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WILLIAM GEMMELL COCHRAN July 15, 1 909-March 29, 1980 BY MORRIS HANSEN AND FREDERICK MOSTELLER WILLIAM GEMMELL COCHRAN was born into modest circumstances on July 15, 1909, in Rutherglen, Scot- land. His father, Thomas, the eldest of seven children, had begun his lifetime employment with the railroad at the age of thirteen. The family, consisting of Thomas, his wife ican- nie, and sons Oliver and William, moved to Gourock, a hol iciay resort town on the Firth of Clyde, when William was six, and to Glasgow ten years later. Oliver has colorful recollections of their childhood. At age five, Willie (pronounced Wully), as he was known to family and friends, was hospitalizecI for a burst appendix, and his life hung in the balance for a day. But soon he was home, wearying his family with snatches of German taught him by a German patient in his nursing-home warcI. Willie hacI a knack for hearing or reacting something and remembering it. Oliver recalls that throughout his life, Willie wouIcl walk or sit around reciting poems, speeches, advertisements, mu- sic hall songs, and in later life oratorios anc! choral works he was learning. Until Willie was sixteen, the family livect in an apartment known in ScotIand as a "two room and kitchen" a parIor- cum-ctining room (usect on posh occasions, about twelve times a year), a bedroom usect by the parents, and a kitchen. . . 61

62 BIOGRAPHICAL MEMOIRS In the kitchen foot! was prepared, cooked, served, and eaten; dishes were washed, laundry done, friends entertained, anct homework accomplishecI. It was also the boys' bedroom in the form of an alcove-with-bed known as "the-hole-in-the- wall." The boys had a happy chilc~hoocI, with mile-Ion" walks to and from public school twice a clay (lunch was eaten at home) anc! play at the oceanside. Willie was a great achiever in school, usually coming in first. Oliver feels he hac! an irresistible urge to be first, often calculating closely just how much he would have to do to gain that enct. Oliver recalls being worrier! about passing a profes- sional exam and having Willie say to him: "l don't know what on earth you're worrying about; you only have to pass, ~ have to be first." He was referring to the Bursary Competition, open to all scholars in Scotland. And he was first, winning his fees to Glasgow University. Later he was in an even larger competition for the George A. Clark Scholarship, which pro- videcI support for four years and paid his Cambridge fees. Without winning these competitions, he almost assuredly wouIcl not have been able to attend either Glasgow or Cam- bridge. Willie tract no absorbing hobbies as a boy, although he dabble(1 in many things. Cycling, hiking, ant! walking in the hills were his chief physical activities. Later, studying and reading became primary. His scholastic prowess won him many books as prizes and created an extensive home library. Cochran graduated with the M.A. from Glasgow in 1931 with first class honors in mathematics and natural philosophy (physics) and shared the Logan Medal for the most distin- guishe(1 graduate in the Arts Faculty. That same year he en- terec] St. John's College, Cambridge, anct stuclied for the mathematics tripos (mathematics major) as a prelude to be- coming a research student. As an elective, he chose a new

WILLIAM GEMMELL COCHRAN 63 course, Mathematical Statistics, taught by John Wishart. A fellow student believes that the Great Depression had inter- estect him in the work of a Dr. Mess, who advocated thorough mathematical investigation of economic problems. He was doubtless also influenced by R. A. Fisher's work at this time. By now he hac} dropped the use of Willie, and among his colleagues he was known as Bill. Bill was persuaclect by Frank Yates to leave Cambridge without his doctorate to accept a position, a rare opportunity in the depression year of 1934, to do practical research at Rothamstec] Experimental Station. Cochran never die! re- ceive an earned doctorate, although he received honorary degrees from The University of Glasgow (1970) ante Johns Hopkins University (19751. During his six years at Rothamstect, Cochran pioneered with Yates in cleveloping techniques for analyzing replicatecl anc! long-term agricultural experiments and for assessing the effects of weather patterns on crop yielcis. They also studied selection effects in non-ranclom sampling. At RothamstecI, Cochran gainer! a great deal of practical experience anct became well known in his fielcI. In 1937 he marries! Betty I. M. Mitchell, a plant pathologist. After visiting Iowa State College (now University) in Ames in 1938, Cochran agreed to return there the following year to teach. The imminence of war in 1939 made him hesitate to leave Europe, but he felt he must keep his worct. Uncler George Snedecor, in 1939 {owe State was a center for statis- tical treatment of experimental work at a time when mocl- ern applied statistics had little foothold in America. The em- phasis in applied statistics at Iowa was then on sample surveys anct experimental design. Cochran lectured on both topics in his first quarter, and these lecture notes matured over the next ten years into his two well-known texts on these topics.

64 BIOGRAPHICAL MEMOIRS Two of Cochran's three children were born in Ames, Eliza- beth in 1940 and Alexander Charles in 1942. In 1943-44 Cochran took leave to join the Princeton Sta- tistical Research Group at Princeton University as a research mathematician. He was to work on Army-Navy research problems, including naval warfare and a survey of bomb ef- ficiency, for the Office of Scientific Research and Develop ment. At Iowa State, Cochran ant! Gertrude Cox initiated their collaboration, which culminated in their book Experimental Design, published in 1950. In 1946, at Cox's instigation, Coch- ran left Iowa to organize and head the graduate program in experimental statistics at North Carolina State College at Ra- leigh. Cox envisioned this program as half of the Institute of Statistics, the second part consisting of a Department of Mathematical Statistics at the University of North Carolina at Chapel Hill, headed by Haroict Hotelling. The Cochran's third child, Theresa, was born in North Carolina in 1946. In January 1949 the Cochrans moved to Baltimore, where Bill became head of the Department of Biostatistics in the School of Hygiene and Public Health at The Johns Hopkins University. Here his interest in medical and health problems increased. Bill published a second book, Sampling Techniques (19531. His two books along with his 1967 revision, at Sne- decor's request, of Snedecor's Statistical Methods-became im- portant reference texts and were widely translatecI. Statistical Methods is one of the most wiclely cited books in the scientific literature. In 1957 the Department of Statistics was organized at Harvarc! University, and Cochran joined the stab, remaining nineteen years until he became professor emeritus in 1976. During his time at Harvarcl, his continual interest in biosta- tistics was reflected in his interaction with the Department of Biostatistics in the Harvard School of Public Health.

WILLIAM GEMMELL COCHRAN COCHRAN S WORK 65 In discussing Cochran's scientific work, we open with his most famous theorem, anc! follow with selections of his work on the design and analysis of comparative investigations, with both experiments ant! comparative observational studies. After an overview of his work on counter! ciata, we present some of his contributions to the theory ancT practice of sample surveys, follower! by brief mention of other areas of work. With a few exceptions, we emphasize his advice and philosophy rather than the details of his technical work. Cochran's first paper (1934), a mixture of algebra and analysis, brought into mathematical statistics an extremely valuable and widely used result, now called Cochran's Theo- rem: ~ Let Xj, j = ~ ,2, . . . ,p, be inclepenclent stanciarc! normal ranclom variables with sum of squares Q. Let Q be clecom- posect into the sum of k quadratic forms Q2, where Qi has rank ri, i = I,2, . . . ,k. Then if one of the following three k conditions hoIcis, so clo the other two: (a) ~ ri = p, (b) each i = ~ Qi has a chi-squarecl distribution, and (c) each Qi is indepen- dent of every other. Cochran (1934) himself exploited this result to show that analysis of variance can be extenclec] to a variety of situations requiring adjustment for covariates. DESIGN AND ANALYSIS OF COMPARATIVE INVESTIGATIONS Agriculture. Over the years, sets of Cochran's papers fo cusecI on methods of value to many applied areas, including agriculture and biomeclical research. At Rothamstec! he ex ' The form cited is suggested by Maurice G. Kendall and Alan Stuart, The Ad- vanced Theory of Statistics, 2d ea., vol. 1 (New York: Hafner Publishing Company, 1 963), 360-6 1.

66 BIOGRAPHICAL MEMOIRS posited new clevelopments in lattice designs, attributing the general method to Frank Yates. These clesigns help breeders of wheat, soybeans, corn, and small grains by permitting comparisons of large numbers of varies es (squares being preferable, such as 49, 64, 8l, 100, ...~. He compares the performance of these designs with that of others (1941a, 194 1 b, 1 943b). Along with the descriptions of the methods and their strengths and weaknesses, Cochran continually emphasized the computational effort required in the analysis and the im- portance of being able to communicate the ideas to the in- vestigator. Why shouIc] the half-day or day of calculation required for the analysis be of much concern when an agri- cultural investigation has already required considerable land for much of a season and several workers to carry it out? Perhaps Cochran realizect that a computation that took him half a clay might leave a breeder helpless. He was therefore eager to reassure the breeder of its feasibility. Indeed, he said (1941a, p. 355), "Extra complication in the statistical analysis may be a drawback to the widespread use of a design in other respects. If the experimenter floes not clearly unclerstand the assumptions involved in the statistical manipulations, or the reasons for them, he loses confidence in the final results of the calculations." In several papers, Cochran gave substantial reviews in- tended to guide experimentation in specialized subject mat- ters. For example, just before leaving the United Kingdom for the U.S.A., he presented a major review paper (1939a) on the design and analysis of long-term agricultural experi- ments that won plaudits during discussion from Sir John Russell, R. A. Fisher, l. Wishart, F. Yates, M. S. Bartlett, M. G. Kendall, and H. O. Hartley. Cochran clealt not only with for- mal design and analysis considerations but also with impor- tant features of the practical execution of these trials in the

WILLIAM GEMMELL COCHRAN 67 fielct: size and shape of plots, numbers of replications, choices of stratification or blocks, heacIlands and guard rows between plots, ant] the value of a year or two of a uniformity trial prior to a long-term field experiment, especially for a new crop. And he warner! the statistician, "It is not sufficient for him Ethe statistician] to provide the best possible clesign to suit the size of the experiment; it is also his duty to acivise whether he thinks the experiment as clesignecI is worth cloing, or whether it should be postponed until more re- sources are available" ~ ~ 939a, p. ~ 061. With Gertrude Cox (1946a), he summarized the principal sources of variation in greenhouse experimentation (temper- ature, moisture, and shading gradients) and major designs ~c Luau eo~ for such sometimes nearly overwhelming variables. Curiously, in 1946 they reported that they had no information about the possible benefits of moving pots around, although this is one advantage of the greenhouse over fielcI conditions. His article in the International Encyclopedia of Statistics (197Sb) on experimental design contains an instructive post- script on the rise of the use of experiments in the social sci- ences and the encouragement given to this movement by the Social Science Research Council. That postscript relates more generally to his stucly (1976) of the history of experimenta- tion. After introducing us to Arthur Young's total intolerance for any method but comparative experiments, Cochran notes (1976, p. 5), "This issue persists today. In reviewing the pres- ent state of knowledge about the relative merits of two ther- apies for hospitalizect patients, we may finct a few well- controlled experiments and a larger number of doctor's observations on their experiences with one or the other ther- apy. Young wouIc! seem to suggest that to consider the latter group is a waste of time." Cochran used the history article to include a little instruc ,] ~, ~

68 BIOGRAPHICAL MEMOIRS tion on experimental design, as well as to get in a few licks about some consulting problems he tract su~erecl. He sug- gested that most consulting statisticians will have had expe- rience with an investigator who begins "'I want to do an ex- periment to show that....' He knows the answer." Cochran user! this remark as a springboard to discuss double-blind experiments. In a similar aside, Cochran used James John- ston's book on agricultures to make an aclctitional point. After describing Tohnston's position that a bad investigation wastes money and leads to incorrect results in standard! textbooks, as well as to the neglect of further research, Cochran said (1976, p. 9), "I have hear(1 this point made recently with re- gard to medical experiments on seriously ill patients, where there is often a question for the doctor if it is ethical to con- duct an experiment, but from the broader view-point fit is] a question of whether it is ethical not to conduct experi- ments." Cochran used history to console the young scholar. Upon recalling that after Student's t tables had been available for fourteen years anti practically no one used them, he said, "Young research workers who feel that the world is very slow to appreciate their results might be heartened by this ex- ample. The world is indeec! a little slow at times to realize how brilliant we are" (1976, pp. 13-141. He sums up the history of statistics in agriculture by saying that it took a cen- tury to take two major steps: (~) to begin applying probability theory (already available in astronomy) to interpret quanti- tative experiments and (2) to establish efficient practical methods for the concluct of field experiments. Bioassay. A sequence of papers (three with Miles Davis: 1963 1964, 1965a; and 1973) reported on Cochran anti Dav 23. F. W. Johnston, Experimental Agriculture, Being the Results of Past and Suggestions for Future Experiments in Scientific and Practical Agriculture (Edinburgh: W. Blackwood and Sons, 1849).

WILLIAM GEMMELL COCHRAN 69 is's studies of bioassay, where the investigator wants to find the LD50, the dosage that kills 50 percent of the animals or insects. They stucliect sequential approaches using a grid of dosages. Animals are tested at an initial close, ant! the out- come at that dose guides the choice of the next close up or clown. In one version, if the first close kills, the second close is one step smaller; if it does not, the next animal gets a close one step higher. This process continues. They recommendec! a two-stage approach. The first stage uses few animals with large steps until it locates a reversal, and the second stage uses the Robbins-Munro method with smaller steps. Clinical Trials. His papers on the design of clinical trials (1961a, 1977b) had a rather general nature. In the first (1961a), he emphasized heavily the value of precise protocol, power, blindness, randomization, and design. The biostatis- tician of the ~ 980s-with special survival analyses, sequential designs, ant! balancing approaches-might be surprised, even affronted, to react ~ ~ 96 ~ a, p. 7 ~ ): "The planning and conduct of a clinical trial cloes not involve any difficult or esoteric intellectual principles. It is mainly a matter of hare! work anc! attention to detail." The second paper (1977b) was a group effort focused on surgical experiments in cluoclenal ulcer. Although Cochran hac! suffered a substantial illness, he was essentially recov- erecI, but he did not want to take on any extra tasks. Conse- quently he refused to take part in a working group in the Faculty Seminar on Health and Medicine at the Harvard School of Public Health. But students and friends pleaded with him to change his mincI, and in the end he chaired the Working Group on Protocol Issues. After two years of dis- cussions in depth of the principal experiments in surgery for duodenal ulcer, the group proclucect a comprehensive list of medical and statistical criteria for consideration in further experiments. Most of the criteria have value for design, anal

70 BIOGRAPHICAL MEMOIRS ysis, anc! reporting of comparative medical investigations generally, notjust for surgery for cluo(lenal ulcer. Again, care and precision in protocol were emphasized. The lists cannot be reproclucect here, but a remark on follow-up to obtain information on nearly 100 percent of patients treated is worth quoting (1977b, p. 1911: "A search produced few ref- erences to available techniques for guarding against follow- up tosses. There seems to be no substitute for determination." They reported that at the Mayo Clinic high rates of follow- up have been "achieved by writing letters ctirectly to patients and not going through their doctors; if no reply is forth- coming, the telephone is used. If the patient is not found, a vigorous search is undertaken, inclucting use of bill-collecting agencies, who apparently have experience with similar prob- lems" (1977b, p. 191). Observational Studies. Program arrangers often asked Cochran to provide a substantial general paper on the con- cluct of comparative studies intended to decicle causation. In discussing the advantages of matching subjects or materials as compared with the use of covariance adjustment in obser- vational studies, he first notes! that the methods perform al- most equally well. "A clifficulty which ~ have occasionally en- counterec! with covariance is that some scientists have an inborn suspicion of adjustments to the ciata, and although the adjustments made in the covariance analysis are entirely objective, they may finct a rather gruciging acceptance" ~1953, p. 6871. (Although Cochran correctly stated that, given least squares, the adjustment itself is objective, the decision to make it usually is not; when many covariables are available, many subsets can be selected. The suspicious scientist has a right to some skepticism because an investigator couIcl adjust for the subset that gave results most pleasing to him or her. Nevertheless, when the covariables for adjustment are cho- sen in advance of the investigation, the method is objective.)

WILLIAM GEMMELL COCHRAN 71 Possibly his Journal of the Royal Statistical Society, Series A paper on observational studies (1965b), followocl by papers in 1967 and 1972, formed the basis for his program to pre pare a book on the planning and analysis of comparative observational studies. At his death, he left six ant! one-half of seven planned chapters completed. Moses and Mosteller edited it for posthumous publication (Cochran, 19831. The 1965b paper itself offers a substantial introduction to such investigations. Some quotations may be appropriate. The opening reminds us of HaroIc! Dorn's3 dictum to ask "How would the stucly be concluctec! if it were possible to do it by controllecI experimentation?" (1965b, p. 2361. In reviewing the ciangers of loading a stucly with so many research ques- tions that it may fall of its own weight, he confessed, "But when clearing with an imaginative investigator ~ clo not finct it easy to determine at what point one should adamantly op- pose all further questions, however ingenious ant! interest- ing" (1965b, p. 2401. In before-after studies of health, for example some investigators note that the initial question- naire may alert participants to behavior they should beware of, and thus bias the study. Cochran said (1965b, p.249), "My own view is that an eclucational programme that cannot im- prove health practices more than can a single questionnaire is not wrongly considered a failure...." When faced with a collection of studies yielding contra- clictory results, the applied scientist "cannot avoid an attempt to weigh the evidence for and against, since some results are so vulnerable to bias that they should be given low weight.... He should state such judgements forthrightly, remembering his duty to maintain even stanciarcts and, if possible, an air of calm detachment" (1965b, pp. 253-541. This last remark is a bit of tongue-in-cheek humor; Cochran was about to sug ~ H. F. Dorn, "Philosophy of inferences from retrospective studies," American Jour- nal of Public Health, 43(1953):677-83.

72 BIOGRAPHICAL MEMOIRS gest that someone else, while doing a good job, may have sometimes strayed just a bit from even standards. C O U NT E D D ATA Among Cochran's several systematic research programs, the analysis of counted data stands out (1936a, 1936c, 1937, ~ 938a, ~ 940a, ~ 942b, ~ 943a, ~ 950, ~ 952, ~ 954b). Maxwell, in his introduction to the first organized text on counted data, Analysing Qualitative Data,4 said "l am indebted to . . . Professor W. G. Cochran from whose work ~ borrowed freely" (p. 91. In studying both the distribution of diseased plants in rows of a field (1936a) and the persistence of one kind of weather (193Sa), Cochran had occasion to derive and use the distribution of the number of runs in a binomial sequence where the probability of success on a single trial differed from A, thus generalizing the work of Marbe and others. He also investigated the power of the sign test (19371. The problem of chi-squared tests and the correction for continuity (1942b) come up in various ways. How small shall the observed counts in cells be before we abandon the at- tempt to use chi-squared, or pool cells, or find some correc- tive device? Repeatedly Cochran returns to this question (1936c, 1942b, 1952, 1954b). In the 1942b paper he gives a special formula and tables for handling the problem, tables still not widely used, we believe. In addition to these, the use of transformations (1940a) and the analysis of variance for data that come as percentages (successes divided by totals; 1943a) and data from matched samples (1950) produced ma- jor contributions to the field. The large papers concerning goodness-of-fit tests (1952) and strengthening the common chi-squared tests (1954b) offer a small education in them iA. E. Maxwell, Analysing Qualitative Data (London: Methuen and Company, 1961).

WILLIAM GEMMELL COCHRAN 73 selves. The 1952 paper (p. 324) lists rules for handling chi- squared with small numbers in the cells, and the 1954b paper (p. 420) offers some slight revision of these rules based on further research. Incleed, these ten papers would form a small textbook on the analysis of counted data. The 1954b paper presents a large number of methods for strengthening the chi-squared tests and includes the essentials, together with a (derivation in the appendix of the now-popular tech- nique, sometimes called the Mantel-Hacnszel method for combining results of several contingency tables. One (lifficulty in reacting Cochran's papers is that it is hard to know what may be original with him anct what he regards as helpful exposition of known results. He often said of sta- tistical research workers, "we all deserve more credit than we get for results others publish, and a little less for those we ourselves publish." His grounds for this remark were that many ideas in statistics float around for a long time before someone actually sets them down in good order and pub- lishes them. Often we cannot nail clown just exactly who had the original idea. The utility of the common chi-squared test for goodness of fit has been much debated, partly because most statisticians including Cochran (1952, p.336) agree with Joseph Berkson. He arguer! that, given enough observations, we would be sure to reject the normal distribution (and presumably any other clistribution) as a motley. in any particular situation. (Amus- ingly enough, when Berkson gathered an enormous body of data to check whether radiation counts followoc! a Poisson process, theory and data agreed extremely well. On the other hand, Berkson's work on counting blood corpuscles showocl that no stanciarct distribution applied.) Cochran pointed out that Karl Pearson was aware of this ctifficulty, even when he invented the chi-squared test. Cochran struggled to suggest new approaches in these situations. He proposed that per

74 BIOGRAPHICAL MEMOIRS haps instead of testing a point null hypothesis we should be testing whether a quantity falls into an interval; or that we shouIct consider as the null hypothesis a broacler family, near the one being assumed. SAMPLE SURVEYS Cochran's initiation into sample-survey theory and prac- tice came when he joined Frank Yates at RothamstecI. R. A. Fisher, with Yates and other colleagues at Rothamsted, had made remarkable acivances in the theory of statistics as a too} of applied research in agricultural experiments. Modern theory and methods for sample surveys were substantially advances] by these developments, including the use of ran- domization in sample selection, aIreacly used to some extent in sample surveys. Cochran's first paper directly relatect to sample surveys (1936b) demonstrated the importance of randomization in the selection of samples as distinguished from purposive or judgmental selection. Yates hacI earlier clone an experimental demonstration of biases that resulted by allowing a judgmen- tal selection of a "random sample" of plants. At a conference of the observers of the crop-weather scheme (for crop fore- casting) in 1935, an experiment was planned to see to what extent the kinds of biases observed by Yates are common to all observers who make deliberate selections. Cochran ana- lyzec! the results of the experiment ant! concluded (1936b, pp. 74-751: It is obvious that samples that are picked by a process of randomization which gives every sample in the population an equal chance of being picked, must be representative of the population from which they are drawn and give an unbiased estimate of the quantity which it is desired to measure. Those who have little experience of the technique of sampling might, however, be unwilling to admit that they could not do as well, or better, by choosing the samples themselves. In this experiment, out of

WILLIAM GEMMELL COCHRAN 75 twelve observers, all of whom have had some training in sampling, not one managed to pick a sample that could be called representative of the ma- terial from which they were sampling.... What is even more serious and striking is that the individual observers were not consistent throughout the experiment; the positive bias in selection increased regularly as the mean height of the sampling-unit decreased. This work helpec! establish the importance of randomization in both sample surveys and experiments. In his work at Rothamstect, Cochran took advantage of the opportunities to be involved in practical studies in design and analysis of experiments and sample surveys. The sample- survey experience inclucled, for example, evaluation of crop- forecasting methods based jointly on sample-survey infor- mation on the crop and on weather data (193Sc). it also re- sultec! in empirical analyses of survey data to evaluate the efficiency of alternative sample designs for agricultural stuct- ies (1938b). As was his usual practice, this paper included a rather exhaustive analysis, including one of the early efforts to balance the amount of work involved against statistical ef- ficiency. He also cleveloped procedures for making approxi- mate advance speculations on sampling variances before re- sults are available for analysis, as is essential in practical work on sample-survey design. In another study (1940b) he eval- uated the gains that would result in estimating cereal yields by estimating the ratio of grain to total produce from the sample and applying the ratio to known information on the total produce. In 1939 he publishect a paper entitled "The Use of the Analysis of Variance in Enumeration by Sampling," based primarily on his work at Rothamsted, but published after he hacI movect to the Statistical Laboratory at Iowa State College in 1939. In this pioneering paper he applied the analysis of variance to finite-population sampling by regarding the finite population as a sample from an infinite superpopulation. He

76 BIOGRAPHICAL MEMOIRS conditions on the finite population and obtains estimators appropriate to the finite population that" with minor excep- tions agree exactly with those arrived at by direct applica- tion of probability-sampling theory. He illustrated the great convenience and power of the application of the analysis of variance to data available from a particular sample in evalu- ating the appropriate use of subdivision (now generally re- ferrec! to as stratification), subsampling, choice of sampling units, and double sampling. He conclucled: The results of a properly planned sampling investigation, in addition to providing an estimate of the accuracy of the sample, often provide es- timates of the accuracy of various alternative methods of sampling which might have been used. These estimates are helpful in increasing the effi- ciency of sampling in future studies on similar material.... The estimate of the relative accuracy of two methods of sampling is shown to be in most cases a simple function of the variance-ratio, so that its sampling limits are easily obtainable. (p. 510) In 1942 Cochran contributed an especially interesting re- sult for sample-survey applications concerned with "Sam- pling Theory When the Sampling-Units Are of Unequal Sizes." The procedure is applicable in estimating a population average, yp, or total for a variable y where information on a correlates! variable, x, is available for the total population and for each unit in the sample. Among others he considerec! a linear regression estimator of yp of the form ye = Ys + burp - xs), where Ys ant! xs are the sample means, b is the usual esti- mate from the sample of the linear regression coefficient, and xp is the known population mean of the x characteristic. It was well known that this estimator is the minimum variance estimator of yp if the population regression of y on x is linear and if the conctitional variance of y given x is constant. Coch- ran, however, showed the exceedingly useful result that V(ys)~! -r2), the well-known estimator of the variance of y, for this particular case, is asymptotically valid in large

WILLIAM GEMMELL COCHRAN 77 samples tor any population; that is, it is a consistent estimator of the variance no matter what the form of the regression of y on x. He consiclerec! weighted as well as unweighted regres- sion estimators ant! compared these and other alternative es- timators for varying sampling designs, as well as (liscussing the conditions under which each estimator is most efficient. As he pointed out, the regression estimator is relatively clif- ficult to compute. While the regression estimator has been extensively used, its applications are limited by the difficulty of computing. In addition, in sample surveys that measure many characteristics the results for multiple characteristics are not additive; that is, an estimate for males plus an esti- mate for females will not necessarily be equal to the estimate for both sexes combined. Nevertheless, it has proved highly useful in many applications. It has also contributed to uncler- standing the principles of estimation from sample surveys. Systematic sampling, of which the simplest form is select- ing every kth unit from some kind of an orclerec! sequence, has long had intuitive appeal and has been wiclely used as a sample-selection procedure. The estimation of summary measures from such a sample, such as means, ratios, or regressions, is straightforward, but theory is not available for making consistent estimates of variances. Often variances are estimated by treating a systematic sample as equivalent to a stratified random sample. Some empirical studies have shown this to provi(le a reasonable approximation in many circum- stances, but far from a satisfactory approximation in others. In 1944 W. G. and L. H. Maclow iclentifiecT systematic sam- pling as a special case of cluster sampling, anti providers theory and examiner! its characteristics under some alterna- tive models.5 Cochran extencled these results in a paper en- titlecl "Relative Accuracy of Systematic and Stratified Ran 5 William G. and Lillian H. Madow, "On the theory of systematic sampling," Annals of Mathematical Statistics, 1 5( 1944): 1-24.

78 BIOGRAPHICAL MEMOIRS dom Samples for a Certain Class of Populations,' published in 1946. He observed that numerous studies of real popula- tions had revealed that the variance among the elements in any group of contiguous elements increases steadily as the size of the group increases, and he constructed a model ap- propriate to such populations. In formulating the model, he regarded the observed finite population as a sample from a superpopulation in which (in what follows, E is the expecta- tion operator): E(xi) = lo, E(Xi ~)2 = ~2, E(xi - ~)(Xi+u- if) = PU ~ ' where Pa ' Pv ' O whenever u < v. He obtained average variances for samples from the possible finite populations from such a superpopulation. For this class of populations he showed that: The stratified random sample is always at least as accurate on the av- erage as the random sample and its relative efficiency is a monotone in- creasing function of the size of the sample. No general result is valid for the relative efficiency of the systematic sample. In fact, there are popula- tions in the class in which the systematic sample is more accurate than the stratified sample for one sampling rate, but is less accurate than the ran- dom sample for another sampling rate. If, however, the correlogram is in addition concave upwards, the systematic sample is on the average more accurate than the stratified sample for any size of sample. (1946b, p. 164) He pointed out that while no unbiased or consistent estimate of the variance of the estimated mean is available from a systematic sample, an unbiased estimator can be obtained if one can properly make an assumption concerning the form of the population being sampled. Its validity will depend, of course, on the validity of the assumed population model. Cochran published numerous other papers concerned with various aspects of sample surveys as he encountered them in consulting or otherwise became interested in them. For example, in a 1961(b) paper he examined alternative

WILLIAM GEMMELL COCHRAN 79 rules for establishing strata boundaries by comparing them empirically for several different forms of populations with varying amounts of skewness. In 1962 he jointly authored, with J. N. K. Rao and H. O. Hartley, a paper that proposed a simple procedure for unequal probability sampling without replacement. This approach hacI the advantages of simplicity of calculation and the ability to provide unbiased estimates of the variance of the estimators. This was a topic that re- ceivec! considerable attention at the time, and a number of different procedures were proposer! by various authors. The problem of nonsampling errors in surveys is one that has received extensive attention, and in 1968 Cochran pre- parect a review paper and extenclect some of the earlier work that had been done in this area. He concluded, as clo others, that errors in measurement can sometimes seriously vitiate most stanciard statistical techniques anti at other times have only trivial effects clepencting on the size of the relevant response variances ant! covariances. He adclec! that what seems neecled at the present state of development of this area are many studies that permit the estimation of these vari- ances and covariances, and that most of these studies should be embedcled in ongoing surveys. "When an 'errors of mea- surement' stucly has to be conducted separately, as will some- times be necessary because of the complexity of such studies, it is always difficult to reproduce the working conditions of an actual survey" (1968, p. 665~. In "Laplace's Ratio Estimator" (1978a), Cochran took an engaging historical tour. He reviewed the well-known esti- mate made by Laplace in 1802 of the total population of France. Laplace took a sample (by purposive sampling pro- ceclures) of communes in France and persuacled the govern- ment to have a population census taken in each of these. Births were registered throughout France, and therefore were known for each commune as well as the country as a

80 BIOGRAPHICAL MEMOIRS whole. He then estimated the total population of France with the ratio estimator Y = Xy/x where X is the known total reg- isterecI births, x is registered births for the sample communes, and y is the total population for the sample communes. The estimate was 28.4 million. L~aplace then estimated the stan- ciard error of this estimate to be 108,000. In computing the estimated sampling error, Laplace assumed that the birth rate in each commune (and of course in all of France) was the consequence of sampling births and population at ran- dom with equal probability from the same urn, a finite su- perpopulation. Cochran reported: "He found the large-sample ctistribu- tion of his error of estimate to be approximately normal, with a small bias and a variance that he calculates" (1978a, p. 31. Cochran then points out that in computing the sampling~er- ror Laplace failecI to recognize that the birth rates in the sample and in all of France were not independent, ant! states in a summary remark: It is unfortunate that Laplace should have made a mistake in proba- bility in a book on the theory of probabilities. In his application, however, the mistake was of little consequence. His working out of the large-sample distribution of the ratio estimator and his concept of the superpopulation as a tool in studying estimates from samples are pioneering achievements. (1978a, p. 10) Cochran wrote a number of review papers related to sample-survey topics (1938b, 1947, 1951, 1956) that pro- vided lucid summaries of the state of the art at the time the papers were written ant! gave acictitional interpretations. Of course his textbook, Sampling Techniques, is a substantially comprehensive summary, with extensions of theory to round out topics and with reporting of empirical results to provide better guidance on practical implications of some of the methods. It is unctoubtedly the most wiclely used textbook in

WILLIAM GEMMELL COCHRAN 81 teaching sample surveys, as is attested by the printing of sec- ond and third editions in 1963 and 1977. COCHRAN S OTHER CONTRIBUTIONS TO STATISTICS AND TO SOCIETY Cochran suggested that statisticians might profitably con- duct a survey to finct out how scientists use statistical tech- niques and how they are helped by them. He thinks it "might be very illuminating to statisticians if it couIcT be carried out despite the obvious difficulties. Statisticians are, ~ think, rather quick to jump to conclusions about the kinds of prob- lems which scientists in other fields are supposec! to face, anc! about their presumer! uses and misuses of statistical methods and ideas" ( 1952, pp. 334-351. Because he was writing in the Annals of Mathematical Statistics, he probably felt he was speak . . . . sing only to the statisticians. Having illustratecl Cochran's propensity for returning to problems repeatecIly, we shall not review all the topics where he carried on such a program. Instead we merely mention that these incluclecI: (a) the problem of weighting to combine results from several comparable experiments (for example, when the effects in the different experiments did not neces- sarily have the same true means or previsions and when pre- cisions neeclec! to be estimated); (b) the problems associates! with both qualitative ant! quantitative ctiscriminant functions; (c) the use of covariates in experiments and observational studies; (cI) the effect of errors of measurement on regres- sion, analysis of variance, and the analysis of counted ciata; anti (e) special analyses for detecting outliers, for handling missing observations, for acIding or removing a variable in regression, or for comparing scales of measurement. Cochran was an exceptional teacher, beloved by his stu- dents. He directed four dissertations at North Carolina, fifteen at Johns Hopkins, ant! nineteen at Harvard. In acicli

82 BIOGRAPHICAL MEMOIRS tion he greatly influenced a large number of other students. They recall his clarity, wit, willingness to help, and use of practical examples culled from his experience. As one said, Bill "pulled it all together in a way that made it fun to cal- culate coefficients ant! to invert matrices. We wanted to do it because Bill would have been clisappointect if we failed." Bill had a great ability to get to the heart of any statistics problem with virtually no time lost. He was succinct and clear in his teaching and writing. He worked with his graduate students to try to make them understand where the problem formulation and inductive statistics enclect and the clecluctive mathematics began. Bill displayer! the great knack for linking the theoretical and the applied that Americans associate with statisticians trained in the United Kingdom, and he was able to explain complicatect statistical information to investiga- tors in language they could unclerstancl. Consequently he was a much sought-after consultant anct an excellent com- mittee member or head. His calm fairness anct down-to earth attitude assurer! attention to dealing with the core problem. Cochran limitecI his committee participation to the amount of work he could hancIle. He chaired the committee appointed by the American Statistical Association at the re- quest of the National Academy of Sciences to review the Kin- sey, Pomeroy, and Martin stucly of sexual behavior in the hu- man male, work that resulted in a book (1954a). He served as chairman of the Pane! of Statistical Consultants, U.S. Bu- reau of the Census. He server! on the committee to consider the effect of battery additives on the life of batteries, on the Academy Committee to the Atomic Bomb Casualty Commis- sion, and on the Committee on Epidemiology and Biometry at the National Institutes of Health. The Subcommittee on National Morbidity Survey of the U.S. National Committee on Health Statistics, of which he was a member, submitted a

W I L L I A M G EM M E L L C O C H RA N 83 report to the Surgeon General that was the basis, with little change, of the National Health Survey Act. A smoker, Bill was the only statistician on the Surgeon General's Committee on Smoking and Health. Bill received many honors. He was at various times pres- iclent of four major statistical organizations: the Institute of Mathematical Statistics in 1946, the American Statistical As- sociation in 1953, the Biometric Society (which he helpecI found as a member of the organizing committee) in 1954- 55, ant! the International Statistical Institute in 1967-71. He was elected to the American Academy of Arts and Sciences in 1971 ant! to the National Academy of Sciences in 1974. He was a fellow of the American Association for the Acivance ment of Science; honorary fellow of the Royal Statistical So- ciety; ant! Guggenheim fellow, 1964-65. He received the Guy Medal of the Royal Statistical Society in 1936, the S. S. Wilks Memorial Medal (American Statistical Association) in 1967, and the "Outstanding Statistician" Award (Chicago Chapter, American Statistical Association) in 1974. He was editor of the Journal of the American Statistical Association from 1945 to 1950. Personally, Bill was an unpretentious man with Scottish wit and humor. He was a believer in the fellowship of man, and one of the few things sure to elicit his anger was a bigoted comment. Although he preferred to work by himself rather than to collaborate with others, he was friencIly to everyone and liked by all. He and his wife Betty, to the delight of col- leagues anti students, entertained frequently, and enjoyed square ciancing, theater, music, and travel. Hundreds of sta- tisticians from far-flung places attended Bill's retirement clin- ner in ~ 976. The last several years of Bill's life were plagued with a series of medical problems. Nonetheless, after his retirement anc! his move to his Cape Cod home, he continued to travel,

84 BIOGRAPHICAL MEMOIRS to teach, and to write. He died in OrIeans, Massachusetts, on March 29, 1980. WE APPRECIATE THE ADVICE AND SUPPORT of his wife Betty Cochran and brother Oliver Cochran, and of colleagues Arthur P. Dempster, John Emerson, Katherine Godfrey, David C. Hoaglin, Augustine Kong, Erich Lehmann, Lincoln E. Moses, Marjorie O1- son, Katherine Taylor-Halvorsen, and Cleo Youtz. We have also benefited from correspondence with Richard L. Anderson and Geoffrey Watson and from their writings about Cochran cited in the references. REFERENCES Anderson, R. L. William Gemmell Cochran 1909-1980, A Per- sonal Tribute. Biometrics, 36(19801: 574-78. Dempster, Arthur P., and Frederick Mosteller. In Memoriam. Wil- liam Gemmell Cochran 1909-1980. The American Statistician, 35, no. 1~19811:38. Dempster, Arthur P., Margaret Drolette, Myron Fiering, Nathan Keyfitz, David D. Rutstein, and Frederick Mosteller (chairman). Faculty of Arts and Sciences-Memorial Minute, W. G. Coch- ran. Harvard Gazette (3 December 19821:4. Watson, G. S. William Gemmell Cochran 1909-1980. The Annals of Statistics, 10~1982~: 1-10.

WILLIAM GEMMELL COCHRAN SELECTED BIBLIOGRAPHY 1934 85 The distribution of quadratic forms in a normal system, with ap- plications to the analysis of covariance. Proc. Cambridge Philos. Soc., 30:178-91. L11 1936 a. The statistical analysis of field counts of diseased plants. i. R. Stat. Soc., Ser. B (Suppl.), 3:49-67. L4] b. With D. }. Watson. An experiment on observer's bias in the se- lection of shoot-heights. Emp. J. Exp. Agric., 4~13~:69-76. F5] c. The of distribution for the binomial and Poisson series, with small expectations. Ann. Eugen., 7:207-17. F61 1937 The efficiencies of the binomial series tests of significance of a mean and a correlation coefficient. l. R. Stat. Soc., Ser. A, 100:69-73. [9] 1938 a. An extension of Gold's method of examining the apparent persistence of one type of weather. Q. }. R. Meteorol. Soc., 64:631-34. b. The information supplied by the sampling results. Ann. Appl. Biol., 25:383-89. L12] c. Crop estimation and its relation to agricultural meteorology. I. R. Stat. Soc., Ser. B (Suppl.), 5:1-45. t151 1939 a. Long-term agricultural experiments. i. R. Stat. Soc., Ser. B (Suppl.), 6:104-48. L18] b. The use of the analysis of variance in enumeration by sampling. }. Am. Stat. Assoc., 24:492-510. F191 1940 a. The analysis of variance when experimental errors follow the Poisson or binomial laws. Ann. Math. Stat., 11 :335-47. L22] NOTE: The numbers in brackets at the end of each entry correspond to the number given that paper in Contributions to Statistics, 1982.

86 BIOGRAPHICAL MEMOIRS b. The estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce. I. Agric. Sci., 30:262-75. F23] 1941 a. Lattice designs for wheat variety trials. }. Am. Soc. Agron., 33:351-60. [24] b. An examination of the accuracy of lattice and lattice square experiments on corn. Iowa Agric. Exp. Stn. Bull., 289:397-415. F27] 1942 a. Sampling theory when the sampling-units are of unequal sizes. I. Am. Stat. Assoc., 37:199-212. L28] b. The X~ correction for continuity. Iowa State Coll. I Sci.,16:421 36. L29] 1943 a. Analysis of variance for percentages based on unequal num- bers. }. Am. Stat. Assoc., 38:287-301. L331 b. Some additional lattice square designs. Iowa Agric. Exp. Stn. Bull., 318: 729- 48. L34] 1946 a. With Gertrude M. Cox. Designs of greenhouse experiments for statistical analysis. Soil Sci., 62: 87-98. F361 b. Relative accuracy of systematic and stratified random samples for a certain class of populations. Ann. Math. Stat., 17: 164-77. t38] 1947 Recent developments in sampling theory in the United States. Proc. Int. Stat. Conf., 3:40-66. F401 1950 The comparison of percentages in matched samples. Biometrika, 37:256-66. L43] 1951 Modern methods in the sampling of human populations. Am. I. Public Health, 41:647-53. F46]

WILLIAM GEMMELL COCHRAN 1952 The x'2 test of goodness of fit. Ann. Math. Stat., 23:315-45. F491 1953 87 Matching in analytical studies. Am. I. Public Health, 43:684-91. F52] 1954 a. With Frederick Mosteller and John W. Tukey. Statistical Problems of the Kinsey Report on Sexual Behavior of the Human Male. Wash- ington, D.C.: American Statistical Association. b. Some methods for strengthening the common x2 tests. Biomet- rics,10:417-51.~591 1956 Design and analysis of sampling. In: Statistical Methods, ed. George W. Snedecor, pp. 489-523. Ames: Iowa University Press. L631 1957 . With Gertrude M. Cox. Experimental Designs, 2d ed. New York: John Wiley. 1961 a. Designing clinical trials. In: Evaluation of Drug Therapy, ed. F. M. Forster, pp.71-77. Madison: University of Wisconsin Press. L701 b. Comparison of methods for determining stratum boundaries. Bull. Int. Stat. Inst., 38:345-58. t721 1962 With i. N. K. Rao and H. O. Hartley. On a simple procedure of unequal probability sampling without replacement. J. R. Stat. Soc., Ser. B. 24:482-91. L75] 1963 With Miles Davis. Sequential experiments for estimating the mean lethal dose. In: Le Plan d'Experiences, pp. l 81-94. Paris: Editions du Centre Nationale de la Recherche Scientifique. t781 1964 With Miles Davis. Stochastic approximation to the median effective dose in bioassay. In: Stochastic Models in Medicine and Biology, ed.

88 BIOGRAPHICAL MEMOIRS John Gurland, pp. 281-300. Madison: University of Wisconsin Press. L82] 1965 a. With M. Davis. The Robbins-Munro method for estimating the median lethal dose. I. R. Stat. Soc., Ser. B. 27:28-44. F841 b. The planning of observational studies of human populations. J. R. Stat. Soc., Ser. A, 128:234-65. t85] 1967 Planning and analysis of non-experimental studies. In: Proceedings of the Twelfth Conference on the Design of Experiments in Army Re- search and Testing, AItO-D Report 67-2, pp. 319-36. Durham, N.C.: U.S. Army Research Office. L88] 1968 Errors of measurement in statistics. Technometrics, 10:637-66. t89] 1972 Observational studies. In: Statistical Papers in Honor of George ~ Snedecor, ed. T. A. Bancroft, pp. 77 - 90. Ames: Iowa State Uni- versity Press. L971 1973 Experiments for nonlinear functions (R. A. Fisher Memorial Lec- ture). l. Am. Stat. Assoc., 68:771-81. L991 1976 Early development of techniques in comparative experimentation. In: On the History of Statistics and Probability, ed. D. B. Owen, pp. 3-25. New York: Marcel Dekker. F1051 1977 a. Sampling Techniques, 3d ed. New York: John Wiley & Sons. b. With Persi Diaconis, Allan P. Donner, David C. Hoaglin, Nich- olas E. O'Connor, Osler L. Peterson, and Victor M. Rosenoer. Experiments in surgical treatment of duodenal ulcer. In: Costs, Risks, and Benefits of Surgery, ed. ~ohn P. Bunker, Benjamin A.

WILLIAM GEMMELL COCHRAN Barnes, and Frederick Mosteller, pp. 176-97. New York: Ox- ford University Press. F1061 1978 89 a. Laplace's ratio estimator. In: Contributions to Survey Sampling and Applied Statistics, ed. H. A. David, pp. 3-10. New York: Aca- demic Press. t1071 b. Experimental design. I. The design of experiments. In: Inter- national Encyclopedia of Statistics, ed. William H. Kruskal and Ju- dith M. Tanur, pp. 285-94. New York: The Free Press. t110] 1980 With George W. Snedecor. Statistical Methods, 7th ed. Ames: Iowa State University Press. 1982 Contributions to Statistics. New York: John Wiley & Sons. (A collection of the 116 papers published by William G. Cochran.) 1983 Planning and Analysis of Observational Studies, ed. Lincoln E. Moses and Frederick Mosteller. New York: John Wiley & Sons.