249 which these comparisons tend to confirm. Indeed, it might well be that using it first would improve the results, and using it last would degrade them, although we do not have any direct comparisons to check this. Finally, we come to the sweeping process. Here we have very little to guide us. If we accept our present pickup rule, then how many cate- gories should we use? Presumably, we want to use as many categories as we can before the variance caused by the categories' being too small becomes greater than the bias reduction from separating unequal death rates. But looking at the variance estimates from analyses I and II, which differ only in the number of categories, we see that using only 12 categories in analysis II gave almost exactly the same variance as did the 24 categories in analysis I. Our criterion had values of 0.0164 and 0.0214 for analyses I and II, respectively, and so it also preferred having more categories. It seems that one would not want very many more than 24 categories, for even with this number the estimated exposed in some of the high-death-rate categories is very small, and hence the death-rate estimates in these categories become unstable. This brings us back to the question of whether the use of equal numbers of deaths was, in fact, a sensible pickup rule. We have very little to go on here; in fact, no other pickup rule was tried with this analysis. By analogy with stratified sampling, perhaps the ideal rule would be one that gave equal values of the square of the population proportion times the variance of the death-rate estimate in the categories. Unfor- tunately, we do not know how to realize such a pickup. It is clear that any rule that worked on the basis of equal numbers of estimated exposed would be further from this ideal than our current rule is; for example, in the middle-death-rate operations about three-fifths of all estimated ex- posed correspond to combinations of the vari- ables with death rates below 0.2 percent. Some pickup rule based on counting estimated exposed but producing decreasing numbers in the cate- gories with higher death rates would be interest- ing to try, but at present we do not know how they should decrease. One simple modification that would improve the current pickup rule would be to pick up as we do to satisfy a requirement that there be a minimum number of deaths, then to pool those categories with fewer than a mini- mum number of estimated exposed. SCOPE OF ANALYSIS In interpreting our results, it is important that we be very clear about the exact question or questions to which they respond. The aspect about which we are most likely to be confused is the extent to which institutional differences have been dealt with. The jackknife is a new technique; accordingly, it is easy to fail to note some of its detailed workings. What the jackknife has done for us is to al- low a proper test of significance, oriented toward the question: In a population of institutions from which these 34 are a reasonable sample, what reliable differences are there among death rates that are pooled over institutions but standardized over categories? What the jackknife has not done for us is to make any allowance for indirect effects spring- ing from differences in institutional death rates over and above those accounted for by standardi- zation over categories. Chapter IV-6 shows that such differences exist. Because we are keenly aware that institutions use different agents with different relative frequencies, the possibility of such indirect effects is well established. Appen- dix 7 makes exploratory and illustrative calcu- lations of how large these indirect effects might be, and concludes that they are likely to be small and unlikely to alter the conclusions found in this chapter, although there is a bare possibility that they might be shown to do so. CONCLUSIONS The most evident conclusion from the re- sults of this analysis is that, for the middle- death-rate operations, there appears to be a significant difference in the standardized death rates associated with the various agents. Halo- thane and nitrous oxide-barbiturate have lower rates, and cyclopropane and Other have higher ones. This conclusion appears to be valid, whether we talk about death rates standardized to the over-all population or death rates standardized to the individual populations to which the various anesthetics are customarily applied. Looking at one of these populations seems merely to change all five of the standardized death rates by a mul- tiplicative factor. Another important conclusion is that, owing to their high variance, very little can be said about the ether death rates. This may be obvious, inasmuch as ether is so infrequently used, but it is well to keep it in mind. 296

The significant differences between hos- pitals, commented on elsewhere, are interesting; perhaps more interesting is the apparent long- tailedness of the distribution of death rates, the presence of so many outliers. Expected, but reassuring, was the significant difference between the populations exposed to the various agents, for it meant that this information had not been lost in the smearing and sweeping. With respect to what we have learned about smear-and-sweep itself through this study, not very much can be said. The practical details of applying it as a technique have been solved, per- haps, but further theoretical insight into its be- havior seems needed. The very consistency of the results, regardless of the choice of the para- meters, which increases our confidence in the foregoing conclusions, means that we have diffi- culty using this analysis to learn the choices of parameters to be preferred. For instance, does this consistency imply that in general the choice of parameters is unimportant, or does it imply that the results with the halothane data were so obvious that the parameter choice had little ef- fect? (One result of this consistency was the de- cision not to try any other parameter combina- tions, because it was felt that not enough new would be learned to warrant the additional effort required as a result of the departure of several members of the staff.) It is perhaps useful to point out that through- out this analysis we have been dealing with quan- tities only barely detectable above the noise. No study much smaller than the National Halothane Study could have been successful in resolving the observed differences in the death rates as sig- nificant. REFERENCES 1. Brillinger, D. R. The asymptotic behaviour of Tukey's general method of setting ap- proximate confidence limits (the Jackknife) when applied to maximum likelihood esti- mates. Rev. Inst. Internat. Statist. 32:202- 206, 1964. 2. Gentleman, W. M. Robust estimation of mul- tivariate location by minimizing pth power deviations. Ph.D. thesis, Princeton Univer- sity. 1965. 3. Jones, H. L. The jackknife method, pp. 185- 201. In Proceedings of the IBM Scientific Computing Symposium on Statistics, Octo- ber 21-23, 1963. White Plains, N. Y.: IBM Data Processing Division, 1965. 4. Miller, R. G., Jr. A trustworthy jackknife. Ann. Math. Statist. 35:1594-1605,1964. 5. Mosteller, F., and J. W. Tukey. Data anal- ysis, including statistics, pp. 80-203. In G. Lindzey and E. Aronson, Eds. The Handbook of Social Psychology. Vol. 2. (2nd ed.) Reading, Mass.: Addison-Wesley, 1968. 297

APPENDIX 1 TO CHAPTER IV-4 (EQUALLY WEIGHTED) JACKKNIFING IN UNEQUAL-SIZED BLOCKS Consider an estimator f(xj, ..., xN) such that ..... xN)] = e+±. We wish to find the expectation of the jackknifed version of this estimator, where we use B blocks, of size If we may assume that all the kj are small, we can expand the denominator so that the bias is approximately ,- Bki2 Nk 2 ..... B The ith pseudo-value is B-f(xj, ... , xN) - (B-l)-[f calculated from all data but the ith block]. The expectation of the average of the pseudo-values, therefore, is: B-E[f(x1. ... , xN)] - (B-l)-ave E[f calculated from all data but the ith block] NB -k = - £ var(k1), which we may prefer to think of as „ rvar(ki)| - N [ave(k1)J - One example we might think of considering is all operations, with hospitals as blocks. In this case B = 34, N = 856,865, and var(ki) = 0.00045723; hence, jackknifing multiplied the original bias by the factor -0.0156, reducing it by a factor of more than 60. NOTE: Had we chosen weights in the jackknife, which we did not, we could have made the 1/N bias zero, as occurs with equal-sized blocks. 298

APPENDIX 2 TO CHAPTER IV-4 STANDARDIZATION IN BROAD GROUPS The purpose of standardizing the death rates in the smear-and- sweep analysis is to ad- just the death rates to correct for the effects of applying the agents to different populations. When we standardize in broad groups, however, such as the categories in the smear-and-sweep anal- ysis, we can correct only for the group- to- group differences; we are unable to correct any im- balances within groups. This leads us to wonder about the magnitudes of such within-group biases. Could the high cyclopropane death rate be due to cyclopropane's being given to worse patients than was halothane, within each category? We can try to estimate the magnitude of such an effect. Let us assume that there is some variable 0, over which these categories have been defined. In our case this is some composite of age, risk, and operation, for example. The distribution of the true values of 6 for patients in category i is fj (6). We wish to compare the agents at the same value of 0, but the only proportions we can ob- serve are for the entire category. Thus, although the proportion of those patients having variable value 8 who are given agent a is y(a, 8), we can observe only the average proportion of category i who receive agent a, cj - j y (a, 0)f j(0)d 0. Sim- ilarly, although the proportion of patients having variable value 8 and receiving agent a who die is 8 (a,0), we can observe only the average proportion who are given a and then die, f8(a.,0)y(a.,0)tv(9)d8, or, normalizing by cj, the average proportion of those receiving agent a who die, J8(a,0)y(a, = If the imbalances are small, we can study them by expanding y and 8 about (ij , the mean value of 8 in the category, getting y(a,0)= y0(a) S(a,0) = 80(a) / w., yv M ~ ' '" +... Then, using aj2= J(0-Hj)2 fl(B)d6, the vari- ance of 6 in the category, we find that the propor- tion in category i exposed to a is approximately = y0(a) + and that the proportion in category i exposed to a who die is approximately {y2(a)S0(a) y0(a) 82(a)} a2. The estimated death rate associated with agent a in category i is thus j = 80(a) + y0(a)82(a) y0(a) y2(a)ff1 I f we regard the final categories of the smear- and-sweep as an approximation to such a variable 6 , we can use the data in Appendix 6 to estimate this second term and its cumulative effect. The observed proportion exposed, c, and death rate, d, can be plotted against category number, from which yj, y2, y0, 81, and 82 can be estimated, and then the observed death rate can be corrected 2 to 8Q. The variances al must, however, be chosen, and we took lr2 = 1. This choice corre- i spends to having over 95 percent of the patients in the correct category, or in the one next to it, or in the one next to that. Carrying out the computations, as on work sheets 1 and 2, we find that the total effect would be that the halothane death rate was estimated 0.000195 too high, and the*cyclopropane death rate 0.000420 too high. Moreover, the greatest part of this was the influence of 82, the curvature of the death rate across categories. This analysis suggests that in our problem the effect of stand- ardizing in broad groups is negligible. 299

WORK SHEET 1 ESTIMATING BIAS OF CYCLOPROPANE DEATH RATE Correction Observed Observed term to * _* '0 death i ., Yjl V2 death rate *0 Category propor- 1 1 tion (c) rate (c [),% l*l+7O*2 c 1 0.1575 0.0002 0.0004 0.157 0.188 0.90 0.000 0.00018 0 0.0011 0.187 2 0.122 0.0010 0.0004 0.122 0.884 0.90 0.000 0.00090 0 0.0074 0.877 3 0.150 0.0020 0.0005 0.149 2.047 0.90 0.000 0.00180 0 0.0120 2.035 4 0.162 0.0030 0.0006 0.161 2.803 0.90 0.013 0.0027 0.0020 0.0271 2.776 5 0.1325 0.0044 0.0006 0.132 3.697 0.95 0.025 0.0042 0.0033 0.057 3.640 6 0.2025 0.0056 0.0006 0.202 3.736 1.00 0.013 0.0056 0.0025 0.040 3.696 7 0.132 0.0070 0.0006 0.131 6.090 1.00 0.013 0.0070 0.0017 0.066 6.024 8 0.168 0.0082 0.0006 0.167 6.760 1.05 0.025 0.0086 0.0042 0.076 6.684 9 0.157 0.0094 0.0006 0.156 8.316 1.10 0.025 0.0103 0.0039 0.090 8.226 10 0.233 0.0105 0.0005 0.232 6.552 1.15 0.025 0.0121 0.0058 0.077 6.475 11 0.134 0.0116 0.0006 0.133 13.233 1.20 0.038 0.0139 0.0051 0.142 13.091 12 0.278 0.0130 0.0006 0.277 7.167 1.30 0.050 0.0169 0.0138 0.110 7.057 13 0.166 0.0141 0.0005 0.165 12.665 1.40 0.075 0.0197 0.0124 0.193 12.472 14 0.156 0.0150 0.0004 0.156 17.860 1.60 0.100 0.0240 0.0156 0.254 17.606 15 0.214 0.0158 0.0002 0.214 13. 014 1.80 0.125 0.0284 0.0268 0.258 12.756 16 0.210 0.0158 -0.0002 0.210 13.471 2.10 0.200 0.0332 0.0420 0.358 13.113 17 0.260 0.0150 -0.0006 0.261 16.034 2.60 0.288 0.0390 0'.0752 0.440 15.594 18 0.352 0.0132 -0.0010 0.353 17.558 3.25 0.378 0.0429 0.1332 0.502 17.056 19 0.304 0.0110 -0.0011 0.305 22.006 4.10 0.438 0.0451 0.1336 0.588 21.418 20 0.296 0.0088 -0.0011 0.297 27.315 5.00 0.513 0.0440 0.1523 0.663 26.652 21 0.326 0.0065 -0.0011 0.327 29.844 6.15 0.600 0.0400 0.1960 0.724 29.120 22 0.188 0.0044 -0.0009 0.189 38.727 7.40 0.688 0.0326 0.1300 0.865 37.862 23 0.340 0.0030 -0.0006 0.341 37.836 8.90 0.788 0.0267 0.2685 0.869 36.967 24 0.341 0.0018 -0.0006 0.342 58.855 10.50 0.978 0.0189 0.334 1.035 57.820 25 0.540 0.0007 -0.0004 0.540 70.385 12.80 1.088 0.0090 0.588 1.105 69.280 ESTIMATING BIAS OF HALOTHANE WORK SHEET 2 DEATH RATE served Ob- Observed Correction Cate- 1 >2 70 death h * -V !i V2 to death | gory pro- rate 71* 1 rate *0 portion (c) (a),* 71*1+^0 "2 c 1 0. 352 -0.0002 -0.00026 0 .352 0.134 0. 60 0.00 -0.00012 0 -0.0003 0.134 2 0. 356 -0.0007 -0.00026 0 .356 0.685 0. 60 0.00 -0.00042 0 -0.0012 0.687 3 0. 331 -0.0012 -0.00026 0 .331 1.498 0.60 0.025 -0.00072 0.0083 0.0230 1.475 4 0. 329 -0.0018 -0.00026 0 .329 2.394 0.65 0.025 -0.00117 0.0082 0.0213 2.373 5 0. 372 -0.0023 -0.00026 0 .372 2.284 0. 70 0.025 -0.00161 0.0093 0.021 2.263 6 0. 296 -0.0028 -0.00026 0 .296 2.518 0. 75 0.025 -0.0021 0.0074 0.018 2.500 7 0. 320 -0.0033 -0.00026 0 .320 3.975 0. 80 0.025 -0.0026 0.0080 0.017 3.958 8 0. 312 -0.0038 -0.00026 0 .312 5.339 0. 85 0.025 -0.0032 0.0078 0.015 5.324 9 0. 334 -0.0044 -0.00026 0 .334 6.023 0. 90 0.025 -0.0040 0.0084 0.013 6.010 10 0. 306 -0.0049 -0.00026 0 .306 6.454 0. 95 0.025 -0.00455 0.00765 0.010 6.444 11 0. 362 -0.0054 -0.00026 0 .362 5.490 1.00 0.038 -0.0054 0.0138 0.023 5.467 12 0. 254 -0.0059 -0.00026 0 .254 9.443 1. 10 0.050 -0.0065 0.0127 0.024 9.419 13 0. 316 -0.0064 -0.00026 0 .316 8.401 1. 20 0.075 -0.0077 0.0246 0.054 8.347 U 0. 349 -0.0070 -0.00026 0 .349 10.165 1. 35 0.075 -0.0094 0.0262 0.048 10.127 15 0. 302 -0.0075 -0.00026 0 .302 11.310 1. 50 0.088 -0.0112 0.0268 0.052 11.258 16 0. 215 -0.0080 -0.00026 0 .215 24.749 1.70 0.125 -0.0136 0.0269 0.062 24.687 17 0. 268 -0.0085 -0.00026 0 .268 16.303 2.00 0.150 -0.0170 0.0402 0.086 16.217 18 0. 237 -0.0090 -0.00026 0 .237 17.546 2.30 0.212 -0.0207 0.0503 0.125 17.421 19 0. 251 -0.0096 -0.00026 0 .251 16.124 2. 85 0.362 -0.0273 0.0909 0.253 15.871 20 0. 298 -0.0101 -0.00026 0 .298 22.168 3.75 0.512 -0.0378 0.1525 0.384 21.784 21 0. 204 -0.0106 -0.00026 0 .204 27.963 4.90 0.912 -0.0520 0.1860 0.657 27.306 22 0. 279 -0.0111 -0.00026 0 .279 23.157 7.40 1.35 -0.082 0.377 1.057 22.100 23 0. 189 -0.0116 -0.00026 0 .189 39.200 9.30 2.00 -0.108 0.378 1.428 37.872 24 0. 263 -0.0122 -0.00026 0 .263 31.665 11.40 1.00 -0.139 0.263 0.472 31.193 25 0. 039 -0.0127 -0.00026 0 .039 100.000 13.00 1.00 -0.165 0.039 -3.23 100.00 300

APPENDIX 3 TO CHAPTER IV-4 ESTIMATION BIAS IN SMEAR-AND-SWEEP The basic step in smear-and-sweep is the construction of a relatively effective, simple classification based on several background vari- ables while consciously and completely neglect- ing the variable under study. From the point of view of estimation of the effect of the variable under study, this procedure tends to make dif- ferences found smaller than differences present (i.e., nearer to zero, but of the same sign). The reason for this reduction is easily seen in an unrealistically extreme example. Suppose that only two anesthetics are being studied, that they have quite different death rates, and that only one of them is used in each combination of opera- tion and age. Here the OA smear-and-sweep, which we may suppose to be the first step in the proce- dure, will sweep cases treated with one anesthetic almost entirely into low-death-rate categories and the other almost entirely into high-death-rate categories. Comparison within categories, whether individual or based on standardized death rates, will now show only a trace of the true dif- ference between anesthetics. Let us consider a purely hypothetical ex- ample in which this effect is present to a sub- stantial but not extreme degree. Suppose, first, that agent B is always twice as dangerous as agent A, while eight age-operation cells (all high- risk) have risks differing by rather smaller ratios and the two agents are preferentially used, as indicated in Table 1. If these death rates were to apply exactly, without sampling fluctuation, the observed death rates per 1000 by cell (without subdivision by agent) would be as shown in Table 2, so that, if two categories are to be formed from the eight cells, the asterisked cells will be combined into the higher-risk category (the other four forming the lower-risk category). If 1000 administrations were to occur in each of the eight cells, and the observed numbers of deaths happened to match exactly the average numbers corresponding to the assumed death rates, the results by category would appear as shown in Table 3. As a result, the mean ratio of death rates would be 1.58, notably reduced from the ratio of 2.00, which, according to the assumptions of this hypothetical example, actually applies in every cell. In more realistic situations, we may expect much weaker effects of the same kind. In the present study, where imbalances in the use of anesthetics are much less extreme, judgment suggests a pulling together of estimated death rates by a few hundredths of the true differences, which is surely not important. Rather than trying to obtain bounds for such an effect, the reasonable approach in studies in which such an effect is a matter of concern is to redo the analysis in such a way as to remove the effect, at least in large part. Although the effect is not here a matter of concern, a brief descrip- tion of how such a computation might proceed in the National Halothane Study may be helpful in other studies. Two approaches for the near elimination of the effect of imbalance are possible: Approach 1: Calculate the death rate for each cell of the smear to be used in the pickup rule for determining the sweep as follows: (la) Fix a single set of weights for the agents. (Ib) Calculate separate estimated death rates by agent for each cell. (Ic) Form weighted'mean death rates, to be used in the pickup rule, from these es- timates, using the same single set of agent weights for all cells. Approach 2: Calculate an adjusted death rate for each cell as follows: (2a) Use a preliminary analysis to estimate over-all death rates (i.e., for all cells combined) by agent (which will be ap- plied within individual cells as if ap- propriate there). (2b) Adjust the estimated numbers exposed in each cell to equivalent numbers exposed to a standard agent, using the relative over-all death rates from (2a). (2c) Use these equivalent estimated exposed, in combination with observed deaths, to form the death rates used in the pickup rule. (2d) Iterate steps (2a), (2b), and (2c) if the relative over-all death rates change ap- preciably. Why does approach 1 work? By being waste- ful, we could avoid bias of pickup (due to differ- ing distributions by agent within cells) by win- nowing to get balance, discarding cases (randomly selected within agent) until the proportions by agent are the same in each cell. When we have done this, the cell death rate for the winnowed cells will be the sum of the agent-by-cell death rates weighted by these fixed proportions. Ap- proach 1 replaces-these agent-by-cell death rates based on winnowed samples with corre- sponding rates based on all available cases, and 301

TABLE I.— DISTRIBUTION OF CASES AND DEATH RATES IN THE EIGHT *3E- OPERATION CELLS FOR THE EXAMPLE Age/ True agent death rates % Cases by agent * operation X I Y X y 20-49 11A, 22B 16A , 32B 10A, 90B 70*. 3OB 50-59 12A, 24B 17A , 34B 30A, 70S 90A, 10 B 60-69 13A, 2bB ISA , 36B 50A, 50B 100A 70- 15A, 3OB 201 , 40B 70A, 30B 100A TABLE 2.—OBSERVED DEATH RATES PER 1000 FOR THE EXAMPLE Age/operation 20-49 50-59 60-69 70- 195 195 208* 187 200* TABLE 3.--OUTCOME OF EXAMPLE WHEN GROUPED BY DEGREE OF RISK Higher-risk category Operation Age A's exposed A's dead B's exposed B's dead Z 20-49 100 11 900 198 X 50-59 300 36 700 168 Y 20-49 700 112 300 96 Y 70- 900 180 100 40 Total 2000 339 2000 502 Death rate 17 -0* 25 .1* Ratio of death rates » 1.48 Lower-risk category Operation Age A's exposed A's dead B's exposed B's dead X 60-69 500 65 500 130 X 70- 700 105 300 90 I 50-59 1000 170 none none Y 60-69 1000 180 none none Total 3200 Death rate Ratio of death rates= 1.69 16.2* 520 800 27.5* 220 is equally resistant to all forms of bias due to unbalanced use of agents within cells. It is, of course, nice to use approach 1 when- ever one can, because it does not use strong as- sumptions. To use it we require reasonable amounts of data from each agent in each smear cell. In the halothane situation this requirement seems likely to be too stringent. Approach 2, which can be used with far fewer data, is based on a more restrictive assumption, namely, that agent-by-cell death rates can be satisfactorily taken as products of an agent factor and a cell factor. Under these circumstances, the agent factors can be taken (iteratively) as proportional to the over-all agent-by-agent re- sults of a previous analysis, and can be applied, by altering, for the smear-and-sweep pickup stages only, the numbers of estimated exposed to equivalent numbers of estimated exposed. If we wished to improve our estimates in the present study, it would be natural to adopt ap- proach 2. Here the natural choices would seem to be: (1) set all ether cases aside during smear- and-sweep, (2) treat halothane and nitrous oxide-barbi- turate as equivalent, (3) treat cyclopropane and Other as equiva- lent, and (4) treat the latter as producing about 5/4 the death rate of the former. These decisions can be implemented, in the notation used above, by: (1) eliminating all ether deaths from D, (2) eliminating all ether exposures from EE and calculating an equivalent EE from* equiv EE = -5- (estimated exposed to cyclopropane or Other) Q + -g (estimated exposed to halothane or nitrous oxide-barbiturate), and (3) carrying out the entire calculation as before. If the estimated death rates are changed in a substantial way, altering the 5/4 ratio noticeably, a further iteration would be called for. ,r, o •The numbers -5- and —need to be in the assumed ratio at 5 to 4. The values suggested have the added virtue of producing an equivalent EE close to the unadjusted EE (also with ether cases omitted). 302

APPENDIX 4 TO CHAPTER IV-4 PSEUDO-VALUES OF OVER-ALL DEATH RATE BY AGENT (From analysis VIII) Institution Agent H N-B C E 0 1 0.021532 0.011965 0.028392 0.025808 0.025037 2 0.027236 0.019895 0.021463 0.027521 0.026733 3 0.022129 0.013047 0.023398 0.023658 0.026837 4 0.010629 0.015829 0.019115 0.020435 0.015473 5 0.034197 0.058306 0.038189 0.022394 0.034358 6 0.013869 0.018885 0.024388 0.019380 0.015489 7 0.026538 0.025576 0.017467 0.045790 0.043684 8 -0.062582 0.029812 0.026021 -0.005365 -0.008451 9 0.033034 0.030567 0.019238 -0.087503 0.012409 10 0.028128 0.021329 0.045466 0.041052 0.033164 11 0.011825 0.024606 0.007382 -0.070818 0.020790 12 0.028894 0.018269 0.026418 0.025706 0.033941 13 0.025234 0.022877 0.047625 0.021613 0.028055 14 0.018598 0.021994 0.018027 0.023803 0.015777 15 0.026865 0.016277 0.022735 0.003114 0.027657 16 0.023574 0.001588 0.025405 0.027627 0.028795 17 0.019178 0.022324 0.022966 0.000938 0.022143 18 0.016668 0.018239 0.012821 0.008850 0.024626 19 0.031989 0.009613 0.034783 0.025210 0.033696 20 0.024192 0.013915 0.018139 0.063541 0.020941 21 0.023809 0.012258 0.009164 0.038633 -0.001531 22 0.023425 0.032466 0.019681 0.027288 0.041386 23 0.018175 0.020125 0.022730 0.023725 0.024531 24 0.027003 0.011354 0.043400 0.044703 0.037946 25 0.024010 0.024885 0.032217 0.034557 0.045610 26 0.021036 0.010147 0.036192 0.026593 0.022822 27 0.016733 0.023318 0.017219 0.025333 0.022545 28 0.020854 0.022067 0.025623 0.025880 0.024496 29 0.018345 0.023480 0.025447 0.023532 0.020832 30 0.018108 0.018971 0.026594 0.023725 0.022686 31 0.024152 0.024683 0.027649 0.024366 0.028690 32 0.013649 0.021564 0.027465 0.023568 0.024410 33 0.000899 -0.005100 0.012118 0.000600 -0.001623 34 0.021096 0.022878 0.037142 0.010878 0.025825 303

APPENDIX 5 TO CHAPTER IV-4 PSEUDO-VALUES OF STANDARDIZED DEATH RATES FOR ALL FIVE AGENTS WITH ALL FIVE POPULATIONS ON MIDDLE-DEATH-RATE OPERATIONS (From analysis VIII) Agent H N-B C E 0 PSEUDO-VALUES FOR INSTITUTION H 0.006212 -0.004051 0.010313 0.000677 0.008590 N-B 0.000530 -0.011564 0.000379 -0.004111 -0.000877 C 0.028049 0.019531 0.037071 0.039126 0.032281 E 0.015335 0.009000 0.024672 0.016521 0.023890 0 0.024195 0.011165 0.029029 0.036293 0.021949 PSEUDO-VALUES FOR INSTITUTION H 0.043230 0.041478 0.039770 0.049101 0.044291 N-B 0.044528 0.034175 0.037453 0.041904 0.042935 C 0.086529 0.064842 0.083562 0.097391 0.093941 E 0.027462 0.020579 0.030048 0.026975 0.035106 0 0.050336 0.039299 0.047762 0.062543 0.054300 PSEUDO-VALUES FOR INSTITUTION H 0.016027 0.008877 0.018023 0.019107 0.020268 N-B -0.001355 -0.005483 -0.007818 -0.003188 -0.002248 C 0.030497 0.012619 0.032396 0.038507 0.036213 E 0.021126 0.011640 0.024424 0.018034 0.026192 0 0.012796 0.001209 0.010203 0.012202 0.015774 PSEUDO-VALUES FOR INSTITUTION H 0.003424 0.001230 0.005305 0.004611 0.002819 N-B 0.013690 0.022417 0.030502 0.018003 0.026450 C 0.054505 0.061246 0.072565 0.070956 0.061397 E 0.027975 0.040248 0.037124 0.040818 0.030573 0 0.052725 0.056824 0.063395 0.069185 0.064219 PSEUDO-VALUES FOR INSTITUTION H N-B C 0.044735 0.038319 -0.003421 0.034696 0.051975 0.062913 0.058937 0.030478 0.063248 0.094979 0.045792 0.042088 0.005591 0.038318 0.068201 0.038282 0.034271 -0.020656 0.038673 0.051132 0.045135 0.046411 -0.019559 0.035675 0.058982 E 0 304

Agent H N-B C E 0 PSEUDO-VALUES FOR INSTITUTION H 0.010684 0.005387 0.022811 0.025222 -0.011442 0.012091 0.010000 0.026197 0.036544 -0.007794 0.016541 0.013929 0.034132 0.031426 -0.002228 0.022915 0.020485 0.019273 0.032750 -0.034856 0.008509 0.004057 0.027529 0.021035 -0.009808 N-B C E 0 PSEUDO-VALUES FOR INSTITUTION 7 H N-B C E 0 0.027889 0.022754 0.053576 0.038516 0.042341 0.025702 0.022583 0.044800 0.031858 0.044908 0.022126 0.016806 0.038158 0.026772 0.029962 0.045306 0.038607 0.079202 0.043449 0.072296 PSEUDO-VALUES FOR INSTITUTION 8 PSEUDO-VALUES FOR INSTITUTION PSEUDO-VALUES FOR INSTITUTION 10 PSEUDO-VALUES FOR INSTITUTION 11 H N-B C E 0 0.008604 0.003993 0.036700 0.054372 0.031550 0.019670 0.014977 0.060288 0.075574 0.040416 0.007658 0.001717 0.031412 0.060693 0.022814 -0.053831 -0.080737 -0.105100 0.037868 -0.070181 0.041627 0.036592 0.082198 0.054932 0.063527 H -0.007255 0.031351 0.047649 0.029725 0.006339 N-B -0.053767 0.004585 0.002159 -0.008684 -0.029360 C -0.122940 0.051241 0.036970 0.009528 0.004899 £ -0.147924 -0.103404 -0.094693 -0.097941 -0.143637 0 -0.177590 -0.021082 -0.034305 -0.140306 -0.061509 H N-B C E 0 0.025287 0.012987 0.053372 0.068877 0.047147 0.023053 0.008918 0.052248 0.097425 0.016168 0.003148 0.031687 0.079991 0.031390 -0.088070 -0.084866 -0.114412 -0.000169 -0.089932 0.008199 -0.008783 0.037874 0.069747 0.021766 0.044936 H 0.022189 0.033876 0.067091 0.029260 0.038791 0.013852 0.031817 0.052409 0.025914 0.036026 0.041952 0.053355 0.080505 0.049175 0.056126 0.040505 0.045908 0.104087 0.028841 0.054024 0.027734 0.042639 0.065628 0.031999 0.047643 N-B C E 0 0.018957 0.008874 0.046910 0.075203 0.044303 305

Agent H N-B C E 0 PSEUDO-VALUES FOR INSTITUTION 12 H N-B C E 0 0.020728 0.021509 0.023399 0.026154 0.043097 0.013451 0.013609 0.006693 0.018993 0.025481 PSEUDO-VALUES FOR INSTITUTION 13 H N-B C E 0 0.018686 0.020456 0.025907 0.024503 0.022269 0.015971 0.017498 0.023481 0.022148 0.020322 0.020199 0.017821 0.019766 0.021366 0.045261 0.039761 0.041916 0.046819 0.048470 0.046071 0.021456 0.019716 0.011151 0.018771 0.040533 0.014100 0.016626 0.022433 0.018653 0.015709 PSEUDO-VALUES FOR INSTITUTION 14 PSUEDO-VALUES FOR INSTITUTION 15 H N-B C £ 0 0.019382 0.021056 0.020626 0.054652 0.040593 0.011544 0.014363 0.002955 0.033925 0.022692 0.015699 0.016096 0.017167 0.043755 0.039923 0.004814 -0.002178 -0.015860 0.033498 0.036468 PSEUDO-VALUES FOR INSTITUTION 16 PSEUDO-VALUES FOR INSTITUTION 17 0.027878 0.025647 0.022407 0.033576 0.046492 0.020629 0.022786 0.028758 0.027551 0.024904 H 0.010540 0.013392 0.008240 0.012812 0.007654 N-B 0.006870 0.008855 0.003204 0.009729 0.001451 C -0.001634 0.002247 -0.002715 -0.000554 -0.005622 E 0.018542 0.023129 0.020710 0.019774 0.016840 0 0.009552 0.014086 0.009129 0.012491 0.007876 0.018897 0.017645 0.021708 0.050320 0.047369 H N-B C E 0 0.020663 0.033265 0.040245 0.022486 0.024900 0.012806 0.036308 -0.012239 -0.012628 -0.009518 0.022414 0.034261 0.046230 0.021366 0.026383 0.023173 0.031583 0.047529 0.019805 0.032192 0.025236 0.037408 0.050399 0.024769 0.031747 H N-B C 0.016144 0.010861 0.036285 -0.039223 -0.004652 0.018003 0.010958 0.038701 -0.038131 0.002397 0.020284 0.012615 0.033682 -0.047883 -0.002837 -0.004261 -0.009544 0.014454 -0.042730 -0.031147 0.016534 0.011836 0.042309 -0.044098 -0.001578 E 0 306

Agent H N-B C E 0 PSEUDO-VALUES FOR INSTITUTION 18 H 0.019750 0.023945 0.019586 0.017484 0.028107 N-B 0.018456 0.021123 0.015111 0.013639 0.024708 C 0.047742 0.048006 0.049766 0.041772 0.060611 E 0.013051 0.013376 0.008903 0.007312 0.021364 0 0.028506 0.027621 0.021964 0.018006 0.037502 PSEUDO-VALUES FOR INSTITUTION 19 H N-B C E 0 0.022403 0.025578 0.042336 0.032523 0.033513 -0.000190 0.006384 0.020732 0.006550 0.009706 0.025188 0.034534 0.047377 0.029206 0.037533 0.016738 0.020336 0.035545 0.019449 0.025279 PSEUDO-VALUES FOR INSTITUTION 20 PSEUDO-VALUES FOR INSTITUTION 21 H N-B C -0.000067 0.021812 0.020657 0.013342 0.030853 -0.009809 0.009240 0.015259 0.014674 0.011781 -0 0.019050 0.009246 0.001806 004113 PSEUDO-VALUES FOR INSTITUTION 22 H N-B C E 0 0.019316 0.016101 0.026976 0.021284 0.020506 0.025528 0.024034 0.036886 0.028800 0.032306 PSEUDO-VALUES FOR INSTITUTION 23 H N-B C E 0 0.014735 0.014467 0.018856 0.017478 0.021292 0.016105 0.014752 0.025593 0.019473 0.024308 0.013191 0.017579 0.016529 0.017515 0.017773 0.015995 0.018837 0.018575 0.024647 0.021362 0.027309 0.005061 0.029447 0.036106 0.022186 0.057844 0.021412 0.024625 0.033309 0.016748 0.029994 0.019546 0.019275 0.028405 0.018894 0.032232 0.023347 0.030851 0.048270 0.030697 0.035841 H 0.017879 0.024222 -0.045153 0.017212 0.028056 0.009238 0.015270 -0.056012 0.016887 0.021133 0.010423 0.018549 -0.051805 0.011813 0.022042 0.047164 0.052100 0.017674 0.025841 0.084773 0.017988 0.027104 -0.068155 0.015627 0.027955 N-B C E 0 -0.021970 -0.001055 -0.025623 -0.006411 -0.003094 0.036124 0.036208 0.042270 0.039063 0.041552 0.020544 0.019921 0.025464 0.025213 0.028756 307

Agent H N-B C E 0 PSEUD0-VALUES FOR INSTITUTION 24 H N-B C £ 0 0.029415 0.044751 0.061214 0.024122 0.045120 0.019877 0.037993 0.041107 0.008500 0.021240 PSEUDO-VALUES FOR INSTITUTION 25 H N-B C 0.031205 0.025621 0.059174 0.024682 0.057492 0.028167 0.026836 0.059478 0.027252 0.059131 0.048139 0.063068 0.080438 0.043611 0.065255 0.036743 0.034906 0.069852 0.035091 0.066393 0.044287 0.064979 0.083088 0.032214 0.071823 0.033389 0.035534 0.087628 0.031862 0.075231 PSEUDO-VALUES FOR INSTITUTION 26 PSEUDO-VALUES FOR INSTITUTION 27 PSEUDO-VALUES FOR INSTITUTION 28 PSEUD0-VALUES FOR INSTITUTION 29 H N-B C E 0 0.018486 0.016648 0.030197 0.018065 0.025689 0.023237 0.018826 0.038682 0.023587 0.033012 0.026176 0.022306 0.039346 0.024831 0.036304 0.023131 0.020339 0.041425 0.018591 0.035409 0.042582 0.056128 0.076309 0.038080 0.054433 0.047418 0.048702 0.085506 0.045966 0.087060 H -0.004866 -0.011323 0.003383 -0.003510 -0.005050 N-B 0.030586 0.014955 0.039998 0.035928 0.033562 C 0.024196 0.006134 0.044486 0.030006 0.020531 E 0.017257 0.004957 0.029656 0.018081 0.018571 0 0.018816 0.010267 0.035948 0.027661 0.021672 H 0.001336 0.005769 -0.003103 0.007892 0.003100 N-B 0.007045 0.011590 0.004184 0.016332 0.011009 C 0.013860 0.025778 0.017961 0.033002 0.019149 E 0.014180 0.019604 0.018983 0.015861 0.015864 0 0.012643 0.024034 0.010497 0.020918 0.018713 H 0.013465 0.014728 0.017789 0.017098 0.016878 N-B 0.011016 0.011239 0.014085 0.013641 0.013583 C 0.030920 0.032509 0.034747 0.041120 0.033618 E 0.016592 0.017405 0.020081 0.016129 0.020577 0 0.026677 0.028650 0.032346 0.035400 0.030118 0.021405 0.017983 0.036025 0.020463 0.029434 308

Agent H i N-B C E 0 PSEUDO-VALUES FOR INSTITUTION 30 H N-B C £ 0 0.023366 0.019331 0.027784 0.016106 0.024809 0.025350 0.020718 0.029075 0.019331 0.022857 PSEUDO-VALUES FOR INSTITUTION 31 H N-B C E 0 0.015527 0.016706 0.024215 0.022369 0.028621 0.016053 0.016142 0.024819 0.024630 0.028793 0.033474 0.030819 0.037574 0.025726 0.033947 0.017268 0.019519 0.031241 0.024034 0.032195 0.028548 0.025206 0.035937 0.018894 0.031972 0.015082 0.017427 0.022107 0.018668 0.029404 PSEUDO-VALUES FOR INSTITUTION 32 PSEUDO-VALUES FOR INSTITUTION 33 PSEUDO-VALUES FOR INSTITUTION 34 H N-B C E 0 0.015088 0.017142 -0.012145 0.018950 0.006803 0.016526 0.017613 -0.011357 0.020456 0.001335 0.029600 0.030600 0.001830 0.035261 0.013053 0.004563 0.003512 -0.025872 0.018879 -0.017912 0.029129 0.025721 0.034923 0.021463 0.029806 0.019526 0.021500 0.027327 0.027687 0.031892 H N-B C E 0.013681 0.008301 0.019691 0.011248 0.012685 0.021248 0.013518 0.037472 0.019435 0.023412 0.024948 0.021307 0.035863 0.024126 0.026706 0.021463 0.015668 0.034127 0.016879 0.026712 0.021242 0.016233 0.037158 0.022727 0.023764 0 H 0.025856 0.028276 0.041161 0.023115 0.029052 N-B 0.003361 0.014865 0.0131/44 0.012024 0.004225 C 0.044377 0.019782 0.049858 0.035136 0.040570 E 0.004084 0.003639 0.033949 0.049658 0.018457 0 0.032414 0.019905 0.044963 0.029576 0.026907 0.019141 0.019503 -0.010890 0.022836 0.005043 334-553 O-69—21 309

APPENDIX 6 TO CHAPTER IV-4 MIDDLE-DEATH-RATE OPERATIONS: DEATHS, ESTIMATED EXPOSED, AND DEATH RATES BY AGENT FOR FINAL CATEGORIES (From analysis VIII) D: Deaths £: Estimated exposed rt Death rate H fr iti O Agent C ho Total •H CD (^ HJ 0) H N-B C E 0 O 1 D 116 80 73 34 70 373 1 E 86661 60576 38839 24305 35821 246204 1 R 0.00134 0.00132 0.00188 0.00140 0.00195 0.00152 2 D 123 88 54 26 91 382 2 E 17947 11831 6111 5695 8884 50470 2 R 0.00685 0.00744 0.00884 0.00456 0.01024 0.00757 3 D 120 77 74 27 76 374 3 E 8011 5765 3614 3151 3639 24182 3 R 0 . 01498 0.01336 0 . 02047 0.00857 0.02088 0.01547 4 D 125 72 71 30 74 372 4 E 5221 3803 2532 1568 2692 15818 4 R 0.02394 0.01893 0.02803 0.01913 0.02748 0.02352 5 D 128 81 75 27 70 381 5 E 5604 3504 2028 1410 2479 15026 5 R 0.02284 0. 02312 0.03697 0.01914 0.02823 0.02536 6 D 89 81 85 47 71 373 6 E 3534 2452 2275 1529 2119 11910 6 R 0.02518 0.03303 0 . 03736 0.03072 0.03350 0.03132 7 D 108 81 68 38 79 374 7 E 2716 2108 1116 1097 1447 8485 7 R 0.03975 0.03842 0.06090 0.03464 0.05459 0.04407 8 D 113 73 77 40 79 382 8 E 2116 1587 1139 853 1080 6777 8 R 0.05339 0.04597 0.06760 0.04687 0.07310 0.05636 9 D 131 79 85 53 82 430 9 E 2175 1101 1022 983 1221 6503 9 R 0.06023 0.07173 0.08316 0.05390 0.06715 0.06612 310

Final category Agent Total H N-B C E 0 10 D 109 68 84 27 90 378 10 E 1688 1154 1282 391 1000 5516 10 R 0.06454 0.05891 0.06552 0.06892 0.08999 0.06852 11 D 104 94 93 23 69 383 11 E 1894 1470 702 523 647 5239 11 R 0.05490 0.06391 0.13233 0.04391 0.10653 0.07310 12 D 96 83 80 41 85 385 12 E 1016 841 1116 172 858 4006 12 R 0.09443 0.09859 0.07167 0.23734 0.09899 0.09610 13 D 110 58 87 38 105 398 13 E 1309 655 686 645 838 4135 13 R 0.08401 0.08849 0.12665 0.05890 0.12522 0.09624 14 D 111 81 87 40 77 396 14 E 1091 817 487 122 615 3133 14 R 0.10165 0.09914 0.17860 0.32763 0.12517 0.12638 15 D 106 73 86 40 65 370 15 E 937 447 660 489 564 3099 15 R 0.11310 0.16319 0.13014 0.08178 0.11509 0.11938 16 D 136 94 72 19 74 395 16 E 549 678 534 234 548 2544 16 R 0.24749 0.13861 0.13471 0.08112 0.13499 0.15523 17 D 109 87 104 36 73 409 17 E 668 579 648 98 496 2492 17 R 0.16303 0.15014 0.16034 0.36406 0.14689 0.16409 18 D 89 59 132 32 76 388 18 E 507 295 751 243 341 2138 18 R 0.17546 0.19998 0.17558 0.13145 0.22278 0.18142 19 D 78 76 128 31 76 389 19 E 483 367 581 114 375 1922 19 R 0.16124 0.20684 0.22006 0.27034 0.20247 0.20230 20 D 102 61 123 30 72 388 20 E 460 227 450 166 235 1540 20 R 0.22168 0.26782 0.27315 0.18007 0.30554 0.25188 311

Final category Agent Total H N-B C E 0 21 D 81 53 138 39 95 406 21 E 289 96 462 218 348 1415 21 R 0.27963 0.54936 0.29844 0.17861 0.27228 0.28676 22 D 86 46 97 32 110 371 22 E 371 192 250 98 414 1327 22 R 0.23157 0.23858 0.38727 0.32593 0.26523 0.27946 23 D 73 44 123 37 104 381 23 E 186 117 325 61 295 984 23 R 0.39200 0.37539 0.37836 0.60563 0.35210 0.38681 24 D 64 36 154 22 105 381 24 E 202 83 261 27 193 767 24 R 0.31665 0.43334 0.58855 0.80209 0.54321 0.49637 25 D 52 28 147 27 102 356 25 E 15 68 208 44 49 387 25 R 1.00000 0.40740 0.70385 0.60067 1.00000 0.91878 312

APPENDIX 7 TO CHAPTER IV-4 APPROXIMATE INQUIRY INTO PLAUSIBLE INDIRECT EFFECT OF DIFFERENCES IN STANDARDIZED DEATH RATES BETWEEN INSTITUTIONS The existence of differences in standardized death rates from institution to institution is made clear in Chapter IV-6. The purpose of this ap- pendix is to explore the mechanism, and crudely assess the magnitudes, of the indirect effects of such differences on the results of Chapter IV-4. Be- fore discussing the limited calculations that were actually feasible for this purpose, it will be ad- vantageous to discuss what computations might have been made if we had been in a position to start this chapter's computations from the begin- ning with the intent of compensating for dif- ferences in SMR = actual deaths/standard deaths from institution to institution. One approach would have been to arrange the institutions in groups of three or four, stratifying on the basis of SMR values, and then to bring in these institutional groups in a further stage of smear-and-sweep. The result would be a set of categories that involved institutional groups. It would then be quite feasible to estimate death rates by agent standardized on these categories, although the estimation of a standard error for this estimate would be a little complicated, inas- much as it would have to be based on institutions within institutional groups (and hence within cate- gories). Such an approach is likely to have proved illuminating, had it been feasible. It has the ad- vantage of making no assumption about the ways in which institutional differences combine with differences from operation to operation, or those from age to age, etc. It has the disadvantage of exposure to inadequate control from the allot- ment of too few categories for the over-all vari- ation that requires control. At the opposite extreme stand computations based on a very specific rule of combination be- tween the institutional effects and other effects. To understand the essentials of such calculations, we need to consider what would happen if the variation of death rate for institutions did, or did not, arise only because of the differing distribu- tions of patients over the various categories. (1) If the variation is only according to the distribution over categories: (la) The number of ste, Hird deaths will equal the number 01 actual deaths, so that SMR = 100. (Ib) Institution will have no effect on death rate within category, so that no indirect effect of institution can be transmitted into the smear-and-sweep estimates of mortality rates by agent. (2) If this variation is not only according to the distribution over categories, the death rate in an institution for a category may be (but need not be) representable as a product of an institu- tion factor and a category factor. If this is the case: (2a) The natural estimate of the institu- tion factor would then be the ratio of actual deaths to standardized deaths, where the standardization is carried out for approximately the factors that enter into the smear-and-sweep esti- mate. (2b) Instead of accumulating estimated ex- posed, EE, by institution, category, and agent (to obtain estimated exposed by category and agent, used as a basis for calculating death rates by category and agent), one should accumulate equiv- alent estimated exposed, EEE, by in- stitution, category, and agent, thus forming equivalent estimated exposed by category and agent, to serve as denominators for equivalent death rates by agent and category. Here: EEE = equivalent estimated exposed - actual deaths (estimated standard deaths - exposed) = (SMR) • (EE). Further procedures, including appli- cation of equivalent death rates to suitable populations, jackknifing, and robust estimation, could then proceed just as in the actual analysis. Regrettably, it did not prove feasible to carry out this computation, either. Although subject to reservations about the adequacy of the assumed multiplicative effect of institution, such computa- tions, especially if used to assess the shift in final comparisons associated with a change from estimated exposed to equivalent estimated ex- posed, would have high plausibility and serve as strong evidence. It has been feasible to carry out a very over- simplified analogue of this latter computation, one that involves the estimated distribution of ex- posure by institution and agent (rather than the estimated distribution of exposure by institution, 313

agent, and category). Accordingly, the over- simplified calculation is not protected from ef- fects of differential distribution of agents as- sociated with institution-category combinations (rather than with institution-to-institution or category-to-category changes). In this calcula- tion we ask about the effect of equivalent esti- mated exposed, rather than estimated exposed, on the "death rates" resulting from prorating institutional deaths and institutional estimated exposed over all five agents in proportion to the total number of uses of each agent in that institu- tion. Table 1 shows the results of converting the percentages and totals of Table EE-20 in Chapter IV-2 into thousands of EE (some rounding errors are inevitable, but their effect should not be serious). Table 2 shows the results of multiplying these EE by the SMR for the institution (stand- ardizing for age-shock strata) as given in Table 21 of Chapter IV-6, thus converting them into TABLE 1.—ESTIMATED EXPOSED IN THOUSANDS BY AGENT AND INSTITUTION (CALCULATED FROM TABLE EE-20 IN CHAPTER IV-2) TABLE 2.—MODIFICATION OF TABLE 1 EE BY EEE (BY MULTIPLYING ENTRIES IN THAT TABLE BY SMR FROM TABLE 21 OF CHAPTER IV-6) Institution H N-B CEO 1 9.8 6.4 0+ 0.3 1.3 2 2.5 0.9 1.0 0.3 0.6 3 1.0 7.3 0.6 0.0+ 1.9 4 7.0 2.9 2.4 0.6 2.4 5 3.9 3.1 7.8 0.3 1.5 6 4.8 5.0 0.5 1.0 3.9 7 1.7 1.0 1.6 0.2 1.2 8 22.7 5.5 0.5 15.5 7.7 9 4.2 10.2 2.1 8.1 2.2 10 6.6 3.2 2.6 0.9 0.7 11 5.5 1.9 1.3 2.7 2.2 12 4.8 1.4 3.9 0.2 1.2 13 1.6 0.3 1.2 0.1 1.8 14 2.6 1.7 3.3 0.1 2.2 15 1.6 4.3 4.6 1.3 1.9 16 0.9 4.6 0.5 0.1 2.6 17 1.6 1.3 0.3 4.5 3.8 18 1.6 1.2 1.6 1.1 0.5 19 4.4 4.2 0.9 0+ 0.8 20 1.5 2.2 9.2 2.9 5.0 21 11.4 7.4 5.0 0.4 5.1 22 8.3 1.4 3.5 0.5 2.1 23 1.2 1.8 0.9 0 0.3 24 1.6 6.2 0.5 0.1 0.3 25 1.7 0.5 1.0 0.1 1.2 26 9.9 4.9 1.0 0.3 1.3 27 5.8 2.0 2.7 0.5 1.6 28 1.5 1.2 0+ 0.3 0.1 29 1.5 0.1 0.1 0.0+ 0.5 30 1.3 0.5 0.2 0 0.3 31 2.0 1.0 1.6 0.1 0.2 32 4.2 1.1 1.0 0.1 1.6 33 3.9 3.6 1.3 0.9 3.1 34 1.0 0.1 3.4 0.3 4.0 TOTAL 145.6 100.4 68.1 43.8 67.1 Institution SMR H N-B C E 0 1 0.564 5.4 . 3.5 0.0 0.2 0.7 2 1.67 4.2 1.5 1.7 0.5 1.0 3 0.871 0.9 6.2 0.5 0.0 1.6 4 1.17 8.2 3.4 2.8 0.7 2.8 5 1.28 5.0 4.0 10.0 0.4 1.9 6 0.819 3.9 4.1 0.4 0.8 3.2 7 1.55 2.6 1.6 2.5 0.3 1.8 8 0.695 15.8 3.8 0.3 10.8 5.4 9 0.972 4.1 9.9 2.0 7.9 2.1 10 1.25 8.2 4.0 3.3 1.1 0.9 11 1.13 6.2 2.1 1.5 3.1 2.5 12 1.12 5.4 1.6 4.4 0.2 1.3 13 1.44 2.3 0.4 1.7 0.1 2.6 w 0.307 0.8 0.5 1.0 0.0 0.7 15 0.960 1.5 4.1 4.4 1.2 1.8 16 1.16 1.0 5.3 0.6 0.1 3.0 17 0.819 1.3 1.1 0.2 3.7 3.1 18 1.10 1.8 1.3 1.8 1.2 0.6 19 1.24 5.5 5.2 1.1 0.0 1.0 20 0.760 1.1 1.7 7.0 2.2 3.8 21 0.635 7.2 4.6 3.2 0.3 3.2 22 1.25 10.4 1.7 4.3 0.6 2.6 23 0.734 0.9 1.3 0.7 0.0 0.2 24 1.74 2.8 10.4 0.8 0.2 0.5 25 1.59 2.7 0.8 1.6 0.2 1.9 26 0.902 8.9 4.4 0.9 0.3 1.2 27 0.839 4.9 1.7 2.3 0.4 1.3 28 0.353 0.5 0.4 0.0 0.1 0.0 29 1.31 2.0 0.1 0.1 0.0 0.7 30 1.52 2.0 0.8 0.3 0.0 0.5 31 1.34 2.7 1.3 2.1 0.1 0.3 32 0.862 3.6 0.9 1.2 0.1 1.4 33 1.05 4.1 3.8 1.4 0.9 3.3 34 1.08 1.1 0.1 3.7 0.3 4.3 Total 139.0 97.4 69.8 38.0 63.2 values of EEE first by institution and agent, and then by agent. Table 3 shows the changes in estimated death rates corresponding to changing from EE to EEE (basic death-rate estimates taken from Table 8 of Chapter IV-4). The changes are all small, especially when compared with the change for the total. Except for ether, the changes are by less than 0.0015. The three differences in death rate found significant in Table 8 of Chapter IV-4, namely, H-C, H-O, and N-B - C, are changed in magnitude by+ 0.0016,+0.0005, and -0.0012, respectively. If the confidence intervals were to retain their widths, two of these three differences would con- tinue to be significant at 5 percent (and the N-B - O difference, which increases in magnitude by 0.0009, would become significant). Although it would be very desirable to have at hand the results of one of the more appropriate and more detailed calculations discussed above, 314

TABLE 3.--APPROXIMATE EFFECTS OF THE PROPOSED ADJUSTMENT H N-B C E 0 Totals Death rate* Total EE** Corresponding deaths Total EEE** Equivalent death rate (Change) Change compared with total 0.0202 0.0195 0.0255 0.0227 0.0248 (0.0219) 145.6 100.4 68.1 43.8 67.1 425.0 2941 1956 1737 994 1664 9292 139.0 97.4 69.8 38.0 63.2 407.4 0.0212 0.0201 0.0249 0.0262 0.0263 (0.0228) (+0.0010) (+0.0006) (-0.0006) (+0.0035) (+0.0015) (+0.0009) +0.0010 -0.0003 -0.0015 +0.0026 +0.0006 *Values from Table 8 of Chapter IV-4. **Expressed in thousands, from Tables 1 and 2, respectively. the computations just described suggest quite strongly that adjustment of agent death rates for nonstandard institutional death rates: (1) would probably not alter the conclusions found above for unadjusted agent death rate, and (2) might well tend to strengthen those con- clusions, rather than weaken them. There is a real possibility of a concealed effect, involving, say, the details of the three-way distributions of deaths and EE by agent, institu- tion, and category. This concealed effect could be larger than, and of opposite sign to, the effect apparent from Tables 1 through 3, but it does not seem likely that that is the case. 315

Abstract of Chapter IV-5 The goals of this chapter are to take account of several interfering vari- ables at one time and to reach conclusions about anesthetic comparisons. Estimated death rates are fitted as a function of such interfering variables as age, sex, physical status, and operation; the data are organized into "strata" homogeneous in terms of such estimated death rates; and then, within strata, the death rates associated with the five anesthetics are computed and combined in a standardized death rate (or indirectly standardized mortality ratio). One set of results is based on the smoothed contingency-table analysis of Chapter IV-3. Roughly, the findings are that cyclopropane and Other have high death rates, ether has low death rates, and halothane and nitrous oxide- barbiturate have intermediate death rates after these multiple adjustments. A second set of results is based on strata constructed by fitting the vari- ables operation, physical status, age, and length with a cubic polynomial. Roughly, the findings are, again, that cyclopropane and Other have high death rates, ether has low death rates, and halothane and nitrous oxide-barbiturate have intermediate death rates. A third set of results is based on strata constructed by a cubic polynomial regression, on the two variables age and shock-likelihood index. These are the only results in this chapter for which all operations are considered simul- taneously, instead of in the three separate groups: high-, middle-, and low- death-rate operations. The results in this analysis are similar to those of the previous two, in that the order of adjusted death rates from highest to lowest is: Other, cyclopropane, halothane, nitrous oxide-barbiturate, and ether. These studies do not carry with them estimates of sampling reliability. Some clue as to the reality of the results was obtained by identifying the hos- pitals in which halothane and cyclopropane were each used in at least 20 per- cent of all surgical procedures (in the middle-death-rate group). There were 11 such hospitals. Adjusted death rates for halothane and cyclopropane were calculated in those hospitals; in 10 of the 11, halothane had a lower adjusted death rate than cyclopropane. That is a good indication of the reality of the difference. Of the same 11 hospitals, eight had lower halothane death rates than cyclopropane death rates for the high-death-rate operations. 316

CHAPTER IV-5. ANALYSIS BY REGRESSION METHODS Jerry Halpern Stanford University Stanford, California Lincoln E. Moses Stanford University Stanford, California Yvonne M. M. Bishop Harvard University Cambridge, Massachusetts Chapter IV-2 gave a general description of the data, exhibiting how death rates depended not only on the anesthetic, but also on such "independ- ent variables" as age, sex, and physical status. Some effort to purge agent contrasts of the inter- fering effects of these other variables was under- taken by computing standardized death rates, standardizing for each variable separately. That chapter explained the desirability of adjusting for several variables at once. Chapter IV-4, dealing with smear-and-sweep analysis, addressed the standardization problem using a particular mode of attack. In Chapter IV-3, the smoothed con- tingency-table analysis fitted a class of statistical models to the data, enabling death rates for in- dividual cells in the complex cross-classification to be estimated; those estimates were not merely based on the cases that belonged to particular cells, but "borrowed strength" from other por- tions of the data. This chapter will partake of some of the nature of both of the immediately preceding ones. It will resemble Chapter IV-3 in that statistical models will be fitted, and it will resemble Chapter IV-4 in that over-all compari- sons among anesthetic agents will be made after adjustment for several variables. The general scheme is as follows: A model for estimating the probability of death as a func- tion of such variables as operation, length, and age is proposed. Then the model is fitted to the data by some estimation procedure. This calcula- tion associates an estimated death rate with each cell of the cross-classification of the data. These estimated death rates tend to be more stable than individual cell rates, because they are based on more of the data. The cells are then coalesced into "strata," in such a way that cells with the lowest death rates fall into the first stra- trum, cells with the next lowest death rates into the second, and so on. These strata are the basis for the calculation of standardized death rates for each anesthetic agent. In this manner, it is to be expected that much of the effect of interfering variables is removed from the comparisons of death rates following the several agents. STRATA DERIVED FROM SMOOTHED CONTINGENCY-TABLE ANALYSIS In Chapter IV-3, the method of fitting death rates for the high-, middle-, and low-death-rate operation groups, as well as for the cholecystec- tomies, was described. Table 3 of that chapter summarizes the fitting procedure. To establish strata, the cells of the contingency table were swept up in the order determined by the fitted rates into 10 categories (or as close to 10 as the data permitted), in such a manner that the cate- gories had equal numbers of deaths plus esti- mated exposed. The results of the standardizing with respect to these categories are displayed in Tables 1 and 2. In Table 1, the directly stand- ardized rates are shown for the low-death-rate operations, for the middle-death-rate operations (both four-variable and five-variable fits have been used), for the high-death-rate operations, and for the cholecystectomies. Table 2 displays indirectly standardized mortality ratios. In the low-death-rate group, nitrous oxide-barbiturate and ether have the low death rates, halothane and cyclopropane have approximately equal rates (higher than for nitrous oxide-barbiturate and ether), and Other has the highest death rate. In the middle-death-rate operations, whether four or five variables are fitted, ether is lowest, halo- thane and nitrous oxide-barbiturate are approxi- mately equal and a little higher, and cyclopropane and Other are much higher, with cyclopropane the highest. In the high-death-rate operations, ether and halothane are nearly equal and lowest, nitrous oxide-barbiturate is next, and cyclopropane and Other are approximately equal and highest. In the cholecystectomies, all rates are between 0.020 and 0.029, increasing in the order halothane, ether, cyclopropane, nitrous oxide-barbiturate, and Other. These analyses are all based on the main fitting procedure used in Chapter IV-3, in which anesthetic agent was one of the variables fitted. We can reasonably ask whether some con- tamination of the findings may result from first 317

TABLE 1.--DIRECTLY STANDARDIZED RATES; STRATA BASED ON CONTINGENCY-TABLE FITTING« Agi -nt H N-B c E 0 All 0.0016 .0025 0 0018 0 .0033 LOT 0.0025 0 0.0022 Middle, model B 0.0195 0 .0195 0 .0279 0 0179 0 .0255 0.0221 Middle, model A 0.01W 0 .0197 0 .0278 0 0177 0 .0257 0.0221 High (A) 0.0844 0 .0914 0 .1097 0.0811 0 .1079 0.0933 Cholt ! 0.0203 0 .0262 0 .0251 0 0230 0 .0284 0.0238 *Strata defined by sweepup (on D+EE Into 10 categories) In order of death rate, as fitted by methods described in Table 3 of Chapter IV-3. TABLE 2.—INDIRECTLY STANDARDIZED MORTALITY RATIOS; STRATA BASED ON CONTINGENCY-TABLE FITTING Agent H N-B C E 0 Low 1.12 0.73 1.13 0.82 1.45 Middle, model B 0.88 0.88 1.26 0.81 1.15 Middle, model A 0.88 0.89 1.26 0.80 1.16 High (A) 0.91 0.98 1.17 0.85 1.12 Chole 0.86 1.10 1.06 0.97 1.20 TABLE 3.--DIRECTLY STANDARDIZED RATES; STRATA BASED ON CONTINGENCY-TABLE FITTING (without anesthetic) Agent H N-B C E 0 All Low 0 0023 0.0016 0 .0026 0 .0018 0. 0034 0.0022 Middl e, model B 0 0195 0.0197 0 .0277 0 .0178 0. 0257 0.0221 Middle, model A 0 0192 0.0197 G .0277 0 .0185 0. 0258 0.0221 High (A) 0 0854 0.0923 0 .1258 0 .0830 0. 1084 0.0933 Chcle 0 0205 0.0257 0 .0253 0 .0244 0. 0318 0.0238 TABLE 4.--INDIRECTLY STANDARDIZED MORTALITY RATIOS; STRATA BASED ON CONTINGENCY-TABLE FITTING (without anesthetic) Agent H N-B CEO Low 1.07 0.73 1.18 0.80 1.48 Middle, model B 0.89 0.89 1.26 0.80 1.15 Middle, model A 0.87 0.89 1.26 0.82 1.16 High (A) 0.91 0.98 1.19 0.80 1.12 Chole 0.86 1.06 1.05 0.93 1.13 using anesthetic in constructing the strata and then making comparisons between anesthetics within strata. Because of unease about this point, the same contingency-table analysis was fitted with agent not being used as a fitting variable. The results appear in Tables 3 and 4. The presen- tation here is of the same form as in the previous tables. The numeric values found are remarkably close to those obtained when anesthetic was used as a fitting variable. Indeed, there are no quali- tative differences worth noting. It is interesting to speculate how this simi- larity of findings occurs. The following plausible explanation may contain the germ of the correct idea. Differences between death rates following the anesthetics are small enough so that even with very numerous data they are hard to exhibit and estimate accurately. At the same time, the other variables used in the fitting are quite in- fluential. Operation, physical status, and age all carry great variability in death rates. It is then to be expected that estimates based on a regres- sion model (or on some other form of fitting) that uses all variables except anesthetic will scarcely be modified when the relatively uninfluential vari- able, anesthetic agent, is adjoined to them as a further predictor. If that is true, it must follow that strata based on the fitted values will have to be very similar for the two modes of fitting, and the standardized death rates based on nearly the same strata will necessarily have nearly the same values. STRATA DEFINED BY REGRESSION ANALYSIS The mode of fitting used above can be re- garded as a regression method of fitting the logarithm of the death rate, but this use of re- gression does not take account of the ordered or metric properties of the predictor variables. Rather, it is regression in the sense that the analysis of variance is regression. Large num- bers of constants had to be fitted, and it required considerable effort to solve the computational problems involved in the fitting. An alternative regression analysis now to be described is a sub- stantially smaller computational task, and there- fore seemed worth developing. There are several steps: (1) define the variables for purposes of regression, (2) choose a model, (3) fit the model, (4) sweep up cases into strata, and (5) construct the standardized rates using those strata. The main body of this section concerns the regression analysis using the variables operation, physical status, length, and age. Of these vari- ables, only length and age are naturally quantita- tive variables as they stand. Physical status con- sists of four ordered categories for elective procedures and three ordered categories for emergency procedures, and the rearrangement of these into a single quantitative variable is, in some measure, arbitrary. The same is even truer of operation. The principle adopted was to arrange the categories of the variable in such a manner that the death rate was an increasing function of the category assignment. That provision, although it may be sensible, certainly introduces a non- standard character to "regression." Later in this 318

section, a "pure regression analysis" is described, in which the two variables used are age and esti- mated propensity for shock; that analysis is free of the novel and arbitrary feature just described. A final discussion gives some consideration to the sampling stability of the findings. The ORLA Analysis Throughout this section we use the variables operation, physical status (risk), length, and age (sometimes the set is abbreviated "ORLA") to construct strata. We have dealt separately with the high-, middle-, and low-death-rate operation fractions of the entire body of data. The general approach has been to find a polynomial regres- sion for one of the fractions using these inter- fering variables; to calculate for each case the estimated probability of death, considering age, operation, physical status, and length, but ignor- ing the anesthetic; and then to collect (sweep up) all the cases in order of this estimated probability into 10 or 15 approximately equal-sized strata. Once the strata have been established, we calcu- late standardized death rates for the anesthetic agents. The summary findings in this section are presented in Table 5, which shows that, whether we consider the standardized rates themselves or the indirectly standardized mortality ratios, cyclopropane and Other have higher death rates than halothane, nitrous oxide-barbiturate, and ether, and that ether is the best, at least in the middle- and low-death-rate groups. Whether these appearances can be trusted depends in part on how uniformly they hold up from hospital to hospital. Thus, the reader should, at this time, reserve his acceptance of the apparent conclu- sions implied by the table. The question of sta- bility across institutions is discussed in the final section of this chapter. It is possible that the category "Other" has been unfairly penalized to the illusory advantage of the remaining agents. It could have happened in this way: Wherever two or more major agents are applied together, the anesthetic has been classified as "Other." If there is a tendency for a patient in trouble on one of these agents then to be switched over to another, the high death rate associated with such troubles would never be charged to the original anesthetic, but rather would be charged to Other. To investigate the degree to which this possible bias may be oper- ating, we have done some calculations in which a patient who ever received halothane is charged to halothane, a patient who ever received cyclo- propane is charged to cyclopropane, and so on.* In these calculations, we used exactly the same strata as in Table 5, but defined anesthetic as just described. The resulting directly standardized rates are displayed in Table 6. The principal effect of using the overlapping definition of anes- thetic agents is to raise the standardized rates *This leads to the association of some cases with two, or even three or four, of the anesthetic agents as now defined. for Other in all three groups, and to affect the rates for any of the agents only a little. It would thus appear that those surgical procedures in which major agents have been used in combina- tion are not especially "rich in deaths," but that, in fact, when they are removed from the category "Other," the residuum in that classification has an even higher death rate than before. Thus, we may dismiss the possibility of bias in question as erroneous; the bias actually favored Other. (A de- tailed presentation of the steps that led to the de- velopment of Table 5 appears as Appendix 1.) Regression on Age and Shock-Likelihood Index Age appeared in seven categories (see Ap- pendix 1). The variable shock-likelihood index (SLI) has five ordered categories of increasing conjectured propensity for shock, plus an "un- known" category. Operation and anesthetic risk (physical status) determine the value assigned to this variable. When anesthetic risk is "un- known" the SLI is also "unknown." "Unknown" was used as the first of the six SLI categories. (Details relating to SLI are given more fully in Chapter IV-7.) Estimated logit (log(p/(l-p)), where p is the death rate) was represented as a cubic poly- nomial (involving all terms) in age and SLI, and a cubic in SLI was fitted separately to the first age category. The 42 cells were then swept up in order of estimated logit into 10 categories ap- proximately equinumerous in terms of deaths- plus-randoms. Standardized death rates were then calculated (Table 7). The results are quali- tatively similar to those we have seen for the middle-death-rate operations: ether lowest, halothane and nitrous oxide-barbiturate next, and cyclopropane and Other highest; Other is higher than cyclopropane. At the same time, the magni- tude of the halothane-cyclopropane contrast is very much lower than it was for the middle- death-rate operations. Halothane is notable in having, to the accuracy shown, the average over- all death rate of the Study. Sampling Stability Estimates of death-rate comparisons, such as those in Tables 5, 6, and 7, necessarily contain a sampling error, and it is desirable to be able to estimate its size. If it is small, compared with the differences, we would be inclined to trust the differences more than if it were large. Unfortu- nately, the methods of analysis used here gen- erally do not admit of known estimates of sam- pling error, but some useful, if fragmentary, in- sight into the problem can be obtained. Consider the difference in rates for halo- thane and cyclopropane, the two most widely used anesthetics in the National Halothane Study. Their standardized death rates differ appreciably, no matter what analysis we refer to, cyclopropane having the higher death rate. On the one hand, if this pattern is not merely an over-all trend, but 319

TABLE 5.--COMPARISON OF ANESTHETIC AGENTS BY STANDARDIZATION ON REGRESSION-BASED STRATA* Agent Direct : H N-B C E 0 All High 0. 0839 0.0903 0 .1166 0.0988 0.1085 0.0934 Middle 0. 0204 0.0194 0 .0275 0.0173 0.0256 0.0221 Low 0. 00228 0.00187 0 .00271 0.00166 0.00324 0.00228 Indirect MR: High 0. 892 0.963 1 .199 0.912 1.128 Middle 0. 921 0.875 1 .236 0.778 1.139 Low 0. 995 0.820 1 .184 0.708 1.395 *The regressions were cubic, on variables operation, physical status, length, and age. Separate regressions were fitted for the three groups: high-, middle-, and low-death-rate operations. In each case, 10 strata, approximately equinumerous in terms of death-plus-ran- doms, were constructed by pooling cases in order of regression estimate of death rate. TABLE 6.— COMPARISON OF AGENTS (OVERLAPPING DEFINITION*) BY DIRECTLY STANDARDIZED DEATH RATES (Regression-based strata) (Corresponding figures from Table 5 in parentheses) Agent H N-B C E 0+ All High 0 (0 .0822 0.0822 .0839) (0.0903) 0.1038 (0.1166) 0.0941 (0.0988) 0.1710 (0.1085) 0.0880 (0.0934) Middle 0 (0 .0209 0.0179 .0204) (0.0194) 0.0261 (0.0275) 0.0193 0.0335 0.0210 (0.0173) (0.0256) (0.0221) Low 0 (0 .00253 0.00205 .00228) (0.00187) 0.00276 (0.00271) 0.00226 (0.00166) 0.00395 (0.00324) 0.00232 (0.00228) *This table uses exactly the same strata as in Table 5; it differs in that any procedure involving halothane (which might be classed as "halothane" or "Other" in the earlier table) is now counted as "halothane," etc. +The frequencies in the modified "Other" category become: high, 506 deaths and 1935 EE; middle, 901 deaths and 30,437 EE; and low, 74 deaths and 14,321 EE. TABLE 7.—DIRECTLY STANDARDIZED DEATH RATES BASED ON AGE-SHOCK REGRESSION STANDARDIZATION* (All operations, "unknown" shock patients included as first of six categories) Agent H N-B C E 0 All Death rate 0.0193 Mortality ratio 1.00 0.0172 0.892 0.0216 1.12 0.0152 0.0233 1.29 0.0193 0.788 *Swept to ID categories on deaths-plus-randoms. recurs in virtually all hospitals, we would be in- clined to conclude that the finding would emerge again if yet other hospitals were studied. On the other hand, if cyclopropane were superior in many hospitals and halothane were superior in many others, we would put little confidence on the over- all difference exhibited in the aggregate figures. Among the 34 hospitals in the Study, 11 used halothane and cyclopropane each in at least 20 per- cent of their operations. These 11 hospitals were studied to see whether the apparent superiority of halothane over cyclopropane was stable across 320

TABLE 8.--ANESTHETIC DEATH RATES* (Standardized by strata from ORLA logit regression in 11** institutions using halothane and cyclopropane each in at least 20 percent of procedures) Standardized death rates Indirectly standardized mortality ratios Inst. Agent Agent H N-B C E 0 All H N-B C E 0 5 0.0379 0.0485 0.0431 0.3081 0.0776 0.0413 0.89 1.04 0.93 2.14 1.18 7 0.0441 0.0437 0.0510 0.2151 0.0657 0.0527 0.79 0.74 0.88 2.40 1.21 12 0.0240 0.0183 0.0260 0.3037 0.0386 0.0247 0.93 0.57 1.00 4.15 1.26 13 0.0342 0.0317 0.0630 0.0161 0.0191 0.0292 1.05 0.93 1.87 0.44 0.61 14 0.0095 0.1530 0.0168 0.4353 0.0107 0.0089 0.79 0.94 1.27 1.55 0.82 18 0.0263 0.0347 0.0366 0.0106 0.0466 0.0312 0.82 1.11 1.15 0.29 1.37 21 0.0143 0.0156 0.0199 0.1198 0.0182 0.0154 0.88 0.86 1.22 2.79 1.04 22 0.0197 0.0272 0.0237 0.0845 0.0931 0.0238 0.79 1.01 0.93 0.52 2.32 23 0.0081 0.0116 0.0242 None 0.0206 0.0114 0.72 0.96 1.13 — 2.15 25 0.0613 0.0761 0.0943 0.2020 0.0929 0.0811 0.74 0.82 1.14 1.75 1.12 31 0.0258 0.0213 0.0185 0.0195 0.1276 0.0205 1.02 0.99 0.86 0.42 2.39 *Middle-death-rate operations only. **Reference population is all middle-death-rate operations in the particular institution. hospitals. Table 8 presents the data on that issue for the middle-death-rate operations. The two halves of the table show directly standardized death rates and indirectly standardized mortality ratios based on 10 strata swept up in terms of deaths-plus-randoms after polynomial regression on risk, length, operation, and agent for the middle-death-rate operations. In all but institution 31, halothane looks better than cyclopropane. It may also be seen that the smoothness in the total data is lacking when we go to the individual hos- pital; indeed, the most regular finding in the table is the halothane-cyclopropane contrast (which is natural, inasmuch as the hospitals were selected for study because the data were relatively ample). When the analysis in Table 9 was done for the same 11 hospitals on the high-death-rate op- erations, there was far less uniformity than in Table 8. The data were sufficiently meager so that nine of the 55 cells did not even permit de- fining standardized death rates. Although these institutions had been selected for the frequency with which they used both halothane and cyclo- propane, there appears to be considerable vari- ability even for these agents. In eight of the hos- pitals, cyclopropane had a higher standardized death rate than did halothane; in the other three hospitals, halothane had the higher rate. It ap- pears that no clear statement can be made from this analysis as to how halothane and cyclopro- pane compare, on the average, for the high-death- rate operations. TABLE 9.--ANESTHETIC DEATH RATES* (Standardized ty strata from ORLA logit regression in 11 institutions using halothane and cyclopropane each in at least 20 percent of procedures) Directly Standardized Death Rates** Institution Agent N-B 5 0 096 0.498 0.144 0 324 0 331 0 .100 7 0 188 0 . . 0.130 0 508 0 272 0 .143 12 0.263 376 0.063 — 0 112 0.066 13 0 007 -. 0.123 0 001 0 018 0 .024 14 0 002 — 0.110 — .. 0 .001 18 0 057 0 021 0.088 0 037 0 136 0 .072 21 0 041 0 256 0.081 0 500 0 272 0 .044 22 0 Ub5 0 140 0.084 0 039 0 166 0 .074 23 0 013 0 O1l 0.110 — 0 006 0 .011 25 0 OK 0 087 0.150 0 802 0 183 0 .137 31 0 138 0 102 0.088 — ~ 0 .089 (b) Indirectly Standardized Mortality Ratios Institution Aeent H N-B C E 0 All 5 0.66 1.51 0.91 0.83 1.31 1.0 7 0.93 0.81 0.87 1.59 1.0 12 1.97 0.84 0.56 .. 0.80 1.0 13 0.34 3.68 77.30X 0.71 1.0 14 0.30 4.04 -- — 1.0 18 0.85 0.17 1.23 0.43 1.95 1.0 21 0.60 1.65 1.48 2.21 0.90 1.0 22 0.85 0.95 0.99 0.43 1.61 1.0 23 0.98 0.93 1.29 M. .- 1.0 25 0.31 0.70 1.05 3.10 1.29 1.0 31 0.89 1.33 0.88 ~ 1.30 1.0 *High-death-rate operations only. **The category any particular observation falls into is based on 10 categories with approximately equal numbers of deaths-plus- randoms made from the estimated logit of the observations of all high-death-rate operations in all 34 institutions. x Undefined. 321

APPENDIX 1 TO CHAPTER IV-5 REGRESSION MODEL USED IN CHAPTER IV-5 Jerry Halpern Stanford University Stanford, California Lincoln E. Moses Stanford University Stanford, California We proposed to develop strata based on the variables age, length, operation, and physical status. The categories that we used for these variables are listed below. The categories for each variable are ordered and the order is ex- hibited in the listing. Age: 0-9; 10-39; 40-49; 50-59; 60-69; 70-79; and 80+ (cases with unknown age omitted). Risk: 1,2,5; unknown; 3,4,6; and 7. Length: 1-28; 29-58; 59-118; 119-238; 239-388; and 389 and over (cases with unknown length omitted). Operation had to be coded differently for high-, middle-, and low-death-rate groups, because dif- ferent operations appear there. For high, the categories were, in order of death rate: large bowel (code 48), exploratory laparotomy (44), craniotomy (12), and heart with pump (33). For middle, the five categories were: 1 - 2, 4, 6, 7, 8, 9, 10, 15, 20, 21, 23, 30, 32, 40, 50, 54, 62, 63, 66, 68, 71, 80, 81, 84, 85, 87, 89, 92, 95, and 99; 2-5, 16, 25, 39, 56, 58, 70, 75, 76, 77, and 98; 3 - 26, 28, 41, 42, 51, 88, and 93; 4 - 13, 17, 27, 34, 36, 43, 45, 46, and 87; and 5 - 22, 47, 52, 57, 59, 67. For low, the categories were, in order of in- creasing death rate: mouth (1), eye (3), and dila- tation and curettage (60); hysterectomy (65); herniorrhaphy (55) and cystoscopy (73); and plas- tic surgery (90). It should be observed that neither risk nor operation is truly a metric variable; in- stead, the value of the independent variable has been assigned after study of its relation to the dependent variable. We set out to fit a polynomial model of no more than third degree in all variables simul- taneously for the expected logit of the death rate. A key question was how many of the possible polynomial terms could reasonably be omitted and which ones presumably should definitely be included. At the beginning, we constructed row and column arrays using two variables at a time and found where there appeared to be important interactions. We then fitted full cubic polynomials to the two-way tables, risk X length and risk X age, and to the three-way table, age X length X risk, and studied which terms appeared to be droppable in the resulting regressions. Age, an important variable, has an erratic relation to death rate, in that there is a sharp drop from the rate at 0-9 to the rate at 10-19, after which death rate steadily increases. Be- cause of that, a separate fit was made for the first age category, a^ Three of the coefficients (of the 24 fitted) were estimated from both the a= a! and the a^ aj data; these were the coef- ficients for r X o, r3, and 1 2r. We finally fitted the following polynomials of the third degree: E(logit p) = bja 12 2 b3r + b *-b24ao 22 where a 4 ai, and E(logit p) = fii + c • cr + co V c,/ where a = aj. In fitting this model we proceeded iteratively and used weights wa = ne^cr(l-^a), where n,, = esti- mated exposed plus 50 (or, in some of the sub- problems, plus deaths). It should be remarked that the weights used could be defended better if the observations had been truly binomial. In the actual case, the value of n is itself an estimated quantity and this should ideally call for the use of different weights. Al- though the theory for fitting with the appropriate 322

TABLE 1.—STANDARDIZED DEATH RATES (ALL AGENTS, MIDDLE-DEATH-RATE OPERATIONS) (Strata from logit regression I on ORLA; various modes of sweepup) Cate- gories Sweepup Agent H N-B C E 0 All (Deaths 0.0205 0.0199 0.0270 0.0177 0.0254 0.0221 EE 0.0198 0.0189 0.0282 0.0169 0.0263 0.0221 D+R 0.0204 0.0194 0.0275 0.0173 0.0256 0.0221 {Deaths EE D+E 0.0208 0.0202 0.0270 0.0179 0.0256 0.0221 0.0201 0.0193 0.0278 0.0166 0.0260 0.0221 0.0205 0.0196 0.0274 0.0173 0.0256 0.0221 TABLE 3.—INDIRECTLY STANDARDIZED M3RTALITY RATIOS, MIDDLE-DEATH- RATE OPERATIONS (ALL AGENTS) (Strata from logit regression I on ORLA; various modes of sweepup) Cate- sweep p Agent gories H N-B C E 0 All (Deaths 0.927 0.899 1.196 0.780 1.133 1.000 10 EE 0.899 0.857 1.275 0.768 1.182 1.000 D+R 0.921 0.875 1.236 0.778 1.139 1.000 f Deaths 0.931 0.903 1.186 0.778 1.133 1.000 15 EE 0.910 0.877 1.252 0.754 1.165 1.000 1 D+R 0.926 0.885 1.228 0.770 1.130 1.000 TABLE 2.--STANDARDIZED DEATH RATES FOR MIDDLE-DEATH-RATE OPERA- TIONS (VARIOUS REFERENCE POPULATIONS) (Strata from logit regression I on ORLA; sweepup to 10 categories on D+RJ Reference population Agent H N-B C E 0 All H N-B C E 0 All 0.0173 0.0180 0.0252 0.0222 0.0244 0.0204 0.0164 0.0171 0.0242 0.0208 0.0232 0.0194 0.0236 0.0244 0.0339 0.0297 0.0326 0.0275 0.0145 0.0151 0.0219 0.0185 0.0209 0.0173 0.0221 0.0229 0.0313 0.0276 0.0300 0.0256 0.0188 0.0195 0.0274 0.0238 0.0263 0.0221 TABLE 4. —INDIRECTLY STANDARDIZED MORTALITY RATIOS, FOR MIDDLE- DEATH-RATE OPERATIONS (VARIOUS REFERENCE POPULATIONS) (Strata from logit regression I on ORLA; sweepup to 10 categories on D+R) Reference Agent population H N-B C E 0 All .345 0.835 1.230 1.083 H 1.000 0.949 1 N-B 1.053 1.000 1 .401 0.890 1.293 1.140 C 0.733 0.698 1 .000 0.624 0.921 0.802 E 1.190 1.128 1 .548 1.000 1.435 1.276 0 0.781 0.745 1 .084 0.673 1.000 0.862 All 0.921 0.875 1 .236 0.778 1.139 1.000 weights was worked out, it was not applied to these data. Each of the pigeonholes (840 for the middle- death-rate operations) could now have associated with it an estimated logit value obtained by com- bining the values of I, a, r, and o for the pi- geonhole with the fitted values of the regression coefficients. The pigeonholes were then listed in estimated logit order, and cumulative deaths, cumulative estimated exposed, and cumulative deaths-plus-randoms were printed out; then by hand the ordered pigeonholes were swept up into 10 categories approximately equinumerous in each of the three modes and again into 15 categories approximately equinumerous in each of the three modes. Then for each of the six methods of sweepup into categories, standardized death rates for all agents were constructed, using as standardizing populations each of the five agent populations and the combined population. Indirectly standardized mortality ratios were also com- puted, using each of the six kinds of category systems and again the six reference populations. Table 1 shows that, as far as the six methods of sweepup are concerned, the findings are all about the same: ether has the lowest stand- ardized death rate, halothane and nitrous oxide- barbiturate are next, with approximately equal rates, and finally cyclopropane and Other are the worst, also with approximately equal rates. It is noteworthy that in all six rows of that table the rank order of the agent death rates is the same. Table 2 involves the 10-category, deaths- plus-randoms sweepup method,* and shows the •Reasons for preferring this sweepup method are sketched in Appendix 2. standardized death rates for the agents, using various reference populations. Qualitatively, the conclusions are as before, i.e., in all six rows of the table the rank order among the five anes- thetic agents is exactly the same, and exactly the same as that found in the first table. How- ever, the numeric values of the standardized death rates are quite different for various ref- erence populations. In particular, it can be seen from the right-hand column that halothane and nitrous oxide-barbiturate have the lowest stand- ardized death rates, and that Other and cyclo- propane have the highest, with ether interme- diate. The position of ether is curious, for it appears to mean that the ether population is a relatively high-death-rate one, a conclusion we must bear in mind (and find contradicted) as we go to the second set of tables. Tables 3 and 4 present findings in terms of indirectly standardized mortality ratios. The former exhibits these ratios with all middle- death-rate operations as the reference popula- tion for the six methods of sweepup. The find- ings are consonant with those for the standardized death rates, i.e., the rank order of the agents is, in every row, identical with what we have pre- viously observed, and there is good qualitative agreement from row to row. Table 4 uses the 10-category deaths-plus-randoms sweepup and various reference populations; again, in every population, the order among the agents is exactly as we have previously found 18 times. But in this case, when we study the right-hand margin, it appears that ether has the most favorable popu- lation, halothane and nitrous oxide-barbiturate the next most favorable, and cyclopropane and Other the least favorable. Note the curious 323

change in the role of ether: it was intermediate when looked at in terms of standardized death rate, and is now most favorable when studied in terms of indirectly standardized mortality ratios. These two sets of tables are a highly con- densed summary of all the middle-death-rate data in terms of regression stratification, and suggest that halothane and nitrous oxide-barbit- urate are anesthetics with lower death rates, that cyclopropane and Other are anesthetics with higher death rates, and that ether may have the lowest rate of all for these operations. Further- more, these conclusions appear to hold for var- ious methods of sweepup, various reference populations, and the use of either direct or in- direct rates. 324

APPENDIX 2 TO CHAPTER IV-5 REMARKS ON SWEEPUP RULES Lincoln E. Moses Stanford University Stanford, California Figures 1 and 2 show (for 10 and for 15 categories) the percentage of "cases"* swept up into categories having predicted logit less than or equal to the value indicated on the abscissa. The graph is interpreted as follows: Look at the 10-categories graph (Fig. 1). Consider the number -3.0, shown on the horizontal axis; that is the logit transform of p = 0.047. The polygonal curve labeled "deaths" has a height of 0.2 at -3.0; this means that 20 percent of the deaths lay in categories (the first two) with estimated death rates smaller than 0.047. The next "curve" (deaths + randoms) stands at 0.6 and implies that the first six categories (containing six-tenths of the observations) had estimated death rates less than 0.047. The highest curve (EE) stands at 0.8 and indicates that 80 percent of the exposures to anesthesia in the Study belonged to combinations of operation, physical status (risk), length, and age estimated to carry a death rate less than 0.047. •"Case" defined for the three graphs; deaths, deaths plus randoms, and EE. From the graphs it appears that: (1) D + R picks cases up in a way intermedi- ate between the other two modes; (2) the three modes compare in about the same way for 10 as for 15 categories; and (3) pickup on deaths gives the smallest vari- ety of estimated death rates, i.e., it ap- pears to lead to coarser strata (the first death stratum corresponds to the first nine EE strata in the case of 15 cate- gories). Primarily because of the last point, sweepup on death may be least desirable. One may prefer D + R to EE, because it gives a finer stratification in the high-death-rate zone, and an equally fine one elsewhere. Table 1 gives the data from which the graphs were made. The sweepup was done by hand from a printout of the 840 ORLA cells (in logit order) that showed cumulative D, D + R, and EE. TABLE 1.--TABULATION FOR SWEEPUP GRAPHS IN ORLA REGRESSION I Deaths Estimated exposed 10 c itegories 15 categories 10 categories 15 categories 1 -4.04 0.1 -4.50 0.065 -7.53 0.087 -7.85 0.071 2 -3.27 0.2 -3.73 0.133 -6.92 0.188 -7.37 0.171 3 -2.72 0.3 -3.27 0.2 -6.47 0.291 -6.81 0.196 4 -2.32 0.403 -2.91 0.27 -5.84 0.396 -6.52 0.282 5 1.99 0.5 -2.61 0.334 -5.29 0.5 -6.23 0.336 6 -1.72 0.601 -2.33 0.402 -4.68 0.596 -5.79 0.405 7 -1.46 0.699 -2.09 0.467 -4.09 0.7 -5.45 0.465 g -1.15 0.801 -1.87 0.535 -3.46 0.8 -5.16 0.533 9 -0.834 0.9 -1.72 0.601 -2.48 0.898 -4.68 0.596 10 -0.067 1.0 -1.53 0.667 -0.067 1.0 -4.37 0.663 11 -1.42 0.735 -3.93 0.733 12 -1.15 0.801 -3.46 0.8 13 -0.971 0.867 -2.85 0.868 -0.655 0.934 -2.05 0.933 15 -0.067 1.0 -0.067 1.0 10 categories Randoms + Deaths 15 categories 1 -7.37 0.12 6 -3.27 0.6 1 -7.5 0.0615 9 -3.27 0.6 2 -6.47 0.203 7 -2.54 0.7 2 -6.9 0.132 10 -2.76 0.665 3 -5.74 0.297 8 -1.87 0.8 3 -6.47 0.203 11 -2.33 0.732 4 -4.75 0.402 9 -1.34 0.898 4 -5.85 0.272 12 -1.87 0.798 5 -4.04 0.5 10 -0.067 1.0 5 -5.34 0.333 13 -1.52 0.867 6 -4.78 0.4 14 -1.11 0.934 7 -4.35 0.464 15 -0.068 1.0 8 -3.80 0.532 334-553 O-69—22 325

10 8 O o VI 4 +- C 10 Categories EE Deaths pius Ran dom>3 Deaths -8 -7 -6 -4 Log it -3 -2 -1 -O Figure 1.--For 10 categories: Percentage of cases swept up into categories having predicted logit less than or equal to value indicated on abscissa. 326

10 8 O> O 6 O t5 Ml i 15 Categories EE Deaths pius Randoms Deaths -8 -7 -6 -5 -4 Logit -3 -2 -1 Figure 2.--For 15 categories: Percentage of cases swept up into categories having predicted logit less than or equal to value indicated on abscissa. 327

Abstract of Chapter IV-6 In this chapter, comparison of anesthetic agents after multiple adjust- ment for interfering variables is attempted by another scheme of stratum construction. In the cross-classification of the data formed from the inter- fering variable categories, the over-all death rate can be calculated for each cell. Cells are then aggregated in death-rate order into a number of strata. Anesthetic death rates within strata can be determined and death rates stand- ardized against strata can be calculated. That was the approach used in this chapter. The first body of results is based on the cross-classifications used in Chapter IV-3. Roughly, cyclopropane and Other had high death rates, ether had low death rates, and halothane and nitrous oxide-barbiturate had inter- mediate death rates. The second body of results began with the cross-classification of the data on the variables age, length, physical status, and operation. The findings were substantially the same. A third body of results used the cell aggregation from the two-way clas- sification by age and shock-likelihood index (SLI); again, Other and cyclopro- pane had high death rates, ether had low death rates, and halothane and ni- trous oxide-barbiturate had intermediate death rates. This was the only set of results in this chapter that treated all the operations as a single body, rather than dividing them into high-, middle-, and low-death-rate groups. The similarity in findings between strata based on aggregation (in this chapter) and findings based on strata obtained by fitting death rates was in- vestigated. The two sets of strata are very similar and must therefore lead to similar numeric conclusions. To assess the sampling stability of the anesthetic contrasts, we identified institutions that used both of a pair of anesthetics "frequently" (at least 10 percent of all operations and again at least 15 percent). We found a strong uniformity in the superiority of ether over halothane, nitrous oxide-barbit- urate, and Other by this analysis, and strong evidence of lower death rates for nitrous oxide-barbiturate and halothane than for Other. The last section of this chapter is devoted to a study of death-rate dif- ferences among institutions. Indirectly standardized mortality ratios based on age-risk-operation aggregation strata for the middle-death-rate operations vary in a 3.1-fold ratio over the 34 institutions. There is strong evidence that these differences are not due to sampling error. For a second way of looking at institutional variation, we chose six commonly used surgical procedures (involving about 15 percent of all the cases in the National Halothane Study) and studied institutional variation in death rates for them only. Again, we found large institutional variation; moreover, we found agreement between the list of institutions with high death rates for these chosen six operations and the list of high-death-rate institutions identified by the more comprehensive but less direct analysis using middle-death-rate operations. Further similar analyses using the four high-death-rate operations, separately and together, corrob- orated these results. We conclude that there are real and important differences in death rates among institutions, and we point to reasons for caution in inter- preting these differences and raise questions calling for further study. 328

CHAPTER IV-6. ANALYSIS FOR "PURE-AGGREGATION" STRATA Lincoln E. Moses Stanford University Stanford, California Jerry Halpern Stanford University Stanford, California Lawrence G. Tesler Information Processing Corporation Palo Alto, California Yvonne M. M. Bishop Harvard University Cambridge, Massachusetts The objective of stratification methods, as used in this study, is to bring together kinds of cases that are homogeneous with respect to death rate and then to compare the agents in such ho- mogeneous classes for their standardized death rates. One way to make this stratification is to assemble the cells with the highest death rates in one stratum and then assemble the cells with the next highest death rates into the second stra- tum, etc. If the samples were sufficiently large, this stratification of observed death rates would agree (almost certainly) with the ideally correct stratification, which would be based on true cell death rates. But if the samples are quite small, the stratification based on the ordering of the cell death rates might be seriously in error. In this chapter, we present the results of stud- ies of death rates in terms of cell aggregation. There are, of course, as many possible ways of forming strata by cell aggregation as there are ways of defining cells. (Indeed, there are more, because there are many possible sweepup rules for any given cross-classification.) It is inter- esting that, if the many cells in a large cross- classification are coalesced into a single stratum, the "standardized rates" then will be merely the crude rates; if they are coalesced into two strata, high- and low-death-rate strata, we will have taken one step toward adjusting for the "interfer- ing variables." In most of what follows, we have used 10 (sometimes 15) strata. AGENT COMPARISONS BASED ON "PURE-AGGREGATION" STRATA Aggregation of Cells Defined in the Smoothed Contingency-Table Analysis In Chapter IV-3, we defined various ways of cross-classifying portions of the data into high-, middle-, and low-death-rate operations and cho- lecystectomies (summarized in Table 3 of that chapter). The pigeonholes corresponding to these classifications underlie the aggregations pre- sented here. The observed values in the cells were summed over the categories of the variable agent, and rates computed from these sums were used for ordering and defining the aggregated strata. Once the strata were defined, the data were again broken down into the five agent cate- gories so that standardized rates could be com- puted. Table 1 shows standardized death rates for various subsets of the data. These standard- ized death rates are based on 10 strata (except that occasionally it was possible to construct only nine), formed by collecting cells of the cross-classification in order of death rate and in such a way as to make the resulting strata ap- proximately equinumerous in terms of deaths plus estimated exposed. In Table 1, we see pat- terns that are remarkably similar to those found in Table 1 of Chapter IV-5, which presented standardized death rates using strata obtained by fitting parameters to the same underlying pigeon- hole system. Table 2 gives information similar to that in Table 1, but based on indirect standardization. This table is comparable with Table 2 of Chapter IV-5 and the figures are quite similar. Results from the ORLA Cross-Classification For each of the low-, middle-, and high- death-rate operation groups, it is possible to construct a system of cells indexed by age, length, risk, and operation, and then to aggre- gate cells into strata in order of their death rate. This leads to standardized death rates. In Table 3, the definitions of the classifications used are set forth; and in Table 4, the standardized death rates are exhibited, with the indirectly standardized mortality ratios. We see that for the low-death- rate operations the most notable features are the high death rates associated with cyclopro- pane and Other. In the middle-death-rate oper- ations, cyclopropane and Other both have death rates of around 0.025, and the three remaining 329

TABLE 1.—DIRECTLY STANDARDIZED RATES BASED ON AGGREGATION OF CELLS USED IN SMOOTHED CONTINGENCY-TABLE ANALYSIS Operation group Agent H N-B C E 0 All Low 0.0023 0.0016 0.0026 0.0018 0.0033 0.0022 Middle, model B 0.0195 0.0196 0.0277 0.0181 0.0255 0.0221 Middle, model A 0.0192 0.0195 0.0277 0.0184 0.0260 0.0221 High 0.0827 0.0931 0.1116 0.0819 0.1060 0.0933 Chole 0.0206 0.0266 0.0248 0.0255 0.0291 0.0238 TABLE 2.—STANDARD MORTALITY RATIOS BASED ON AGGREGATION OF CELLS USED IN SMOOTHED CONTINGENCY-TABLE ANALYSIS Operation group Agent H N-B C E 0 Low 1.07 0.73 1.19 0.81 1.45 Middle, model B 0.88 0.89 1.26 0.81 1.14 Middle, model A 0.87 0.88 1.26 0.82 1.17 High 0.89 1.00 1.20 0.85 1.12 Chole 0.86 1.08 1.01 1.03 1.10 TABLE 3.—DEFINITION OF VARIABLES USED IN ORLA PURE-AGGREGATION ANALYSES Age,« years Length, min Risk Operation 0-9 10-49 1-28 29 - 58 59 - 118 119 + Unknown 1, 2, 5 1 3 55 60 63 73 90 Low 50-69 3, 4, 6 70 + 7 0-9 10-39 1-28 29-58 Unknown 1, 2, 5 5 groups of operations"" 40-49 59 - 118 3, 4, 6 Middle 50-59 119 - 238 7 60-69 70-79 239 - 388 80+ 389 + 0-9 1-28 Unknown 12 10-49 50-69 29 - 58 59 - 118 1, 2, 5 3, 4, 6 33 High M 70+ 119+ 7 48 "Cases with unknown age omitted. ""See Appendix 1 to Chapter IV-5. anesthetic agents have death rates of around 0.020. In the high-death-rate operations, the comparisons among the agents are very much as in the middle-death-rate operations, except that nitrous oxide-barbiturate seems to lie be- tween the two pairs, cyclopropane and Other, and ether and halothane. Considerations Relating to Emergency Operations Cyclopropane was used rather more heavily than halothane in emergency operations in all three death-rate groups. (This statement can be confirmed by looking at Tables EE-4, -5, and -6 of Chapter IV-2.) Because cyclopropane is used more extensively in the higher-risk categories, it is reasonable to inquire whether within any risk category the cyclopropane patients may not be "worse" than the halothane patients. That would happen if there were a general shift in the distribution of physical status of cyclopropane patients compared with halothane patients. If there were such a shift, it would bias the death-rate comparison, causing halothane to compare favor- ably with cyclopropane, partly because of this phenomenon. To check this possibility, a pure- aggregation analysis based on ORLA was done with all emergency-operation patients omitted completely; the corresponding standardized death rates are presented in Table 5. In that table, the rows labeled "Out" correspond to the analyses where emergency patients have been left out, and the rows labeled "In" correspond to analyses where they have been included. Ob- viously, the most striking feature of the data is the substantially lower death rate for every an- esthetic agent that occurs when emergency op- erations are excluded. Looking at the bottom 330

TABLE 4.--DIRECTLY STANDARDIZED+ DEATH RATES BASED ON AGGREGATION OF CELLS IN AGE, LENGTH, RISK, AND OPERATION CROSS-CLASSIFICATION* Operation group Agent H N-B C E 0 All Directly (High standardized < Middle death rate ( Low 0.0862 0.0920 0.0205 0.00185 0.1118 0.0261 0.00302 0.0908 0.0191 0.00174 0.1044 0.0249 0.0934 0.0221 0.00229 0.0200 0.00217 0.00304 Indirectly ( High standardized ^Middle mortality ( Low ratio 0.918 0.910 0.946 0.994 0.925 0.815 1.15 1.16 1.29 0.873 0.850 0.761 1.09 1.11 1.34 1.00 1.00 1.00 + Standardizing population is "all agents." * Aggregation (in order of D/(D+EE)) into 10 strata approximately equinu- merous in terms of deaths-plus-randoms. TABLE 5.--DEATH-HATE COMPARISONS BY ORLA AGGREGATION WITH AND WITHOUT EMERGENCY OPERATIONS (Middle-death-rate operations only) Emergency operation Agent Directly standardized death rate H N-B C E 0 All In Out 0.0200 0.0157 0.0205 0.0163 0.0261 0.0209 0.0191 0.0138 0.0249 0.0192 0.0221 0.0170 Mortality ratio In Out 0.910 0.927 0.925 0.961 1.16 1.23 0.850 0.815 1.11 1.13 1.00 1.00 half of the table, where the directly standardized mortality ratios are exhibited, we see that there is no evidence of an effect of the sort described. The mortality ratio for cyclopropane is 1.16 when emergency patients are included and 1.23 when they are excluded. For halothane, there is also a difference, smaller but in the same di- rection. Generally, there are not large differ- ences in the two patterns of mortality ratios. Analysis Based on Aggregation of the Age- SLI Distribution for~All Operations Shock-likelihood index (SLI) is a numeric variable intended to represent the propensity for a given patient to experience shock on the op- erating table. The SLI score is determined from the operation code of the surgical procedure, and from the emergency status of the procedure (i.e., whether physical status was 1, 2, 3, or 4, rather than 5, 6, or 7). A detailed description of this index appears in Chapter IV-7. Shock includes six categories: unknown and five ordered clas- sifications reflecting increasing probability of shock. The 7X6 two-way classification of shock and age was swept into 10 categories in order of death rates, so as to form categories as nearly equinumerous as possible in terms of deaths- plus- randoms. The directly standardized death rates were computed for each anesthetic agent, the standardizing population being the all-agents population. The directly standardized death rates obtained appear in Table 6, with corresponding mortality ratios. It is notable that the stand- ardized death rate for halothane is almost exactly "average." As other analyses have re- peatedly shown, cyclopropane and Other have higher than average death rates and nitrous oxide-barbiturate lower than average. RELATION OF ANALYSES BASED ON REGRESSION STRATA AND PURE-AGGREGATION STRATA We are able to compare the results arising from the two methods of strata formation in three different bodies of data. They are the contingency- table analysis data, the ORLA classification, and the age-shock (for all operations) data. The three sets of comparisons are not wholly similar. In the case of the contingency-table analysis, the comparison amounts to comparing Tables 1 through 4 of Chapter IV-5 with Table 1 of this chapter. The similarity is striking, and there are no important differences. In the case of the ORLA studies, the high-, middle-, and low-death-rate operation groups' 331

TABLE 6.—DEATH-RATE COMPARISONS BASED ON SHOCK*-AGE AGGREGATION STRATA (10 strata swept on D+R) Directly standardized Agent H N-B C E 0 All Death rate Mortality ratio 0.0195 1.00 0.0169 0.861 0.0216 1.11 0.0156 0.0237 1.22 0.0194 0.804 *A11 operations; unknown shock included. TABLE 7.—DIRECTLY STANDARDIZED DEATH RATES BASED ON CELL AGGREGATION OF ORLA. CROSS-CLASSIFICATION (Middle-death-rate operations) Category sweepup Agent H N-B C E 0 All { Deaths 0.0208 0.0204 0.0256 0.0196 0.0249 0.0221 10 {EE 0.0195 0.0194 0.0273 0.0180 0.0262 0.0221 (D+R 0.0200 0.0205 0.0261 0.0191 0.0249 0.0221 TABLE 8.—DIRECTLY STANDARDIZED DEATH RATES BASED ON ORLA. LOGIT REGRESSION STRATA (Middle-death-rate operations) Category sweepup Agent H N-B C E 0 All {Deaths EE D+R 0.0205 0.0198 0.0204 0.0199 0.0189 0.0194 0.0270 0.0282 0.0275 0.0177 0.0169 0.0173 0.0254 0.0263 0.0256 0.0221 0.0221 0.0221 ( Deaths 15 { EE ( D+R 0.0208 0.0201 0.0205 0.0202 0.0193 0.0196 0.0270 0.0278 0.0274 0.0179 0.0166 0.0256 0.0260 0.0256 0.0221 0.0221 0.0221 0.0173 standardized death rates obtained by regression- based strata appear in Table 5 of Chapter IV-5, and those obtained by aggregation appear in Table 4 of this chapter. The agreement is strong, but less than with the contingency-table analysis. For the low-death-rate operations, the death rates are in close agreement, the only notable difference being a larger cyclopropane death rate in the aggregation-based rates; it is not clear that the difference has meaning, occurring in the fourth decimal place. For the middle-death-rate operations, the death-rate comparisons among anesthetics are qualitatively alike; generally, the differences among the agents are greater in the regression-strata figures. For the high-death-rate operations, the pictures are again qualitatively alike, with the contrasts among agents being gen- erally less great in the aggregation-strata figures. Further comparison of the two modes of analysis for the middle-death-rate operations is made in Tables 7 and 8. The former shows the standardized death rates for sweepup (with 10 categories by aggregation) by deaths, estimated exposed, and deaths-plus-randoms. The latter shows the standardized death rates for sweepup (into 10 and 15 categories by regression) by deaths, estimated exposed, and deaths-plus- randoms. The generally more pronounced dif- ferences for regression strata appear throughout. If we turn to the age-shock data, which cover the full range of operations, the similarity of re- sults for regression and aggregation is again striking. Table 9 presents directly standardized mortality ratios for the shock-age data where the stratification has been done by regression and by aggregation. There are no notable disagreements. To understand the reasons for the similarity, we took one body of data, the middle-death-rate ORLA classification, and inquired into the corre- spondence between the strata arrived at by the stratification schemes. Every pigeonhole can be characterized by a pair of numbers: its regres- sion stratum and its aggregation stratum. In Table 10, we exhibit all the deaths in the middle- death-rate operations in a two-way classification. The row represents the aggregation stratum in 332

TABLE 9.—DIRECTLY STANDARDIZED MORTALITY RATIOS FOR THE SHOCK-AGE DATA, STRATIFIED BY REGRESSION AND BY PURE AGGREGATION (All operations) Mode of stratification Agent I! N-B C E 0 Regression* Aggregation+ 1.00 1.00 0.892 0.861 1.13 1.11 0.788 0.804 1.29 1.22 *Source, Table 7, Chapter IV-5. +Source, Table 6, this chapter. TABLE 10.—DISTRIBUTION OF DEATHS IN TWO-WAY CLASSIFICATION* OF REGRESSION AND AGGREGATION STRATA FOR MIDDLE-DEATH-RATE OPERATIONS Ordered Ordered "regression strata" d£,J,J. U^UW^WU strata" 1 2 3 4 5 6 7 8 9 10 1 7-74 139 33 16 0 0 0 0 0 0 2 163 510 170 60 30 13 0 0 0 0 3 16 236 373 168 130 2 4 31 0 0 4 0 17 214 232 175 180 66 56 22 2 5 1 2 121 271 259 184 49 17 0 13 6 0 36 104 133 206 267 119 71 16 0 7 0 0 0 0 24 115 155 429 259 19 8 0 1 0 22 81 165 346 158 103 27 9 0 4 0 11 3 2 104 119 401 232 10 0 0 0 0 2 9 52 59 86 413 r= 0.88 *Both sets formed by sweeping to 10 strata approximately equinumerous in terms of deaths. TABLE 11.--CORRELATION COEFFICIENTS FOR REGRESSION- AND AGGREGATION- STRATUM IDENTIFICATION OF "CASES" FOR SIX SWEEPUP RULES Number Swept by counting: of strata Deaths Estimated exposed Death + randoms 10 0.88 0.92 0.95 15 0.88 0.94 0.96 which the death belongs, and the column repre- sents the regression stratum in which the death belongs. (The strata have been swept up in terms of deaths.) It is clear that there is a strong sim- ilarity between the two systems of strata. The correlation coefficient between two indices is 0.88. Six such two-way tables were prepared, corresponding to the six modes of sweepup (10 or 15 categories in terms of deaths, or estimated exposed, or deaths-plus-randoms). The correla- tion coefficients for these six tables are shown in Table 11. All are at least 0.88; if sweeping is in terms of deaths-plus-randoms, the similarity of strata is greatest. These considerations lead to the conclusion that, for any given scheme of cross-classification, the similarity of agent com- parisons based on aggregation strata and those based on regression strata is largely to be ex- pected, because the strata developed by the two methods are quite similar. VARIATION AMONG INSTITUTIONS OF THE COMPARISONS BETWEEN ANESTHETICS (IN TERMS OF STANDARDIZED DEATH RATES USING AGGREGATION STRATA) In this section, we shall study the agent com- parisons within separate institutions, rather than lumping the entire body of data across institu- tions. There are two reasons for examining the agent comparisons in this way: (1) to prevent hospital effects from distorting agent compari- sons, and (2) to assess the sampling stability of over-all figures. Each of these reasons deserves elaboration. Chapter IV-2 showed that different institu- tions had very different death rates and patterns of use of the various anesthetic agents. Compari- sons of the agents that simply lump the hospitals together are subject to the risk that the apparent agent differences are partly created by (or partly 333

TABLE 12.—INDIRECTLY STANDARDIZED MORTALITY RATIOS (BASED ON AGE, RISK, AND OPERATION AGGREGATION STRATA) DISPLAYED FOR 34 INSTITUTIONS BY AGENT (Middle-death-rate operations only) Inst. Agent H N-B C E 0 All 1 0.8247 0.4240 92.2436 0.9324 0.7800 0.7286 2 1.2986 1.1399 1.3854 1.9593 1.3726 1.3221 3 0.9142 0.7468 1.0164 0.2500 1.0913 0.8619 4 0.8955 1.0532 1.2038 1.8531 1.0961 1.1324 5 1.2435 1.8274 1.5046 4.6516 1.7303 1.5633 6 0.7483 0.9648 0.5769 1.2044 0.9996 0.8952 7 1.3859 1.1852 1.0960 3.5477 2.3088 1.5979 8 0.6419 0.9000 2.9322 0.6582 0.6365 0.6834 9 1.5906 0.8457 0.8529 0.5005 0.6220 0.7582 10 1.1002 1.0472 1.4540 1.6927 1.6482 1.2738 11 0.5594 1.0859 0.6282 0.5129 1.2663 0.7121 12 1.0749 0.6537 1.2314 10.7208 1.3447 1.1641 13 1.7270 1.3716 2.6247 0.6957 0.8985 1.4877 *1 4 0.3766 0.6437 0.8852 0.2675 0.5233 0.5771 15 1.7181 0.8131 1.1431 0.7303 1.1303 1.0351 16 1.8225 1.2879 1.2910 4.9441 1.2829 1.3380 *17 0.9637 0.5095 0.4845 0.4168 1.2740 0.8583 18 0.6505 1.0058 0.9306 0.2787 1.2133 0.8384 19 1.3888 0.4618 1.2644 6.3394 2.4099 1.0609 20 1.0993 0.4693 1.2409 0.9873 1.0775 1.0213 21 0.7453 0.7394 0.7899 3.2776 0.7377 0.7695 22 0.9032 1.1259 1.0228 0.6107 2.5697 1.1257 *23 0.3817 0.6082 0.6334 -1.0000 0.9434 0.5741 24 1.4478 0.9614 2.0318 7.5223 2.9339 1.4329 25 1.0838 1.1944 1.3809 3.5390 1.5384 1.3756 26 0.8517 0.6435 6.3498 0.9698 0.9033 0.9426 27 0.6575 1.3785 1.0248 1.3128 1.0225 0.9400 *28 0.7392 0.6491 0.1345 2.1928 0.6445 0.7056 »29 0.7674 1.6757 1.1747 0.2500 0.6343 0.8416 30 0.7453 0.5345 1.4036 -1.0000 0.8566 0.7379 *31 1.6282 1.6556 1.5655 7.0894 9.5743 1.7667 32 0.6228 0.5942 1.0281 0.2500 0.9031 0.7576 33 0.6148 0.9210 0.8387 1.1509 0.8609 0.7928 »34 0.8377 3.7092 2.2157 1.1811 0.8835 1.2088 All 0.899 0.907 1.212 0.803 1.140 1.000 "Starred institutions had fewer than 200 deaths in entire Study. -1.0000 denotes that there were no estimated exposed or deaths associated with the cell. TABLE 13.—DISTRIBUTION OF RANK WITHIN INSTITUTION FOR FIVE ANESTHETICS IN 32 INSTITUTIONS--RANKS APPLIED TO MORTALITY RATIOS* INDIRECTLY STANDARDIZED FOR AGE, OPERATION, AND PHYSICAL STATUS USING 15 AGGREGATION STRATA (Middle-death-rate operations) Rank* Agent H N-B C E 0 Average 1 6 8 4 11 3 6.4 2 12 8 4 1 7 6.4 3 7 6 9 2 8 6.4 4 5 7 8 4 8 6.4 5 2 3 7 14 6 6.4 Rank sum 81 85 106 105 103 96 = 7.325 p = 0.122 *Rank 1 given to the smallest mortality ratio, 5 to the largest . 334

masked by) the joint effect of these two factors concerning institutions. For example, one of the agents, say A, might tend to be used largely in hospitals with difficult patient loads and thus high death rates, whereas another agent, say B, might be used largely in hospitals with few difficult cases and thus low death rates. This combination of circumstances would lead to the spurious ap- pearance of A's having a higher death rate than B. In some measure, this kind of difficulty is avoided by standardizing for age, operation, and risk, but if we not only make adjustments, but then look directly at the agent comparisons within the vari- ous hospitals, we need not rely wholly on the ef- ficacy of adjustments. Furthermore, the designa- tion of the physical status of a patient may vary from one hospital to another, and perhaps to some extent the designation of the operation code may, also (consider, for example, the judgment factor in deciding whether a mastectomy is or is not "radical"). Such hospital variations would par- tially offset the corrective effects of adjustments through standardization. But these idiosyncrasies will tend to remain constant for all agents within a given hospital, and not distort comparisons there. We would like to attach some measure of firmness to the statistical contrasts between the agents (differences between death rates associ- ated with the anesthetics). If a contrast is small but consistent from one hospital to another, we may have high confidence that it reflects a dif- ference between the anesthetics, and not merely a fortuitous averaging of conflicting comparisons that happened not to cancel out wholly in our particular sample of 34 institutions. Similarly, a contrast between agents, even if large, may not be trustworthy if it is the average of large swings in both directions. To resolve these matters, it is necessary to see how these agents compare, hospital by hospital. It would certainly be attractive to compare the agents within hospitals over all operations, but we have elected not to. Thirty-eight percent of the deaths arose from the four high-death-rate operations. These had sharply varying patterns of anesthetic use; for example, cyclopropane and ether were rarely used in operation 12, and ether was never used in operation 33. The statistical effects of such great differences in usage are difficult to appraise, and we have chosen to ex- clude the four high-death-rate operations for in- dividual study. Because only 5 percent of the deaths arose from the low-death-rate operations, we have also excluded them. Thus, we focus our attention on agent comparisons within hospitals for the middle-death-rate operations. We have used indirect standardization based on 15 strata (equinumerous in terms of deaths- plus-randoms), obtained by aggregating cells from the age, operation, and physical status cross-classification; these three variables have been categorized as in Chapter IV-5. Table 12 shows the indirectly standardized mortality ratios obtained in the 34 institutions for the middle- death-rate operations. The seven institutions having fewer than 200 deaths in all have been starred. The last row of this table displays the indirect standardized mortality ratios for the five anesthetics where cases in all institutions have been accumulated. There ether has the lowest ratio, 0.803; halothane and nitrous oxide- barbiturate have essentially equal ratios, 0.9; and Other and cyclopropane have the highest ratios, 1.14 and 1.21, respectively. An assessment of the degree to which the over-all results reflect consistent intrahospital differences can be approached by the following procedure: For each institution, replace the five figures for the agents by their ranks, assigning 5 to the highest and 1 to the lowest; omitting in- stitutions 23 and 30, in which no ether was used, sum the ranks for each agent. If such a rank sum is large, it means that in many institutions the death rate for that agent was high; if the rank sum is small, it means that in many institutions the death rate for that agent was small. The rank sums obtained by this method are displayed in Table 13. These sums form the basis for a test of significance called Friedman's rank analysis of variance; application of that test yields a chi- square with four degrees of freedom equal to 7.325, which corresponds to a significance level of 0.122. This result is a very weak indication of con- sistent differences among the anesthetic agents. But the distribution of ranks in the ether column in Table 13 strikes the eye. In more than three- fourths of the hospitals, it was either best or worst; in Chapter IV-2, we learned that it had a very spotty usage pattern, being scarcely used in some hospitals and widely used in others. These are good reasons to omit ether from our present consideration. Furthermore, the presence of ether has absorbed more than one-third of the extreme ranks (i.e., ranks 1 and 5) in the entire table, restricting the opportunity of other agent differences to become manifest. Therefore, in Table 14 we exhibit results of the rank analysis of variance with ether omitted, and with insti- tutions 23 and 30 present (because their nonuse of ether is no longer a reason for their exclu- sion). The value of chi-square is now 10.2, which for three degrees of freedom has a significance level of 0.017 and is probably a reliable indica- tion of real differences among these four agents. This conclusion has some vagueness, in that it fails to answer such more precise and inter- esting questions as: "Which agents are associated reliably with lower death rates than which others?" Instead, it simply gives us the shotgun answer: "There are some real differences." Another shortcoming of the over-all rank anal- ysis of variance just applied is that it treats the institutions with complete symmetry and the agents with complete symmetry. Thus, important distinctions may be ignored, e.g., whether an institution was large or small, or whether some 335

TABLE 14.—DISTRIBUTION OF RANK WITHIN INSTITUTION FOR FOUR ANESTHETICS (ETHER OMITTED) IN V, INSTITUTIONS—RANKS APPLIED TO MORTALITY RATIOS« INDIRECTLY STANDARDIZED FOR AGE, OPERATION, AND PHYSICAL STATUS USING 15 AGGREGATION STRATA (Middle-death-rate operations) Rank» H N-B C 0 Average 1 11 12 5 6 8.5 2 12 9 6 7 8.5 3 7 8 12 7 8.5 4 4 5 11 14 8.5 Rank sum 72 74 97 97 85 Xj = 10.2 P= 0.017 •Rank 1 given to the smallest mortality ratio, 4 to the largest. TABLE 15.— PAIRWISE ANESTHETIC COMPARISONS* FOR MIDDLE-DEATH-RATE OPERATIONS IN INSTITUTIONS IN WHICH BOTH AGENTS IN PAIR WERE USED IN AT LEAST 10 % OH 15 * OF ALL PROCEDURES'* 10* N-B C E 0 H N-B C 17/29 4/22 7/19 5/10 7/23 7/20 11/18 1/6 E 6/11 4/6 15* N-B C E 0 H N-B C 9/20 3/U 3A 4/4 4/13 6/10 7/9 0/2 E 4/12 1/1 • Comparisons based on age-operation-physical status indirectly standardized mortality ratios. **Entries are shown as fractions. The denominator is the number of institutions having 10 (or 15) percent dual use; the numerator is the number of institutions in which ISMR for the row anesthetic exceeded that for the column anes- thetic. Thus, the second entry in the first row of the upper table shows that in four of 22 in- stitutions halothane had a higher ratio than cyclopropane and in 18 cyclopropane had a higher ratio than halothane. agent was scarcely used at all in a particular institution. So it would be good to look at these agent contrasts within institution after multiple adjustments, for a second time. Table 15 offers such a second look. From Table EE-22 of Chapter IV-2, we can identify the in- stitutions inwhichany agent was used "frequently," however we may choose to define that word. We have used two definitions: (1) we have taken an anesthetic agent to be used "frequently" in an in- stitution if at least 10 percent of all the surgical procedures in that institution involved that agent; and (2) we have used 15 percent in a similar way. From Table EE -22, we can now identify for each pair of anesthetic agents the institutions in which both members of the pair were used frequently (10 or 15 percent). In Table 15, the results appear. This table has two parts: the upper panel relates to institutions in which there was dual usage of at least 10 percent, and the lower panel to institu- tions in which there was dual usage of at least 15 percent. Each entry is in the form of a frac- tion; the denominator shows how many institu- tions satisfy the usage requirement, and the numerator shows in how many of those the anes- thetic for the row had a higher mortality ratio than did the anesthetic for the column. Thus, the second entry in the first row of the upper panel shows that in four of 22 institutions halothane had a higher ratio than cyclopropane; in the re- maining 18, cyclopropane had a higher ratio than halothane. The table reinforces no strong con- clusions, except for a superiority of halothane over cyclopropane (the significance level in the upper panel is 0.005, and in the lower panel is 0.06).* We can also see that the data are consistent with a possible lower rate for halothane in com- parison with Other, and of nitrous oxide-barbi- turate in comparison with cyclopropane, although neither of these contrasts is statistically signif- icant. Finally, it is worth observing that ether, on which there is the least information, provides no strong indication of difference from any of the other agents. *The halothane-cyclopropane comparison Just described was looked at in a similar way by F. Mosteller. He defined fre- quent usage in terms of 10 percent dual usage in the middle- death-rate operations (rather than all operations); in place of weighting the deaths and death rates in the various strata by indirect standardization, he has coarsely accumulated the cases in the strata, beginning with the strata with the highest rates, the "top" strata. The findings are, naturally, very similar. Mosteller's report follows: To get still another look at halothane and cyclopropane for the middle-death-rate operations, institution by institu- tion, I considered institutions using each of these two anes- thetics in at least 10 percent of the middle-death-rate oper- ations. Starting at the category with the highest death rate, I counted down until both anesthetics had at least 50 deaths in the institution. (This meant omitting some institutions that had fewer than 50 deaths.) Then death rates were com- pared. If possible, another 50 deaths were considered, and so on. For details, see the table (on the next page). Finally, there was a residual category. In 11 institutions such a com- parison could be made. Halothane had the lower rate in 10 of the 11 top categories. Halothane had the lower rate in both of the second categories, and cyclopropane the lower in the single third category. In the residual categories, halothane 'iad the lower rate in seven of 11 comparisons. in the residual categories the differences are usually tiny, for these are low-death-rate categories. This is a summary of the com- parisons: Lower death rate H 10 Top categories Second categories Third category Residual categories 2 0 7. 336

DEATH-RATE ANALYSIS SUMMARIZED IN FOOTNOTE: COMPARISONS, BASED ON ORA STRATA, OF HALOTHANE AND CYCLOPROPANE TOR FREQUENTLY USING INSTITUTIONS FOR MIDDLE-DEATH-RATE OPERATIONS Insti- tution Cate- gory Halo thane Cyclopropane D EE D/EE D EE D/EE Lower 2 Top 54 100 54.0 78 131 59.5 H 2nd 89 847 10.5 52 415 12.5 H Res. ia 1508 1.2 3 439 0.7 C 5 Top 63 190 33.2 61 136 44.9 H 2nd 51 436 11.7 72 436 16.5 H 3rd 54 953 5.7 60 1415 4.2 C Res. 21 2476 0.8 26 5771 0.5 C 7 Top 53 407 13.0 56 271 20.7 H Res. 25 1377 1.8 8 1378 0.6 C 10 Top 64 584 11.0 141 609 23.2 H Res. 71 5944 1.2 39 1950 2.0 H 12 Top 53 524 10.1 80 733 10.9 H Res. 46 4354 1.1 30 3244 0.9 C 21 Top 68 629 10.8 70 456 15.4 H Res. 61 10734 0.6 41 4513 0.9 H 22 Top 82 338 24.3 52 312 16.7 C Res. 71 7983 0.9 37 3198 1.2 H 24 Top 61 282 21.6 88 205 42.9 H Res. 22 1286 1.7 21 258 8.1 H 25 Top 59 349 16.9 82 350 23.4 H Res. 26 1366 1.9 24 653 3.7 H 26 Top 67 965 6.9 50 96 52.1 H Res. 16 8981 0.2 4 901 0.4 H 33 Top 66 768 8.6 60 467 12.8 H Res. 44 3235 1.4 16 851 1.9 H The foregoing analysis has the strong point that it compares agents only in hospitals in which they were much used. At the same time, it ignores a considerable portion of the data. We now turn to two approaches that utilize all the information more fully. One takes special account of the re- liability of the agent contrasts, and the other takes special account of the size of those con- trasts. We compare the results for the various agents in the 34 hospitals shown in Table 12. Both approaches use a statistic of the form shown be- low: 34 2 sgn(Xal- This expression is the sum of 34 terms, one for each hospital. The expression, sgn(Xal - Xaj), denotes the sign of the difference between tne logarithms of the indirectly standardized mortality ratios for the two agents, a and 3. The expres- sion, r{(a,0), represents the "rank" in institution i for the agent pair a,f}. This rank is an integer from 1 to 34, assigned as described below. For the first approach, rank 1 is assigned to the institution for which the variance of the dif- ference between the logarithms is greatest, and rank 34 to the institution for which it is least. This method of analysis provides a statistic that gives the greatest weight to the differences that are best determined, in the sense of having small variance. The distribution of the statistic, under the hypothesis of no agent differences, is the same as the null distribution of Wilcoxon's signed rank test. There are a few institutions for which the logarithm of the indirectly standardized mor- tality ratio is undefined, either because there were no deaths, or because there were no esti- mated exposed, or both. In such cases, the range of the sum and the range of the ranks are reduced to the number of institutions for which the dif- ferences are defined. The bottom panel of Table 16 shows how many institutions were involved in the various agent comparisons. The top panel of that table shows the value of the statistic To 8 divided by its null standard error. These num- bers are referable to tables of the normal dis- tribution for the assessment of statistical sig- nificance. A positive number means that the indirectly standardized mortality ratio for the agent shown in the row was greater than that for the agent shown in the column. Values exceeding 1.96, the two-sided 5 percent point of the normal distribution, are starred. The second approach leads to the second panel of Table 16. The approach is wholly analo- gous to the first, except that the ranks have been assigned, not in terms of precision of the con- trasts, but in terms of their magnitudes; in par- ticular, the quantities |log ISMRa - log (i = 1, ... ,34) were ranked, with the largest rank being assigned to the greatest such quantity. This method of analysis gives greatest weight to the greatest dif- ferences in logarithms, which means the greatest ratio of indirectly standardized mortality ratios. Study of Table 16 shows, in both panels, lower rates for halothane than for cyclopropane, and lower rates for halothane than for Other. Ether, in the lower panel, gives indication of having higher rates than either halothane or ni- trous oxide -barbiturate, but the corresponding indications in the upper panel are weak, and we cannot conclude that the comparisons are to be trusted. The remaining strong indications in the table are the comparisons of nitrous oxide- barbiturate with cyclopropane and with Other. In the lower panel, both of these are large, and in the upper panel, rather strongly corroborative. In an effort to settle these two comparisons, a more powerful analysis was used, which takes account of both the variability and the size of an agent 337

TABLE 16.—ANESTHETIC COMPARISONS (MIDDLE-DEATH- RATE OPERATIONS) AFTER INDIRECT STANDARDIZATION (FOR AGE, OPERATION, AND PHYSICAL STATUS) OB- TAINED BY APPLYING SIGNED RANK TESTS TO DATA OF TABLE 121 Ranks based on variability of contrast2 N-B C E 0 H 1.05 -3.10* -1.31 -2.40* N-B -1.74 -0.98 -1.72 C -0.27 0.35 E 1.01 Ranks based on size of contrast3 N-B C E 0 H 0.28 -2.85* -2.39* -2.83* N-B -2.21* -2.07* -2.42* C -0.48 -0.04 E 1.26 Number of institutions entering into comparison N-B C E 0 H 34 33 29 34 N-B 33 29 34 C 28 33 E 29 1 A positive number means that the agent shown in the row had a higher ISMR than the agent shown in the column; thus the first entry in the second row of the upper table, -1.74, means that the ISMR for nitrous oxide-barbiturate was lower than that for cyclopropane, and the significance of the difference can be ascertained by referring 1.74 to a normal table. 2 "Variability" means variance of (log ISMRB - log ISMRo), where a and 9 denote the agent pair. Largest rank given to smallest variance. 3 "Size of contrast" means (log ISMRa - log ISMRgl. Largest rank given to largest such absolute difference. contrast within a hospital. The methods are ex- plained in Appendix 3. The results are that the nitrous oxide-barbiturate—cyclopropane com- parison yields a normal deviate of -2.75, and the nitrous oxide-barbiturate—Other comparison one of -2.86. Both are highly significant. In the following section, there is an account of how six frequent operations came to be studied in all 34 hospitals. These six operations had death rates ranging from about one-fourth of 1 percent to about 5 percent. Standardizing for the number of exposures to the six operations en- ables the construction of logarithms of the in- directly standardized mortality ratios for each agent in each hospital. (These data are not dis- played.) The numbers of defined agent contrasts for these six operations are displayed in the bottom panel of Table 17. The upper two panels again show the values of the signed rank statis- tics divided by their null standard errors, and can be interpreted as before. The starred values indicate reliably lower death rates associated with halothane in comparison with both cyclopro- pane and Other, and with nitrous oxide-barbiturate in comparison with Other. In addition, there is some suggestion that halothane has a lower death rate than nitrous oxide-barbiturate for these six operations. The two analyses corresponding to Tables 16 and 17 are not statistically independent. Table 16 is based on all middle-death-rate operations after standardization on age-operation-physical status aggregation. It is thus a rather highly de- rived and heavily adjusted set of statistics that affords the base. The figures in Table 17 have been subjected to no adjustment whatever (other than standardization for frequency for these six N-B C E 0 H -1.80 -1.98* -0.86 -3.40* N-B -1.79 0.26 -2.07* C 1.29 -0.09 E -1.83 N-B C E 0 H -1.98* -2.98* -1.08 -3.81* N-B -1.63 -0.31 -2.58* C 1.45 -0.01 E -1.68 N-B C E 0 H 28 31 20 32 N-B 25 17 26 C 19 30 E 20 TABLE 17.—AGENT COMPARISONS1 (FOR SIX SELECTED OPERATIONS2) OBTAINED BY APPLYING SIGNED RANK TESTS TO STANDARDIZED MORTALITY RATIOS ANALOGOUS TO P* IN TABLE 24 Ranks based on variability of contrast3 Ranks based on size of contrast* Number of institutions entering into comparison 1 A positive number means that the agent for the row had a higher ISMR than the agent for the -column. 2 The description of these six operations and the direct standardization are explained later in this chapter. 3 "Variability" means variance of (log ISMRa - log ISMRp), where a and 0 denote the agent pair, Largest rank given to smallest variance. * "Size of contrast" means |log ISMRB - log Largest rank given to largest such absolute difference. 338

TABLE 18.--PAIRWISE ANESTHETIC COMPARISONS* FOR FOUR HIGH-DEATH-RATE OPERATIONS IN INSTITUTIONS IN WHICH BOTH AGENTS IN PAIR WERE USED IN AT LEAST 10% OR 15% OF ALL PROCEDURES** 10* 15* N-B C E 0 N-B C E 0 H 13/18 1/1 3/4 10/15 H 9/11 0/0 3/4 6/9 Operation 12 JJ~B 0 0/0 2/3 0/1 4/10 0/2 N-B C 0/0 2/3 0/0 2/4 0/1 E 0/2 E 0/1 H 4/12 2/7 0/0 4/10 H 2/7 1/4 0/0 2/5 Operation 33 !J~B o 3/5 0/0 0/0 5/6 3/5 N-B C 2/3 0/0 0/0 3/4 0/0 E 0/0 E 0/0 H 8/22 6/19 6/9 9/19 H 7/14 4/13 2/4 5/12 Operation 44 JJ-B u 5/13 6/7 4/5 4/13 8/16 N-B C 5/9 2/3 2/6 5/9 E 3/5 E V1 V2 H 12/21 3/18 4/6 7/17 H 8/13 3/13 1/3 4/6 Operation 48 J!-B 4/14 V7 2/5 8/14 7/16 N-B C 2/7 1/2 V1 2/5 4/9 w 2/4 E 2/2 E "Comparisons based on age-physical status indirectly standardized mortality ratios. **Entries are shown as fractions. See footnote 2 to Table 15. operations) and are very directly interpretable, although liable to distortions from imbalances in risk and age; the data represent about one-sixth of all the cases in the Study. There is a reason- able concordance between the conclusions from the two analyses. Let us summarize our findings for the three types of analysis we have reported: (1) that halothane was associated with lower death rates than cyclopropane is sepa- rately demonstrated in the case of the frequent-usage analyses, in the case of the signed rank tests on indirectly stand- ardized rates for the middle-death-rate operations, and on the panel of six se- lected operations; (2) that halothane was associated with lower rates than Other is separately demon- strated in the case of the signed rank analyses and on the panel of six opera- tions; the frequent-usage analysis is con- sistent with this, but only weakly con- firmatory; (3) that nitrous oxide-barbiturate was asso- ciated with lower rates than cyclopropane is separately demonstrated by the signed rank tests (after supplementary analysis) on middle-death-rate operations after adjustment, and weakly confirmed both by the frequent-usage analysis and on the panel of six operations; and (4) that nitrous oxide-barbiturate was asso- ciated with lower rates than Other is separately demonstrated by the signed rank tests (after supplementary analysis) and on the panel of six operations; there is no confirmation of this finding from the frequent-usage analysis, although there is no incompatibility either. The above conclusions lead us, with consid- erable confidence, to regard halothane as having had lower rates than cyclopropane and lower rates than Other and, with only slightly less con- fidence, to regard nitrous oxide-barbiturate as having had lower rates than cyclopropane and lower rates than Other. The only "positive" indications not summa- rized in the above statement are suggestions that ether may have had higher rates than either halothane or nitrous oxide-barbiturate* in the middle-death-rate operations after adjustment for age, operation, and physical status. We note these indications, but are unprepared to assert «Reference to Appendix 3 shows that the analysis of that sec- tion (which affirms the halothane-cyclopropane, halothane- Other, nitrous oxide-barbiturate—cyclopropane, and nitrous oxide-barbiturate—Other contrasts) yields the following contrasts; /ISMRH\ log — £)= -0.319 with s.e. 0.167 -0.319 = -1.91 /ISMR.,R\ log Ml = -0.380 with s.e. 0.177 \ISMRj. j -0.380 339

confidence in them, inasmuch as the pattern of usage of ether is both sparse and erratic. In concluding this section, we take a glimpse at the results for the four high-death-rate oper- ations. The findings are essentially negative, but the data are shown in order to complete the rec- ord. Table 18 is constructed in the same way as Table 15. It shows for each of the high-death- rate operations the number of institutions with 10 and 15 percent dual usage, with the number of institutions in which each agent had a mortality ratio exceeding that of each other agent. The most striking feature of this table is the paucity of large denominators; this shows the rarity of institutions whose experience with these oper- ations was sufficiently extensive to permit com- puting indirectly standardized mortality ratios for both anesthetics. In the entire table, only one indication appears strong, the halothane-cyclo- propane contrast in operation 48 (large bowel). Because there have been so many opportunities to "find something," i.e., so many comparisons to examine, we refrain from taking this result as conclusive, but it poses the interesting ques- tion of whether halothane is actually superior to cyclopropane for this operation. Table 19 sum- marizes the results for operations 12, 33, 44, and 48 in the same way that Table 16 does for the middle-death-rate operations. In the entire table, three comparisons exceed 2: halothane—nitrous oxide-barbiturate for operation 12, and halothane- cyclopropane and cyclopropane-ether for oper- ation 44, where ranks are assigned by the mag- nitudes of the differences of the logarithms of the indirectly standardized mortality ratios. Each gets some corroboration from the other signed rank analysis. COMPARISON OF INSTITUTIONAL DEATH RATES AFTER STANDARDIZATION BY AGE-RISK-OPERATION AGGREGATION AND AFTER DIRECT COMPARISON FOR SD£ SELECTED COMMON SURGICAL PROCEDURES Variation of death rates among institutions before and after standardization (for one variable at a time) was discussed in Chapter IV-2. The observed death rate among institutions varied from a low of 0.00268 to a high of 0.06405. The ratio of these extremes is about 1:25. Standard- ization for operation did the most to equalize the indirectly standardized mortality ratios. The ef- fect of standardization by that variable was to reduce the ratio of largest to smallest indirect mortality ratios from about 25:1 to about 9:1. In this section, we discuss the death-rate ex- perience in the various institutions after stand- ardization for several variables simultaneously, reaching over various groups of operations, to a total of seven analyses. We have applied cell aggregation to the cross-classification of age, risk (physical status), and operation as those variables are defined in Appendix 1 to Chapter IV-5. The seven analyses are based on the middle-death-rate operations (five classes), the high-death-rate operations (four classes), the middle- and high-death-rate operations (nine classes), and the four high-death-rate operations taken separately (with operation disappearing as a classification variable). In the seven analyses, age and risk always are used with seven and four classes, respectively. The cross-classification, whatever it was, was swept in death-rate order into 15 categories equinumerous in terms of deaths-plus-randoms. These strata were then used for the construction of indirectly stand- ardized mortality ratios for each agent within each hospital (the reference population being "all agents" in the particular hospital). Table 12 dis- plays these standardized mortality ratios for the middle-death-rate operations. Interest attaches to two kinds of features in that table. From the first five columns it is possible to make a de- tailed study, institution by institution, of dif- ferences in death rates by anesthetics, and so to assess their sampling reliability. This investiga- tion was done in the preceding section. The sec- ond kind of investigation is the study of the in- stitutional rates themselves (as opposed to the comparisons among anesthetics)—the topic of this section. The largest mortality ratio in Table 12 is 1.77 for institution 31, and the smallest is 0.574 for institution 23. The ratio of these two num- bers is about 3.1.* The interpretation of these figures is that, after we have taken account of the variables operation, physical status, and age, there remains for the middle-death-rate oper- ations a three-fold variation in institutional death rates for which no further explanation is at hand, and for which explanation would be de- sirable. It is notable that this institutional vari- ation is definitely smaller than was obtained by adjustment for any single variable.** This is partly because only the middle-death-rate oper- ations are involved here, whereas standardiza- tion in Chapter IV-2 was applied to the data for all operations, and partly because the efficacy of multiple adjustment is involved. In the follow- ing analyses, we have not adjusted for anesthetics, because the order of magnitude of their differ- ences is very much smaller than the observed differences between institutions. «Because some of the institutions are much smaller than others, it is wise to inquire how strongly indications referring to the entire set of institutions may depend on possible peculiarities of the smallest institutions. A way to do this is to look at in- stitutional variation ignoring the smallest institutions. For this reason we have starred, in Table 12, all those (seven) with fewer than 200 deaths. For the unstarred institutions, the extreme mortality ratios are 1.60 and 0.683, which still differ by a factor of 2.3. So we conclude that the variation is not an artifact of small-institution fluctuation. **If the seven small institutions are omitted, it is still true that standardization for no single variable reduces the insti- tutional variability as substantially as does the multiple ad- justment. 340

TABLE 19.—ANESTHETIC COMPARISONS FOR FOUR Him-DEATH-RATE OPERATIONS AFTER ADJUSTMENT FOR AGE AND PHYSICAL STATUS OBTAINED BY APPLYING SIGNED RANK TESTS TO INDIRECT STANDARDIZED MORTALITY RATIOS WITHIN INSTITUTIONS* Ranks based on variability of contrast** Operation 12 Operation 33 N-B C E 0 N-B C 0 H 1.55 -0.53 1.96 -0.12 H -0.59 -0.25 0.03 N-B -1.34 -0.37 -1.59 N-B -0.26 0.77 C -1.00 0.45 C 0.41 E Operation 44 -1.36 Operation 48 N-B C E 0 N-B C E 0 H -0.44 -1.77 0.15 -0.05 H 1.68 -1.06 0.52 -0.04 N-B -1.65 0.41 -1.49 N-B -0.23 -0.28 -0.04 C 0.91 0.44 C 0.23 -0.70 E 0.11 E -0.93 Ranks based on size of contrast*** Operation 12 Operation 33 N-B C E 0 N-B C 0 H 2.21 0 1.68 0.12 H -0.87 -1.07 -0.80 N-B -1.34 0 -1.36 N-B 0.36 0.65 C -1.00 -0.45 C 0.30 E Operation 44 -1.57 Operation 48 N-B C E 0 N-B C E 0 H -0.80 -2.12 0.81 -1.27 H 1.08 -1.34 0.25 -0.86 N-B -0.12 1.73 -0.44 N-B -1.45 -0.91 -0.78 C 2.04 0.47 C -0.06 -0.06 E -1.67 E -0.52 Number of institutions entering into comparison Operation 12 Operation 33 N-B C E 0 N-B C 0 H 21 3 8 19 H 13 10 13 N-B 2 4 15 N-B 10 9 C 1 2 C 9 E Operation 44 6 Operation 48 N-B C E 0 N-B C E 0 H 22 26 18 27 H 20 21 13 20 N-B 19 14 19 N-B 21 15 20 C 15 25 C 15 23 E 18 E 16 * A positive number means that the agent shown in the row had a higher ISMR than the agent shown in the column; the significance of the difference can be ascertained by referring to a normal table. ** "Variability" means variance of (log ISMRa- log ISMR^), where a and P denote the agent pair. Largest rank given to smallest variance. *** "Size of contrast" means jlog ISMRa- log ISMR^j . Largest rank given to largest such absolute difference. 334-553 O-69—23 341

TABLE 20.—INDIRECT STANDARDIZED MORTALITY RATIOS (BASED ON AGE-RISK-OPERATION AGGREGATION STRATA) AND RELATED STATISTICS FOR 34 INSTITUTIONS (MIDDLE- DEATH-RATE OPERATIONS ONLY) lust. Indirect SMR Deaths D Standard deaths D D-D ir2 = Var a D-D A (D-D) a 1 0.7286 185 254 -68a 1109.36 33.31 -2.041 2 1.3221 446 337 109 991.59 31.49 3.255* 3 0.8619 122 142 -19 398.03 19.95 -0.953 4 1.1324 452 399 53 1943.11 44.08 1.202 5 1.5633 724 463 261 2549.52 50.49 5.169* 6 0.8952 219 245 -25 888.91 29.81 -0.839 7 1.5979 318 199 119 705.83 26.57 4.478* 8 0.6834 632 925 -292 7023.43 83.81 -3.484 9 0.7582 519 685 -165 5129.44 71.62 -2.304 10 1.2738 501 393 108 1619.86 40.25 2.683* 11 0.7121 296 416 -119 1546.03 39.32 -3.026 12 1.1641 297 255 42 864.68 29.41 1.478 13 1.4877 150 101 49 196.22 14.01 3.498* *M 0.5771 90 156 -65 472.83 21.74 -2.990 15 1.0351 367 355 12 1916.01 43.77 0.274 16 1.3380 296 221 75 688.52 26.24 2.858* *17 0.8583 106 124 -17 284.46 16.87 -1.008 18 0.8384 191 228 -36 493.47 22.21 -1.621 19 1.0609 231 218 13 623.94 24.98 0.520 20 1.0213 444 435 9 2067.50 45.47 0.198 21 0.7695 459 596 -136 4526.47 67.28 -2.021 22 1.1257 384 341 43 1552.70 39.40 1.091 *23 0.5741 48 84 -35 190.09 13.79 -2.538 24 1.4329 441 308 133 1065.23 32.64 4.075* 25 1.3756 398 289 109 818.95 28.62 3.808* 26 0.9426 243 258 -14 1339.71 36.60 -0.382 27 0.9400 142 151 -8 510.21 22.59 -0.354 *28 0.7056 13 18 -4 24.44 4.94 -0.810 *29 0.8416 65 77 -11 127.06 11.27 -0.976 30 0.7379 90 122 -31 178.69 13.37 -2.319 *31 1.7667 104 59 45 154.40 12.43 3.620* 32 0.7576 139 183 -43 490.86 22.16 -1.940 33 0.7928 379 478 -98 2902.38 53.87 -1.819 *34 1.2088 132 109 23 257.23 16.04 1.434 Negative numbers are smaller in absolute value by 1 than they should be because of the computer's treatment of rounding. These are not important because they are not used in the line of argument. Generally, the institutional variation appears to be least for that part of the data comprising only the middle-death-rate operations. The largest-to-smallest ratios for the other analyses are: high-death-rate operations, 6.8; middle- and high-death-rate operations, 3.3; operation 12 (craniotomy), 20.5; operation 33 (heart with pump), 10.2; operation 44 (exploratory laparot- omy, etc.), 12.4; and operation 48 (large bowel), 16.5. Consider now Table 20. For the middle- death-rate operations, it shows for each institu- tion the indirectly standardized mortality ratio (agreeing with that in Table 12), the number of actual deaths, and the figure labeled "standard deaths," calculated by applying the whole-Study death rates for the 15 age-risk-operation strata to the number of estimated exposed in those strata for the particular institution. The next column is the difference between the actual deaths and the "standard deaths." It is a positive number where there were more actual deaths. From that column, we are able to get some idea of how many deaths were involved in institutional variation. To assess the statistical significance of entries in this column, we need an estimate of the standard error of such a statistic. The next two columns, labeled "a2" and "a," present the square of the standard error of D - D and the standard error itself. The last column is computed as D - D divided by its standard error. Where this num- ber Is large, we may suspect a real institutional effect;* if it exceeds 2, it is starred. s Table 21 displays the values of (D - D)/CT for all seven analyses, again with values exceed- ing 2 starred. Before discussing this summary «Appendix 1 presents a derivation of the estimate of a. The discussion refers to the estimation of a for Table 25, but the argument is the same for both cases. 342

TABLE 21.--STAIOARDIZED DIFFERENCES BETWEEN DEATHS AND "STJ»«D" DEATHS BASED ON AGE-RISK-OPERATION STRATA FOR SEVEN PARTS OF THE DATA D-fl Middle- death-rate operations iligh-deal.il- rate operations Middle- and high-death- rate operations Inst. Operation 12 Operation 33 Operation 44 Operation 48 1 -2.041 0.712 -1.555 0.962 1.000 0.661 -0.500 2 3.255* -0.692 2.160* 1.103 -0.244 -0.440 -0.598 3 -0.953 -0.335 -0.844 3.125* -0.467 -0.331 -1.636 4 1.202 -0.480 0.425 -1.128 -0.565 1.212 0.995 5 5.169* 2.614* 5.572* 1.070 0.266 1.803 3.505* 6 -0.839 0.317 -0.310 1.240 0 -0.268 -0. 102 7 4.478* 1.848 4.404* -0.612 4.245* 2.166* 0.830 8 -3.484 -1.166 -3.413 -0.486 -0.043 -1.855 -0.359 9 -2.304 -0.162 -1.683 1.585 0.102 -0.811 0.623 10 2.683* -3.268 -1.165 -2.144 -2.647 -0.501 -0.699 11 -3.026 -0.543 -2.651 0.251 0.400 -0.680 -0.927 12 1.478 2.493* 2.536* 2.742* 0.618 0.965 0.329 13 3.498* 4.591* 5.866* 2.823* 0.265 3.393* 3.088* *14 -2.990 -1.398 -3.557 1.305 -0.631 -2.002 1.200 15 0.274 1.045 1.234 0.504 0.592 0.691 2.629* 16 2.858* -0.814 1.514 -1.942 -0.129 -0.268 1.587 *17 -1.008 -0.369 -1.257 -1. 110 0.640 0.446 -0.448 18 -1.621 -1. 173 -1.904 -1.826 0.796 -0.568 -0.444 19 0.520 1.374 1.379 0.533 2.588* -0.208 -1.138 20 0.198 -0.785 -0.298 -0.888 6.851* -0.725 -1.650 21 -2.021 -1.935 -2.621 -0.939 -1.851 -0.418 -0.468 22 1.091 2.822* 1.865 3.108* 0.229 5.196* 0.450 *23 -2.538 0.362 -1.675 0.227 -1.044 1.018 0.647 24 4.075* 4.461* 6.187* 0.061 4.223* 2.243* 2.706* 25 3.808* 1.359 3.604* 2.161* 0.513 0 0.660 26 -0.382 1.023 0.294 0 1.671 3.203* -0.706 27 -0.354 1.524 0.491 1.253 3.000* 0.391 0.126 *28 -0.810 -1.034 -1.571 -0.938 0 0 0 *29 -0.976 1.550 0.560 0.571 1.747 1.100 0 30 -2.319 -2.580 -3.351 -2.516 -1.804 0.175 0.926 *31 3.620* 3.386* 5.281* 2.519* 0.610 1.546 2.187*' 32 -1.940 -1.223 -1.903 -0.758 -1.552 0.686 -0.273 33 -1.819 -1.669 -2.340 4.088* -2.677 -1.879 -0.907 34 1.434 2.100* 2.099* 0 0 1.781 1.073 table further, we turn to a second approach to studying institutional variation in death rates. The analyses of this section thus far have been based on rather complicated statistical procedures. To examine more directly the in- dicated institutional fluctuations, six operations were chosen* that generally defined particular procedures, rather than collections of proce- dures. The chosen operation codes were: 40, gall bladder; 45, gastric resection; 65, hysterec- tomy; 73, cystoscopy; 86, open reduction of the femur; and 99, thoracic or lumbar laminectomy. As a set, these operations involved 1844 deaths (about 11 percent of the 16,840 deaths in the entire Study) and 141,914 estimated exposed (about 16 percent of the 856,000 in the entire Study). Table 22 shows the estimated exposed for these six operations, by institution; Table 23, the deaths similarly classified; and Table 24, the death rates. The last two columns of Table 24 are labeled "P" and "P*"; the former is the John P. Bunker and William H. Forrest. crude death rate (calculated as D/(D+EE)) for the six operations, and the latter is the death rate after standardization for frequency of the six operations in the entire Study. This standardi- zation evens out the effects on the death rates arising frorn the different frequencies of the more serious operations in individual institu- tions. Table 24 shows that the standardized death rate P* for these six operations has a large range of variation, reaching below 0.007 in in- stitutions 8, 14, 23, 27, and 28, and above 0.0500 in institution 25. These standardized death rates for these six operations are plotted in Fig. 1 against the logarithm of hospital size, as meas- ured in EE for all operations. Slightly detached from the graph, at the left, is the marginal dis- tribution of death rates, and below the graph, the marginal distribution of EE (logarithmically scaled). Study of the death rates, on the left-hand scale, shows strikingly that they are (in the main) rather uniformly spread between 0.0047 and 0.0181, with rates for eight remaining institutions appearing to be "out of the distribution," ranging 343

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TABLE 25. —COMPARISON OF OBSERVED AND FITTED DEATHS FOR SIX OPERATIONS IN EIGHT INSTITUTIONS IN WHICH THE SIX-OPERA- TION DEATH-RATE EXCEEDED 0.02 Inst. ^Actual deaths i-Standard deaths D-D Var f) '(D-D) D-D *D-fi 2 102 32.2 69.8 10.6 10.6 6.58 5 176 69.1 106.9 73.7 15.8 6.78 7 92 32.2 69.8 14.6 10.3 6.79 10 79 54.6 24.4 54.2 11.5 2.12 24 79 36.2 42.8 18.1 9.9 4.32 25 139 46.5 92.5 16.5 12.5 7.40 30 8 2.35 5.65 0.41 2.9 1.95 33 72 43.6 28.4 33.6 10.3 2.76 Total 747 316.8 from 0.02 to 0.05. These eight are institutions 2, 5, 1, 10, 24, 25, 30, and 33. Table 25 shows for the eight institutions information like that in Table 20. We are now in a position to tie together the diverse information about the statistical signifi- cance of the larger discrepancies between observed and "standard" deaths. We have iden- tified institutions with large differences in seven different ways: each of six analyses* based on age-risk-operation ^strata yielded some institu- tions for which (D-D)/ir exceeded 2; the panel of six operations (four of which belong to the middle-death-rate group and two of which belong to the low-death-rate group) identified eight in- stitutions with large-looking death rates for that panel of operations. We can now, by turning to Tables 21 and 25, list all institutions among the 34 that have ever been indicated as "high," and display the values of (D-D)/a leading to the in- dication. Table 26 summarizes these findings. The first column shows the institutions that have come to notice in any of the analyses. The next three columns are the most important. The fourth is based on data independent of the second and third, which are themselves somewhat inter- dependent through sharing four operations, but otherwise are quite different. The last four columns supplement the fourth for those readers who may prefer not to combine the four high- death-rate operations. The panel of six oper- ations involves no adjustments for age or risk. The reason for displaying these figures is that we have some diffidence about accepting a large value of (D-D)/u as indicating statistically sig- nificant institutional effects. But if the same in- stitutions are indicated in different and independ- ent analyses, then considerable strength is gained. *We Ignore the middle- and high-death-rate analysis because of its overlap with other analyses. Table 26 shows that 20 of the 34 institutions have in some place exhibited a high value of (D-D)/a. Of these, eight have appeared but once; their institution identification numbers are cir- cled. We dismiss these as probably signifying nothing of importance. More data are represented in the middle-death-rate group than in either the panel of six operations or the high-death-rate group. Of the nine institutions where (D-D)/ci exceeds 2 for the middle-death-rate operations, only one (institution 16) was dismissed above for noncorroboration. Six of the remaining eight institutions are also members of the eight out- liers for the panel of six operations. Finally, in four cases (five if we count institution 7), there is a match with identification by the high-death- rate operations analysis as an outlier. In all, eight institutions are identified by the middle- death-rate operations and also by the high-death- rate operations or the panel of six, or both: in- stitutions 2, 5, 7, 10, 13, 24, 25, and 31 (these are starred in Table 26). We conclude that the con- sistent appearance of these institutions as out- liers demonstrates heterogeneity of indirectly standardized mortality ratios to an extent calling for some explanation other than sampling error. Several questions present themselves. Given that some large part of the institutional variation is real, of what practical consequence is it? Did dozens, hundreds, or thousands of deaths occur in the few hospitals with notably high death rates? After gauging the practical size of the apparent institutional effects, what conclusions, if any, are warranted about appropriate steps to take? We turn now to an investigation of these -questions. The sum of the positive values of D-D for the middle- and high-death-rate operations to- gether was about 1750, as was the sum of the negative values. (The number expected by chance 348

TABLE 26.— SUMMARY OF EXTREME POSITIVE VALUE OF D-D FROM TABLES 21 AND 25 Panel of six operations Middle- death-rate operations High-death- rate operations Operation 12 Operation 33 Operation 44 Operation 48 Inst. *2 6.58 3.255 © 3.125 *5 6.78 5.169 2.614 3.505 *7 6.79 4.478 4.245 2.166 *10 2.12 2.683 (-3.268) (-2.1M) (-2.647) 12 2.493 2.742 *13 3.498 4.591 2.823 3.393 3.088 © 2.629 © 2.858 © 2.588 © 6.851 22 2.822 3.108 5.196 *24 4.32 4.075 4.461 4.223 2.243 2.706 *25 7.40 3.808 2.161 © 3.203 © 3.000 30 1.95 (-2.319) (-2.580) (-2.516) *31 3.620 3.386 2.519 2.187 33 2.76 4.088 (-2.677) © 2.100 is in the range 435 to 525.) Involved here, in each direction, is about one-tenth of the number of deaths in the whole Study. Of course, a total of 1750 deaths is enormous in comparison with the number of deaths attributable to hepatic necrosis, but may or may not be regarded as importantly large when related to a total of 490,000 surgical procedures in the middle- and high-death-rate groups. At this point, we are forced to conclude that the differences between observed deaths and estimates of what they might be are, in many in- stitutions, real and importantly large. But these differences are very difficult to interpret because "standard deaths," our estimate of "what they might be," cannot receive our unreserved con- fidence. These estimates are based on statistical adjustments, taking some account of operation, age, and physical status, but the adjustments may be incomplete and, more importantly, no account has been taken of other variables that might be very important. Among such possible important considerations are a differential willingness to undertake operations that look risky and a differ- ential selection of difficult cases in certain hos- pitals. We have not looked at medical deaths and do not know how to. In some hospitals with sur- gical death rates that look high in our data, there might be a degree of surgical enterprise resulting not only in many more surgical deaths, but also in an unseen substantial decrease in what would otherwise be recorded as mortality belonging to the medical wards. The fact is that we can point to unexplained differences, but we do not have the evidence to claim, where one institution is "high" and another is "low" in our data, that the latter is "better." That, of course, is one possibility, but other possibilities are that these two institu- tions would be found to be equal, or even in re- verse order, if comparisons on thoroughly equiv- alent surgical cases were to be made, or if net medical gain to the total population of patients— medical and surgical—were measured. We are able to summarize our findings on institutional differences in the following way: (1) There are numerous institutions in which the numbers of actual deaths differ from the numbers of estimated deaths by sev- eral times their standard errors, and these differences cannot be regarded as due to randomness in the data. 349

(2) For each institution where such a large difference occurs, there is posed a ques- tion: "What accounts for this difference?" (3) The presence of this question does not necessarily indicate that the particular institution has inferior surgical perform- ance. It is naturally important to under- stand better whether this is or is not the case for the outlying institutions. (4) The question of what steps, if any, should be taken in the further elucidation of these unexplained institutional peculiarities is difficult to answer. Some indications of how further statistical analysis may il- luminate the problem are given in Ap- pendix 2. Questions of administrative feasibility and related matters are dis- cussed in Chapter IV-8. 350

APPENDIX 1 TO CHAPTER IV-6 EXAMINING THE STANDARD ERRORS Lincoln E. Moses Stanford University Stanford. California Tables 20, 21, and 25 of Chapter IV-6 present values of Dj - DJ and of estimated standard errors for such quantities. The formula used for computation of the values in those tables appears below as Eq. 11. That formula is an approximation to the variance. Its derivation uses the assumption that the deaths for each operation in each institution, and the randoms for each operation in each institution, are mutually independent Poisson Vandom variables. The derivation also neglects certain terms in the correct expression for the variance. (The neglected terms are of opposite effect and will in some measure offset each other.) Finally, expected values of Poisson random variables are replaced by their observed values. This appendix presents the derivation, considers the sizes of the neglected terms, and exhibits them for the data of Table 25; it also presents evidence on the assumption that the deaths and randoms behave as Poisson random variables. The formulas are developed in terms of six strata, one corresponding to each of the special operations. In the case of the age- ope ration- risk strata used in Tables 20 and 21, there are 15 strata. DEVELOPMENT OF FORMULAS Define RIJ as the number of patients in the random sample from institution j who proved to be exposed to operation i. Similarly, define Dj, as the number of deaths from operation i observed in institution j. Let k, denote the "blowup factor" in institution j. In each case, in our example, j = 1.....34 and i = 1,...,6. Then we define: the estimated exposed for operation i EEjj = kjR(J, in institution j as the estimated death rate for operation i . 1J . over all institutions as p = - ' the observed deaths for institution j Dj = £ Dy , ,.i (for the six operations) as i * s\ the "estimated" deaths for operation i Djj = PjEEj, = PjkjRj, , and in institution j as the "estimated" deaths for all six opera- Dj = z Dj, . tions in institution j as i It is our aim to evaluate Var (Dj - D,). In the sequel we shall affix a superscript asterisk to the symbol for a random variable to denote its expectation. Thus, D*j means E(D{.). Var (Dj - Dj) = Var Dj - 2 cov (Dj,Dj) + Var Dj . (2) 351

The first term is equal (under the Poisson assumption) to D,*. We now develop an expression for the third term under the Poisson assumption. Var (Dj) = Var (r P^Rl}) = kj2 £ [ (pp2 Var (Rjj) + 2p*R* cov (pj,^) + R* 2 Var (pj)] (3) = k2 i [(P*)2RJS + 2p*R* covfopRjj) + R*2 Var (p.) ] . To proceed further, we need to evaluate Var (pj) and cov (p^Rj,). rDjj z DIJ Var fo) = Var = Var r ZEEj) i jjj J J Applying the usual formula for the variance of a ratio and observing that the vari- ance of a Poisson random variable equals its mean, we obtain: Now define Wj. = kJR*j , noting that i Wj, = 1, and (5) kJRf] 1 SWykj. (6) These enable us to rewrite the expression for Var (p ) as: Var (p) = (p*)2 J We must now evaluate cov (p , cov (pliRlJ) = cov - = y v R / I i / \ ^ 1 il ' kjR« / / V j / *>"*!-«• (8) Substituting Eqs. 7 and 8 into Eq. 3, we have: Var (ftp = k2 s (p*)2R* . afe^R* Wlj + (R* )2)2 - + -_ . (9) i L * J2^)2 f-1 + _i_ 1 1 . [> DjJ kjRjj J J 352

Now we replace expectations by their observed values and, using the definitions in Eqs. 1, obtain the estimated variance: Var (Dj) = 2 j kjpjDjj - 2p2k2 Now, because 2 Dy = 2 DJJ, we may write: (10) Dy = ij 2 DJJ and note 2 jj = 1 , J J which enables rewriting Eq. 10 as: Var (Dj ) = 2 Dy (kjPl - Zp^^ + £y + Kjp^l ] ). (11) The last three terms in the parentheses are small, compared with the first. kj is a number like 20 or 25 (and so is Kj, a weighted average of the k, , the weights depending on i); p. is a number like 0.01. The 34 values of ljj add to 1, and they are numbers like 0.03. The same is true of Wjj . In all, we conclude that omission of the three later terms should result in an error that is not serious. The estimate of Var (D, - £),) used in Table 26 was: Var (Dj - Dj) = Dj + kj 2 Pj2EEy , (12) which is obtained from Eq. 2 by neglecting the terms in Eq. 11 as just discussed, by estimating D? the variance of Dj by its observed value, D=, and by ignoring the covariance term in Eq. 2. We now study the actual magnitude of the covariance term. Recall that = 2 EEjjPl = 2 2DjJ By defining JJ J 2 J (note 2 Ujj = 1) and we may write: = 2 Djj , D]= Then cov (Dj , Dj ) = cov ( 2 Di] , 2 Uy D+) = 2 U* Var Djj . 353

This we estimate as: cOv(D],D])= S UyDy . (13) i A Because it is easily shown that V^ = f jj , we write this as: cov (Dj,Dj) = SlyDy . (14) This neglected term is comparable in size with what was neglected in Eq. 10; it has the coefficient -2 in the true expression for Var (Dj - D.). Entering into Eq. 2 the expression Dj for the first term, the covariance term fromEq. 14, and the value of Var (D. ) given in Eq. 11, we obtain the full expression for the estimated variance: ' £ar (Dj - DJ) = Dj - 2 Z lyDi] + 2 DjjtkjPj- apjkjWy + (. + Kj^) , (15) and, by writing 2 Dykjpj as k, £ p2EEjj , we may exhibit the expression above as two parts—the two-term approximation used in Tables 21 and 26, plus the neglected terms: Var (Dj - Dj) = JDj + kjS P^EE - {2 Z lyDy + 2 DiJ(2pikjWy - |jj - Kjpjly) j . (16) In Table 1 of this appendix, we display all the terms in Eq. 16 for the 34 institutions on the six-operation data. The third column displays the approximation; the last column, the complete expression. In general, the approximate expression slightly exceeds the complete sum, and we conclude that the approximation is good. THE POISSON ASSUMPTION The occurrence of a death in a particular hospital from a specified operation is in any short period an event of small probability. It is plausible that the proba- bility of such an event is proportional to the brief time of observation, and that the occurrence of more than one such death in a brief period has, in comparison, a negligible probability. If, further, events in nonover lapping intervals are inde- pendent, we should have a correspondence with the axioms for a Poisson process, and it would be reasonable to expect the deaths in a fixed period to be observa- tions on a Poisson random variable. In the case of the randoms, we took, in each month for each hospital, a fixed (almost) number of cases at random (almost) from the month's surgical load. With any particular operation, there is associated a small probability that a random case will be exposed to that operation. Altogether, our six operations involved 16 percent of the exposures. The typical monthly sample was 25. For N = 25, p = 0.16, the binomial distribution (which is, at least plausibly, applicable) is fairly well approximated by the Poisson. Thus, we have some reason to expect approximate Poisson behavior of the randoms. But, because experience often shows overdispersion in what might be hoped to be Poisson data, we checked both series, Ry and D[j , in two ways, for the six operations. The first approach was to study yearly totals for each of the six op- erations in each of the 34 hospitals. If the deaths (or randoms) were occurring as a Poisson process during the 4 years, and if in addition the Poisson process was 354

the same in all 4 years, then the statistic below should be distributed as x3 (chi- square with 3 degrees of freedom): 4 k=l *»!• where i= 1,...,6 indexes operation; j = 1,...,34 indexes hospital; k = 1,...,4 in- dexes year; and x represents deaths or randoms. Assessing the validity of the Poisson assumption in this way may be unneces- sarily stringent, for if xjji, xjj2, xjj3, and x^ are independent Poisson random variables with different parameters (reflecting a change in the process over time), it would be true that their sum was a Poisson random variable (the assumption with whose validity we are concerned), but the statistic proposed above would be stochas- tically greater than x2 casting doubt on the Poisson assumption. The second approach taken is nearly free of the difficulty of the first. The variation from one month to the next allows the assessment of over dispersion. But the classification simultaneously by month, operation, and institution is too fine; we therefore took monthly figures for the six operations combined in each institu- tion, defining xjk to be the number of deaths (or randoms) for all six operations in hospital j for the kth one of the months 1, 3, 5,...,47 and xjk to be the corre- sponding figure for the kth of the months 2, 4, 6,...,48. Then if xjk and xjk are inde- pendent Poisson random variables with the same parameter, the statistic below is distributed approximately as x2: xjk+xjk The expected value of the statistic under the Poisson assumption is exactly 1.0. This statistic also will be stochastically larger than X2 in the presence of a time- varying Poisson process. But a slow variation will induce only a small inflation in the contrast between adjacent months. The yearly total data (Tables 2 and 3) show a degree of dispersion exceeding Poisson variation by about 40 percent for the randoms, and by about 13 percent for the deaths. Neither of these is important so far as gauging the standard error of Dj - 6j is concerned, first, because in the standard error we are dealing not with 1.40 and 1.13, but with their square roots; and, second, because no qualitative con- clusion in Chapter IV-6 would be altered if all standard errors were changed by as much as half their value, let alone by less than one-fifth. Furthermore, we have already pointed out that the degree of overdispersion relevant to our standard errors may possibly be exaggerated by using results based on yearly totals. The odd-even monthly comparisons, shown in Table 4, indicate variance ex- ceeding the Poisson norm by 5 percent for the randoms and 10 percent for the deaths. In terms of the standard errors, these imply corrections of 2.5 percent and 5 percent to standard errors calculated on the Poisson assumption. 355

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TABLE 2.--DISTRIBUTION OF YEARLY TOTALS xf- STATISTIC FOR SIX OPERATIONS IN 34 INSTITUTIONS (RANDOMS) TABLE 3.—DISTRIBUTION OF YEARLY TOTALS xf- STATISTIC FOR SIX OPERATIONS IN 34 INSTITUTIONS* (DEATHS) Percentile interval Interval of values Observed frequency Expected frequency if Poisson Pe re entile interval Interval of values Observed frequency Expected frequency if Poisson 0-10 0-0.584 13 20.4 0-10 0-0.584 6 17 10-20 0.584-1.00 15 20.4 10-20 0.584-1.00 16 17 20-30 1.00-1.42 U 20.4 20-30 1.00-1.42 2 17 30-40 1.42-1.87 20 20.4 30-40 1.42-1.87 6 17 40-50 1.87-2.37 21 20.4 40-50 1.87-2.37 36 17 50-60 2.37-2.95 12 20.4 50-60 2.37-2.95 4 17 60-70 2.95-3.67 27 20.4 60-70 2.95-3.67 51 17 70-80 3.67-4.64 11 20.4 70-80 3.67-4.64 16 17 80-90 4.64-6.25 28 20.4 80-90 4.64-6.25 17 17 90-95 6.25-7.81 16 10.2 90-95 6.25-7.81 7 8.5 95-99 7.81-11.3 16 8.2 95-99 7.81-11.3 7 6.8 99-100 13. 3- 00 11 2.0 99-100 13.3- 00 2 1.7 204 204 170 170 The sum of 204 values of the statistic was 853.10, yielding an average of 4.182. This is 139.4 per- cent of the null expected value, which is 3. The sum of 170 values of the statistic was 576.12, yielding an average of 3.39. This is 113 percent of the null expected value, which is 3. *Instead of 204 = 6 x 34 values, there are 170, because in certain institutions there were no deaths associated with some of the operations dur- ing the 4 years. In 13 institutions there was one such operation, in six there were two, and in three there were three. TABLE 4.—DISTRIBUTION OF 34 INSTITUTIONS' AVERAGE VALUES OF SUCCESSIVE-MONTHS' X*-STATISTICS (RAN- DOMS, DEATHS) Midpoint of Frequencies class interval Randoms* Deaths+ 0 .55 2 0 0 .70 2 1 0 .85 7 11 1 .00 8 6 1 .15 6 13 1 .30 5 1 1 .45 3 1 1 .60 1 1 *Unweighted average of the 34 values is 1.05. Of the 801 individual values of X*, 44 exceeded 3.84, the upper-5-percent point of the X\ distribution, and four exceeded 6.61, the upper-1-percent point. +Weighted average of the 34 values is 1.10. Of the 595 individual values of X\, 21 exceeded 3.84, the upper-5-percent point of the X* distribution, and none exceeded 6.61, the upper-1-percent point. 357

APPENDIX 2 TO CHAPTER IV-6 A FURTHER INQUIRY INTO INSTITUTIONAL DIFFERENCES BY MEANS OF SUPERSTANDARDIZATION (A REGRESSION ADJUSTMENT BEYOND STANDARDIZATION) John W. Tukey* Princeton University, Princeton, New Jersey and Bell Telephone Laboratories, Murray Hill, New Jersey This appendix develops and discusses the technique of superstandardization, as applied to institution-institution comparisons. Most of the discussion is based on an earlier standardization of death rates that is no longer considered satis- factory and that has disappeared from the text of Chapter IV-6. Because the results of this ap- pendix were influential in precipitating the use of a better standardization, and because the ex- istence of the better standardization allows a partially objective check on the conclusions of the earlier analysis, I have reproduced the original superstandardization analysis in es- sentially the form it stood in when the new standardization became available. The original versions of the last two sec- tions of Chapter IV-6 used a standardization based on a combination of age and shock- likelihood index (SLI), described in an earlier version in these words: We have applied cell aggregation to the age X SLI classification. This involves ad- justment for three interfering variables, age and, through SLI, operation and whether the procedure was elective or emergency (a part of anesthetic risk classification). As in earlier applications in Chapter IV-6, the age X SLI cross-classification was swept in death-rate order into 10 categories equi- numerous in terms of deaths-plus-randoms. These strata were then used for the con- struction of indirectly standardized mor- tality ratios for each agent with hospital (reference population being "all agents" in the particular hospital). Most of this appendix (ending with "Possi- bilities of Further Studies") is based on the earlier standardization, which covered all op- erations. OLD INTRODUCTION The age-SLI analyses of (the old version of) the last section of Chapter IV-6 are based entirely on simple standardization over strata combining three background variables: age, operation, and the emergency or elective character of the anesthetic risk. We know that these variables, although clearly helpful, are incomplete. The riskiness of an individual ad- ministration does not depend only on what is ex- pressed by age or class of operation and whether the operation is considered an emer- gency. There is also, for example, what has been summarized into other aspects of anesthetic risk. The various stages of standardization produce large and important reductions in institution-to-institution differences of mor- tality, but there is no reason to suppose that these reductions are complete and final. Ad- justments of this sort are never perfect, and usually fall short of complete adjustment, rather than going beyond it. In a situation, like the present one, where unadjusted differences are large in comparison with adjusted ones, failures of adjustment to be complete will typically leave accidental differences of at least the same order of magnitude as any true dif- ferences that may be present. Although these failures may make apparent differences between certain institutions smaller than the correspond- ing true differences, that is quite unlikely to happen for all ^(34)33 = 561 differences, either individually or in terms of some convenient summary (such as the sum of squares of devia- tions). Accordingly, we expect that, over-all but not necessarily individually, true differences in quality of performance between institutions will tend to be somewhat overstated by apparent death rates, under nearly any adjustment. Thus, we strive to reduce such overstatement either to negligibility or as far as we reasonably can, so that any remaining differences are largely true differences. MEANS OF FURTHER ADJUSTMENT Standardization--adjustment based directly on differential death rates applicable to identi- fiable groups of administrations—can take us only so far. We will always have only a limited number of variables, and can expect to expend only a limited amount of effort in adjusting for •Prepared in part in connection with research at Princeton University sponsored by the Army Research Office (Durham). 358

their effects, apparent or real. This being so, what more can we do easily? If we were able to identify a measurable characteristic of hospitals that (1) is certainly not influenced directly by the average quality of anesthesia and surgery at that hospital and that (2) equally certainly does not directly influence the quality of surgery and anesthesia, but that (3) is clearly associated with the difficulty of the cases brought to operation, we could reason- ably eliminate from detailed consideration as much of the hospital-to-hospital differences as could be accounted for by regression on this new variable. (Measures of operative performance and measures of financial strength would be ruled out as likely to be influenced by, or to influence, quality of anesthesia and surgery.) In the language of general statistics, we would be making a regression (or covariance) adjustment with the aid of this variable. Such an identification would aid us by di- versifying our problems, provided that this regression absorbs a reasonably large fraction of apparent institution-to-institution differences. We would then have two phenomena instead of one: (1) Explicit differences between institutions (at least on the average) would be greatly re- duced, but would probably continue to deserve some attention; and (2) those differences as- sociated with (i.e., eliminated by the fitting of) the regression would become a specific, broadly occurring phenomenon that both could and should be studied as such. Thus, we would have made important progress in two ways: we would have gained a much better hold on (and a much greater hope of understanding) the dif- ferences assembled into (2); and we would have learned that differences due to quality of surgery, anesthesia, and aftercare can only have contrib- uted to the generally smaller differences visible in (1), unless the identified characteristic, which neither influences nor is influenced by per- formance quality, is somehow correlated with this quality for more complex reasons. What variable or variables can we consider in the present situation? What measures of over-all riskiness of patients presented do we have? In every case of indirect standardization there is one variable that seems peculiarly ap- propriate for such purposes. In indirect stand- ardization for any set of strata, we calculate standard deaths = 2 (stratum death rate) (stratum size) and could equally well calculate: SM = standard death rate = 2 (stratum death rate) (stratum frac- tion), which is the average death rate for an unselected body of cases (drawn from the whole experience) with the distribution over strata of the group of cases studied (here a hospital). This latter variable clearly measures, about as well as information about the strata considered can, the over-all riskiness of the cases in a given group, here those operated on in a given hospital. It is usually relatively free of influences of quality of anesthesia and sur- gery, and equally free of influences on this quality. We discuss below the possibility and relevance of influences in the present situation. The use of SM, or of some other measure of an adjustment already made, for regression adjust- ment of group results is sufficiently general to deserve a name. We shall call it "superstand- ardization." Just as standardization can proceed in two ways--"directly" by adjusting the observed death rates, and "indirectly" by adjusting the figures to which the numbers of observed deaths are compared—so too can superstandardization proceed in two ways. We shall here find it con- venient to adopt indirect superstandardization, so that we are still comparing the number of deaths actually observed with a reference value— now, we trust, one that is still more suitable. We must also decide just how we should carry out the regression. On the one hand, it is rather customary so to compute any sort of standardized deaths as to make their sum over all strata equal (or nearly equal) to the corre- sponding sum of actual deaths. On the other hand, linear regression of the logarithm of SMR on the logarithm of SM seems most likely to give a generally close fit in a wide variety of circum- stances. Accordingly, the recommended general procedure is: (1) to do linear regression of log SMR on log SM; (2) to calculate the correspond- ing number of (indirectly) superstandardized deaths for each stratum, (3) to calculate the common factor by which these should be multi- plied to make total superstandardized deaths equal total actual deaths, and (4) to apply this fac- tor to the raw fitted values for each stratum. How large a fraction of the part of institution- to-institution variability that is associated with differences in intake populations but not dealt with by standardization itself can we expect such a regression adjustment to absorb? One-half? Two-thirds? Three-fourths? Many would think all these numbers too large. We shall consider the consequences of such judgments below. APPLICABILITY TO THE NATIONAL HALOTHANE STUDY In the National Halo thane Study, why is superstandardization reasonable, and why is it plausible that superstandardization should be effective in reducing apparent differences be- tween hospitals? Without claim or hope of com- pleteness, it is easy to give a few possible reasons. In the National Halothane Study, the pro- posed carrier of (additional) regression, SM, is determined by two sorts of information: (1) over-all death rates, assessed without regard to the institutions in which the deaths occurred, according to categories determined by age group, type of operation, and elective or 359

emergency character assigned to individual operations; and (2) the distribution, over these same categories, of all patients brought to operation in particular institutions, as assessed from the randoms. Because SM is unaffected by transferring deaths in a given category from one institution to another, it is at least a plausible candidate for use in a regression adjustment of standardized death rates for the 34 institutions. Standardization, according to less-than- adequate strata, frequently leaves an opportunity for further adjustment on the same basis. The simplest case arises when adjustment for a single variable is based on broad groups, as when a continuous variable is dichotomized. Here, the larger the fraction of cases above the cutting point, the higher the center of gravity of these upper-class cases. Consequently, ad- justment for percent upper class, although very helpful, still leaves a need for further adjust- ment in the same direction as the initial adjust- ment. When we standardize for some but not all of a set of correlated background variables, a somewhat related phenomenon is likely to occur. In the prototype situation, where variable x l is stratified on but variable x2 is not, displacement of both xj and x2 along—or nearly along—a structural regression line common to these two variables will increase x2 more than would be forecast by the ordinary regression of x2 on xj. When differentiation between groups corresponds more or less closely to shifting along a struc- tural regression, standardization of xj, while allowing for the ordinary regression of x2 on xj, will underestimate the correction required to allow for both x j and x2. The strata used for standardization in the old version of the last section of Chapter IV-6 might have included the age-SLI analysis of the numeric details of "anesthetic risk" in their definition. (Had this been so, better standardiza- tion might have ensued.) It is to be expected that the definition of this variable will be at least partly relative to the over-all experience of the hospital concerned. Thus, if two completely equivalent cases are rated, one in a hospital in which the distribution of ages and operation types makes the over-all death rate low, and one in a hospital in which these variables lead to more frequent deaths, it is to be expected that a some- what higher anesthetic risk will tend to be as- signed at the hospital with the lower death rate. As a consequence, the average anesthetic risk in different hospitals would vary less than it should if anesthetic risk were to reflect the same standards in all hospitals. Accordingly, if the details of anesthetic risk did enter, that would cause standardization to underestimate the ap- propriate adjustments. Fortunately for the simplicity of interpretation of the present analysis, only the emergency-vs.-elective character of anesthetic risk enters into the age- SLI standardization. Although a very small ef- fect on which operations are considered "emer- gency" is plausible, it is hard to believe that such an effect is in any way important for the present analysis. Another related effect arises if the decision as to whether marginal patients should be op- erated on is influenced in a similar direction by the hospital's over-all death-rate experience. RESULTS OF SUPERSTANDARDIZATION Let us now turn to the actual situation. The first few columns of Table 1 (in which the hospi- tals have been arranged in order of the values of SM) give values of both SM and SMR, the standardized mortality ratio, defined by: SMR = deaths standard deaths actual death rate. SM The tendency of the SMR to increase as the SM increases is quite obvious in the table. Roughly speaking, the SMR doubles as the SM quadruples. Consider the regression of the logarithm of the SMR on the logarithm of the SM. Calculation indicates the use of the regression: log(SMR)~0.60 log(SM) + constant. When the constant is adjusted to make total superstandard deaths equal total actual deaths (within rounding error), the results shown in the SSMR column are obtained, and hence the indi- cated number of unbalanced deaths. Other col- umns show the same figures for the unsuper- standardized SMR. The reduction in unbalanced deaths from 2297 to 1738 consequent on super- standardization is quite striking (these numbers are the averages of the two column totals in Table 1. which disagree because of rounding errors). (The detailed computations for Table 1 proceeded as follows: Linear regression of loggSMR on logeSM gave: logeSMR = 0.60 logeSM + 2.346. Superstandardized mortality ratios, SSMR, were calculated from: log SSMR = log SMR - 0.60 log SM - 2.346, 66 C and converted into roughly superstandard (ab- breviated "r-superstd") deaths through: r-superstd deaths = deaths/SSMR. The sum of all 34 r-superstd deaths was found to be 17364.4, in comparison with the total 360

TABLE 1.--COMPARISON OF SMR WITH SM, SUPERSTANDARDIZED RESULTS, UNBALANCED DEATHS Deaths Unbalanced deaths Institution SM SMR SSMR Act. Std Superstd Standard Superstd + - + - 28 0.0076 0.3519 0.6293 19 54 29 35 10 3 0.0093 0.8701 1.3788 201 233 141 32 60 31 0.0108 1.3433 1.9459 180 135 90 45 90 17 0.0115 0.8194 1.1431 186 229 158 43 28 23 0.0115 0.7345 1.0247 83 113 79 30 4 26 0.0115 0.9011 1.2571 392 438 302 46 90 6 0.0127 0.8198 1.0715 332 408 299 76 33 27 0.0129 0.8385 1.0919 296 355 263 59 33 1 0.0131 0.5643 0.7280 294 524 392 230 98 34 0.0147 1.0778 1.2977 180 168 135 12 45 22 0.0167 1.2505 1.3946 614 494 427 120 187 14 0.0170 0.3072 0.3390 141 462 404 321 263 9 0.0179 0.9721 1.0399 976 1010 910 34 66 12 0.0179 1.1200 1.1981 504 453 408 51 96 11 0.0187 1.1303 1.1779 555 494 457 61 98 21 0.0197 0.6352 0.6416 815 1291 1232 476 417 15 0.0201 0.9607 0.9586 586 613 593 27 7 20 0.0201 0.7605 0.7589 727 962 929 235 202 32 0.0207 0.8625 0.8456 276 322 316 46 40 4 0.0210 1.1697 1.1369 717 616 612 101 105 8 0.0213 0.6948 0.6697 1348 1953 1953 605 605 19 0.0214 1.2398 1.1915 486 395 396 91 90 10 0.0219 1.2504 1.1852 799 642 654 157 145 33 0.0242 1.0493 0.9367 681 652 705 29 24 13 0.0248 1.4386 1.2655 246 172 189 74 57 16 0.0277 1.1611 0.9558 519 450 527 69 8 5 0.0282 1.2802 1.0426 1197 941 1114 256 83 7 0.0307 1.5486 1.1985 542 352 439 190 103 18 0.0334 1.1010 0.8101 327 299 391 28 64 30 0.0334 1.5170 1.1162 223 148 194 75 29 29 0.0336 1.3053 0.9570 171 132 173 39 2 24 0.0384 1.7602 1.1911 1035 591 843 444 192 2 0.0386 1.6685 1.1256 614 370 529 244 85 25 0.0434 1.5879 0.9985 578 365 561 213 17 Total +2299 -2295 +1737 -1740 Note: SM is = A-SM, SMR is = A-SMR (in later notation). actual deaths of 16840. To balance this out, all r-superstd deaths were multiplied by 16840/ 17364.4 = 0.968. Thus, superstd deaths = (0.968) (r-superstd deaths). For institution 28, the actual values were: log SSMR, 0.463; SSMR, 0.6293 (as given); r- superstd deaths, 30.1913; and superstd deaths, 29.23 (rounded to 29). The standard deaths used in Table 1 are taken from the old version of Table 21 of Chapter IV-6, and involve standardization for age-shock categories. We will henceforth make this explicit by a prefixed "A-," referring to A-SMR, A-SM, A-superstandardized deaths, etc. This analysis shows a convincingly strong dependence of A-SM on A-SMR and a substantial reduction in the number of unbalanced deaths consequent on changing from standardized to A-superstandardized deaths. The effect is so large that we must be careful to consider all the actions and decisions that could contribute to the values of A-SM. To change A-SM, we would have to change either the distribution of patients operated on (as to age, sex, and/or operation code), or the distribution of emergency or elective character assigned to their operations. Accordingly, A-SM is influenced by: (1) the decisions of specific referring physicians as to when and where their patients should be hospitalized, (2) decisions at the hospital as to whether or not the patients shall be operated on, and (3) assignment by the hospital of an emergency or elective character to each opera- tion. Although we are reasonably clear that (1) and (2) do not directly reflect hospital per- formance, we dare not be quite so sure of (3). We therefore seek extra security by repeating our superstandardization analysis, using standardization for operations alone—regressing 361

log O-SMR on log O-SM. Table 2 shows the values of O-SM and O-SMR and various regres- sion coefficients. Again a substantial dependence of log SMR on log SM is clear, although the regression is weaker than for A-SMR on A-SM. We are now at the following position: Superstandardization could at best be hoped to remove a fraction of that part of the institution- to-institution variance in apparent performance (as measured through the standard mortality ratio, by death rates for crudely equivalent cases) associated with nonperformance varia- bles. It has very substantially reduced the total variance. We have inquired into whether direct effects of institutional differences in quality of anesthesia, surgery, or aftercare could account for the efficacy of Superstandardization, and have concluded that they could not, unless an unexpected association were produced in some complex way. Accordingly, the fraction removed by Superstandardization must, unless this hap- pens, correspond to nonperformance variables. Because not all nonperformance contributions will be removed, we are led to conclude, with the same caveat, that the portion of institution- to-institution variance in standardized mortality ratios associated with factors other than quality of anesthesia, surgery, and aftercare is greater than the fraction removed by Superstandardiza- tion, plausibly considerably greater. There may well be real differences in the quality of performance, as measured by com- parative death rates for truly equivalent cases, from one hospital to another within the 34 in- cluded in the Study, but we cannot use the data to assert that this must be so. More specifically, we cannot set a lower bound on the variance of performance from institution to institution. However, although the results of Superstand- ardization do estimate a reasonable upper bound for this variance, there are strong reasons to believe that this upper bound, far reduced as it is below the results of mere standardization, is larger than the variance of performance itself by what is likely to be a quite substantial factor. Institution O-SM* O-SMR 28 0.477 OA n si i l-O3C A SUE- deathS - death rate >CU U • -*iJ- 1 0.523 0.716 14 0.561 0.472 n ,„ 6 0.573 0.974 ~ constant (A-SM) 31 0.581 1.261 3 0.598 0.688 23 0.663 0.651 27 0.708 0.780 34 0.719 1.122 n OMH deaths death rate 22 0.770 1.301 " 0-std deaths O-SM 20 0.800 0.778 15 0.805 1.223 17 0.860 0.557 ~constant(0-SM) 1 9 0.998 0.897 21 1.015 0.627 12 1.045 0.976 7 1.046 2.309 32 1.070 4 1.086 n A£*5 n -^ft n 1149 A-SMR ~~ constant (O-SM) (A-SM) 25 5 1.143 19 1.203 1.620 ,-, ,, 1.121 = constant(0-SM)u-DJ(A-SM/0-SM) .13 11 1.212 0.887 10 1.247 1.115 is equivalent to 2 1.271 33 1.272 2.577 1.291 O-SMR ~ constant(0-SM)-0-62(A-SM)1-25 8 1.302 25 1.442 0.577 0 ,, 2.413 = constant (O-SM) (A-SM/0-SM) 87 16 1.512 1.086 18 1.685 1.109 13 1.854 0.979 29 1.982 1.121 24 2.331 1.477 30 2.752 0.935 TABLE 2.--COMPARISON OF O-SMR WITH O-SM AND OF THE ADDITIONAL REGRESSIONS *These entries differ from O-SM by a multiplicative constant (the same for all). 362

THE VARIABILITY OF "UNBALANCED DEATHS" Let us turn to the question of the actual and estimated variability of: unbalanced deaths = actual deaths - superstandard deaths. It is easy to isolate three major contributions to the variance of each such value, namely: (1) the variance in actual deaths; (2) the variance in standard deaths, due to the calculation of estimated exposed from sample results (the randoms), as amplified or diminished by the effects of superstandardiza- tion; and (3) uncertainties in the fit of the coef- ficients involved in the superstandardization. The formulas of Appendix 1 to Chapter IV-6 indicate a means of calculating reasonable values for the variances. If Poisson variability applied, we would have: variance {deaths} ~ deaths, and variance {standard deaths} ~ Q-standard deaths, where Q has to be calculated as discussed in Appendix 3. The corresponding formulas for logarithms (to the base e) are: variance {logedeaths} - deaths and variance {loge superstandard deaths} deaths} 2.56 Q standard deaths ' where 1.60 = 1 + 0.60, the 0.60 being the regres- sion coefficient, and where we neglect, for the moment, contributions from the variance of the superstandardizing regression coefficient. Be- cause logeSSMR = logedeaths - logesuperstandard deaths, we have: variance{logeSSMR 1 + 2.56(A-SMR)Q . . . . . .- deaths When we allow for the over-Poisson variation also assessed in Appendix 1, and for the un- certainties due to fitting (in the superstandardiz- ing regression) three constants to 34 data points, it is reasonable to accept 1.1 + 1.54(2.56) (A-SMR)Q 1.1 + 3.94(A-SMR)Q deaths deaths as an estimated variance for logeSSMR. The best available set of Q's correspond to a closely related but somewhat different basis of standardization. Their use may affect com- parisons for individual institutions appreciably, but the over-all picture should not be disturbed by such borrowing. Having considered regression on both log A-SM and log O-SM separately, it is natural to consider joint regression on both together. Be- cause log O-SM + log O-SMR = log A-SM + log A-SMR (both sides being equal to the actual death rate), the residuals after regression of either O-SMR or A-SMR on both O-SM and A-SM will be the same, as will the corresponding numbers of superstandardized deaths. We will use "AO-" to designate such results of joint superstandardiza- tion. The results of AO-superstandardization are set forth in Table 3 with the results of the varia- bility calculations. The ratios of log AO-SSMR to its standard error are obviously too large to be entirely accounted for by sampling fluctuations. The mean square of this ratio is 3.86, in com- parison with the 1.0 anticipated for pure sampling fluctuations. The corresponding ratios for logeSMR are noticeably larger, with a mean square of 7.00. The reduction of excess varia- bility by superstandardization thus appears to be about (7.00 - 3.86)/(7.00 - 1.0) = 52 percent. Study of the 34 ratios via modified proba- bility plots indicates that three of the logeAO- SSMR values, those for institutions 14, 8, and 21, are out of line with the other 31, which appear like a sample from a normal (i.e., Gaussian) distribution. A similar analysis for logeSMR shows that the same three institutions provide three of the lowest five values of "(ratio*)." Omitting those three institutions, the mean square ratios become 2.12 for log AO-SSMR and 5.31 for log A-SMR, corresponding to a reduc- tion of about (5.31 - 2.12)/(5.31 - 0.8) = 71 per- cent. (An analysis based on the modified proba- bility plot and insensitive to both outliers and long tails yields a reduction of 74 percent in "central variance.") If we believe that the fraction of the variance associated with quality-unrelated variables that 363

TABLE 3.—ASSESSMENT OF INSTITUTIONAL VARIABILITY IN COMPARISON WITH SAMPLING VARIABILITY Institution AO-SSMR log AO-SSMR Ster* Ratio A-SMR log A-SMR (Ratio*) 1 0.799 -0.224 0.133 -1.68 0.564 -0.573 -4.30 2 1.296 0.259 0.112 2.30 1.67 0.513 4.56 3 1.280 0.247 0.156 1.58 0.871 -0.138 -0.88 4 1.122 0.115 0.135 0.86 1.17 0.151 1.12 5 1.118 0.112 0.121 0.92 1.28 0.247 2.04 6 1.132 0.124 0.150 0.83 0.819 -0.200 -1.34 7 1.370 0.315 0.130 2.42 1.55 0.438 3.37 8 0.620 -0.478 0.112 -4.28 0.695 -0.364 -3.26 9 1.001 0.001 0.137 0.01 0.972 -0.028 -0.21 10 1.126 0.119 0.120 0.99 1.25 0.223 1.86 11 1.070 0.068 0.136 0.50 1.13 0.122 0.90 12 1.134 0..126 0.135 0.94 1.12 0.113 0.84 13 1.083 0.079 0.126 0.63 1.44 0.365 2.90 14 0.397 -0.924 0.155** -5.97 0.307 1.181 -7.63 15 1.041 0.040 0.136 0.29 0.960 -0.041 -0.30 16 0.916 -0.088 0.124 -0.71 1.16 0.148 1.19 17 0.996 -0.004 0.167 -0.02 0.819 -0.200 -1.20 18 0.796 -0.228 0.125 -1.83 1.10 0.095 0.76 19 1.138 0.129 0.151 0.85 1.24 0.215 1.42 20 0.826 -0.191 0.130 -1.47 0.760 -0.274 -2.12 21 0.634 -0.456 0.134 -3.41 0.635 -0.454 -3.39 22 1.443 0.367 0.128 2.87 1.25 0.223 1.74 23 0.984 -0.016 0.199 -0.08 0.734 -0.309 -1.55 24 1.089 0.079 0.102 0.78 1.76 0.565 5.56 25 1.142 0.133 0.119 1.12 1.59 0.464 3.91 26 1.334 0.288 0.163 1.77 0.902 -0.103 -0.63 27 1.067 0.065 0.153 0.42 0.839 -0.176 -1.15 28 0.594 -0.520 0.290 -1.79 0.353 -1.041 -3.58 29 0.890 -0.116 0.157 -0.74 1.31 0.270 1.72 30 0.915 -0.088 0.140 -0.63 1.52 0.419 2.99 31 1.918 0.651 0.168 3.88 1.34 0.293 1.75 32 0.834 -0.181 0.143 -1.27 0.862 -0.149 -1.04 33 0.915 0.089 0.136 0.66 1.05 0.049 0.36 34 1.321 0.278 0.157 1.77 1.08 0.077 0.49 Ster* = estimated standard error of logeAO-SSMR. (Ratio*) = loggA-SMR/Ster (calculated for comparison with ratio only). **Arbitrary Q-value of 2.00 assigned. is likely to be absorbed by superstandardization is no more than one-half, two-thirds, or three- fourths, we are then led to the following judgment bounds on the fraction of variance in log SMR that may be associated with institutional varia- tion in quality of performance: (supe r standardization reduction) (52 percent) (71 percent) less than one-half recovery less than two-thirds recovery (impossible) <22 percent (impossible) (impossible) less than three-fourths recovery <31 percent <5 percent (Here, for example, if 52 percent is no more than a two-thirds reduction, the whole reduction is no less than 78 percent, hence no more than 22 per- cent remaining.) The evident conclusion is that only a small fraction of the variation in (unsuper- standardized) institutional standard mortality ratios is likely to be due to institutional dif- ferences in performance. (A fraction of 0 per- cent seems a quite reasonable possibility.) MORE DETAILED CONSIDERATIONS The modified-probability-plot examination of logarithmic values of AO-super standardized mortality rates indicated that three institutions were apparent outliers in the direction of low death rates. The lowest of all, institution 14, had a very distinctive and relevant feature: all cases were classified as "unknown anesthetic risk." 364

Because these cases could not be divided between the emergency and elective categories, some special treatment must have been required during age-shock standardization. The second lowest, institution 8, contributed to the Study more than one-third of all the ether administrations in the middle- and high-death- rate operation groups. This fact may well be related to some of the idiosyncratic behavior of ether noted in the course of other analy- ses. item mean square (all 34) mean square (31 only) "central variability" reduction (all 34) reduction (31 only) reduction ("central variability") However, this small change in over-all numbers is associated with a considerable change in the appearance of the 34 individual SSMR's. The three low values of AO-SSMR are clearly dis- tinguished from the other 31, whereas the three low values of A-SSMR are much less clearly delineated (except for institution 14). In view of the idiosyncrasies (for two of these three) just noted, the analysis indicating these three as distinguished is considered more likely to be in- formative. POSSIBLE INTERPRETATIONS OF THESE RESULTS We have found that, although a few individual differences between institutions appear to re- main after super standardization, the greatest part of institutional differences in standardized mortality ratio (SMR) may be accounted for in terms of regression on one or both forms of standard mortality (A-SM or O-SM). Moreover, the sign of this regression is opposite to that which would arise, in the sort of situation at hand, from fluctuations in estimating the death rates generating the values of SM. The regres- sion effect is large, accounting for most dif- ferences between institutions. How could it arise? What approaches might be considered in a later study of its origin and nature? In view of the very substantial regression of O-SMR on O-SM, neither of whose definitions involves anesthetic risk (physical status), we cannot assign the greater part of this effect to institutional differences in classifying patients on this scale. It would not be unnatural if in- stitutions with a higher fraction of more serious operations, and thus with a more seriously ill patient population, were to classify patients as somewhat lower anesthetic risk than would in- Neither of these facts leads to firm con- clusions; both offer food for thought. COMPARISON WITH A-SUPERSTANDARDIZATION The summary effects of double superstand- ardization (with A-SM and O-SM) are not greatly different from the summary effects of simple superstandardization with A-SM alone. Ex- plicitly: A-superstd AO-superstd 4.13 2.19 2.48 48 percent 69 percent 73 percent 3.86 2.12 2.43 52 percent 71 percent 74 percent stitutions whose typical patients were less severely ill. Such an effect, if extended to the distinction between elective and emergency op- erations, would contribute to a positive regres- sion of A-SMR (or O-SMR) on A-SM. Such an effect may well exist, but could not produce the positive regression of O-SMR on O-SM that we have observed. Differences in judgment as to which patients should be operated on and which are too sick to risk operating on are to be expected. It might well be that institutions in which the patient population is more seriously ill on the average are more likely to operate on equivalent cases. If this ef- fect existed, it would contribute to the observed positive regression. How large such an effect might be must, at this point, be a matter of judgment, but it seems quite unlikely that it would be large enough to account for the size of the observed effect. In addition to the decision of whether or not to operate, there is an earlier decision by the referring physician as to what institution he will send his patient to. In general, we may expect considerable consensus among referring phy- sicians as to which accessible hospitals are better fitted to deal with more serious cases. In such circumstances, it would be surprising if hospitals receiving a larger fraction of cases scheduled for more serious operations did not also receive, on the average, more difficult cases scheduled for a given operation, even after allowance for age, etc. This would, of course, lead to higher death rates by operation in such institutions. The resulting high SMR would go with the supposedly high SM, and we would again have a positive regression. A numeric example showing the magnitude of the effects that might be produced in this way may be helpful. Let us suppose an over-all death 365

rate of 2 percent, with half the deaths coming from more serious cases (whose death rate is 10 percent). For 100,000 cases this requires: 90,000 cases with a 1.11 percent death rate and 10,000 cases with a 10 percent death rate. If equal numbers of the less serious cases are sent to each of two hospitals, but 85 percent of the more serious are sent to hospital A, the mortality rates become: 500 + 850 45,000 + 8500 500 + 150 45,000 + 1500 = 2.52 percent and . Percent. Thus, a ratio approaching 2:1 can arise from what, to the eyes of a layman, seems a not im- possible degree of selectivity by referring physicians. There is a third possibility. The available data in this Study refer to deaths in hospitals. Differences in institutional practice in returning terminal cases to other care (at home or at another institution) could have a noticeable effect on observed mortality rates. This effect could prove to be associated with the value of SM, although both the strength and the direction of such an association are hard to assess. A fourth possibility deserves mention. Dif- ferences in the socioeconomic status of the population served by an institution may be re- flected in differences in tendency to delay or avoid entry into hospital. A reasonable conse- quence could be association of avoidance of minor operations with delay in hospitalization for major ones. If delay corresponded to in- creased risk of death, this would produce a positive regression of SMR on SM. Differences in socioeconomic composition might also associate poor general health and nutrition with avoidance of minor operations. This would be a fifth possibility for generating a positive regression coefficient. Beyond these recognized possibilities, there are undoubtedly other unrecognized ones involving medical phenomena. In addition, of course, there remain a variety of purely numeric phenomena that might contribute to explaining the observed additional regression on SM. The treatment of age in 10- year classes, for example, in which crude death rates double every decade for the older classes, implies that an appreciable part (perhaps 3 to 10 percent) of the age effect can never be ad- justed or standardized for in terms of age classes. Similar phenomena occur for most other variables. POSSIBILITIES OF FURTHER STUDIES If it were desired to gather further data that might help to answer the question of which of these recognized possibilities contributes to the observed regression, and how much is thereby contributed, what might be done? Data relevant to differences in the return of terminal patients to other care could certainly be found, although considerable expenditure of effort and money would probably be necessary. Some light could be thrown on the plausible ex- tent of this effect by studying the termination of stay in hospital according to two competing causes: dismissal and death. More light could be gained by a follow-up study of a substantial sample of patients who underwent operations with relatively high death rates. Investigation of this possibility thus requires considerable effort, but can follow clearly understood approaches. The second possibility, differential referral, requires a new form of inquiry, and should not be attacked on a large scale without considerable exploratory study and inventive discussion. It seems that some effective approach ought to be possible, but none has yet been identified. Approximately the same thing can be said about the first, fourth, and fifth possibilities. THE IMPROVED STANDARDIZATION We now turn to the improved standardization discussed in the current version of the last two sections of Chapter IV-6. Three standardizations based on cell aggregation of the cross- classification of age, risk (physical status), and operation are relevant here, -covering middle- death-rate operations, high-death-rate opera- tions, and middle- and high-death-rate opera- tions combined. Because of a change in interest and under- standing, no such standardization was made for all operations. Thus, direct comparison with results of the original (age-SLI) standardization is not possible. However, comparison of the earlier results for "All" with the newer results for middle- and high-death-rate groups should not be too seriously distorted for our purposes, inasmuch as the low-death-rate operations pro- vide very few deaths and very few standard deaths. The apparent regression coefficients found for the three analyses were: Middle: +0.15 High: -0.35 Middle-and-high: + 0.07 These are greatly reduced from the +0.60 for the old standardization. Indeed, if we had not had 366

experience with the old standardization, we would probably pay little attention to such regression coefficients. COMPARISON OF RESULTS Table 4 presents various standardized and superstandardized mortality ratios by institu- tion. It is clear that there have been substantial changes, particularly in SMR, in passing from the old standardization (for all operations) to the new standardization (for middle- and high-death- rate operations). Table 5 presents the sums of squared differences between one mortality ratio based on the old standardization and one mortality ratio based on the new standardization, which is ac- cepted to be better. Whichever new standardized mortality ratio we chose as a reference, it is clear that superstandardizing the old analysis brought it closer to the best values now avail- able. The individual adjustments made by super- standardization are not all in what would appear to be the right direction, but we would, of course, be quite surprised if they were. As far as the range of values is concerned, Table 6 shows the three lowest and three highest mortality ratios for each of the four types of standardization. Again there is a tendency to better agreement between old superstandardization and either new result. TABLE 4.—COMPARISON OF VARIOUS STANDARDIZED (SMR) AND SUPERSTANDARDIZED (SSMR) M3RTALITY RATIOS New Old Old Inst. SMR SMR SSMR (M+H) (All) (All) 28 14 30 8 23 21 33 11 32 1 18 9 17 3 10 6 20 4 26 27 29 19 16 15 2 22 12 34 25 7 24 5 0.59 0.60 0.73 0.74 0.74 0.77 0.78 0.78 0.80 0.80 0.84 0.84 0.84 0.88 0.91 0.96 0.97 1.04 1.04 1.06 1.06 1.13 .U .u .18 .20 1.26 1.29 1.31 1.47 1.49 1.52 1.77 1.98 0.35 0.31 1.52 0.70 0.73 0.64 1.05 1.13 0.86 0.56 1.10 0.97 0.82 0.87 1.25 0.82 0.76 1.17 0.90 0.84 1.31 1.24 1.16 0.96 1.67 1.25 1.12 1.08 1.59 1.55 1.76 1.28 1.44 1.34 0.63 0.34 1.12 0.67 1.02 0.64 0.94 1.18 0.85 0.73 0.81 1.04 1.17 1.38 1.19 1.07 0.76 1.14 1.26 1.09 0.96 1.19 0.96 0.96 1.13 1.39 1.20 1.30 1.00 20 19 04 27 New New New SSMR SSMR SSMR (M+H) (MID) (HI) 0.64 0.62 0.67 0.74 0.76 0.77 0.75 0.76 0.80 0.84 0.81 0.83 0.89 0.92 0.88 .00 .98 .02 .08 .11 .02 .13 .11 .14 1.09 1.21 1.27 1.37 1.22 1.41 1.42 1.50 1.80 2.09 1. 0. 1. 1. 1. 1. 1. 1. 1. 0.85 0.61 0.65 0.71 0.56 0.78 0.74 0.68 0.76 0.78 0.77 0.74 0.96 0.93 1.23 0.94 1.03 1.11 1.00 1.03 0.79 1.07 1.31 1.01 1.13 13 13 31 1.95 17 32 18 50 34 52 51 1.93 0.30 0.57 0.85 0.81 1.00 0.68 0.83 0.93 0.84 1.02 0.93 1.00 0.81 0.88 0.62 1.11 0.84 0.96 1.45 1.17 1.44 1.28 0.99 1.30 1.13 1.49 1.29 1.67 1.35 1.50 1.63 1.47 1.84 1.94 367

TABLE 5.—SUMS OF SQUARES OF DIFFERENCES BETWEEN MORTALITY RATIOS BASED ON OLD AND NEW STANDARDIZATIONS Old SMR Old SSMR (All) (All) New SMR 2.4893 1.9499 (M + H) New SSMR 3.2759 1.8711 (M + H) TABLE 6.—COMPARISON OF THREE LOWEST AND THREE HIGHEST MORTALITY RATIOS Three lowest Three highest Old SMR 0.31, 0.35, 0.56 1.55, 1.59, 1.76 (All) Old SSMR 0.34, 0.63, 0.64 1.38, 1.39, 1.95 (All) New SMR 0.59, 0.60, 0.73 1.52, 1.77, 1.98 (M + H) New SSMR 0.62, 0.64, 0.67 1.50, 1.80, 2.09 (M + H) BEHAVIOR OF THE MIDDLE-DEATH-RATE GROUP The most positive regression coefficient is for the middle-death-rate operations. In judging its significance, we need to take account of the fact that log SMR and log SM are subject to cor- related sampling fluctuations, so that the natural null-hypothesis value of this regression coeffi- cient is negative, rather than zero. The analysis of Appendixes 4, 5, and 6 to Chapter IV-6 indi- cates that the negative value for this particular instance is about -0.06.* Accordingly, because the estimated standard error of our regression coefficient is about 0.11, the proper t-ratio is 0.15 - (-0.06) - l Q 0.11 which is significant at a one-sided 5 percent point, rather than 0.15 - (0.00) = 1.4. 0.11 *See note to Table 2 of Appendix 6 to Chapter IV-6. OVER-ALL CONCLUSIONS In discussing the results of the super- standardization based on the old standardization, we concluded that it was reasonable to believe either that better standardization (involving among other things allowance for unmeasured variables) would remove the apparent differences among institutions, or that it would not. Do we now have cause to alter that view? Table 7, which shows the ratios of highest to lowest and of second-highest to second- lowest for mortality ratios obtained by various standardizations indicates one set of reasons. The new standardization has reduced the highest:lowest ratio from 5.8 (it started at 25.4) to 3.4. Further improvement in standardization might, but need not, reduce this ratio still further. A reduction to, say, 2.0 seems not out of the question. Because a ratio of 1.8 would be expected by chance, it would not require a tremendous further improvement to make the observed ratio quite consistent with the results expected by chance. However, the large increase in the mortality ratio for institution 31 consequent on super- standardizing the old analysis leads to an old SSMR that agrees quite well with both the new SMR and the new SSMR. One would be inclined to believe that mortality rates in institution 31 are indeed somewhat higher than for other institu- tions. Concerning individual institutions, it may help to compare the extreme (D-D)A values for the two standardizations. This is done in Table 8. The first thing that strikes the eye is the tremendous reduction in these ratios consequent on using a better standardization. The next thing noticed is the striking behavior of institution 30, which migrates from a critical ratio of +6.0 to one of -3.4. With as weak a degree of association as this between the results of a standardiza- tion once thought good enough to use and the results of the present choice, it is hard to be very sure about present results for individual institutions. Both standardizations were, however, ef- fective in important ways, showing that true institutional differences were much smaller than those among unadjusted death rates, properly discouraging excessive excitement about institu- tional differences, but preserving an active interest in their cautious exploration. In summary, then, both views of the eventual results of careful study of institutional differ- ences--no real differences, or small real differences—remain reasonable. 368

TABLE 7.—HIGHEST AND LOWEST MORTALITY RATIOS AT VARIOUS STAGES OF STANDARDIZATION (INSTITUTION NUMBERS IN PARENTHESES) Observed Operations Age-SLI* Age -risk-operation* IV-2, Table 5 IV-2, Table 5 IV-6, App. 2, Table 1 IV-6, App. 2, Table 4 Highest 2nd-highest 2nd-lowest Lowest Mortality ratios 3.480(25) 2.577(2) 1.760(24) [1.946(31)] 1.976(31) [2.085(31)] 3.443(24) 2.413(25) 1.668(2) [l.395(22)] 1.770(13) [l.806(13)] 0.265(14) 0.472(14) 0.352(28) [0.625(28)] 0.596(14) [0.644(23)] 0.137(28) 0.287(28) 0.307(14) [0.339(14)] 0.588(28) [0.619(14)] Ratios of mortality ratios Highest:lowest 25.4 9.0 5.7 [5.7] 3.4 [3.4] 2nd-highest:2nd-lowest 13.0 5.1 4.8 [2.2] 3.0 [2.8] Expected by chance** Highest:lowest [1.8] [1.8] 2nd-highest:2nd-lowest [l.6] [l.6] *Superstandardized values in brackets. Values for age-risk-operation standardization are for middle- and high-death-rate groups of operations combined. **Based on mean values of "ster" for all 34 institutions (probably leads to over-small values) and mean ranges of 4.189* (full range) and 3.353o (1st quasirange) for n = 34. TABLE 8.—EXTREME VALUES OF (D-D)A COMPARED (INSTITUTION NUMBERS IN PARENTHESES) Old standardization (original version of Table 21 of Chapter IV-6) Ne,; standardization (current version of Table 21 of Chapter IV-6) Positive extremes +12.3(24*) +6.2(24*) +8.9(2) +5.9(13) +7.6(25*) +5.6(5*) +6.4(7*) +5.3(31) +6.0(30**) +4.4(7*) +5.3(5*) +3.6(25*) Negative extremes -14.4(14*) -3.6(14*) -7.0(1) -3.4(8*) -6.4(8*) -3.4(30**) -6.0(21*) -2.7(11) -4.2(20) -2.6(21*) -3.0(6) -2.3(33) *Repeating at same extreme. **Repeating at opposite extreme. 369

APPENDIX 3 TO CHAPTER IV-6 THE VARIABILITY OF THE DIFFERENCE BETWEEN DEATHS AND STANDARDIZED DEATHS Lincoln E. Moses * Stanford University Stanford, California The two kinds of signed rank analyses in the next-to-last section of Chapter IV-6 are aimed at assessing the statistical significance of the 10 possible agent contrasts, giving more weight to intra-institutional contrasts that (1) are larger or (2) have smaller standard errors. It would be desirable simultaneously to take both these prop- erties of an intra-institutional agent contrast into account; it would further be desirable to give numeric estimates of suitably weighted average contrasts. This appendix exhibits a method de- veloped to meet these two purposes and displays the resulting figures. Consider a pair of anesthetics a and 0. Let j index institution. Define djui*>, the difference in the jth institution between the two indirectly standardized mortality ratios, i.e., d/"-" = log(ISMR(o))j - log(ISMR«3))j . For convenience now drop a and 0 from the nota- tion. We regard dj as an estimate of a parameter A, which is the expected value over institutions of the difference of the logarithms of the indi- rectly standardized mortality ratios for the two anesthetics. The argument in Appendix 1 to Chap- ter IV-6 enables calculation of the variance of dj due to Poisson variation in the deaths and ran- doms. We denote this variance of o-J. Least squares theory suggests the estimate: A = where l/o Wj = We do not choose to use the estimate A because it proceeds from a model that postulates no source of variation in the dj other than intra- institutional uncertainty. It is plausible that, if no variation of this kind were present at all (e.g., owing to "infinitely large" samples), there would still be variation in the values of the dj from one institution to another. So we consider the model: dj = where Cj is a fluctuation associated with peculi- arities at institution j and Sj is the intra-insti- tutional Poisson-induced fluctuation in that insti- tution. We further postulate that: Var and and that all the ej and 8j are uncorrelated. Fi- nally, and reluctantly, we suppose the Cj and Sj to be normally distributed. These assumptions yield the likelihood function: L = (dj -A)' J " " l-'o2*'? Differentiating log L = I with respect to the un- known parameters CTO and A, we obtain the equa- tions: - A)' = Q and 0=2 J (dj -A) (1) (2) +CT For given values of the dj and a j, these equations are readily solved, iteratively, by try- ing a value of v2 that yields a value of A in Eq. 2, and then putting both that A and a^ into the ex- pression Q. If that quantity is negative, a smaller value of a ^ is tried next. By successively dividing the intervals where Q is positive and negative, the root ffQ is quickljr found, and with it the solu- tion A. We may take A to be approximately nor- mally distributed. Its standard error is estimated in the ordinary way as: A + 370

The estimates CTO, A, and s.e.(A) were computed for those 28 of the 34 institutions where dj and a? were defined for all anesthetic pairs. TABLE 1 A. Values of A for all anesthetic pairs (middle- death-rate operations)(Figures in parentheses correspond to all-institution Ki'H's from Table 12 of Chapter IV-6)1 N-B C E 0 H 0.088 -0.202* -0.319 -0.224* (-0.009) (-0.299) (0.113) (-0.220) N-B -0.271* -0.380* -0.280* (-0.290) (0.122) (-0.211) C -0.096 0.019 (0.412) (0.088) E 0.079 B. Values of s'Te.(A) (0.333) N-B C E 0 H 0.091 0.063 0.167 0.082 N-B 0.099 0.177 0.098 C 0.185 0.122 E 0.149 Table 1 shows in part A the value of A. Each value of A has beneath it a figure in parentheses, which is log(ISMRRow) - log(ISMRCol) computed from the bottom row of Table 12 of Chapter IV-6. It is notable that the present analysis shows ether as less favorable in comparison with each of the other agents than do the figures in Table 12 of Chapter IV-6. This is similar to what happened with the two signed rank analyses, where ether compared adversely with halothane, nitrous oxide-barbiturate, and cyclopropane in contrast to the showing in Table 12. Part B of Table 1 ex- hibits the estimated standard errors of the values of A in part A. Where the magnitude of £ exceeds twice s\e.(A), it is starred in part A. 2 Table 2 displays the estimated values of aQ for the various pairs of anesthetics. It is nota- ble that all the largest values are associated with ether. It is also notable that for the halothane - cyclopropane comparison CTQ is estimated as zero (a negative value would be necessary to satisfy Eqs. 1 and 2). These facts indicate the instability of ether comparisons from one institution to another, and the stability of the halothane- cyclopropane contrast. Both of these are probably reflections of usage patterns. TABLE 2.—VALUES OF £n FOR ALL ANESTHETIC PAIRS N-B C E 0 1 Each figure is positive if the value of log (I3ffi) for the row anesthetic was greater than that for the column anesthetic. H N-B C E 0.100 0 0.104 0.436 0.495 0.558 0.058 0.096 0.248 0.281 371

APPENDIX 4 TC CHAPTER IV-6 SHOULD WE SUPERSTANDARDIZE AGENT COMPARISONS? GENERAL CONSIDERATIONS AND APPARENT CONCLUSIONS W. Morven Gentleman Bell Telephone Laboratories Murray Hill, New Jersey John W. Tukey* Princeton University Princeton, New Jersey Bell Telephone Laboratories Murray Hill, New Jersey Appendix 2 to Chapter IV-6 introduced the idea of superstandardization, regression adjust- ment beyond ordinary standardization of death rates, as a general tool, and as a specific means of improving the basis for comparing one hos- pital with another. Such objective checks as we were able to make in that appendix indicated that superstandardization was efficacious. Once we recognize superstandardization as a routine tool, it is natural to ask why it should or should not be applied to the comparison of anesthetic agents. It is the purpose of this appendix to explore the possibilities of that application, developing spe- cific techniques to the extent necessary. The general pattern of our discussion is to consider first what effect the sort of super- standardization thought to be appropriate for institutional comparisons, here called "complete superstandardization," has on agent comparisons, and why we should probably not adopt that ap- proach for agent comparisons. Then we turn to the question of less violent adjustment: how we might estimate how much adjustment to make, and what the consequences of an apparently rea- sonable adjustment prove to be. We shall conclude by indicating considerable changes in earlier results. As did the present versions of the last two sections of Chapter IV-6, this appendix developed relatively suddenly at a very late stage in the preparation of the report. As a result, it has not had the benefit of either continuing thought or high interstatistician discussion to a degree com- parable with many other parts of the report. Moreover, many of the issues that it raises are at a deeper level than seems reachable with the specific sorts of data gathered in this Study. Ac- cordingly, both its techniques and its conclusions are good objects for further inquiry. COMPLETE SUPERSTANDARDIZATION The simplest approach would take five values of standardized mortality rates for the five anes- thetic agents and superstandardize these values, calculating corresponding standard mortalities by dividing crude death rates by standardized mortality ratios. (The same thing can be done to otherwise adjusted values.) The adjustments thus made in the standardized mortality ratios tend to set a practical upper bound on the adjustments appropriate to reasonable forms of incomplete superstandardization. Accordingly, we look first at them and their consequences, as a preliminary to asking whether they are themselves appro- priate as a guide to whether any reasonable ad- justment of this sort could make meaningful changes in the results of our other analyses. The results of such computations for high- and low-death-rate operations and for all opera- tions combined are set out in Table 1. The effect of even complete superstandardization on the high- and low-death-rate groups is quite small, amounting, when we consider median results, to: low std: superstd: H 8.4 8.5 N-B 9.2 9.1 11.2 11.2 8.3 8.7 10.8 10.4 H 0.23 0.24 N-B 0.16 0.17 0.26 0.26 0.18 0.17 O 0.33 0.32 •Prepared In part in connection with research at Pririceton University sponsored by the Army Research Office (Durham). 372

TABLE 1.—STANDARDIZED (OR OTHERWISE ADJUSTED) AND COMPLETELY SUPERSTANDARDIZED DEATH RATES BY AGENT FOR HIGH-DEATH-RATE, LOW-DEATH-RATE, AND ALL OPERATIONS Source Standardized death rates, % Regr. Superatandardized death rates*** table* H N-B C E 0 Coeff** H N-B C E 0 High-death- rate opera- IV-8,3 \ IV-5,1/ 8.4* 9.14 10.97 8.11 10.79 +0.28 8.59 9.13 11.22 9.23 10.35 tions IV-5,3 8.54 9.23 12.58 8.30 10.84 -0.08 8.13 9.06 11.37 8.64 10.31 IV-5,5 8.39 9.03 11.66 9.88 10.85 -0.15 8.54 9.32 11.51 9.31 11.11 IV-6,1 8.27 9.31 11.16 8.19 10.60 +0.20 8.50 9.36 12.40 8.14 10.97 IV-6,4 8.62 9.20 11.18 9.08 10.44 +0.05 8.30 8.76 11.21 8.70 10.44 (Median of 5) (8.44) (9.20) (11.18) (8.30) (10.79) (-0.05) (8.50) (9.13) (11.37) (8.70) (10.44) Low -death- rate opera- IV-8,3\ . IV-5,1/ 0.25 0.16 0.25 0.18 0.33 -0.19 0.255 0.168 0.254 0.169 0.323 tions IV-5,3 0.23 0.16 0.26 0.18 0.34 -0.29 0.243 0.172 0.262 0.164 0.326 IV-5,5 0.228 0.187 0.271 0.166 0.324 +0.19 0.219 0.183 0.271 0.174 0.329 IV-6,1 0.23 0.16 0.26 0.18 0.33 -0.22 0.240 0.169 0.261 0.167 0.321 IV-6,4 0.217 0.185 0.302 0.174 0.304 +0.01 0.216 0.185 0.302 0.174 0.304 (Median of 5) (0.23) (0.16) (0.26) (0.18) (0.33) (-0.19) (0.24) (0.17) (0.26) (0.17) (0.32) All opera- tions IV-8,11 1.95 1.69 2.16 1.56 2.37 +1.16 2.00 1.91 1.75 1.80 2.17 IV-6,6/ IV-5,7 1.93 1.72 2.16 1.52 2.33 +1.15 1.98 2.00 1.77 1.72 2.11 (Median of 2) (1.94) (1.70) (2.16) (1.54) (2.35) (+1.16) (1.99) (1.96) (1.76) (1.76) (2.14) "Chapter, table number. **Regression of y = log (stdized death rate) on x = log(erude death rate) - log(stdized death rate). Note that y = log SUR + log over-all death rate (for group concerned), and x = log SM + log over-all death rate (for group concerned). ***Log(entry) = y - b(x-x), where b = regression coefficient and x = mean of the 5 x's. At most, there may be a. slight lowering for Other and a slight increase for ether in the high-death- N-B E rate group. For all operations combined, the effects are somewhat large, namely (again for medians): spread omitting O spread other std: superstd: 1.94 1.70 2.16 1.54 1.99 1.96 1.76 1.76 Here we see a considerable pulling together of the results, suggesting a still larger effect of com- plete super standardization on the death rates of the middle-death-rate operations. When we examine Table 2, we see that com- plete superstandardization of the death rates of H N-B std: superstd: 1.95 1.95 2.56 2.19 2.27 2.24 1.85 Here all agents but ether give completely super- standardized death rates of within 0.04 percent of 2.23 percent. Although the median value for ether is 1.85 percent, the completely superstand- ardized death rates for the two calculations the authors regard as most trustworthy (the robust estimators of Tables 7 and 8 of Chapter IV-4) give values of 2.37 and 2.48 percent. Accordingly, there is little if any evidence to deny (or to con- firm) that ether has a completely superstand- ardized death rate for middle-death-rate opera- tions that is also close to 2.23 percent. 2.35 2.14 0.81 0.38 0.62 0.23 the middle-death-rate operations effectively eliminates almost all agent-to-agent differences in death rates. Thus, there is no doubt that super- standardization could have meaningful effects on agent-agent comparisons. For the medians of the first five lines we find: 1.81 2.48 2.24 0.75 0.42 spread omitting ether 0.61 0.08 Because the data for the low-death-rate operations are of limited precision, whereas those for the high-death-rate operations involve a large amount of tailoring of specific anesthetics to specific operations, the main comparison of one agent with another must, for many purposes, be the comparison for middle-death-rate opera- tions. Superstandardization could make very large changes in agent comparisons, so we ask: "Is complete superstandardization appropriate in comparing agents? If not, how much superstand- ardization is appropriate?" 334-553 O-69—25 373

judgment might well correlate selection of agent with severity of case in such a way that stand- ardized death rates by agent would become highly correlated with the standard mortalities by agent that sum up the riskiness of specific operations on patients of specific age and anesthetic risk. This might well happen, whatever the true relative quality of the various agents and the correspond- ing appropriate adjustment for regression on log SM. A situation not far different from the one we actually face would arise if the agents fell into two subgroups such that all agents within a subgroup were approximately the same, in terms of both log SMR and log SM. Inasmuch as simple linear regression always removes one degree of free- dom from comparisons, complete superstandard- ization in such a situtation would lead to almost complete removal of agent differences, whatever the true state of affairs. (In Appendix 2 to Chapter IV-6, by contrast, our task was simplified by the existence of 34 hospitals, a large enough number for the removal of one additional degree of free- dom to be nonessential.) We cannot be sure that such things are not going on. Accordingly, we dare not use the naively apparent regression coefficient, and must seek out other ways to assess a more appropriate regression coefficient. (The results of complete superstandardization are not necessar- ily wrong, but merely subject to serious doubt.) If we are to estimate such a more appro- priate regression coefficient, it seems highly probable that we should turn to assessing a re- gression coefficient involving some other compar- ison of differences in log SMR with differences in log SM. At least three possibilities seem natural: (1) comparison of institution-to-institution differences (combined across agents), (2) comparison of institution-by-agent inter- actions, and (3) comparison of institution-to-institution differences within agents (which, when averaged over agent, turns out to be a combination of the first two comparisons). HOW SHOULD WE ESTIMATE? Before turning to the actual calculations, we must prepare for the fact that all the quantities with which we are concerned, including log SM, are measured with error. Thus, apparent and structural regressions will differ, and the actual amounts of error involved will affect our results. We will therefore not be able to borrow overt, apparent regression coefficients. Instead, we must try to estimate so-called structural regres- sion coefficients, in which the effect of more or less inevitable, meaningless fluctuations common to log SM and log SMR has been eliminated or, more realistically, greatly reduced. For a long time, nearly all discussions of simple linear regression between two variables considered only two components for each vari- able, one rigorously following the linear regres- sion, the other consisting of independent errors and fluctuations. Most discussions, indeed, re- stricted the possibility of the second component to only the y-variable, so that "regression with both variables subject to error" was often re- garded as a recondite subject. Already in this latter case there are two different regressions of y on x with different interpretations and dif- ferent areas of usefulness. If we have observed pairs (x,y) that consist of strictly linearly related values (f,i), modified by additive fluctuations x-f and y-i, then the observed variance of x and the covariance of x and y lead to an observed (or apparent) regres- sion coefficient cov(x,y)/var x. This is the re- gression coefficient most appropriate to predict- ing or approximating a value of y given the corresponding value of x. There is also the regression coefficient of TJ on f. If we could obtain estimates cov(f,i)) and var T) of the covariance and variance of these structural variables, their ratio cov(f ,*j)/var v would estimate the structural regression coef- ficient of y on x, which is the same as the actual regression coefficient of TJ on f. If, starting from a randomly selected (x,y) pair, we make a change in f and observe the corresponding change in x, the best estimate of the corresponding change in y is that based on the structural regression coef- ficient. This is so because in that situation, the best estimate of the change in ( is given by the change in x, whereas the average change in y is equal to the change in *). Thus, the regression coefficient of i on f is the appropriate regression coefficient to use. Notice that the best estimate of the difference in f 's is not given by the difference in x's when we are concerned with two randomly selected (x,y) pairs. Our present situation is complicated in two ways, beyond the usual "error in both variables" situation just described: (1) the relation of ij to f need not be precisely linear (TJ has a linear re- gression on f that may take up little or much of the variation, but not all); and (2) the independent fluctuations basic to the definition of f and i? are not in x and y, but rather in v = y + x and x. These complexities are partially compensated for by the fact that we have more than just pairs (x,y) to work with; indeed, we can estimate (rather crudely) what the fluctuation or sampling vari- ance of x appears to be. Appendix 5 to Chapter IV-6 treats the mathematical-statistical details involved in ob- taining an appropriate estimate for such a struc- tural regression coefficient. The formula reached there is: structural-b = cov(log SMR, log SM) + $ j var (log SM) - 5* 374

TABLE 2.—STANDARDIZED (OR OTHERWISE ADJUSTED) AND COMPLETELY SUPERSTANDARDIZED DEATH RATES BY AGENT FOR MIDDLE-DEATH-RATE OPERATIONS Source table* Standardized death rates, % Regr. Coeff** Superstandardized death rates*** H N-B C E 0 H N-B C E 0 IV-4,6 1.92 1.99 2.54 1.81 2.41 +0.64 2.13 2.27 2.18 1.85 2.17 IV-4,7 2.01 1.95 2.56 2.17 2.47 +0.53 2.19 2.10 2.21 2.37 2.24 IV-4,8 2.02 1.95 2.55 2.27 2.48 +0.44 2.16 2.06 2.24 2.48 2.28 IV-8,3\ 1.93 1.97 2.78 1.77 2.57 +0.99 2.22 2.33 2.34 1.74 2.28 IV-5,lf IV-5,1 1.95 1.95 2.79 1.79 2.52 +1.02 2.27 2.29 2.35 1.78 2.23 (Median of 5) (1.95) (1.95) (2.56) (1.81) (2.48) (+0.64) (2.19) (2.27) (2.24) (1.85) (2.24) IV-6,12 1.98 2.00 2.68 1.77 2.52 0.73 2.24 2.30 2.31 1.75 2.28 IV-5,3 1.93 1.97 2.78 1.77 2.57 +0.96 2.25 2.32 2.33 1.76 2.28 1.95 1.95 2.79 1.79 2.55 +1.09 2.21 2.35 2.27 1.89 2.25 IV-5,5 IV-6,1 IV-6,4 2.04 1.95 1.92 2.00 1.94 1.96 1.95 2.05 2.75 2.77 2.77 2.61 1.73 1.81 1.84 1.91 2.56 2.55 2.60 2.49 +0.74 +1.00 +1.15 +0.63 2.36 2.26 2.23 2.22 2.18 2.40 2.30 2.33 2.32 2.32 2.25 2.23 1.68 1.81 1.87 1.97 2.33 2.23 2.28 2.24 2.08 2.04 2.56 1.96 2.49 +0.52 2.30 2.25 2.22 2.03 2.27 IV-6,7 1.95 1.94 2.73 1.81 2.62 +1.04 2.27 2.27 2.23 1.81 2.35 2.00 2.05 2.61 1.91 2.22 +0.40 2.16 2.24 2.39 1.96 1.99 2.05 1.99 2.70 1.77 2.54 +0.67 2.34 2.24 2.35 1.74 2.31 1.98 1.89 2.82 1.69 2.63 +0.93 2.32 2.14 2.44 1.60 2.41 IV-6,8 2.04 1.94 2.75 1.73 2.56 +0.74 2.36 2.18 2.40 1.68 2.33 2.08 2.02 2.70 1.79 2.56 +0.63 2.36 2.26 2.36 1.77 2.34 2.01 1.93 2.78 1.66 2.60 +0.70 2.29 2.15 2.48 1.57 2.42 2.05 1.96 2.74 1.73 2.56 +0.69 2.35 2.20 2.40 1.68 2.34 (Median of 15) (2.00) (1.96) (2.75) (1.79) (2.56) (+0.74) (2.29) (2.25) (2.35) (1.77) (2.31) *Chapter, table number. **Regression of y = log(stdized death rate) on x = log(orude death rate) - log(stdized death rate). Note that y = log SMR + log over-all death rate (for group concerned), and x = log SM + log over-all death rate (for group concerned). ***Log(entry) = y - b(x-x), where b = regression coefficient and x = mean of the 5 x's. POSSIBLE DIFFICULTIES IN COMPLETE SUPERSTANDARDIZATION Once we have chosen the variable on which we are to regress the logarithm of the stand- ardized (or otherwise adjusted) death rate, we have separated the differences among the five agents in two complementary parts: a one- dimensional part corresponding to differences that can be explained (fitted) by regression, and a three-dimensional part that is entirely unaf- fected by the choice of coefficient in a regres- sion adjustment. Together these parts can de- scribe any instance of the 3 + 1 = 4-dimensional manifold of possible sets of differences among the five agents. The principal conclusion of the last section may be put thus: For the middle-death-rate operations, the three-dimensional part of the differences among agents contributes essentially nothing to either observed or adjusted differ- ences among agents. Thus, the whole question of agent comparison for the middle-death-rate op- erations is reduced to: "How much of the one- dimensional part (i.e., of the differences that regression accounts for) should be taken to be real and how much should be taken to be an arti- fact of the original analysis and removed by an appropriate regression adjustment?" The basic questions become questions about the numeric values of the regression coefficient to be used in the adjustment. And the concluding questions of the preceding section become: "Are we sure that we should use the regression coef- ficient based only on the standardized death rates for the five agents (which corresponds to com- plete superstandardization)? If not, where should we seek a better regression coefficient for this purpose?" Matters would be simpler if we could an- swer "yes" to the first question, but it is rela- tively clear that we dare not. Agents are se- lected with regard to the clinical pictures presented by individual patients. Anesthesiologic 375

where cov(log SMR, log SM) = the sample (or esti- mated) covariance between log SMR and log SM (across what- ever comparisons are being borrowed from), var(log SM) = the sample (or esti- mated) variance of log SM (across the same comparisons), and 2 £x = an estimate of the part of the variance of log SM that is wholly unrelated to log SMR (being mainly due to sampling fluctuations). This formula is easily modified (see Appendix 5) to apply to weighted analyses. In any case, we have to give serious attention to the relation of the numbers that we use for a^ to the numbers calculated on the assumption that deaths and randoms are subject only to Poisson sampling variability (see Appendix 1 to Chapter IV-6). We have chosen to state our results for two extreme values of the factor ffx/(Poiss-var log SM), namely, 1.0 and 1.5 (for whose choice see Appen- dix 5), thus allowing both mental adjustment to a chosen value of this ratio and clear insight into its importance in affecting the values of struc- tural-b. RESULTING VALUES OF STRUCTURAL-b The results of a number of different ap- proaches to the estimation of structural-b are collected for comparison in Table 3 of Appendix 1 to Chapter IV-6. We can hope that these diverse approaches will reach similar results, but we must be prepared for any sort of complexity. All these approaches refer to the middle-death-rate operations and are based on the 15-aggregation- strata standardization used in the last section of Chapter IV-6 and in the later analysis of Appen- dix 2, the best standardization for which detailed figures are available. The computation of these results is dis- cussed in detail in Appendix 6 to Chapter IV-6. The one-way analyses involve, respectively: (1) values of log SMR and log SM for institutions (based on pooled results for all agents) and un- weighted regression; (2) similar values for in- stitution-agent combinations, regressed sepa- rately within each agent (using weights appropriate to that agent), and the estimated structural re- gression coefficients averaged with equal weight for different agents; and differences, 8 log SMR in log SMR and 8 log SM in log SM, for a partic- ular pair of agents regressed over institutions (using weights appropriate to that pair), the re- sulting estimates of structural regression coef- ficients being averaged with equal weight for different pairs of agents. The two-way analyses are based on a condensation of the 34 institutions and five agents to 15 compound institutions and three compound agents, so chosen as to bring together cells with similar values of log SM and to make the weights appropriate to each of the 15X3 new cells close enough so that unweighted two-way analysis is relatively efficient. Making such a two-way analysis, collecting sums of squares for both log SMR and log SM, as well as sums of cross-products, enables us to calculate structural regression coefficients based on: (1) differences between composite agents (which we discard for varied reasons), (2) differences between composite institutions, or (3) interac- tions between these two composite variables. The results of the latter two choices are given in Table 3. Table 3 has two columns and five rows. The columns represent extreme alternatives concern- ing the actual variability (of logarithms) of stand- ard mortality. (The authors felt, before this table was prepared, that a factor much closer to 1.5 than to 1.0 was likely to be appropriate.) In inter- preting the table, we should interpolate between these columns, being guided both by the analyses of Appendix 1 and by the effect of this choice on the consistency of our results from row to row. We would hope that the results for the five rows would be in reasonably close agreement, because it is then that our problems of interpretation are simplest. When we look at the table we do see reason- able consistency. The three types of one-way analysis give, for a factor of, say, 1.5, estimates of 0.21, 0.18, and 0.16. (A factor of 1.6 would give 0.21, 0.19, and 0.20.) The two types of two-way analysis give, for a factor of 1.2, estimates of 0.28 and 0.31. It would thus be hard to exclude any value between 0.2 and 0.3, and it seems quite reasonable to adopt a value of 0.25. (As in so many other analyses, this is somewhat pred- icated on omitting ether, which behaves quite distinctively in the within-agent and between-agents analyses based on institutional differences.) Ac- cordingly, we shall shortly trace the consequences of adopting this value for structural ft. WHY MIGHT ADJUSTMENT BE SOUND? Before beginning to trace these consequences, we ought to ask whether there is any excuse for a super standardization-like adjustment in connec- tion with agent-to-agent differences. As usual, there are a variety of possible reasons. As noted in Appendix 2 to Chapter IV-6, we know that any standardization by strata, even one based on 15 aggregation strata, is far from per- fect. There will be need for further adjustment 376

TABLE 3.—COLLECTED ESTIMATES OF STRUCTURAL-b Source of estimate d£/(£oiss-var log SM) 1.0 1.5 0.13(0.10) 0.18(0.15) -0.04(-0.18) 0.16(-0.03) One-way analysis (based on 34 institutions and four or five agents): Institution-to-institu- tion differences for all agents combined 0.19 0.21 Institution-to-institu- tion differences within agent (Weighted; ether omitted except in parentheses) Insti tution-to-insti tu- tion differences in agent-to-agent differ- ences (Weighted; ether omitted except in parentheses) Two-way analysis (based on 15 compound institu- tions and three compound agents*): Institution-to-institu- tion differences 0.27 0.29 Interactions of insti- tutions and agents 0.20 0.47 *The 15 compound institutions were:17+28+30, 3+27+31+34, 1+6+14+26, 8+13+23, 20+21, 12+19+22+32, 15+16, 4+9, 5+10, 11, 7+29, 24+33, 18, 25, and 2; the three compound agents were: halothane alone, nitrous oxide-barbiturate combined with ether, and cyclopropane combined with Other. that, under plausible assumptions, will correlate quite highly with the adjustment already made by the use of the strata (i.e., with log SM). Notice, in particular, that we have not yet allowed for the fact that, for low- and middle-death-rate opera- tions, a patient x-ty-9 years old is subject to almost twice the death rate of patient x-ty-1 years old for x = 4, 5, 6, 7, or 8 (and probably for x = 3 or 9, also). In another direction, many variables not measured in this Study may contribute to the riskiness of operations. To the extent that these unmeasured variables are correlated, among agents, with those that were measured, there is a real possibility (as discussed , i Appendix 2) that the effects of the unmeasured variables will be correlated with the effects of the measured vari- ables in such a way as to require super standardi- zation-like adjustments. In a third direction, preferential assignment of higher-risk patients to certain agents, whether or not all the variables relevant to the risks were measured in this Study, can produce a need for a superstandardization-like adjustment. All in all, we find it quite reasonable that further adjustment according to structural ft = 0.25 compensates for real effects whose can- cellation is desirable. FIRM CONSEQUENCES OF STRUCTURAL ft = 0.25 Table 2 gives agent comparisons for a wide variety of analyses. We have selected three of these for careful consideration of the effects of partial superstandardization: the 15-aggregation- strata analysis of Table 12 of Chapter IV-6 and the robust smear-and-sweep analyses of Tables 7 and 8 of Chapter IV-4. All these analyses in- clude careful assessments of significance; all show that differences between certain pairs of agents are significant. Although based on different approaches, the results are rather similar. They are certainly among the most sensitive analyses of significance so far available. Table 4 shows the effect of adjusting stand- ardized death rates for a structural ft of 0.25. The changes are perhaps not strikingly large, but, as we shall soon see, their consequences are sub- stantial. Table 5 shows the more crucial effects of adjustment on the differences in death rates that are our prime concern. We see that agent- to-agent differences are reduced to between two-thirds and one-half of their previous values. If such a reduction were to occur without change in the "error variances" against which these dif- ferences are compared (actually or effectively) in significance tests, then most instances of weak, moderate, or quite strong significance would be converted into nonsignificance. The effect of adjustment for a structural ft of 0.25 on "error variance" is far from certain. Either no reduction or a quite substantial reduc- tion is quite possible. The most natural basis for a crude estimate is the reduction in error term in the two-way analysis involving compound in- stitutions and compound agents. Here a reduction of some 25 percent seems to be indicated. This would decrease standard deviations in the ratio (100 percent-25 percent)1/2 /(100 percent)1/2 = 0.866, corresponding to the change from "ratio" to "ratio*" in Table 5, as far as the ratio of dif- ferences to their standard errors is concerned. This still leaves us with reductions in (differ- ences in death rates)/(estimated standard error of same) by factors of three-fifths to four-fifths. What are the consequences of such a reduction? GUESSED (PLAUSIBLY INFERRED) CONSEQUENCES OF ADJUSTMENT Our proposed adjustment in "error variance" is itself subject to considerable uncertainty. Moreover, its application to any of the analyses 377

TABLE 4.--EFFECTS OF ADJUSTMENT FOR A STRUCTURAL & OF 0.25 ON DEATH RATES FROM THREE ANALYSES Source table* Standardized death rates, % H N-B C E 0 IV-6,12** IV-4,7 IV-4,8 (raw***) 1.98 2.01 2.00 1.95 2.68 2.56 1.77 2.17 2.02 1.95 2.55 (1.727) (1.709) (3.401) 2.52 2.47 2.27 2.48 (1.855) (2.988) Adjusted death rates , * IV-6,12** 2 .05 2 .08 2 .52 1 .75 2 .42 IV-4,7 2 .09 2 .03 2 .38 2 .25 2 .36 IV-4,8 2 .10 2 .03 2 .37 2 .16 2 .37 Relative standard mortalities IV-6, 12 1 .15 1 .17 0 .79 0 .95 0 .84 IV-4,7 1 .16 1 .14 0 .75 1 .17 0 .83 IV-4,8 1 .17 1 .14 0 .75 1 .22 0 .83 *Chapter, table number. **The over-all death rate of middle-death-rate operations, 2.207 (see Table DR-5 of Chapter IV-2), applied to the over-all standard mortality ratios at the foot of Table 12 of Chapter IV-6. ***From Chapter IV-2; for comparison and as used in adjustment. of the next-to-last section of Chapter IV-6 or the third section of Chapter IV-4 is subject to further question. Accordingly, any conclusions we may attempt to reach about the effect of adjustment on the assessed significance of agent-to-agent dif- ferences must be regarded as guesses, or per- haps plausible inferences. The applications to the analyses of Chapter IV-4 are relatively direct, so that only questions of applicability are important. However, because of the detailed nature of the analyses of Chapter IV-6, applications to these, to which we turn first, are quite indirect and subject to additional questions. Ratio* for these analyses is close to 0.77. Table 14, after selective elimination of ether, reaches a chi-square of 10.2 on three degrees of freedom. Although not justifiable, it is not unreasonable to guess by multiplying this by about (0.77)2, reaching a chi-square of 6.05, corresponding to p = 0.11. This is our nearest guess as to what reanalysis might give. In the rank analyses of Tables 16 and 17 of Chapter IV-6, there are 40 critical ratios derived from paired analyses with the aid of ranks. Some 15 (2+6+3+4) of these are starred as exceeding 2.00 in magnitude. If these critical ratios were to be shrunk in the ratio 0.77 by a reanalysis taking account of a structural ft of 0.25, which is not implausible, six (1+2+1+2) of them would still exceed 2.00. In 40 null trials we must find an average of 2 beyond the 5 percent point. Would 6 be strong evidence, where 2 must occur on the average? Even if the occurrences were independ- ent, 6 or more would occur in 1.4 percent of all cases. In the present situation, where there is a large amount of dependence, we could hardly re- gard 6 as strong evidence of the differences. (If we try to reduce the effects of dependence by going to individual sets of 10 comparisons, we find 1 or 2 observed when 0.5 must occur on the average. The corresponding tail areas are p= 0.40 and p = 0.086, again far from clear evidence.) Turning to the robust smear-and-sweep analyses of Tables 7 and 8 of Chapter IV-4, we can guess at the effect on the confidence inter- vals found there by (1) replacing the standardized differences that determine the centers of the confidence intervals with adjusted differences, and (2) shortening these confidence intervals to 87 percent of their former length. The results are shown in Table 6. Whereas the original analyses agreed that three agent-to-agent differences in standardized death rate were significant, and that the fourth came 378

TABLE 5.—EFFECTS OF ADJUSTMENT FOR A STRUCTURAL 0 OF 0.25 ON DEATH-RATE DIFFERENCES FOR THREE ANALYSES Difference in death rates Source table Item C minus H 0 minus H C minus N-B 0 minus N-B Chapter IV-6, Table 12 Std 0.70% 0.68% 0.52% Adj 0.47* 0.37* 0.44* 0.34* Ratio 0.67 0.69 0.65 0.65 Ra*tio* 0.77 0.80 0.75 0.75 Chapter IV-4, Table 7 Std 0.55* 0.29* 0.46* 0.27* 0.61* 0.35* 0.52* 0.33* Ratio Adj 0.53 0.59 0.57 0.63 Ratio* 0.61 0.68 0.66 0.73 Chapter IV-4, Table 8 Std Adj 0.53* 0.27* 0.46* 0.27* 0.60* 0.34* 0.53* 0.34* Ratio 0.51 0.59 0.57 0.64 Ratio* 0.59 0.68 0.66 0.74 *Ratio adjusted for a possible reduction in error variance of 25 percent associated with adjusting the death rates. very close to significance, the guessed results for a similar analysis based on adjusted rates are very different. Only one difference barely reaches significance, and that is only one of two analyses. (A further notable aspect is the sub- stantial guessed reduction in the upper limits of the confidence intervals. The averages for the two original analyses are 1.02, 0.78, 1.18, and 1.08 percent. The guessed replacements are 0.70, 0.55, 0.85, and 0.85 percent.) It is fair to say that our best guess as to the effect of adjustment for a structural ft of 0.25 on all three analyses is that it would bring the assessed significance of all differences below the usual levels. (Some readers may prefer some other value of structural ft, larger or smaller than 0.25. The note on Table 6 may help in assessing the con- sequences.) WHERE DO WE STAND? Adjustment for a structural ft of 0.25 (which we do not claim to have assessed with high pre- cision, and whose appropriateness is subject to discussion) reduces agent-to-agent differences for the middle-death-rate operations, and seems likely to reduce them below the usual levels of significance, insofar as can be guessed from the computations presented here. In preparing to summarize the over-all posi- tion, it is especially important to recognize that the general agreement of unsuperstandardized agent comparisons, as to order of apparent death rates and their apparent significance, is hearten- ing, because it indicates that different approaches have educed quite similar results from the same data, thus raising our confidence in their com- mon answer to a certain question. If, however, adjustment for an appreciable structural ft is appropriate, all these analyses are answering an inappropriate question. Only appropriately ad- justed analyses would then be relevant. A reasonable summary, then, would seem to be as follows, considering that we regard a limited superstandardization-like adjustment as appro- priate: (1) The sizes of the apparent agent-to-agent differences in death rates found for the middle- death-rate operations elsewhere in this report should be reduced to about three-fifths of the values stated, although one cannot be entirely sure of the appropriateness of this reduction. (This leaves cyclopropane and Other with apparent 379

TABLE 6.--GUESSED EFFECTS OF ADJUSTMENT FOR A STRUCTURAL 0 OF 0.25 ON THE CONFIDENCE INTERVALS OF THE ANALYSES OF TABLES 7 AND 8 OF CHAPTER IV-4 Confidence interval, % Source table Difference Original Modified (guessed) Chapter IV- C minus H 0.03 to 1.06(») -0.14 to 0.74 4, Table 7 0 minus H 0.10 to 0.82(«) -0.04 to 0.58 C minus N-B 0.03 to 1.18(») -0.15 to 0.85 0 minus N-B -0.05 to 1.08 -0.16 to 0.85 Chapter IV- 4, Table 8 C minus H 0.09 to 0.98(») -0.12 to 0.66 0 minus H 0.17 to 0.74(*) +0.02 to 0.52(») C minus N-B 0.05 to 1.17(») -0.15 to 0.83 0 minus N-B -0.02 to 1.08 -0.14 to 0.82 Note: A rough idea of the consequences of choosing different structural fi's is obtained if we tate changes in agent-to- agent differences proportional to structural 8. Thus, if we take 0 = 0.10, the lower limits of the modified (guessed) confidence intervals above would be, in order: -0.04, +0.04, -0.04, -0.09, +0.01, +0.11, -0.03, and -0.07 percent, so that three of the eight comparisons would prove significant. death rates for middle-death-rate operations of about 7/6 of those for halothane and nitrous oxide - barbiturate.) (2) The significance, at the usual levels, of all differences between agents for middle-death- rate operations is subject to question. (3) Further analyses could throw some light on the situation (as by replacing the guesses of the preceding section with facts), but questions about plausibility of different possible sources for a need for the further adjustments made in this appendix are likely to require a more subtle and comprehensive study for their elucidation. (4) If the limited superstandardizing adjust- ment presently considered reasonable proves to be unsound, it is possible either that the results obtained elsewhere should be accepted without adjustment or that larger adjustments than here contemplated would be appropriate. (5) Although the high-death-rate operations are distinctive enough to require individual con- sideration, we do need to recall that, in the pro- portions used during the period of this Study, halothane does appear to show a more favorable death rate than, for example, cyclopropane, what- ever superstandardization-like adjustment we may contemplate. What conclusions concerning the relative safety of, for example, halothane and cyclopropane would be justified if the adjustment reached in this appendix were accepted and the guessed ef- fects on other analyses proved to be correct? There is only one: that the apparent difference in death rates after the use of, for example, halo- thane and cyclopropane, as applied with much guidance from anesthesiologic judgment, appears to be smaller than might otherwise have been thought, and in fact is no longer statistically sig- nificant at the usual levels of significance (although it is rather close to significance). In inter- preting these changes, we need to be keenly aware that we are not directly measuring the effect of changing the anesthetic given to a patient, either individually or on some average (a retrospective study cannot do that), but rather are measuring something relevant to this effect but subject to disturbances of unknown size and amount. The discussions of this appendix lead to smaller (and less statistically significant) suggested differ- ences; they do not, however, change the direction of any suggested difference or appreciably reduce the uncertainties about the gap between what is suggested by our analyses and what changing anesthetics would do in practice. What death-rate evidence is at hand still favors halothane over cyclopropane. The scope of the Study and the delicacy of its analysis still go far beyond anything previously available. The uncertainties in interpretation remain. 380

APPENDIX 5 TO CHAPTER IV-6 DEVELOPMENT OF CERTAIN FORMULAS W. Morven Gentleman Bell Telephone Laboratories Murray Hill, New Jersey John W. Tukey* Princeton University, Princeton, New Jersey and Bell Telephone Laboratories, Murray Hill, New Jersey In this appendix, we develop a number of for- mulas needed in the considerations of Appendix 4 to Chapter IV-6. These formulas all relate to the regression of log SMR on log SM, where SMR is a ratio of observed deaths to standard deaths and SM is a standard death rate. THE EFFECT OF ERRORS IN ALL VARIABLES Let us define v, x, and y as follows: y = log SMR (or something differing from this by a constant), x = log SM (or something differing from this by a constant), and v = log observed death rate (or something differ- ing from this by a constant). Then v = y + x, and fluctuations due to sampling will affect v and x (not y and x) nearly independently. That will be so because x depends on (1) the distribution of cases brought to operation with the cell or stra- tum involved, and (2) deaths in all strata or cells, whereas v depends on deaths in only the stratum or cell concerned. Let, then, a2 = contribution to the variance of v from sampling fluctuations, CTx = contribution to the variance of x from sampling fluctuations (and from other sources not described by r2 and y), 2 T - contribution to the variance of v from long-run differences in strata or cells, and 7 = rate (regression coefficient) by which these long-run differences in v contribute to x. Because y = v - x, it follows that 2 22 var x = o + y T , x v cov(x,v) = yr2 , and 2 2 var v = o- + r . V V Consequently, cov(y,x) = cov(v-x,x) = yr2 - y2r2 . CT2 , and the apparent regression coefficient of y on x is 22 2 (7-7 )r - a 22 2 y T + a V x 222 2 22 Now, if we put Tx = T Tv and 0r^ = (y-y )ry, so that 0 = (y-y )/y2, these become: 0T2 - erx = apparent covariance of y and x, 2 2 T + a = apparent variance of x, and xx = apparent regression coefficient. 2 2 If we had a nearly infinite amount of data for each stratum or cell, the CTx terms could be neg- lected, and the apparent regression of y on x would be 0. Thus, 0 is the structural regression of y on x. If ax cannot be neglected, but ax is a reasonable estimate of it, then an appropriate estimate of 0 is: structural-b = Sov(y,x) + &j var x - &? which is worth considering only when var x - & > 0 (by an appreciable fraction of &2). Notice that, •Prepared in part in connection with research at Princeton University sponsored by the Army Research Office (Durham). 381

because a negative value of cov(y,x) can be out- weighed by a positive value of &£, structural-b may have the opposite sign to observed regression coefficient = ' A var x where rw and WEIGHTED ANALYSES When, as is so often the case, the variabi- lities of the different y differ in a known or esti- mable way, and when these differences are sub- stantial, unweighted combination of various differences is of little merit, because of low ef- ficiency. In such situations, we wish to take advantage of weighted combinations, which will be helpful (as far as efficiency goes) if the ratios of the weights are even roughly correct. To deal with the extension of the formula just given to weighted analyses, we need to know a few algebraic-arithmetic facts about the usual expressions arising in a weighted analysis. Let WSS WSSx = - yw) and where sw, and xw = are the similarly weighted means of the y's and x's. Also, let WSP = WSV= j (yj - yw)(xj - x^ and for which the requirement on &2 will be stated shortly. If we return to the sort of model considered above, allowing the a2 's to depend on i and replac- ing, for example, rx by individual T i 's, one for each i, we may write: ave x = TT, var = cr xl , ave v j = Ti + 2 var vj = ffyl and are the corresponding weighted means. Algebra, postponed to the last section of this appendix, then shows that ave 'jtf xl = var x is a sufficient condition for structural-b = WSP + WSV WSSx - WSV to be an appropriate estimate of the structural 0 relating y to x. This, then, is the weighted rel- ative of structural-b = Sov(y,x) var(x) - &2 Notice that, in the weighted case, we have altered both numerator and denominator by the same fac- tor, thus replacing means by weighted sums and slightly simplifying the arithmetic. PROPER ADJUSTMENT FOR AGENT MEANS Having found an estimate for the structural 0, how should we use this estimate in adjusting agent death rates or agent SMR's to allow for differences in the corresponding SM's (standard death rates)? This question is not a trivial one. It can be answered easily in quite specific situ- ations, but the answer in different specific situ- ations is different. Suppose, first, that yj = log SMR for unit i and x i = log SM for unit i, and that the sort of detailed model used above in connection with the weighted analysis applies di- rectly in the sense that the set of units (for this analysis, agents) is fixed. We can again put ave x and where (TI; yr^ represents the related part of V and xj , and ici is an additional part that we can appropriately assume to satisfy (Tj - = 0, so that ave vj = T+ 1-7 382

is the reflection among the x's and y's here studied of the structural 0 among the x's and y's from which such an estimate is borrowed. It need not, however, now be true that the *j, which are the quantities we are trying to estimate by adjust- ment, are unrelated to the TI in any particular sense. Now, ave y = ave v - ave x = T + - yr - I-T — and ave fe -¥,»)-.., so that adjustment using structural-b, which esti- mates (l-Y)/7, is appropriate. If, instead, the i's had corresponded to a sample from a bivariate (Gaussian) population in which var then var >PI = <P , and .V) = 0, ave yl = (1-7) ^ + ^ and the average of iPI among the i's yielding a given observed yj would be As more detailed analysis would show, we would be led to use, not structural-b itself, but structural-b reduced by a factor estimating In dealing with SMR's for agents, we are con- cerned with quite specific units, so that the first result "Use structural-b directly" applies. ESTIMATING ffxl We turn now to the calculation of estimates ffxl of the individual a^ , which in our case is c*xl = sampling variance of log SM (sampl-var log SM). Because log SM = log(std deaths) - log EE, it is clear that a large part of the assessment of sampl-var log SM is taken care of by the assess- ment of sampl-var log (std deaths), which was considered in Appendix 1. The analysis there was conducted in two stages: first, the assessment of the sampling variance of D = std deaths on the assumption that all the observations followed Poisson distribu- tions; and second, the assessment of the factor by which this result needed to be increased to make it applicable to the observed data. To make our calculations a little more transparent, we shall use "Poiss-var" for variances based on the Poisson assumption and "Poiss-cov" for analo- gous covariances, thus distinguishing these from "sampl-var" and "sampl-cov," which apply to actual sampling fluctuations (and from "var" and "cov," which would apply to all forms of difference among the quantities considered). Simple algebra leads to this relation: Poiss-var log SM = Poiss-var log 6 # of randoms' where SM, D, and the number of randoms all apply to the unit with which we are concerned (whatever its nature). In Appendix 1 to Chapter IV-6 various esti- mates of are made, ranging from 1.40 to 1.05. As a broad basis for analysis, we might well consider a range of factor from 1.0 to about 1.4. This is conveniently done by considering = (1.0 to 1.5) (Poiss-var log D # of randoms where Poiss-var log D is a reasonable estimate of the Poisson-sampling variance of log D. Here, the use of 1.5 corresponds to expansion factors ranging from 1.35, when the negative term is 30 percent of the preceding term, to 1.45, when it is 10 percent. (Most of our instances fall between 10 and 30 percent.) THE POISS-VAR OF LOG SM Using a notation somewhat analogous to that used in Appendix 1, we shall write, for any one unit that concerns us. provided that it is part of a single institution and thus shares a common blowup factor: h = stratum identifier, Rh = number of randoms observed in stratum h for the unit concerned, 383

k = "blowup factor" for the given unit, ph = the estimated death rate for stratum h, which we will treat as given (thus making the approximation found to be reasonably satisfactory in Appendix 1), E = estimated number of exposed (or administrations) in the unit concerned, and D = number of standard deaths for the unit. The usual relations apply, namely: E = kZRhi . D = kZphRh, and Poiss-var Rh = Rh, where = means "equal on the average." Accordingly, Poiss-var D = k Zph-Poiss-var Rh Poiss-var E = k 2-Poiss-var Rh 2 = k22Rh= kE, and Poiss-cov(D.E) = k22p -Poiss-var R l b = k22p R = kD, h h so that we may use the right-hand values as na- tural estimates of the left-hand quantities. Using the usual relations for "propagation of error," we find: Poiss-var log D Poiss-var log E * ?oiss-cov(log D, log E) = = $. and E' Accordingly, Poiss-var log SM = Poiss-var(log D-log E) = £oiss-var(log D) - 2 & and, because k k E k2Rh # of randoms for whole unit ' we obtain the desired result: Poiss-var log SM = Poiss-var(log D) # of randoms THE WEIGHTED ALGEBRA Returning ^o the second section of this ap- pendix, where we introduced WSSx = and WSP = zwj (yj - yw) it is easy to show, by simple algebra, that these single- sum expressions for WSSx and WSP are identically equal to suitable double- sum expres- sions, namely: WSSx= and wsp = where i j w, = 2w. These identities make it easy to see how the average values of various weighted expressions are related to the weights and to the assumed constituents of x and y, as also given above. Building backward from later to earlier expressions, to keep the algebraic steps more clearly visible, we have, using the vanishing of covariances among the x's and v's, and the re- lation yi = x| - vj! £- £ ave(x!-xj) = [ave(xj-xj)] + var(xj-xj) ,222 = (yr- - yr.) + a + a - + "xI+ ffxj 384

ave[(Vj -Vj) (xj-xj)] = avefy - vj)ave(xl -xj) y(Tl - TJ ) + (TI - Tj)2; and ave[(yj -y^ (xj - xj )] = ave[(vj -vj) (xi -x^] - ave )2 - v2( -T,)2 -a2, j' xl 2 22 - (axl +axj Moreover, relating the double-sum forms to the single-sum forms, we have: 2 2 . 1 1 J ZW i W^ and + .r where we have used the defining constraint on the >P'S. Accordingly, using the double-sum forms and the constancy of the w's, w i J i j and and ave WSP = -g^- ^ ^ WjWj-ave [ fy - y] ) (xj - " i j Whence, setting w+T2 = 2Wi(Tl-rw)2 385

we have: ave WSSV = y2w+r2 + w a2 , * T x ave WSP = (y-y2)w+ 2 - w+ a2 , and 22 2 ave WSP (y "y )T - ffx Again we see that the structural 0 is (y -y2)/y2 and that, if aveWSSx 22 , then ave (WSP + w+d2)= (y -y2)w+ r2 and ave (WSSx- w+62)=y2w+T2 , so that, WSP + w£2 WSSV - wi(r2 x + x is an appropriate estimate of the structural ft as long as its denominator is not too small. If d2 is an estimate of a2 , in the sense that xl xi ave S2 =ff2 , xl xl' and if we put then ,,,<„, "2 2 2 ave WSV = ave 2wiffxl = ^WJCTxI = w+ax, so that we can put A2 -WSV . and use WSP+WSV . . , = structural-b wssx-wsv as a natural estimate of structural p. 386

APPENDIX 6 TO CHAPTER IV-6 ASSESSMENT OF VARIOUS STRUCTURAL REGRESSION COEFFICIENTS W. Morven Gentleman Bell Telephone Laboratories Murray Hill, New Jersey John W. Tukey* Princeton University Princeton, New Jersey and Bell Telephone Laboratories Murray Hill, New Jersey This appendix discusses the calculation of various structural regression coefficients for the regression of log SMR on log SM, where SMR is a ratio of observed deaths to standard deaths and SM is the corresponding standard death rate. These coefficients are needed for the considera- tions of Appendix 4 to Chapter IV-6. The calculations used here involve formulas developed and discussed in Appendix 5, partic- ularly £ov(log SMR, log SM) + a I structural-b = var(log SM) - a ^ (and its relatives) and cJx = (1.0 to 1.5) (Poiss-var(log std deaths)) 1 (# of randoms) - For the particular case where we are concerned with a weighted one-way analysis, the relative of the first of these formulas of greatest convenience structural-b = where WSP + WSV WSS, - WSV ' WSP = sum of (weights) X (log SMR - *) (log SM - **), WSSx= sum of (weights) X (log SM - **)2, WSV = sum of (weights) X (S2. for individual unit), * = weighted mean of log SMR, and ** = weighted mean of log SM. Note that WSV still contains the adjustable factor of 1.0 to 1.5. In giving details for various approaches, we shall follow a standard order for rows or col- umns, as set out in Table 1. TABLE 1.—STANDARD ORDERING AND NUMBERING FOR RO»S OH COLUMNS OF TABLES LEADING TO ESTIMATES STRUCTURAL^ OF THE STRUCTURA1 /9 Row or column Wsighted sum of (1) Unity* (2) Product of deviations of log SMR and log SM from thsir means (3) Squared deviation of log SM from its mean (4) Estimated Poisson variance of log CM (5) Numerator "for 1.0" = (2) + W (6) Denominator "for 1.0" = (3) - (4) (7) structural-b "for 1.0" = (5)/(6) (8) Numerator "for 1.5" = (2) + 1.5(4) (9) Denominator "for 1.5" = (3) - 1.5(4) (10) structural-b "for 1.5" - (8)/(9) »In equally wsighted analyses (often called "unwsighted"), an entry of n indicates the use of simple sums of squared deviations or cross-products. "for 1.0" means J* = Raiss-var log SM (see Appendix 5 to Chapter IV-6). "for 1.5" means , t2 = 1.5 X Poiss-var log SM (see Appendix 5 to Chapter To assess how much of the naive superstand- ardization it is appropriate to apply to agent death rates, we face a nontrivial problem in se- lecting a source from which to borrow. Several approaches are discussed. FIRST APPROACH One place from which to borrow regression coefficients for use in adjusting agent compari- sons for SM values is institution-to-institution differences. In Appendix 2 to Chapter IV-6, these differences have been studied for two basic standardizations: (1) the rather weak standardi- zation based on age and shock-likelihood index, and (2) the stronger standardization based on 15 aggregation strata. Only the second is available «Prepared in part in connection with research at Princeton University sponsored by the Army Research Office (Durham). 387

TABLE 2.--CALCULATION OF STRUCTURAL b BASED ON (UNWEIGHTED) INSTITUTION-TO-INSTITUTION DIFFERENCES (AS TREATED IN APPENDIX 2 TO CHAPTER IV-6 (1) unity cross-product of deviations squares of deviations for log SM Poiss-var (log SM) 34 (2) (3) (*) 1.256 8.631 0.308 (5) numerator = (2) + (4) denominator = (3) - (4) structural -b for 1.0 = (5)/(6) numerator = (2) + 1.5(4) denominator = (3) - 1.5(4) structural -b for 1.5 = (8)/(9) 1.564 8.323 0.19 1.718 8.169 0.21 (6) (7) (8) (9) (10) Note: If structural 0 were zero, and 1.5 were an appropriate factor, the regression of y on x would be estimated by - (l.5)(0.308)/(8.169) = -0.06. THIRD APPROACH If we are willing to direct our attention to the comparison of only two agents at a time, we can take the view that we have 34 values of the log SMR difference, & log SMR = (log SMR for agent 1) - (log SMR for agent 2), which are to be adjusted in view of the corre- sponding 34 values of the log SM difference, b log SM = (log SM for agent 1) - (log SM for agent 2). We now need to use for the middle-death-rate operations separately; it provides the results set out in Table 2. There is, however, reasonable ground for suspecting that the structural regressions be- tween institutions and between agents may not be the same. Accordingly, we would not like to give too much weight to the results of borrowing a regression coefficient from institution-to- institution differences alone. SECOND APPROACH Let us select any one agent, say, halothane, and consider the 34 sets of results for the 34 institutions. We can try to assess a regression coefficient of log SMR on log SM here, and it may be appropriate to borrow such a regression coef- ficient to adjust agent SMR's, especially if such assessments within different agents lead to con- sistent regression coefficients. In doing this, it will be essential (1) to take account of the exist- ence of a2. and try to estimate structural/S's, and (2) to take account of differences in (estimated) variance of the various values of log SMR. We have learned (Appendix 5 to Chapter IV-6) how to cope with the first problem. Because we have only a one-way table, a weighted analysis can cope effectively with the second problem. Table 3 sets out the results found for each of the five agents using wt = 170 estimated variance of log SMR The results vary somewhat from agent to agent, so that, because we will want to treat ether sepa- rately in the next approach, it seems reasonable to give a pooled result for all agents except ether, as well as for all agents including ether. var 6 log SMR = £ var (log SMR for agent i) and i =1 var & log SM = £ var(log SM for agent i), i =1 thus leading to a new set of weights and a new adjustment for a2. The results for all 10 pairs of agents, separately and combined, are given in Table 4. We see that any combination including ether is persistently negative, whereas almost all other pairs give a positive regression, at least when 1.5 is used. Accordingly, it seems desirable to present separate results with and without ether, as well as for all pairs combined. FOURTH APPROACH We now turn to the question of a reasonable two-way analysis. The central unsolved problem is what to do with a rather irregular pattern of weights. The situation we must face is shown in Table 5, where the rough weights wt = nearest integer to are shown, with zeroes replaced by hyphens for emphasis. Clearly, something must be done if we are to have a relatively straightforward analysis. The simplest thing that we can do is to com- bine both rows and columns in a reasonable way. For the columns (agents), we would like to keep halothane separate, combine nitrous oxide - barbiturate with ether (as the apparently better two of the remaining four), and combine cyclo- propane and Other (as the apparently worse two). This gives total weights of, approximately, 123, 122(= 91 + 31), and 143(= 64 + 79), which are 388

TABLE 3.— RESULTS FOR THE VARIOUS AGENTS ANALYZED SEPARATELY H N-B C E 0 Combined (all) Combined (omitting E) fWeie ited sums) 95.7 (1) unity 130.5 91.7 34.7 82. 9 435 5 400.8 (2) £oiss-var log SM 2.884 2.611 2.894 2.064 3. 169 13 622 11.558 (3) cross-product 1.554 3.212 -6.128 -3.864 5. 127 -0 099 3.765 (4) (log SM-mean)2 34.951 27.236 46.015 ing values -3.234 16.608 28. 000 152 81 136.20 ( 5 ) numerator* 4.438 5.823 -1.800 8. 396 13 623 (15.423) (6) denominator* 32.067 24.625 43.116 14.544 24. 831 139 183 (124.639) (7) structural-b (for 1.0) 0.14 0.19 -0.07 -0.12 0. 33 0 .10 (0.13) (8) numerator** 5.880 7.1285 -1.787 -0.768 9. 880 20 334 (21.102) (9) denominator** 30.625 23.3195 41.674 13.512 23. 247 132 .377 (118.863) (10) structural-b (for 1.5) 0.19 0.31 -0.04 -0.06 0. 42 0 .15 (0.18) *0f expression for **0f expression for estimated estimated structural-b, using 1.0. structural-b, using 1.5. TABLE 4.— RESULTS FOR ALL 10 AGENT PAIRS, SEPARATELY AND TOGETHER (COLUMN NUMBERS ASSIGNED ACCORDING TO TABLE 3, BUT INVOLVING 8LOGSMRAND8LOGSMIN PIACE OF LOG SMR AND LOG SM, RESPECTIVELY; VALUES OF STRUCTURAL-b IN COLUMNS (7) AND (10) FOR FACTORS OF 1.0 AND 1.5, RESPECTIVELY) Agent pair (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) H-N-B 134.44 -8.54 26.24 8.00 -0.54 18.24 -0.03 3.46 14.24 0.24 H-C 135.01 -8.64 40.33 8.34 -0.30 31.99 -0.01 3.S7 27.82 0.14 H-E 61.10 -13.36 35.13 6.32 -7.04 28.81 -0.25 -3.88 25.65 -0.15 H-0 126.61 -7.46 29.77 9.07 1.61 20.70 0.05 6.15 16.16 0.38 N-B-C 104.05 -9.48 42.89 7.95 -1.53 34.94 -0.04 2.45 30.96 0.08 N-B-E 53.07 -19.90 28.48 6.23 -13.67 22.25 -0.61 -10.55 19.13 -0.55 N-B-0 100.09 -5.40 21.81 8.68 3.22 13.13 +0.24 +7.62 8 .79 0.87 C-E 47.58 -17.78 36.63 6.10 -11.67 30.53 -0.38 -8.63 27.48 -0.31 C-0 113.12 -17.49 34.21 8.78 -8.71 25.43 -0.34 -4.32 11.09 -0.39 E-0 54.77 -12.41 27.95 6.70 -5.71 21.25 -0.26 -2.36 17.90 -0.13 Combined* No E 713.32 -57.01 195.25 50.82 -6.25 144.43 -0.04 19.16 119.02 0.16 E 216.52 -63.45 128.19 25.35 -38.09 102.84 -0.37 -25.42 90.16 -0.28 All 929.84 -120.46 323.44 76.17 -44.34 247.27 -0.18 -6.26 209.19 -0.03 *Weighted sums merely added together; corresponds roughly to weighting of mean squares, etc., accordance with column (1). No E = sums for all pairs not including ether; E = sums only for pairs including ether. in 334-553 O-69—26 389

TABLE 5.—ROUGH WEIGHTS, BASED ON EST VAR LOG SMR, ASSOCIATED WITH AGENT-INSTITUTION CELLS FOR MIDDLE-DEATH-RATE OPERATIONS (ONLY RELATIVE VALUES ARE IMPORTANT; - = 0) ins u. Agent H N-B C E 0 1 5 3 1 2 10 4 4 1 3 3 1 4 . - 2 4 3 3 3 1 4 5 5 4 4 - 2 6 3 4 . 1 3 7 4 2 2 1 4 8 8 3 - 6 3 9 4 4 1 5 1 1D 2 4 3 1 1 11 5 1 2 4 4 12 5 1 4 - 2 13 3 1 4 - 5 U 1 1 2 - 2 15 2 3 2 2 2 16 2 7 1 - 4 17 2 1 1 1 4 18 4 3 4 1 2 19 7 4 1 - 2 20 1 2 4 3 4 21 5 4 1 - 2 22 2 1 2 - 1 23 1 2 1 - - 24 4 11 2 1 1 25 6 3 4 1 7 26 4 3 - — - 27 2 1 2 1 1 28 1 - - - 29 4 1 1 - 1 30 6 2 1 - 1 31 2 1 3 - - 32 4 1 1 - 2 33 4 2 1 1 3 34 1 3 - 5 much more nearly balanced. For the rows (in- stitutions), it seems natural to rearrange in order of values of SM (or log SM) for the institu- tion as a whole before combining institutions and to avoid making too many combinations. These decisions lead to the results set out in Table 6, where we have shown in each cell the sum of the entries in the cells of Table 5 that have contributed to it. (This will be, roughly, the weight of this cell when properly calculated.) The over-all efficiency, for unweighted anal- ysis, of this aggregation can be estimated as: (# of cells) /(total wt) = sum of (1/wt) = ?8 pe ce ' The 15 compound institutions involved seem to be enough to make the analysis otherwise satisfac- tory. Accordingly, we adopt the aggregation set out in Table 6 as the standard for this approach. To deal with an analysis based on these ag- gregated cells, we have to sum the following quantities for the constituent cells of each ag- gregated cell: EE = estimated administrations, D = actual deaths, D = standard deaths, var 6 = estimated variance of D, and R = number of randoms involved. We can then calculate, for each aggregated cell, the various other quantities of interest, such as: SMR = D/6, SM = D/EE, var log D = , var log SMR = var log D + var log D, and var log SM «- var log 6 - (1/R). Note that the use of 1/R as an adjustment term in the expression used for var log SM is only an approximation, because we are combining across institutions, and hence across k's. Test calculations indicate, however, that the approxi- mation is good enough. Institutions H N-B/E 0/0 (Range of lo ge 100 ad 17+28+30 9 4 7 (0.52 to 0.97) 3+27+31+34 6 7 16 (1.20 to 1.29) 1+6+14+26 13 12 8 (1.41 to 1.61) 8+13+23 12 12 13 (1.79 to 2.03) 20+21 6 9 11 (2.04 to 2.10) 12+19+22+32 18 7 U (2.12 to 2.28) 15+16 4 12 9 (2.50 to 2.58) **» 7 13 9 (2.60 to 2.65) 5+10 7 9 10 (2.76 to 2.81) 11 5 5 6 (3.02) 7+29 8 4 8 (3.40 to 3.48) 24+33 8 15 7 (3.59 to 3.68) 1t 4 4 6 (3.84) 25 6 4 U (6.35) 2 10 9 7 (6.40) TABLE 6.—CHOSEN COVBINATIOKS OF HOIS AND COLUMNS, AND RESULTING HOUGH WEIGHTS Note: f of cello = 45, total wt - 388, sum of (1/wt) - 6.72. Over-all efficiency = i|5)' , = 78 percent. 390

We can now calculate various sums of squares and cross-products, and various sums for the unweighted two-way analysis of three compound agents by 15 compound institutions. The resulting values are shown, in mean form, in Table 7. The same data, with the addition of values of var log SMR, can be used to estimate the reduc- tion in variance when "log SMR" is replaced by "log SMR adjusted for log SM." The resulting reductions are 25 and 30 percent for compound institutions and 5 and 25 percent for interactions, the two values in each pair corresponding to fac- tors of 1.0 and 1.5, respectively. Accordingly, an estimated reduction of 25 percent seems to be ample. TABLE 7.—RESULTS OF TWO-WAY ANALYSIS ON AGGREGATED CELLS Compound agents Compound institutions Interactions (1) Degrees of freedom (2) Mean cross-product (loggSMR with loggSM) (3) Mean square (of loggSM) (4) Mean est Rjiss samp var (for log SM) (5) Numerator (for 1.0)( = (2) + (4)) (6) Denominator (for 1.0)( = (3) - (4)) (7) Structural-b (for 1.0)(= (5)/(6)) (8) Numerator (for 1.5)( = (2) + 1.5(4)) (9) Denominator (for 1.5)( = (3) - 1.5(4)) (10) Structural-b (for l.5)( = (8)/(9)) 2 14 28 0.4411 0.7182 0.0186 0.4597 0.6996 0.66 0.4690 0.6903 0.68 0.1720 0.7239 0.0186 0.1906 0.7053 0.27 0.1999 0.6960 0.29 -0.0084 0.0690 0.0186 0.0102 0.0504 0.20 0.0195 0.0417 0.47 391

Abstract of Chapter IV-7 Because shock is such an important variable in determining both a patient's prognosis and the choice of his anesthetic, an index of the estimated likelihood of shock was devised. This index was based on individual operations and on whether the operation was done as an emergency. One hypothesis suggested for explaining the higher death rate associated with cyclopropane is that it is used more frequently when the patient is in danger from shock. The observation that cyclopropane has a higher death rate than halothane for all categories of shock likelihood casts doubt that this differential usage alone can explain the higher death rates. CHAPTER IV-7. ANALYSIS OF MIDDLE-DEATH-RATE OPERATIONS USING A SHOCK-LIKELIHOOD INDEX John P. Gilbert* Harvard Computing Center Cambridge, Massachusetts The analysis presented here attempts to group individual operations into medically meaningful categories. This method differs from most of the other analyses of the middle-death- rate operations, in which the operations were grouped somewhat arbitrarily into 25 classes according to their observed death rates. There are as many ways of grouping by medical criteria as there are criteria for rating operations. Thus, operations can be grouped ac- cording to the part of the body or according to the medical service under which the operation was performed. The criterion used here was the shock-likelihood index (SLI), the likelihood that a patient undergoing a given operation would suffer from shock. To sharpen the criterion, the likelihood of shock associated with each opera- tion was assessed twice, once when the operation was elective (physical status 1, 2, 3, or 4) and once when it was an emergency procedure (physical status 5, 6, or 7). This criterion was chosen, not only because of the importance of shock to the patient's prognosis, but also because of the suspicion that the high death rate observed after administra- tion of cyclopropane in the Study was due in part to its selective use for patients in shock. The operations were assigned to index categories;** the breakdown is shown in Table 1. The index runs from 1 to 5, with 1 signifying the smallest likelihood of shock and 5 signifying the greatest likelihood. It should be emphasized that these assignments were based not on the condi- tion of the patient, but only on the operation and whether it was an emergency. If the operation or the physical status was unknown, SLI was coded as zero. These assignments were made for all the operations in the Study. It is much more ef- fective, however, in the case of the seven low- death-rate and the four high-death-rate opera- tions, to study the individual operations them- selves. Given only a few operations having quite different death rates and also different usage patterns, what seems to be an agent comparison may really be an operation comparison. Conse- quently, this investigation is based only on the middle-death-rate operations. Table 2 shows for each anesthetic agent the distribution of cases over the categories of SLI, as well as the observed death rates. It is clear that, although cyclopropane was used pro- portionally more often in SLI categories 4 and 5, this differential usage is not sufficient to ac- count for the higher death rate associated with cyclopropane. Indeed, cyclopropane seems to suffer most in comparison with halothane and nitrous oxide-barbiturate in the lowest SLI categories. There is the possibility that even in the low-likelihood categories some patients will have suffered from shock. If these patients were also more likely to receive cyclopropane, this would explain, to some extent, the observed differences. It is one of the disadvantages of the nonrandomized study that, after all corrections have been made, there is no answer to such an argument, except to question whether the selec- tion effects could have been so large. Two fac- tors that tend to decrease the chance that the selection of poorer-risk patients might account for all the excess deaths after cyclopropane are: (1) cyclopropane was rarely used in several institutions, and (2) cyclopropane was used in •Part of the work reported here was done while the author was L. L. Thurstone Distinguished Fellow, University of North Carolina. This fellowship was partially supported by PHS research grant M-10006 (National Institute of Mental Health). **By John P. Bunker and William H. Forrest, Jr. 392

TABLE 1.—OPERATIONS ASSIGNED TO THE FIVE SHOCK-LIKELIHOOD INDEX (SLI) CATEGORIES If elective SLI category 1 SLI category 2 SLI category 3 SLI category 4 SLI category 5 1 2 3 4 10 13 15 20 5 12 17 23 33 34 46 52 6 7 8 9 21 22 26 28 27 32 36 43 57 59 16 30 50 54 31 39 40 41 44 45 48 51 55 60 71 87 42 47 58 62 56 67 68 70 89 90 92 95 63 65 66 73 86 93 75 76 77 80 81 84 85 88 98 99 If emergency SLI category 1 SLI category 2 SLI caI egory 3 SLI category 4 SLI category 5 2347 9 30 31 54 56 20 21 1 10 12 13 33 34 36 45 8 50 71 87 80 81 84 85 26 55 60 73 15 16 17 22 48 51 57 59 89 90 92 95 88 76 77 98 99 23 27 28 32 67 39 40 41 42 43 44 46 47 52 56 58 62 63 65 66 68 70 75 86 93 some operations for which it is contraindicated. Such factors are very difficult to evaluate, how- ever, and it is interesting to note that standardiz- ing on the six SLI categories has done as much as any other single variable to eliminate the dif- ferences between agents. To explore further the effects of controlling for SLI, we performed a version of the analysis described in Chapter IV-4, using the variables SLI and age. Age was grouped into five categories so that the cross-classification of SLI and age defined 30 categories on which to standardize the common practices. These were then jackknifed to produce over-all death rates comparable with those in Table 2 of Chapter IV-4. These death rates are shown in Table 3. The variables in analysis VTII are operation (coded into 25 categories), physical status, and age. Analysis VIII gives a smaller range of dif- ferences between death rates than does the SLI X age analysis. The significance of the agent differences for analysis VIII is discussed in Chapter IV-4. To conclude, the SLI analyses suggest that the selective use of cyclopropane for patients in shock is a contributing factor to the high cyclopropane death rate. They also indicate, however, that that factor could hardly account for all the observed differences in death rates. 393

TABLE 2.—AGENT VS. SLI MIDDLE-DEATH-RATE OPERATIONS Distribution, SLI category Agent H N-B C E 0 All Unknown 16.3 16.5 15.0 16.6 17.9 16.4 1 25.0 28.8 24.8 23.9 23.6 25.5 2 39.5 34.7 33.7 34.3 35.7 36.3 3 14.0 14.9 12.8 18.9 14.4 14.6 4 4.7 4.6 10.8 5.5 7.1 6.1 5 0.7 0.5 2.8 0.8 1.3 1.1 Total EE (in thousands) 145.7 100.8 68.1 44.2 67.2 426.0 Death rates Unknown 0.01781 0.01512 0.02937 0.01920 0.02342 0.01999 1 0.00405 0.00419 0.00582 0.00386 0.00801 0.00492 2 0.01018 0.01158 0.02192 0.01237 0.01793 0.01368 3 0.02968 0.02856 0.03786 0.02233 0.03797 0.03089 4 0.07758 0.07976 0.08818 0.07466 0.10723 0.08625 5 0.16867 0.19932 0.17865 0.17483 0.23408 0.18941 Over-all 0.01727 0.01709 0.03401 0.01854 0.02988 0.02207 Standardized death rate 0.01922 0.01969 0.02781 0.01900 0.02793 0.02207 Standard (ratio) 0.87096 0.89232 1.26010 0.86073 1.26539 Indirect MR 0.85973 0.87653 1.17859 0.84430 1.28205 TABLE 3.—OVER-ALL DEATH RATES ESTIMATED FROM SLI X AGE AND FROM ANALYSIS VIII OF CHAPTER IV-4 FOR THE MIDDLE-DEATH-RATE OPERA- TIONS Agent H N-B C E 0 SLI X age 0.0205 0.0129 0.0281 0.0136 0.0258 Analysis VIII 0.0192 0.0199 0.0254 0.0181 0.0241 394

CHAPTER IV-8. AFTERWORD FOR THE STUDY OF DEATH RATES Lincoln E. Moses Stanford University Stanford, California Frederick Mosteller Harvard University Cambridge, Massachusetts COMPARISON OF ANESTHETICS This section summarizes the results of the study of death rates. It begins with the raw re- sults of Chapter IV-2, continues with the stand- ardization of death rates on one variable, and then turns to the analyses for several variables of Chapters IV-3 through IV-7, including some measures of reliability of differences. It closes with a discussion of the limitations of the study of death rates. In speaking of death rates following the ad- ministration of an anesthetic and surgery, brevity of language is likely to suggest erroneously that a given death rate is wholly attributable to the anesthetic agent used, whereas the study asks whether any of it is. The cause-and-effect ques- tion could well be held in abeyance until the summary is completed. To eliminate the con- venience of referring to a death rate by its as- sociated anesthetic would make the following description longer and cloudier. The Unadjusted Death Rates (Chapter IV-2) In all the analyses of death rates, the many administrations of anesthetics were grouped into categories labeled "halothane," "nitrous oxide- barbiturate," "cyclopropane," "ether," and "Other." Frequently, also, the analyses were ap- plied separately to three groups of operations: the seven low-death-rate operations, the many middle-death-rate operations, and the four high- death-rate operations. Table 1 displays the compositions and sizes of the groups. Only about 5 percent of the deaths followed the low-death- rate operations, although they accounted for about 43 percent of the cases. The middle-death- rate group accounted for about 57 percent of the deaths and about 50 percent of the cases. The high-death-rate group accounted for the re- maining 38 percent of the deaths and 7 percent of the cases. The raw death rates corresponding to the anesthetic agents showed considerable dif- ferences, as displayed in Table 2. Naturally, such rates cannot be accepted at face value because of sources of bias, such as the preferential use of some anesthetics for some hazardous opera- tions or for patients in poor physical status. Nevertheless, the findings of the whole study are similar to the findings of this unadjusted table, except that the differences given here exaggerate the results found after adjustment. Generally speaking, cyclopropane and Other had higher death rates than halothane, nitrous oxide- barbiturate, or ether, and the latter three had similar death rates over-all. Adjustments for One Variable (Chapter IV-2) To make allowances for some of the sources of bias, individually, Chapter IV-2 showed the effects of adjustments allowing for the different frequencies of use of the several anesthetic agents. We summarize in Table 3 the effects of these one-variable adjustments for physical status, age, institution, and operation. Ratios greater than 1 imply higher death rates than average. This table gives the same general im- pression as the unadjusted table (Table 2). In every comparison, Other has an indirectly standardized mortality ratio higher than average. In the low-death-rate operations, ether and nitrous oxide-barbiturate have ratios lower than average, and halothane and cyclopropane ratios a bit higher than average. In the middle-death-rate operations, halo- thane, nitrous oxide-barbiturate, and ether have similar lower ratios, and cyclopropane and Other have higher ratios. In the high-death-rate operations, ether has the lowest ratios, halothane the next-lowest ratios, and nitrous oxide-barbiturate and cyclo- propane slightly higher ratios. Both ether and cyclopropane vary a great deal from one ad- justment to another, which makes us want to look at rates adjusted for several variables simultaneously. No adjustment method is entirely satisfac- tory. In Table 3, every starred number is an indirectly standardized ratio (for definitions, see 395

TABLE l.--SUlyMARY OF HIGH-, MIDDLE-, AND LOW-DEATH-RATE GROUPS OF OPERATIONS Operation codes included Estimated exposed Deaths Death rate, 56* Operation group Number (in thousands ) * Number * High-death-rate 12,33,44,48** 61.7 7.2 6,354 37.8 9.34 Low-death-rate 1,3,55,60,65, 366.3 42.9 844 5.0 0.23 73,90*** Middle-death-rate All others**** 426.0 49.9 9,615 57.2 2.21 All — 854.0 100.0 16,813 100.0 *Death rate computed as D/(D+EE). **12: c rani o tomy, 33: heart and great vessel with pump, 44: exploratory laparotomy, and 48: large bowel. ***Mouth and dental; eye, all; dilatation and curettage, etc.; hysterectomy; herniorrna- phy; cystoscopy; and plastic surgery. ****Except mammoplasty, which was omitted. Source: Table 1 of Chapter IV-2; Appendix 1 to Chapter IV-5. TABLE 2.—UNADJUSTED DEATH RATES IN PERCENT AND STANDARD MORTALITY RATIOS FOR EACH ANESTHETIC AGENT AND LOW-, MIDDLE-, AND HIGH-DEATH-RATE GROUPS OF OPERATIONS Unadjusted death rates, % Group of operations Anesthetic agent H N-B C E 0 All Low-death-rate 0.27 Middle-death-rate 1.76 High-death-rate 9.5 0.20 1.74 11.4 0.26 3.52 0.13 1.89 6.5 0.29 3.08 13.3 0.23 2.26 10.9 10.3 All 1.91 1.51 2.61 1.37 2.57 1.97 Source: Table 1 of Chapter IV-2. Group of operations Standard Mortality Ratios Anesthetic agent H N-B C E 0 Low-death-rate 1.2 Middle-death-rate 0.8 High-death-rate 0.9 0.9 0.8 1.1 1.1 0.6 0.8 0.6 1.3 1.4 1.3 1.6 1.1 All 0.97 0.77 1.33 0.70 1.30 the second section of Chapter IV-2) that is con- siderably lower than the directly standardized ratio. The disagreement probably means that the number should be viewed with a little more sus- picion than usual. The indirect ratios are less sensitive to large death rates based on few cases, but may mislead by seeming to be ap- plicable when they are not, as in the ether ratio for high-death-rate operations; ether was rarely used in the heart operation (code 33). 396

TABLE 3.--INDIRECT MORTALITY RATIO BY ANESTHETIC AGENT BASED ON "ALL ANESTHETICS" POPULATION Standardized tor: Agent H N-B C E 0 Low-death-rate operations Physical status 1.08 0.84 0.97 0.73« 1.47 Age 1.11 0.72 1.28 0.71 1.42 Institution 1.10 0.91 0.98«* 0.64 1.32* Operation 1.01 0.88 1.21 0.58 1.32 Middle-death-rate operations Physical status 0.87 0.87 1.22 0.92 1.15 Age 0.82 0.77 1.54 0.83 1.27 Institution 0.80 0.76 1.32»» 1.05* 1.40 Operation 0.83 0.91 1.32 0.74 1.26 High-death-rate operations Physical status 0.99 1.07 0.97 0.78 1.11 Age 0.95 1.14 1.03 0.61 1.23 Institution 0.96 0.99" l.Cl*» 0.72** 1.32 Operation 0.8? 1.04 1.24 0.74** 1.23 *The directly standardized ratio is more than 20 percent higher. **The directly standardized ratio is more than 50 percent higher. Source: Tables 2, 3, and 4 of Chapter IV-2. Adjustments for Several Variables Smoothed Contingency-Table Analysis (Chapter IV-3). The smoothed contingency-table analysis uses data from several variables to make esti- mates of death rates in cells adjusted simulta- neously for such variables as age, physical status, and sometimes operation and sex. Selected lines from the many large tables in Chapter IV-3 illustrate in Table 4 the detail of the estimates for cells where more data are available. In the low-death-rate operations, ad- justed for age, physical status, and sex, the re- sults presented are rather similar for the several anesthetic agents. The largest difference in the upper panel, 0.08, would represent four deaths in 5000 operations. In the middle-death-rate operations, ad- justed for operation category (operations were grouped into five categories according to death rate), physical status, and age, ether usually appears to have had the lowest rate, nitrous oxide-barbiturate and halothane to have had higher and similar rates, and cyclopropane and Other usually to have had the highest rates. In the high-death-rate operations, adjusted for operation code, risk, and age, halothane, ether, and nitrous oxide-barbiturate had similar rates (after operations 48 and 44, for which the comparison is readily made). Halothane's high rate for craniotomy in the 50-69 age group dis- agrees with its lower rate for the 10-49 age group. For heart and great vessel operations with pump, halothane had a lower rate than nitrous oxide-barbiturate. Cyclopropane and Other had somewhat higher rates for opera- tions 48 and 44, and they were little used in operations 12 and 33. By considering rankings of the anesthetics on observed death rates for subclasses of pro- cedures controlled on several variables, we can get one summary of these results, including evidence from rows of the tables not presented in this section. Table 5 shows the results of such rankings. It must be recalled that we have set Other aside because it so systematically has a high death rate and because of the heterogeneity of its anesthetics. The implications of the rankings are as follows: (1) For the low-death-rate operations, cyclopropane usually had the highest death rates among the four anesthetic agents, and nitrous oxide-barbiturate and ether the lowest, with halothane having about the average rank. The differences between the rankings are significant beyond the 0.5 percent level. (2) For the middle-death-rate operations, cyclopropane systematically had higher ranks, and halothane, nitrous oxide-barbiturate, and ether had similar sets of ranks. The differences between the agents are significant far beyond any of the usual levels (see the cautionary remark below, however). (3) In the high-death-rate operations, cyclo- propane also ranked high, except for operation 33, where the ranking data are sparse and the number of cyclopropane cases low, and opera- tion 12, for which it was rarely used. Halothane had the lowest summed ranks for operations 48 and 44, was below average for operation 33, and was one point above average for operation 12. A fair summary would say that for the high-death- rate operations halothane had the best record, based on ranks. Nitrous oxide-barbiturate was a close second, and ether, as usual, was variable. Taking the operations one at a time, the anes- thetic differences are not significant at any of the usual levels. The significance tests reported on the ranks may be subject to some correlation owing to the appearance of the same institutional com- ponents in different rankings. The effect of ad- justing for institutional correlation, if it could 397

TABLE 4.—ILLUSTRATIVE DEATH RATES IN PERCENT FRO! THE SMOOTHED CONTINGENCY-TABLE ANALYSIS LOW-DEATH-RATE OPERATIONS Female; known physical status; at least 5000 estimated exposed Physical status Age Agent H N-B C E 0 1,2 1,2 1,2 0-9 0.12 0.06 0.04 10-49 0.06 0.04 0.04 0.06 0.06 50-69 0.20 0.20 0.18 Source: Tables 7, 9, 11, and 13 of Chapter IV-3. All ages; both sexes; at least 2000 estimated exposed Agent H N-B C 3,4 2.2 1.9 2.4 5 0.11 0.12 0.13 Source: Tables 8 and 10 of Chapter IV-3. MIDDLE-DEATH-RATE OPERATIONS Risk 1, 2, 5; at least 1000 estimated exposed Operation category Age Agent H N-B C E 0 1 10^19 0.08 0.08 0.14 0.07 0.13 1 50-69 0.40 0.33 0.46 0.27 0.54 2 10-49 0.43 0.62 0.47 0.22 0.85 2 50-69 1.2 1.3 1.7 1.3 1.8 3 10-49 0.61 0.56 0.93 0.24 0.86 3 50-69 1.9 2.1 2.6 2.4 2.6 4 10-49 1.0 1.1 1.3 0.36 1.7 4 50-69 3.0 3.0 3.7 2.2 3.7 5 10-49 2.7 2.6 5 50-69 4.1 6.1 Source: Table 16 of Chapter IV-3 (also mainly available in Table 19 of Chapter IV-3). HIGH-DEATH-RATE OPERATIONS Risk 1, 2, 5; at least 500 estimated exposed Agent Operation Age H.- N-B 1 C I 0 48 (large bowel) 10-49 1.3 1.2 1.5 0.9 0.8 44 50-69 2.2 2.1 3.1 2.0 3.6 (exploratory 10-49 1.3 1.8 2.4 1.8 2.1 laparotomy) 50-69 3.5 3.3 6.4 3.2 9.4 12 ( craniotomy) 10-49 3.9 9.6 3.4 33 50-69 9.5 6.3 4.4 (heart and 10-49 7.7 12.2 great vessel Risk 3,4,6 with pump) 10-49 12.7 20.8 Source: Table 27 of Chapter IV-3. be done, would probably be to reduce the signifi- cance of a set of differences. The significance of the middle-death-rate results would probably still stand, but that of the low-death-rate dif- ferences might be weakened from 0.5 to 5 per- cent. We defer the over-all death-rate sum- maries in percent and the special study of cholecystectomies based on the smoothed contingency-table analysis to a later discussion. The Smear-and-Suieep Analysis of the Middle- Death-Rate Operations (Chapter IV-4). A second way to adjust death rates for many background variables simultaneously is offered by the smear -and -sweep analysis. In this analy- sis, a test of the significance of the differences between anesthetic death rates is based on the consistency of the differences in rates from one institution to another. By exploring many possi- ble related analyses, it was shown that the re- sults (except for ether) were not very sensitive to various choices in the method. We concentrate here on the preferred analysis VIII, which ad- justed for age, physical status, and operation category. Table 6 gives means and standard errors for analysis VIII. Ether's large standard error reflects its limited use in some institu- tions. Because the extreme variation in ether death rates disturbed the analysis of variance of the institutional effects, a reanalysis was re- quired with ether omitted. Table 7 shows the differences at the 5 percent level of significance. A ranked analysis of variance, like that carried out for the data in Chapter IV-3, ranks the death rates (actually pseudo-rates) asso- ciated with the five anesthetic agents within each hospital. Then these ranks are summed. Low ranks were given to low death rates. The sums of the ranks were: H N-B 81 83 CEO 121 109 116 The differences were significant beyond the 0.5 percent level. A further, especially sensitive analysis (Table 8 of Chapter IV-4) found significant differences at the 5 percent level in the death rates for the pairs of anesthetics halothane- cyclopropane, halothane-Other, and nitrous oxide- barbiturate — cyclopropane, where cyclopropane and Other had the higher rates. Analysis by Regression (Chapter IV- 5). Five different analyses provide summary statistics for the death rates. (1) Smoothed contingency tables: Using the analyses of Chapter IV-3, new strata were formed within each operation group (low-, middle-, and high-death-rate). For the middle- death-rate group, the strata in one study (model A) were operation category, physical status, age, and sex, and in another (model B), operation category, physical status, and age. We present here the model B results. (2) ORLA regression: Using the variables operation, risk (physical status), length of op- eration, and age (the four variables are abbre- viated "ORLA"), a cubic polynomial was fitted, strata were created, and summaries were com- puted. 398

TABLE 5.--SUKMARY OF SUMMED RANKS OF OBSERVED DEATH RATES Low totals Imply low death rates LOW-DEATH-RATE OPERATIONS TABLE 6.—MEANS AND STANDARD ERRORS OF THE DEATH RATES IN PERCENT BASED ON SMEAR-AND-SWEEP FOR MIDDLE-DEATH-RATE OPERATIONS Physical status Agent H N-B C E Total Unknown 25 13.5 26.5 15 80 1,2 22 18 23 17 80 3,4 20 19 24.5 16.5 80 5 21 19.5 21.5 18 80 6 19 21 26.5 13.5 80 Total ranks 107 91 122 80 Source: Table 15 of Chapter IV-3. MIDDLE-DEATH-RATE OPERATIONS 400 Operation Agent category H N-B C £ Total 1 27 23 42 28 120 2 27 28 40 25 120 3 19 17 30 24 90 4 23 32 42 23 120 5 23 26 34 17 100 Total ranks 119 126 188 117 550 Source: Table 20 of Chapter IV-3. HIGH-DEATH-RATE OPERATIONS Agent Operation H N-B C £ Total 48 11 13 18 18 60 (large bowel) 44 17 19 27 17 80 (exploratory laparotomy ) 12 11 13 « 6 30 I craniotomy) 33 5 9 4** * 18 (heart and great vessel with pump) *Too few cases to rank. **Based on small numbers of cases. Source: Table 28 of Chapter IV-3. (3) Overlapping definition: The ORLA re- gression was redone to see whether the category Other had been treated unfairly by including in it mixtures of anesthetics containing halothane, nitrous oxide-barbiturate, cyclopropane, and ether. To adjust for this, every administration of an anesthetic was counted in every major category in which it appeared and excluded from Agent H N-B C 1 : o Death rate 1.92 1.99 2.54 1. 81 2.41 Standard error 0.27 0.18 0.17 0. 48 0.20 Source: Analysis VIII of Tables 2 and 3 of Chap- ter IV-4. TABLE 7.—TWO-WAY ANALYSIS OF VARIANCE OF HOSPITAL AND ANESTHETIC DIFFERENCES IN DEATH RATE (ETHER OMITTED) FOR MIDDLE-DEATH-RATE OPERATIONS Degrees of Mean 5% critical value freedom square F Hospitals 33 0.000273 2.56 1.57 Agents 3 0.000314 2.94 2.70 Error 99 0.000107 Source: Table 5 of Chapter IV-4. Other, which then contained no procedures using halothane, nitrous oxide-barbiturate, cyclopro- pane, or ether. For example, if a mixture of halothane and cyclopropane were administered, the case would be counted in the halothane and cyclopropane categories and omitted from Other. (4) Age-SLI regression: The shock-likelihood index (SLI) of Chapter IV-7 was combined with age, and again a cubic polynomial was fitted to give strata. All operations were considered simultaneously. (5) Cholecystectomies: A special study of cholecystectomies based on the analysis in Chapter IV-3 is presented because of its interest in the question of hepatic necrosis. In Table 8, the results of these five studies are assembled for easy comparison. They show, as usual, that nitrous oxide-barbiturate and ether had the low death rates in the low-death- rate operations, that cyclopropane and Other had the high death rates in the middle- and high- death-rate operations, and that the rates for halothane and nitrous oxide-barbiturate are closely matched in the middle- and high-death- rate operations. As explained elsewhere, rates for ether are somewhat unreliably determined. Over all operations, the smoothed contingency-table analysis and the ORLA re- gression agree that halothane, nitrous oxide- barbiturate, and ether had much the same low rates, and that cyclopropane and Other had higher rates. The age-SLI regression gives 399

TABLE 8.—DEATH RATES IN PERCENT FROM THE THREE REGRESSION STUDIES OF CHAPTER IV-5 Agent H N-B C £ 0 LOW-DEATH-RATE OPERATIONS Smoothed analysis 0.25 0.16 0.25 ORLA* regression 0.23 0.19 0.27 Overlapping definition 0.25 0.21 0.28 MIDDLE-DEATH-RATE OPERATIONS 0.18 0.33 0.17 0.32 0.23 0.40 Snoothed analysis ORLA regression Overlapping definition Cholecystectoinies 1.95 1.95 2.79 1.79 2.55 2.04 1.94 2.75 1.73 2.56 2.09 1.79 2.61 1.93 3.35 2.03 2.62 2.51 2.30 2.84 HIGH-DEATH-RATE OPERATIONS Smoothed analysis 8.4 9.1 ORLA regression 8.4 9.0 Overlapping definition 8.2 8.2 TOTAL OPERATIONS 11.0 8.1 10.8 11.7 9.9 10.8 10.4 9.4 17.1 .28 .17 Smoothed analysis** 1.68 1 .69 2 1, .55 2 ,71 .68 .32 1. .17 ORLA regression** 1, 1 2 .62 2 Overlapping definition ^, ,73 1 .56 .15 1. .63 3 .06 2 Age-SLI regression 1, .93 1 .72 2 .16 1, .52 2 .33 .91 .61 ,37 .57 Unadjusted rates 1, 1 .51 2 1, 2 *ORLA means control for operation, risk (physical status), length of operation, and age. **Using weights of 43 percent for low-, 50 percent for middle- and 7 percent for high-death-rate operations, as shown in Table 1. Source: Table 1 of Chapter IV-2; Tables 1, 5, and 6 of Chap- ter IV-5. somewhat different results, and that needs com- ment. This index does not take account of physical status, except for the distinction between emer- gency and nonemergency operations, and it analyzes low-, middle-, and high-death-rate operations simultaneously. The smoothed and ORLA analyses not only stratify on death rate in operations, but break down the high-death-rate operations to single codes, a matter of some importance in that these four operations ac- counted for nearly 40 percent of the deaths. Con- sequently, we regard the age-SLI index as in- ferior for this purpose. The attempt to be more fair to Other by omitting cases in which a patient received two or more major anesthetic agents (a possible indication of a patient in trouble) from that category left Other with noticeably higher rates, up 1 percent over-all, and reduced the rates for cyclopropane and nitrous oxide-barbiturate by about one-seventh of 1 percent. Far from showing high death rates in the cholecystectomies, halothane had the lowest rate of any of the anesthetics. However, the evidence from the rankings for significance of differences is weak, so we must leave the matter as an indication favoring halothane, rather than a firm finding that halothane actually had a lower rate than any of the others. To search further into the sampling sta- bility of the halothane-cyclopropane difference, their death rates adjusted for ORLA were com- pared in some hospitals for the middle-death- rate operations. These hospitals were chosen because each used both anesthetics in at least 20 percent of its operations. In 10 of 11 hospi- tals, halothane had the lower rate, a result significant at about the 1 percent level. A similar analysis on the high-death-rate opera- tions favored halothane in eight of 11 hospitals, which is indicative but not significant. Analysis by Aggregation (Chapter IV-6). In Chapter IV-6, multiple adjustments for interfering variables are made in a manner somewhat different from that in Chapter IV-5. In the cross-classification created by these variables, death rates can be calculated for each cell, and then aggregated in death-rate order. Death rates can then be calculated and standard- ized against the strata. Although the data were treated quite differently, the death-rate results are so similar to those of Table 8 that it would not be helpful to reproduce them. Nevertheless, the close agreement shows the stability of the results when substantially different techniques adjust for similar variables. Because cyclopropane was used more often than halothane in emergency operations, the mortality rates in the middle-death-rate opera- tions were checked. The comparison was ad- justed for ORLA, and, although all death rates are reduced when emergencies are removed, cyclopropane's mortality ratio rises about 6 per- cent, as shown in Table 9, whereas those for halothane, Other, and nitrous oxide-barbiturate rise 2, 2, and 4 percent, respectively. Only ether lowers its ratio when the emergencies are re- moved. Consistency among Institutions of Differences between Anesthetic Agents (Chapter IV-6) We study the stability of anesthetic dif- ferences among the 34 institutions for two reasons: (1) it enables institutional effects to be disentangled from the anesthetic comparisons H N-B C E 0 TABLE 9.—VARIOUS MORTALITY RATIOS OBTAINED BY AGGREGATION FROM MIDDLE-DEATH-RATE OPERATIONS Agent ORA — ORLA In ORLA Out 0.90 0.91 1.21 0.80 1.14 0.91 0.92 1.16 0.85 1.11 0.93 0.96 1.23 0.82 1.13 Source: Tables 5 and 12 of Chapter IV-6. 400

(their entanglement was discussed above, under "Adjustments for One Variable"); and (2) it is more reliable to study the differences from institution to institution that recur consistently. Three approaches to studying transinstitutional stability were undertaken. First, the agent comparisons were studied separately in each institution (for the middle- death-rate operations) after aggregation stratifi- cation for age, risk (physical status), and opera- tion. The indirectly standardized mortality ratio (ISMR) obtained in this analysis for each anes- thetic agent over all institutions appears in the first line of Table 9. Two rank methods, taking account of the variability and the magnitude of the anesthetic differences, were applied, and a more delicate technique, taking account of both factors at once, was developed and applied. The results were four highly significant differences: halothane and nitrous oxide-barbiturate had lower rates than cyclopropane and Other. A less clear, although statistically significant, finding of ether having higher rates than nitrous oxide- barbiturate also emerged, but we put little con- fidence in this conclusion because ether com- parisons typically differ under various methods of analysis. Second, we identified, for each pair of anes- thetics, institutions that used both of the pair "frequently" (in 10 or 15 percent of all opera- tions). When an anesthetic in a pair had the lower ORA-adjusted ISMR for the middle-death- rate operations in an institution, it scored a point in its favor. Winning a large preponderance of institutional comparisons indicated interinsti- tutional stability of the comparison. Halothane compared favorably with cyclopropane, with high significance. Less strong indications were found for halothane having lower rates than Other and nitrous oxide-barbiturate having lower rates than cyclopropane. This analysis did not demon- strate the favorable comparison of nitrous oxide- barbiturate with Other, found in the first analysis. Third, we undertook a simple, easily under- stood study based on operations performed often in nearly every institution, standardizing only for the frequency with which each operation was performed in each institution. For this purpose, six, mostly middle-death-rate, operations were chosen that accounted for about 16 percent of the total estimated exposed. The analysis produced the usual halothane-cyclopropane, nitrous oxide- barbiturate--Other, and halothane-Other dif- ferences at the 5 percent level of significance, plus a strong suggestion that nitrous oxide- barbiturate had a lower rate than cyclopropane. Lack of adjustment for other important variables would be a reasonable criticism of the study; nevertheless, many will find it reassuring that a simple analysis supports the more complicated ones. Finally, the four high-death-rate operations, analyzed individually, by the first two methods exhibited no significant comparisons stable across institutions. The Shock-Likelihood Index (Chapter IV-7) Because cyclopropane is said to be used often for operations on patients in shock and for operations likely to produce shock, a special study investigated the possible bias of such selective use. Each operation code was sub- jectively rated, separately as an emergency and as a nonemergency, into five categories on the basis of the likelihood of association with shock. It must be emphasized that the rating was not based on the condition of individual patients in this Study. A study of the middle-death-rate operations found that, although cyclopropane was used pro- portionately more often in the higher SLI cate- gories, 4 and 5, the major contribution to its higher standardized death rates came from the differentials in SLI categories 1, 2, and 3, which include 77 percent of its cases (Table 10). In a further study of the middle-death-rate op- erations, age and SLI were combined to examine the death rates. The results for age-SLI and for Chapter IV-4 are compared in Table 11; how- ever, it must be remembered that the age-SLI analysis has little control for physical status and that the emergency-nonemergency distinction, although it affected the death rates considerably, did not reduce cyclopropane's mortality ratio in Chapter IV-5. For the purpose of comparing over-all death rates, we do not believe that age- SLI is as effective as the smoothed contingency- table analysis or the ORLA regression or, in the middle-death-rate operations, analysis VIII of smear-and-sweep and the analyses of Chapter IV-6. To what extent have the original death rates been changed by the adjustments? Table 12 shows that Other, cyclopropane, and halothane reduced their rates by 0.4, 0.3, and 0.2 percent respec- tively, and nitrous oxide-barbiturate and ether increased their rates by 0.2 and 0.3 percent, respectively, from the original unadjusted fig- ures. The relative positions of halothane and cyclopropane are scarcely changed. Anesthetic Death Rate In this summary, we have alluded to dif- ferences in death rates of the order of half of 1 percent or more. Although the death rates them- selves cannot be attributed to the anesthetic agent, the differences in the adjusted or stand- ardized death rates do imply rates for which the differences between the anesthetics and their procedures might be held partly accountable. Beecher and Todd (1) found the anesthetic death 401

TABLE 10.--DEATH RATES IN PERCENT FOR SHOCK-LIKELI- HOOD CATEGORIES AND ANESTHETICS FOR MIDDLE-DEATH- RATE OPERATIONS SLI Agent % of all opera- tions in category H N-B C E 0 Unknown 1 2 3 4 5 1.78 1.51 2.94 1.92 2.34 16 0.40 0.42 0.58 0.39 0.80 26 1.02 1.16 2.19 1.24 1.79 2.97 2.86 3.79 2.23 3.80 7.8 8.0 8.8 7.5 10.7 16.9 19.9 17.9 17.5 23.4 36 15 6 1 Standardized death rate 1.92 1.97 2.78 1.90 2.79 100 Source: Table 2 of Chapter IV-7. TABLE 11.--COMPARISONS OF SEVERAL STANDARDIZATIONS OF DEATH RATES IN PERCENT FOR MIDDLE-DEATH-RATE OPERATIONS Agent H N-B C E 0 Age-SLI Analysis VIII Snoothed analysis ORIA regression Unadjusted rate 2.05 1.29 2.81 1.36 2.58 1.92 1.99 2.54 1.81 2.41 1.95 1.95 2.79 1.79 2.55 2.04 1.94 2.75 1.73 2.56 1.76 1.74 3.52 1.89 3.08 Source: Tables 2 and 8; Table 3 of Chapter IV-7. TABLE 12. — SUMMARIES OF OVER-ALL DEATH RATES IN PERCENT Agent H N-B C E 0 Staoothed analysis ORLA regression Unadjusted 1.68 1.69 2.28 1:55 2.17 1.71 1.68 2.32 1.62 2.17 1.91 1.51 2.61 1.37 2.57 Source: Table 8; Table 1 of Chapter IV-2. rate itself to be only 1 per 1560. How are these results to be reconciled? How can the difference between the death rates for halothane and cyclopropane be larger than the anesthetic death rate itself? When there are multiple causes, it is diffi- cult to assign responsibility with any definiteness. To say of a surgical death that it was an "anes- thetic death," one must attribute the death to something very specific about the administration and the response to it in the particular patient. In their general capacity as catalysts for the success of an operation, some anesthetics may be slightly more successful than others, and the differences in death rates may not be due as much to active damage, as implied by the ex- pression "anesthetic death," as to the general prevention of damage. However that may be, it is one thing to suggest that the identifiable oc- currences of an event have a small rate, and a compatible but quite different thing to suggest that all actual occurrences of an event produce a larger rate; for example, we all recognize that the number of serious violators of traffic regulations is much larger than the number con- victed. However, these differences may exag- gerate the anesthetic effects because the ad- justments rest on crude variables crudely grouped, because of sampling error, and because they include the effects of an observational study, rather than of an experiment. Limitations on Generalizations Population Although the institutions participating in this Study do not come from a randomly constructed sample either of institutions or of patients in the United States, it is wise to think of the results as coming from some sampled population and to think of what kinds of hospitals are and are not represented. As to types of hospitals among the 34: 15 are nonprofit corporations; three are church- affiliated; eight are state hospitals, two are county hospitals, and one is a city hospital; two are affiliated with the U.S. Air Force, one with the Veterans' Administration, and one with the U.S. Public Health Service; and one is hard to classify. The institutions in the Study have from 250 to 1500 beds; the mean is 675 and the median is 650. Small hospitals are not represented in the sample, but nine have between 250 and 400 beds. Seven hospitals in the sample have over 1000 beds. Obstetric uses of anesthetics were specif- ically excluded from the Study. Three of the hospitals are primarily for male patients. With respect to age, the sample includes a children's hospital, a large hospital in a retire- ment area, and a hospital specializing in joint 402

diseases (which implies a predominance of older patients). The hospitals joined the Study largely by volunteering when they heard of the plans for it, usually through NAS-NRC news or members. Service institutions were actively encouraged to join, but no attempt was made to publicize the Study on a national scale. Nevertheless, the geographic representation is broad. All the hospitals have a teaching connection, and all have anesthesiologists. Compared with the nation's population of hospitals, the sample has heavy representation of hospitals with research interests, and especially research interests in anesthesiology. All the hospitals in the sample have records suitable for the Study. Hospitals in which anesthesia is adminis- tered largely by nurses or other persons with- out medical degrees cannot regard the results of this Study as directly applicable to their practice of anesthesia. Possibly a study of a sample of small hospitals would be of interest. Data Data on the numbers of hospital deaths should be very sound. The weakness lies in the numbers of deaths that may have occurred out- side the hospital within 6 weeks of operation, for these are not included. The careful investigation of this question appears expensive. Although the recorded background variables may themselves often be in error for individual patients, for purposes of aggregative compari- sons, such as we are making, they are likely to be, if not entirely satisfactory, at least a con- siderable improvement over making no adjust- ments at all. One special kind of weakness does exist in the rating of physical status. If physical status is assigned after the operation, rather than before, as may occur, then the adjustment for physical status is likely to correlate more highly with the outcome of the surgery than it would have if it had been assigned in advance. In such cases, the rating of physical status may explain too much and overadjust the death-rate differences between anesthetic agents, most likely leading to underestimation of those dif- ferences. A few institutions or parts of institutions in this Study do not use the index of physical status, and the presence of this large clump of "un- known" has complicated the adjustments and to some extent reduced their effectiveness. If we were beginning the Study anew, we would re- consider the statistical treatment of the "un- known" group in the hope of strengthening the analysis. We have attempted to defend against quirks of particular forms of analysis by executing several different kinds. Over-all, the results seem gratifyingly consistent. Attention has repeatedly been called to the unreliable state of inferences about ether. Its distinctive pattern of institutional and opera- tional usage has prevented our being able to reach any firm conclusion. Objectivity of Planning and Analysis In reflecting on how to achieve objectivity in such analyses as these, the statisticians noted that naming a study can have a considerable ef- fect on its analysis. For example, this one was called the National Halothane Study. One effect of such a name is that it skews the interest in one direction and restricts, at least during the planning stage and the early analysis, the de- sirable equal scattering of attention that might have been implied by a less focused title, such as Comparative Study of Anesthetics. In at least one instance, the title of this Study led to a de- cision about what data to collect that made the final tabulations difficult to interpret. Fortu- nately, the magnitudes were all so small that it scarcely mattered, but the principle is as clear in a small mistake as in a big one. Just as we use blindness to treatments in experimentation, there could be some advantage in blindness in the statistical analysis of the treatments under study through some definite stage. We realize that several difficulties face such a program; for example, certain compari- sons would be made only between certain treat- ments, and in a large body of data certain treat- ments would identify themselves by their presence, absence, or known effects. On the other hand, we could go further than we have in the direction of blind analyses. One reason that blindness might matter concerns the intensity of follow-up. When a difference is observed that is unfavorable to one known treatment, the in- vestigator may pursue it to the ends of the earth to try to explain it away, whereas the same dif- ference unfavorable to another known treatment may be accepted with stoic indifference. Bias grows. The category "Other" partly illustrates this point in the present Study. Except for the overlapping analysis, no special study has been made to try to search out the reasons for the high death rates in this category, and it would be well if such a study were undertaken. At the same time, it is a study that could not itself be done very blindly, because of the hodgepodge nature of the category. The study of Part IV is deliberately re- stricted to death rates. This restriction has the advantage of giving a rather definite criterion and of evaluating the over-all safety of halothane in terms comparable with those used in studying massive hepatic necrosis. Because small dif- ferences in death rates imply differences in numbers of deaths so much greater than those involved in massive hepatic necrosis, we can 403

say that the choice among anesthetic agents con- sidered in this Study would almost always be made without consideration of deadly hepatic consequences. However, a weakness of the whole National Halothane Study is that it does attend especially to death as a criterion and does not consider explicitly other important factors of safety, comfort, ease, flexibility, and side ef- fects, which the anesthetist must appraise. The anesthetist will find that, as far as the threat of death is concerned, this Study leaves halothane available as a free choice. Finally, the National Halothane Study was a retrospective study and not an experiment. If it were desired to establish firmly the sizes of differences between the death rates associated with the anesthetic agents when used on com- parable patients, a large experiment would be required. INSTITUTIONAL DIFFERENCES Death rates for the 6-week period after surgery varied widely among the 34 institutions cooperating in this Study. They ranged from 0.0027 to 0.0640, a 24-fold ratio. In six institu- tions, the death rate was below 0.0100; in 10, it was above 0.0300. Such variation in so important an outcome of surgery compels attention. It was clear from the beginning that some, possibly large, variation in institutional death rates would be a necessary consequence of the institutions' varied patient populations, surgical loads, etc. So the institutions' death rates were adjusted separately for each of the variables sex, year, previous operation, physical status, age, operation, and SLI. The first three had no appreciable effect, but adjustment for each of the remaining variables substantially reduced the heterogeneity of institutional death rates; the original 24-fold variation came down to about 10-fold with each of the last three. Multiple adjustment for age, operation, and physical status reduced the extreme ratio to about 3 for the middle-death-rate operations only. Thus, there is evidence that a large part of the variation in institutional death rates is at- tributable to differences in age distribution of patients, to differences in the frequency with which high-risk and difficult operations are undertaken, and to the balance between elective and emergency procedures. Moreover, we can be nearly sure that fully adequate allowance for these factors would reduce the apparent hospital variation further, but we cannot say how much. There do remain, after all the analyses we have seen fit to apply, substantial differences in adjusted institutional death rates. We must face several questions concerning them: (1) Are the differences large enough, com- pared with sampling error, to call for explanation as other than plausibly random irregularities? (Are the dif- ferences real?) (2) If they do outweigh sampling error enough to be "real," what do they mean? Are they evidence of "excess" deaths? (3) How important are the deaths quantita- tively? Are thousands, hundreds, or dozens of deaths involved? (4) If the reality of the differences is ac- cepted, their meaning is understood, and they are quantitatively large, what are the reasonable indications for further study or action? The answer to this dif- ficult question must be well proportioned to administrative and legal (and emotional) realities, as well as to medical- scientific and statistical factors. To get a more direct grasp on the. question of the reality of the differences, we chose six operations (as described in Chapter IV-6) that were common in nearly all the hospitals, and studied the death rates for merely those six (about 16 percent of the data). No elaborate ad- justments were made. Strikingly, the institutions with the highest death rates (unadjusted) for the six chosen operations agreed well with the in- stitutions that gave the strongest appearance of being "high" on the middle-death-rate operations after multiple adjustments. The evidence for "real" institutional differences on the six opera- tions is very strong. We conclude that there are real differences in institutional death rates that are explained neither by the data taken in the Study (age, sex, physical status, operation, etc.) nor by sampling error. Some of the institutions had surgical deaths far outweighing the standard value ascribed to them by our statistical methods. In any such case, this may mean that deaths are high, or that the standard value is low. The latter possi- bility could arise because statistical adjust- ments have corrected inadequately for age, operation, etc., because they have taken no ac- count of other important institutional properties, or both. Sometimes statistical adjustments for interfering variables tend to undercorrect, and Appendix 2 to Chapter IV-6 gives strong evi- dence of an important tendency of this kind in some of our data. Furthermore, it is true that many variables relevant to surgical death rates have not been studied, and data have not been taken on them. Such variables include the general nutritional and health level of the institutions' clienteles, the willingness of the institutions' staffs to undertake risky cases, and the tendency of the institutions to be sent (and to accept) a high fraction of problem cases. Accordingly, we must recognize that if an institution seems "high" in our data (with statistical significance), we are not able to conclude that its surgical 404

death rate is therefore "excessive." But the question is certain even if the answer is not. Gauging the quantitative importance of the institutional differences is partly a matter of observing that in several institutions actual deaths exceeded a calculated standard for the institution by more than 200. Indeed, the sum of all "excesses" was about 1750 for the middle- and high-death-rate operations. However, in- stitutions with favorable experience had an ag- gregate sum of 1750 fewer deaths in the middle- and high-death-rate operations than the cal- culated standards for those institutions. The total number of deaths in the study was about 17,000; thus, positive and negative swings of 1750 are not to be ignored. More importantly, the main portions of these sums came from a very few institutions: about half the "excesses" came from three institutions, and about half the "deficits" from three. This tendency for the major part of the un- explained differences to be concentrated in a few places improves the prospects of being able to "do" something, such as to understand what actualities underlie the observed statistics or, possibly, even to identify opportunities for useful action. In assessing the quantitative importance of the heterogeneity of adjusted death rates, it would not only be good to know that institutions x, y, and z had unexplained high death rates, but far more important to know whether these in- stitutions were representative of 1, 5, 10, or 20 percent of hospital practice in the United States. The larger the group they "represent," the greater the quantitative importance attaching to the elucidation of their death-rate experience. Unfortunately, this question of representation is not one on which reliable information can be given. If we accept the unexplained differences as "real," and regard the magnitudes as definitely important, we come to the last issue: what to do? The problem is difficult. First, it must be recognized that we do not know for sure which institutions (if any) are the right ones to study. It is true that our evidence for the existence of real unexplained differences in institutional death rates has been the occurrence of several death rates that appear to be "outliers" in our data, but some of these may have resulted from large random fluctuations, and, similarly, some others deserving study may not have appeared in our sample as definitely high. Second, not only do we not know for sure which institutions should be studied, but we do not have from the statistical study any clear idea of what kinds of things should be studied. Finally, many im- portant and delicate questions must be faced if a useful study of so challenging a question ("Are there many preventable deaths in this institu- tion?") is to be possible. Thus, although we are persuaded by the evidence that some institutions have higher death rates than others, even after adjustment for the variables on which we have data, we are not persuaded that "strenuous efforts to correct the situation" are necessarily called for. With ill- defined targets for study and so sensitive an issue to explore, it would be all too easy to set in motion what could fairly be called a circus. At the same time, indications of such im- portance based on so many data should not be swept under the rug. We feel that, at a minimum, "someone" should recognize that there may be a problem and that it is not trivial. "Someone" should try to ascertain which institutions, al- though inexplicable by our data, are readily understood as "naturally" having high death rates because of, for example, the poverty of their clientele, or other compelling and well- understood reasons. Finally, "someone," after finding a remain- ing set of hospitals whose death-rate experience cannot be dismissed, should cause these to be thought about. Quiet, unofficial, cooperatively oriented inquiries into opportunities for studying the problems should be sought. Perhaps two hospitals, comparable except for death-rate ex- perience, could exchange two or three members of their staffs for a period of a year, for in- stance. Perhaps a cooperative, randomized trial of anesthetics on one or two operations (in- cluded in our set of six) would afford an oppor- tunity for communication and exchange of experi- ence that would lead to better understanding and improved practice. The importance of corrective efforts arises not from their effects (if success- ful) in these hospitals, but from the benefits that may accrue by wider application of similar ef- forts later. In summary, real and important differences in death rates do exist. They are not explainable statistically. Explanation will have to rest on medical-social-biologic procedural information. Getting the relevant understanding will be diffi- cult. The effects chosen for study will have to be large if the hope of useful results is to be more than slight. METHODOLOGY Randomized Trials versus Study of Past Data In discussing the design of an investigation like the National Halothane Study, one's first thought is that the natural and firm study will assign patients to their treatment groups on a random basis. Yet, the National Halothane Study used past data. Consequently, in Chapter II-1 reasons for studying past data were reviewed, together with the liabilities of the method. In addition, views about these pro's and con's in the light of the study performed may be helpful, even though the writers, as participants, are inevitably somewhat biased. 334-553 O-69—27 405

The greatest merit of the randomized clinical trial is its natural defense against sources of bias that cannot be recorded or per- haps even imagined. Its over-all results could be analyzed by straightforward statistical methods (for such purposes as comparing over-all death rates). But the importance of confirming over- all comparisons within each of as many groups as possible increases the desirability of gaining precision. To get increased precision, then, it might well be important to make computations of the kind and complexity undertaken in the present retrospective study, even if the data were drawn from a randomized trial. In the randomized study, there is a great risk that such a trial may collapse under ethical challenge and be a total loss. As it has turned out, ethical challenges can now be met by ex- tensive data; indeed, the whole problem is quite different in the light of the study of the retrospec- tive data, for halothane now looks safer than cyclopropane. In a randomized trial, it might well have been possible, although expensive, at least in parts of the study, to trace patients leaving the hospital within the standard 6-week period so as to study all deaths, rather than only those that occurred in the hospitals. In the study of hepatic necrosis, however, even the randomized clinical trial would have had no way of ensuring that a high rate of complete necropsy could be obtained; therefore, that part of the study would still be subject to a very large rate of nonresponse. (There may also be problems about differential ability to rate the necrosis at different times between death and necropsy.) Unless this non- response rate can be reduced, perhaps through biopsy if full necropsy is not feasible, the more controlled trial may not be worth carrying out as a means of studying massive hepatic necrosis. It seemed valuable to get historical data on death rates. Indeed, the design of a randomized trial of any two anesthetics might be aided by a careful study of a breakdown of the data on op- erative procedures by institution, anesthetic, and operation, together with the corresponding death rates. This would help in selecting operations for which choice of anesthetic might make a meas- urable difference and for which alternatives were clearly available. Of course, these historical data also offer baselines against which surgical teams and hospitals can measure their per- formance and improvement. When one recalls that the evidence against halothane was small, compared with the excel- lence of its reputation and the extent of the studies in its favor, the reluctance to set up a large, long, expensive study becomes clearer, particularly inasmuch as the original fears might well be settled by a study of past data. Because a randomized trial was not done, it is hard to say what it would have been like, but at one time a study comparable in size with the study actually done was under consideration. It might have required 4 or 5 years for the data- gathering process alone. The expense in money and professional talent would have been con- siderably greater, as would the demands on the participating hospitals. Similarly, the staff of the central office would have had to be larger and would have had to participate more con- tinuously in traveling and monitoring operations. In retrospect, considering the effort and training that went into the making of the team that carried out the several phases of the simpler study of past data, some Committee members believe that even the preparatory stage for a randomized clinical trial would have taken an additional year. These remarks merely describe a specific shortcoming of our national medical research picture, one emphasized by Wilfrid Dixon in discussing with us the problems of large-scale, cooperative, randomized trials. He pointed out that good will and hard work, although necessary, are no substitutes for a trained team of medical research workers, statisticians, data analysts, and processors, including scientific administrators and finance experts. Dixon pointed out that few such teams exist, that few persons realize how long it takes to train one, and that in this country we do not yet systematically try to produce such teams and keep them organized. Seen from that point of view, the National Halothane Study cer- tainly had no such team to start with, and at its close had only a partial capability of handling a large, cooperative, randomized trial—a partial capability now largely dispersed. In an era of in- creased governmental support for medicine and increasingly frequent searches for rare occur- rences, the notion of building and maintaining such teams should become more important. If a program for building and maintaining such teams is devel- oped, it should make the execution of randomized trials much more feasible. The existence of such teams mi ght facilitate the more systematic evalu- ation of new drugs and therapies. Because the rates of usage of the several anesthetic agents varied greatly from one hos- pital to another, something other than simply careful choice appropriate to individual patients was behind the usage rates. Either policy or personal preference must have ruled many of the decisions. This fact weakens somewhat the argument against the study of past data that claims that each anesthetic agent was especially chosen from among all agents to be optimum for each patient and his operation. We have used hospitals in which comparison was possible to get more comparative data on anesthetics. These data support further the general findings of the Study. Ironically, but inevitably, it is in the hospitals in which direct comparison is possible that the argument for possible selection biases by the anesthetist is hardest to combat. This discussion is intended, not to depre- cate the randomized trial, but rather to set 406

forth the view that it is not the only tool useful to the investigator with a question relating to the comparison of therapies. Finally, as discussed in Chapter IV-1, very small differences may be unresolvable by any method, for the very actions required to set up a randomized trial may change the effect that was to be measured. Data Analysis The key problem from the point of view of applying existing methods of statistical analysis was that the complete cross-classification natural to the data had so many cells, compared with the number of cases, that most cells had poorly determined death rates. To handle this difficulty, several new methods of analyzing data from multidimensional contingency tables (or even from continuous variables if no harm comes from grouping them) were used. Two basic problems have been treated: (1) getting estimates for counts or rates in cells determined by several variables, and (2) getting summary statistics over the entire set of data to make over-all comparisons. Both problems need treatment in many studies—not only in such fields as medicine and biology, but wherever contingency tables arise, and especially in the social sciences. The smoothed contingency-table analysis was especially designed to obtain improved cell estimates by borrowing strength from the mar- gins of the table, not only from single-factor margins, but also from those of higher dimen- sion. To get such improvements, something had to be given up, and it is the simultaneous effects of four or more variables that have been sacri- ficed here. The simultaneous effects of all possi- ble sets of three variables from among those in- cluded in the investigation have often been re- tained. In other instances, the data were fitted satisfactorily by retaining only the information about pairs of variables. Multiple regression is a device for estimat- ing cell values, but it was not used for this primary purpose in this Study. Instead, the cal- culation of regression values led to the construc- tion of strata used in computing summary sta- tistics for the whole Study. Cell aggregation methods are used for grouping cells homogeneous on the key variable (here, death rates) so as to improve the relia- bility of over-all statistics. These methods can be applied in many different ways. For example, in the smoothed contingency-table analysis we fitted the death rates in the cells without includ- ing the variable "anesthetic agent," and then grouped cells by the fitted death rates so as to collect more cases for basic rates. Rates ac- cording to the separate agents were then com- puted within the aggregated cells, and finally these were applied to some standard population to get summary statistics for the whole Study. Smear-and-sweep (Chapter IV-4) uses re- gression to borrow strength in making cell esti- mates preliminary to sweeping (aggregating) the cells of a two-way array into one new pseudo- variable. These cell estimates then guide the cell aggregation so that homogeneous cells are collected together, just as in the regression method (Chapter IV-5). However, the pseudo- variable may be used to form an additional array, by smearing each cell of the pseudo- variable along the distribution of values of the new variable for cases in the cell. This smear- ing produces a new two-way array. Then a new regression is computed and a new cell aggrega- tion is performed to form a new pseudo-variable, and so on. In the end, a set of summary statistics is computed. A new method, super standardization, of ap- praising incompleteness of the adjustments has been described in Appendix 2 to Chapter IV-6. Because the estimation of cell values and the computation of summary statistics are com- mon and important problems, these methods may find themselves in much wider use in the future. Naturally, owing to their newness, they will pro- duce many theoretical problems for statistical research workers. CONCLUSIONS AND RECOMMENDATIONS FROM THE STUDY OF DEATH RATES (1) Halothane, rather than being a dangerous anesthetic, appears to have a lower death rate than either cyclopropane or Other, and to have a rate about equal to that of nitrous oxide- barbiturate for many classes of operation, in- cluding the more severe ones. (2) Cyclopropane, even after adjustment (undoubtedly incomplete) for interfering varia- bles, shows a higher death rate than halothane in the middle-death-rate operations, and the dif- ferences may be even larger in the high-death- rate operations. (3) The performance of ether poses an im- portant puzzle. It has an excellent record in a few hospitals, but our assessment is unreliable. (4) The mixed category of anesthetics, Other, corresponds to a higher death rate than halothane or nitrous oxide-barbiturate. This finding is difficult to interpret, partly because of the large number of anesthetics and combinations included in this category. Insofar as we can tell, halothane, nitrous oxide-barbiturate, ether, and their combinations have had a much better rec- ord than the less-used anesthetics and their com- binations. (5) Institutional differences in death rates may be large enough, even after allowance for 407

age, operation, and physical status, to suggest the need for studies of the sources of these dif- ferences. However, the administrative, economic, social, legal, and medical problems in embark- ing on such a large study with such a diffuse target hold us back from making a definitive recommendation. The matter should be con- sidered carefully and an alert watch kept for any natural opportunities that would make such a venture more likely to succeed than it appears to be at the moment. The existence of obvious dif- ferences in the background of patient popula- tions—e.g., level of nutrition—and differences in institutional personnel and budget may already be enough to account for much of the variation that our data do not explain, and little in this Study allows for these variables. (6) New consideration should be given to building teams of medical research workers (adequately advised and supported in both sta- tistics and data analysis) for carrying out large cooperative studies. In addition, more effective and more frequently used mechanisms should be developed for evaluating, particularly through large-scale cooperative trials, new therapies during their early years of clinical use. REFERENCE 1. Beecher, H. K., and D. P. Todd. Study of the deaths associated with anesthesia and sur- gery; based on a study of 599,548 anes- thesias in 10 institutions 1948-1952 in- clusive. Publication 254, American Lecture Series; American Lectures in Anesthesi- ology. Springfield, Hlinois: Charles C Thomas, 1954. p. 47. 408

PART V. THE TOXICOLOGY OF DICHLOROHEXAFLUOROBUTENE

Next: THE TOXICOLOGY OF DICHLOROHEXAFLUOROBUTENE, by Ellis N. Cohen »
National Halothane Study: a Study of the Possible Association Between Halothane Anesthesia and Postoperative Hepatic Necrosis; Report. Edited by John P. Bunker [and Others] Get This Book
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