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Estimates of V(k), V (log10b) and Cov (log10b, k) can be obtained using RISK 81 (Krewski and Vany Ryan, 1981).

Rather than discard the 69 data sets for which mle's could not be obtained, we chose to fit a Weibull model to each of these data sets using a fixed value of the shape parameter k. In this regard, we first separated the 69 data sets into two subgroups based on their overall shape. A value of k = 1.7 was used for the 42 data sets that demonstrated clear upward curvature, this being the median value of k observed among the 68 of the 122 data sets for which k > 1. Similarly, a value of k = 0.55 was used for the 27 data sets exhibiting downward curvature, this being the median value of the 54 of the 122 data sets for which k < 1. The variance of log10TD*/50 was then estimated using (A.9), with k treated as an estimated rather than a known parameter. Allowance for some degree of uncertainty in the value of k is desirable in order not to severely underestimate the variance of log10TD*/50 (cf. annex B).

The 26 data sets in which only a control and single dose group were available were not used here since no information on the shape of the dose-response curve is available.

Annex B. Shrinkage Estimators of the Distribution of Carcinogenic Potency

The distribution of TD50 values for a series of chemical carcinogens provides useful information on the variation in carcinogenic potency. Because each estimate of the true TD50 for a specific chemical is subject to estimation error, the distribution of estimated potency values will exhibit greater dispersion than the distribution of true potency values (TD50s). This overdispersion may be eliminated using empirical Bayes shrinkage estimators (Louis, 1984).



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OCR for page 162
APPENDIX F 162 original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Estimates of V(k), V (log10b) and Cov (log10b, k) can be obtained using RISK 81 (Krewski and Vany Ryan, 1981). Rather than discard the 69 data sets for which mle's could not be obtained, we chose to fit a Weibull model to each of these data sets using a fixed value of the shape parameter k. In this regard, we first separated the 69 data sets into two subgroups based on their overall shape. A value of k = 1.7 was used for the 42 data sets that demonstrated clear upward curvature, this being the median value of k observed among the 68 of the 122 data sets for which k > 1. Similarly, a value of k = 0.55 was used for the 27 data sets exhibiting downward curvature, this being the median value of the 54 of the 122 data sets for which k < 1. The variance of log10TD*50 was then estimated using (A.9), with k treated as an estimated rather than a known parameter. Allowance for some degree of uncertainty in the value of k is desirable in order not to severely underestimate the variance of log10TD*50 (cf. annex B). The 26 data sets in which only a control and single dose group were available were not used here since no information on the shape of the dose- response curve is available. ANNEX B. SHRINKAGE ESTIMATORS OF THE DISTRIBUTION OF CARCINOGENIC POTENCY The distribution of TD50 values for a series of chemical carcinogens provides useful information on the variation in carcinogenic potency. Because each estimate of the true TD50 f or a specific chemical is subject to estimation error, the distribution of estimated potency values will exhibit greater dispersion than the distribution of t rue p otency values . This overdispersion may be eliminated using empirical Bayes shrinkage estimators (Louis, 1984).

OCR for page 162
APPENDIX F 163 original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Let Y = log10TD50 and suppose that E(Y) = µ = log10TD50, with V(Y) = σ2. Let Y1,…, Yn denote the logarithms of the estimated TD50 values for a series of n chemical carcinogens. We suppose that Yi is normally distributed with mean µ i and variance σi2. We further suppose that µ i are normally distributed with mean µ and variance τ2 where τ2 reflects the variance among the µ i. Our objective is to estimate µ and τ2, and hence describe the lognormal distribution of unknown TD50 values. Noting that an estimator of τ2 is where and is the estimator of V(log10TD50) based on (A.9). The shrinkage estimator of µ i is given by where represents an estimator of the intrastudy correlation, is an estimator of the overall mean of the log potency distribution, and is designed to protect against overadjustment for overdispersion. In general < 1, so that the estimators of the µ i are obtained by