StatisticalScience 2009,Vol.24,No.2,244–254 DOI:10.1214/08-STS259 (cid:13)c InstituteofMathematicalStatistics,2009 A Conversation with Shayle R. Searle Martin T. Wells Abstract. Born in New Zealand, Shayle Robert Searle earned a bach- elor’s degree (1949) and a master’s degree (1950) from Victoria Uni- 0 versity, Wellington, New Zealand. After working for an actuary, Searle 1 0 went to Cambridge University where he earned a Diploma in mathe- 2 matical statistics in 1953. Searle won a Fulbright travel award to Cor- n nell University, where he earned a doctorate in animal breeding, with a a strong minor in statistics in 1959, studying under Professor Charles J Henderson. In 1962, Cornell invited Searle to work in the university’s 9 1 computing center, and he soon joined the faculty as an assistant pro- fessor of biological statistics. He was promoted to associate professor in ] E 1965, and became a professor of biological statistics in 1970. Searle has M also been a visiting professor at Texas A&M University, Florida State University, Universita¨t Augsburg and the University of Auckland. He . t has published several statistics textbooks and has authored more than a t 165 papers. Searle is a Fellow of the American Statistical Association, s [ the Royal Statistical Society, and he is an elected member of the In- ternational Statistical Institute. He also has received the prestigious 1 v Alexander von Humboldt U.S. Senior Scientist Award, is an Honorary 2 Fellow of the Royal Society of New Zealand and was recently awarded 7 the D.Sc. Honoris Causa by his alma mater, Victoria University of 2 3 Wellington, New Zealand. . 1 0 0 Thefollowinginterview,withMartinWellsofCor- ble such things as 1+2=3 in a book of wallpa- 1 nellUniversity, took place over anumberof visits to per samples used as a scratch pad. And throughout : v the home of Professor Searle in Ithaca, NY in the most of my school days I was occasionally moved i Fall of 2007. up a class because of being good at mathematics. X (Classes were not governed by age, as in the U.S.A., r a 1. THE EARLY YEARS butbyability.) Mindyou,mathematicswasnotpar- ticularly rigorous orconceptual atthekindergarten- Wells: Shayle, tell me a little about your early type school where I was for a year, nor during my education. two years at a grade school. In 1937 I started at a Searle: As a small boy I was, so my mother often boarding school (for 8–14-year-old boys) where the told me, in love with numbers and arithmetic. Ap- teaching was very good, including mathematics. Af- parently even before starting school I used to scrib- terfiveyearsItransferredtoahighschoolwherethe teaching wasgenerally bad,exceptformathematics. Martin T. Wells is Professor, Departments of Biological Wells: Tell me about your undergraduate days. Statistics and Computational Biology, Social Statistics Searle: It was in March 1945 when I started at and Statistical Science, Cornell University, Ithaca, New University. I was to be at Victoria University Col- York 14853, USA (e-mail: [email protected]). lege in Wellington (N.Z.’s capital) 120 miles south of my home town Wanganui. It was a college of the This is an electronic reprint of the original article University of New Zealand, at that time, formally published by the Institute of Mathematical Statistics in Statistical Science, 2009, Vol. 24, No. 2, 244–254. This New Zealand’s only university with students only at reprint differs from the original in pagination and its four colleges and two agricultural colleges dotted typographic detail. around the country—half of them in each island. 1 2 M. T. WELLS I found it to be incredibly dull work. That decided me; I wanted to do mathematics. So at year’s end I quit my job, changed courses to do a B.A., and went home where I could, and did, get some excel- lent tutoring for three months to bone up on the maths I should have done in my third year at high school. All the work was algebra from an old and wonderfully good book by Hall and Knight. Wells: How did you resolve these early career is- sues? Searle:DuringthatfirstyearofmathematicsIstill hadthecrunchquestion:forwhatjobwouldamath- ematics training prepare me? Becoming an actuary came as the answer, surprisingly from a lady who ownedasuccessfuldepartmentstore.Withtherebe- ing only four actuaries in New Zealand in the 1940s, none of whom were anywhere near my hometown of 20,000 people, basically a farming town, it was surprising any resident had even heard of an actu- ary!“Sortof ahigh-level accountant” was aboutthe nearest description. Anyway, I found out about it, had another summer of tutoring and in May 1947 sat and passed the preliminary exam of the London Institute of Actuaries. The exam consisted of three hour papers in English and mathematics. Then for the next two years I concentrated on the B.A. exams coming in 1948, these being two papers in pure maths and two in applied, the latter involv- ing topics like statics, dynamics and hydrostatics: Fig. 1. Shayle Searle, around 1937, school uniform, aged 9. dull, difficult and for me from an agriculturally ori- ented background, of no use whatsoever. In 1949, They are now, and have been for some years, all after weathering a bout of pneumonia, I took the autonomous universities, and the University of New six exam papers for the M.A. in mathematics (no Zealand has disappeared. thesis required), one of which was on matrices. The The difficulty I faced in 1945 was deciding what instructor for that course was senior lecturer J. M. course of study I would follow. My family (no sib- Campbell, using Aitken’s 1948 book “Determinants lings, but a bunch of cousins) knew nothing about and Matrices.” Campbell, a New Zealander as was universities;leastofalldidtheyknowanythingabout Aitken also, had done his Ph.D. in statistics at Ed- careers based on mathematics. Schoolmaster or ac- inburgh where Aitken was having a very eminent countantseemedtheonlyoptionsandbecauseinmy career. third and last year in high school (having the previ- Wells:Atthatpoint,whatpathdidyoupursueaf- ous year passed the nationwide university entrance ter an undergraduateand masters training in math- exam) I took and passed two year-long courses for ematics? theB.Com.degree,soIchoseaccountancyandspent Searle: After the 1949 M.A. exams I took a job as half of that first university year as an office boy in a assistant to the actuary at Colonial Mutual Life As- large accountancy firm in Wellington, and thus was surance Company in Wellington. I had no office of a part-time student. The firm did a lot of auditing myown,butmerelyadeskinalargeroomwithsome work and this led to my first apprenticeship, so to dozen or so retirees who, day in and day out, were speak, of being an auditor: checking the arithmetic checking the weekly premiums paid for what were of long columns of journal entries in the books of calledindustrialpolicies—somethingliketwenty-five a Lever Bros. soap-making plant near Wellington. cents a week. The actuary’s office was but a few 3 SHAYLER. SEARLE steps across the hall. He was a real proper English- to enable me to study statistics. Interest in statis- man and helped me a great deal in preparing my tics had been promoted by the course in Wellington first paper, “Probability: Difficulties of Definition.” and by the Part I actuarial exam. Unfortunately I It was published in the Journal of the Institute of discovered that I should have applied for the schol- Actuaries Students’ Society, 1951, pp. 204–212. Al- arship before, not after, my M.A. exams. I could thoughIhadreadthevonMises book,andVenn’s,I apply after, but I knew I’d be competing with the soon realized after attending my first lecture or two notable New Zealander Peter Whittle (who has re- at Cambridge that my knowledge of probability was cently retiredfromhisCambridgeprofessorship).So I scrubbed that idea. However, I was told that I very na¨ıve and incomplete. (I had not even had a could be supported overseas by a family agriculture course in set theory!) trust (established by my successful maternal grand- Anyway, in 1950, now in the actuarial environ- father), so I proceeded to get myself accepted at ment, I reverted to the actuarial exams, but still both Emmanuel College and the statistics labora- kept an eye on the B.Com. degree to which several tory at Cambridge University. courses in my B.A. degree (e.g., English and eco- nomics) could also be counted. So I took a statistics 2. CAMBRIDGE DAYS course for the B.Com. which also helped in prepar- ing for Part I of the actuarial exams destined for Wells: Let’s chat about your time at the statistics May1951.Theseandthefollowingpartswereknown laboratory at Cambridge University. Tell me about to be difficult; the average time for becoming fully your introduction to Cambridge University. qualified was eleven years! Individual exam ques- Searle: Departure from New Zealand in mid Au- tions dealt with annuities and life insurance premi- gust, by ship, was not easy; three hours before leav- ums (with absolutely horrible notation) and some ing the Wellington wharves I received a phone call statistics. Many questions had such long descrip- from the government actuary (the Institute’s offi- cial representative) telling me that I had failed the tions that a paper took a full ten minutes to read— whole of the Part I exam taken in May. That was and after reading it one had to decide which ques- a bitter pill to add to the emotion of a ship pulling tions to answer to satisfy the instruction “Do three out from its berthage for what was to be its usual from the five questions in each section of the exam 31-day voyage to Britain. My first days after arrival paper”!! As if all this wasn’t going to be difficult in Britain were spent in London during which time enough, there were no lectures available and only I went to see the Institute of Actuaries. Compared two or three books, some of them only in galley to the facilities I’d struggled with in New Zealand proof form. Notational distinction, in these books, for trying to pass their exams, the Institute looked of a population statistic from an estimator of it was wonderful: a variety of lecture courses, some 80–100 sparse: often the same symbol was used for both. students, and very nearby was a big Prudential As- From England (where the exams originated) I was surance building where a large number of actuarial given the name and address of an actuary in an studentswereemployed.Ifbecominganactuaryhad insurance company in Sydney, Australia who was still been my intention, I’d have been very envious. supposed to be available to me to answer questions ComingtotheInstitutewastheobviousthingtodo. and give advice on my attempted solutions to ex- But I was going to Cambridge. And a day or two ercises in the books. But, despite the almost daily after getting there, an easy hour by train, I paid flying-boatservicesbetweenWellingtonandSydney, a visit to the statistics laboratory. After my knock it usually took him a month to get his comments to on the director’s door, I followed “come in” and an- me. Not much use. Anyway, in May 1951 I sat the nounced myself “Shayle Searle, from New Zealand.” exam. “Who are you?” said John Wishart (of Wishart’s Wells: How did you initially get interested in the distribution). “I’ve never heard of you!” That did subject of statistics? not seem to be a very auspicious start. However, Searle:Inthe1949 M.A. exam I’ddonemuch bet- in gentlemanly English manner, Dr. Wishart, said ter than expected.By onemark outof 600 I was top “Well, you’ve come a long way so we can’t send you of New Zealand; however, no kudos in that since back.” there were only four examinees! Nevertheless, as a Wells: What happenedafter this auspicious intro- result,Ibecameinterestedinanoverseasscholarship duction at Cambridge? 4 M. T. WELLS Fig. 2. Cambridge University Statistical Laboratory 1953 personnel. Back row: D. A. East, B. Guss, E. S. Page, G. A. Coutie, K.K. Chaudary, D.J. Newell,F. J. Chatterly, J. R.Bell,R. S. Bawa,J. R. Ashford, P. A.Wallington. Second row: B. Reifenberg, W. L. Smith, D. R. Cox∗, J. Wishart (Director)∗, F. J. Anscombe∗, D. V. Lindley∗, P. A. Johnson. Front row: J. N. Darroch, B. D. Gee, W. S. Townson, K. W. C. de Silva, B. Das, G. B. Aneuryn-Evans, S. R. Searle. Absent: H. E. Daniels∗, Th. Metakides, J. T. Laws. (∗Faculty) Searle:Istayed;andfromamongthestar-studded to me, and as he rightly said, “you don’t want to faculty of F. Anscombe, D. R. Cox, D. Lindley and work for just a Failed Ph.D.” Agreement was clear. H. E. Daniels I was given Dennis Lindley as my tu- Wells: How did you handle this early disappoint- tor and into whose class on probability I was di- ment? rected. Boy, was I lost. But fortunately courses did Searle: During those early years I did have a dis- not have exams; and my learning revolved around appointment or two: mis-timing an application for the customary 2–3 hour tutorial session I had each the overseas scholarship; getting no help preparing week with Lindley. Just he and me. Most of the for the actuarial exams; the “Who are you?” intro- time centered on my attempts at answering ques- duction to statistics at Cambridge; and then being tions that came from previous years’ exams for the discouraged from the Ph.D. degree. You ask “how Diploma. To begin with I was expecting to work for didIhandle”allthis?Certainlyinthosedaysthele- a Ph.D. But after a couple of months or so, Lindley gions of counselors available today for all manner of told me like it was: he recommended that I do the situationsdidnotexist.Onelargelyreliedononeself DiplomaandnotthePh.D.Hisreasoningwasasfol- and learnt to tough it out. lows: statistics has a formal connection to the math Wells: Tell me about your Cambridge Diploma department and mathematicians do not always look project. veryfavorablyatstatistics.Yettheyusuallycometo Searle: The Diploma has stood me well. It con- the final oral exam for statistics Ph.D. candidates. sisted of two papers, one theory and one data anal- And often they decline to award the Ph.D. but in- ysis, and also a written report resulting from being stead award an M.A.—which in this situation has seconded for theacademic year to a data-generating come to mean “Failed Ph.D.” and there was no re- research project within the university: the report to course. Lindley felt that this is what would happen describe the data analysis and its consequences. I 5 SHAYLER. SEARLE was seconded to E. H. Callow’s lab where he was farmer but for the nation also, since New Zealand measuring iodine number in the fat taken from dif- has, for more than a century, lived by its exports of ferent joints of various beef carcasses—and my ef- agricultural products; butter, cheese and milk pow- forts finished up as a co-author (see Callow and ders being important parts thereof. The outstand- Searle, 1956). All this was usually considered a one- ing researcher in this discipline of animal breeding yeareffort,butinmycaseitwassetattwoyears.At wasProfessorC.R.HendersonofCornellUniversity. theendofthefirstyearIsatthetwoexamsforprac- And it was my good fortune that he came to New tice! My second year exam results were not as good Zealand for his first sabbatical, and actually had a as the firstyear; in fact of seven out of nineteen stu- desk in my office for eight months from September dentswhogot“PassedwithDistinction”yourstruly 1955.HisownPh.D.fromIowaStateUniversity was wasnotamongthem.Lindleygavemetheraspberry inanimal breeding,undertheeye of Professor Lush, like I’ve never had it before or since. I’d been enjoy- the father figure of the discipline. But Henderson ing too much the social activities of college end-of- had strong interests and training in statistics, and year festivities! more than a nodding acquaintance with matrices. So we got on well together, especially after I showed 3. THE COMING OF AGE AS A him the formula for the inverse of a partitioned ma- STATISTICIAN trix neededinestimating environmental andgenetic Wells:AfterCambridge,whatwasyournextmove? trends (see Henderson et al., 1959). Searle: I then needed to find a job. Cambridge Wells:Whatwastheconsequenceofyourrelation- ship with Professor Henderson? UniversityhadwhatIbelieveatthetime(1953) was Searle: The result of all this was that in August the early years of its career center. Although their 1956 I went to Cornell and did a Ph.D. with Dr. advisorshadclearlyneverpreviouslydealtwithare- search (graduate) student, let alone one with statis- Henderson.BeforeleavingNewZealand(withaFul- ticsqualifications(andfromNewZealand!),theydid bright travel grant) I knew what my thesis topic find me two interviews in London; one was for a job wouldbe,andbyAugust1958hadfinishedmyPh.D. with the Colonial Service, in agriculture in Kenya That coincided with the New Zealand Dairy Board (I resisted the temptation to ask the interviewer if sendingmedatatheywantedanalyzedtoinvestigate his missing leg (or was it arm?) had been eaten by the possibility of having yearly production records the Kikuyu), and the other was with Royal Dutch estimated from just 3 or 4 months measured (sam- Shell who wanted to employ me in Venezuela. I de- pled)productioninsteadofthethen-usual9months. cided to pursue neither opportunity when I heard Dr.Hendersonwasinterestedinthis,too,andkindly of the possibility of a position in New Zealand, as kept me on as a Research Associate. a statistician at Ruakura Research Station, a large Wells: What did Cornell uniquely offer you as a and comprehensive agricultural research farm. So I graduate student? applied—but the position was canceled. Searle: I cannot describeCornell’s offerings as be- Wells: So much for the Cambridge career center; ing uniquebecause I have no comparison with other whatdidyoudoaftertheRuakuraResearchStation placessinceIappliednowhereelse.ButCornell’stol- job was canceled? erance of my special circumstances was wonderful: Searle: I returned to New Zealand and in Octo- Iarrivedlate,sometwoweeksintothesemester,asa ber 1953 got a newly established post as Research resultofthetravelarrangementsmadebyFulbright. StatisticianwiththeHerdImprovementDepartment Forming my degree committee was greatly aided by of the New Zealand Dairy Board, in Wellington. It Henderson. Animal breeding was to be my major turned out to be a decisive moment for my life’s (with Henderson with his strongly statistical inter- activities. ests); one minor was to be statistics with Federer, Wells:TellmeaboutyourtimeattheNewZealand head of Biometrics. The second minor was trouble- Dairy Board. some because I refused to do mathematics (I felt I Searle: Thework consisted of deriving ways of us- had enough), and I couldn’t do anything related to ing dairy cow milk production records for deciding embryology becauseIhadabsolutely nobackground which cows and bulls would be used for breeding in chemistry or biology or physiology. Henderson offspring (by artificial insemination) that would in- came to the rescue by reassuring the department crease milk production not only for the individual head to take me on with a minor of Animal Science 6 M. T. WELLS doing a few undergraduate courses, in at least two Searle:Dealing withdairycow productionrecords of which (dairying and sheep husbandry) I gave the made me realize that unbalancedness of data can lectures on breeding. So I scrambled through! materiallyaffectthemeaningofmany ofthecalcula- Above all, the greatest benefit of Cornell was the tions (e.g., sums of squares) that were being used in completefreedomandencouragementtogetonwith (at least agricultural) research literature. And this what I wanted to do. I knew what my thesis topic was before the flood of computer software that we wastobe,Iwasgetting thedataforitfromtheN.Z. have today. The Dairy Board work simply started Dairy Board, the department had just got its own me down the path of unbalanced data, matrices and computer (an IBM 650), I wrote my own programs variance components. Genetic studies use a ratio of and worked many nights on the computer from 10 variance components and that prompted estimating pmtill6am.Thefreedomwassuperb—andproduc- those components and that was highlighted by the tive. 1953 Henderson paper in Biometrics. Wells: After writing your Ph.D. with Professor Wells:TellmeaboutHenderson’sinfluenceonyour Henderson what did you do? work. Searle:I finishedthePh.D.at theendof1958 and Searle:Hisgreatestinfluenceonmewashisenthu- was hired as a Research Associate under Hender- siastic encouragement. For example, it was a cus- son, attending to an extension or two of my thesis, tom in the Animal Science Department that each writing several papers for publication, and learn- semester every graduate student had to be part of ing as much as I could about computing facilities a team to give a seminar. In my first doing this I’d needed for this kind of work. Henderson and I gave a semester-long seminar on unbalanced data and I been allocated to talking about the uptake of iodine wrote it up as an extensive set of notes, the proof- in the thyroid—something I knew absolutely noth- reading of which was left to me. ing about. So for my remaining five semesters I sug- Wells: What was your next move? gestedatopiconbreedingtoHenderson,heropedin Searle: In late 1959 I returnedto N.Z. and my po- another graduate student and the job got done. He sition with the New Zealand Dairy Board where a was also very tolerant of my asking questions, and sire-proving scheme was being inaugurated for se- was exceedingly patient of my saying “I still don’t lecting bulls to be sires in the artificial breeding follow you,” and he would try again to placate me. program. For me it was a period of successful pub- He was also greatly helpful in suggesting improve- lication, for example, nine publications in 1961, not ments to whatever I was writing—although when only in The Journal of Dairy Sciences, but in Bio- it came to proofreading a supposedly final draft of metrics, Journal of Agricultural Science, Annals of a paper, a modicum of procrastination and delay Mathematical Statistics. During this time I became would sometimes set in! a one-third-time scientist of the N.Z. Department of Wells: What was the state of random effects mod- Scientific and Industrial Research in their Mathe- eling in the 1950s? matics laboratory where I took part in their intro- Searle: It was quite limited: mostly for balanced ducing computing and programming to the coun- data. And almost the only method of estimating try’s scientists. variance components was what we today call the 1961 was also the year I was asked to reduce my analysis of variance method. In its general terms research and spend time visiting dairy farmer meet- it consists of equating sums of squares (or other ings and giving talks. The happy coincidence was quadratic forms) of data to their expected values, that, withoutmy knowingit, I was beingconsidered in which the random effects give rise to their vari- for a job at Cornell as statistician to their Comput- ances. The trouble was that no real criteria were ing Center. The official offer to me was delayed sev- used for deciding on which sums of squares to use. eral months because two members of the committee deciding to employ me each thought the other had With balanced data, analysis of variance seemed an written to me. When the offer did come I of course “obvious” choice, and usually yielded as many sums accepted it to start on June 1, 1962 after finishing of squares as variances being sought. But for unbal- some responsibilities in New Zealand. anced datatherecould beanexcess numberof sums Wells: How did working at the N.Z. Dairy Board of squares which made a problem for the desired es- influence you? timation. 7 SHAYLER. SEARLE 4. BACK TO CORNELL Searle: My consulting job also came with a cour- tesy appointment as assistant professor in the Bio- Wells: When did you return to Cornell? metrics Unit of Cornell’s College of Agriculture as Searle: At Christmas time 1961 I received a let- it was then named, but with ...“and Life Sciences” ter from a friend at Cornell saying he was glad to added to it later. This was where I had formally hear thatI was tobereturningto Cornell.Thatwas done the statistics part of my Ph.D. under the very newstome;I’dheardnothing.AroundMarch1962I helpful eye of Professor W. T. Federer, head of the wrote to Henderson to find out what the story was. Biometrics Unit. And that helpfulness and encour- It turned out that two members of the university agementre-asserted itselfonmyjoiningtheBiomet- computing committee each thought the other had rics Unit as faculty in 1962. I was enthusiastically written to me, but in fact neither had. So then I did urged to write up whatever I was working on. And get a letter; could I start in six weeks? I pointed out I certainly did; five papers both that year and the I was ninethousand miles from Cornell and my wife next. was expecting our second child, and I was already In 1965, just as computing was becoming a big committedtosomeDairyBoardresponsibilities,but itemoncampus,Iacceptedalineitemassistantpro- yes,IcouldarrangetostartonJune1stofthatyear, fessorship in Biometrics and gave up my responsi- 1962. bilities as consultant at the Computing Center. The Wells: What was your new position at Cornell? College of Agriculture started to have its own com- Searle: In 1962 when I started in Cornell’s Com- puting facilities and I became lightly involved with puting Center there was no commercial software some aspects of that operation. But I had decided I available. Will Dixon and colleagues at UCLA were wanted to be a statistician and not a computer-nic. well on the way with BMDP package; but SAS had That started my thirty years in the Biometrics Unit barely started (its first annual user conference was whichrevolved aroundthreeinterrelated topics:ma- 1976)sopartofmyresponsibilitywastodecidewhat trixalgebra,linear modelsandvariancecomponents statistical packages we should have and to get them estimation.For each ofthesethreeIstartedacourse programmed.The ComputingCenter had a statisti- andwrotea bookor two. Writing, to me,was an en- cal programmerwhocoulddoacredible job,and we joyable form of hard work so I kept at it. proceeded to provide for regression and for analysis of variance of data from well-designed and executed 5. VARIANCE COMPONENTS, LINEAR experiments (i.e., balanced data). MODELS AND MATRICES Wells:Whatwasthestateof“moderncomputing” in 1962? Wells: What researchers showed an early interest Searle: Computing in 1962 was rudimentary com- in variance components? pared to today’s activities. Cornell had begun in Searle:Inthe1950sonlyasmallcoterieofstatisti- 1956 with an IBM 650 (2000 words of 10-digits plus cians(manyofthemwithanimalbreedinginterests) sign) and in 1962 had a 1604 CDC. There was no feltcomfortablewithrandomeffects. Occasional pa- commercial software, no data editing and few pro- pers by such people as Crump, Daniels, Eisenhart, gramming languages: Fortran and Algol. The con- Winsor and Clark, Tippett, and Cochran made in- sultingworkwasoftenquiteelementary,suchascor- teresting butnotearth-shattering contributions and rectingthefollowingmisadventures:regressionanal- mostly dealt with analysis of variance methods for ysisthatusedasdatathe−1sthathadbeenentered balanced data. I remember, as a graduate student, in place of missing observations; the reproduction being at a 6-week research gathering in 1957 called of data so that there were 800 of them because the a seminar on analysis of variance held in Boulder, 400 actual data were too few in number to make Colorado under the direction of Oscar Kempthorne a correlation estimate be significant; the scrutiniz- with such notables as David (now Sir David) Cox, ing of some six pages of data for which a published BillKruskalandJerryCornfieldandothersinatten- analysisofvarianceseemedspurious;itwas,because dance. Following my lecture there on variance com- amongst 300 3-digit data we found two values had ponents I had several people come up to me and ask been entered as 5 digits (only 100 times too big!). me to “really explain random effects,” one such be- Wells:How didyougetaffiliated withtheBiomet- ing Jerry Cornfield. Well, after all, I suppose 1957 rics Unit? is half a century ago! 8 M. T. WELLS Wells: What computational issues were there in variance component modeling those days? Searle: Not only were random effects not widely understood, but the computations were horrendous for unbalanced data. There was a series of papers givingscalarformulaeforsamplingvariancesofvari- ance components estimates obtained from the anal- ysis of variance method of estimation and on un- balanced data, but these formulae were incredibly complicated. And there was no software; indeed, in 1955 before going to Cornell, I struggled with a very small data set to do these calculations with a Powers-Samas punched card tabulator using the method of successive digiting (see Searle, 1993) for Fig. 3. Shayle, 1952, on St. John’s Bridge, Cambridge. obtaining sums of squares and products. It was hor- rible. Sciences; they are considered more fully in Linear Wells: How did you get interested in unbalanced Models (1971). data? Wells: You were an early advocate of using matri- Searle:Mystronginterestinunbalanceddata(hav- ces in statistics; looking back this perspective seems ing unequal numbers of observations in the sub- obvious.Doyouhaveaconjecturewhyearlyprogress classes of the data) arose from dealing with dairy on the application of matrices to statistics was so production records when working for the Dairy slow? Board. Herds do not all have the same number of Searle: The first of my Annals papers of 1956, cows, not all cows give milk every year, and within 1958 and 1961 was “Matrix Methods in Variance aherdvaryingnumbersareofthesameage.Iclearly andCovarianceComponentsAnalysis.”Itstitlebegs rememberbeingpuzzledforalongtimeinstatistical the question: Why has it taken so long for matrices methods giving two different least squares estimates to get widely adopted where they are so extremely of fixed effects in a one-way classification depend- useful?Afterall,matricesaretwohundredandsome ing upon whether one assumed that one effect was yearsoldandtheiruseinstatistics isonlyslowlybe- zero, or that all the effects summed to zero. Even as coming commonplace. But this was not so, even as late as my second year as a graduate student (1957) recently as the 1950s. Even at Cambridge, in lec- when Henderson and I gave a weekly 2–3-hour sem- tures on regression in 1952 there was no use of ma- inar on unbalanced data we were still confused by trices. In Aitken’s two 1939 books, one on matri- this situation. ces and one on statistics, neither mentions the main Wells: How did the notion of the g-inverse change topic of the other! The very first paper in the first your thoughts on linear models? issue of Annals of Mathematical Statistics (Wick- Searle: One of our troubles was we had not kept sell, 1930) is entitled “Remarks on Regression” yet up with the concept of estimability propounded by it has no matrices. And even the Williams (1959) R.C.BoseinNorthCarolina[linearcombinationsof book on regression has only a tiny mention of ma- theparametersβ,sayAβ,aredefinedasestimableif trices. Maybe this tardiness of adoption of matrices the rows of the matrix A belong to the vector space arose from their being treated so much a topic of spanned by the rows of the design matrix; Bose, pure mathematics that they remained hidden from 1949]. Nor were we aware of Penrose’s (1955) gener- their practicalities. alized inverse matrix which, as Rao (1962) demon- Wells: Tell me more about your early efforts in strated, clarified the whole business of solving least teaching linear models using matrices. squaresequationswhicharesooftennotoffullrank, Searle: Around 1960 a visitor to the Biometrics and thus have an infinite number of solutions, but Unittaught acourseoutofGraybill’s excellent 1961 which, with the aid of a generalized inverse, easily book An Introduction to Linear Statistical Models. lead,foreverysolution,tounbiasedestimatorsofes- He made very slow and pedantic progress and never timable functions. Some details of this situation are gotanywherenearthedifficultiesofunbalanceddata. in my 1966 book Matrix Algebra for the Biological A year later D. S. Robson took on the course but 9 SHAYLER. SEARLE after a few weeks had to beabsent atresearch meet- 6. BOOK WRITING ings and I was left with the teaching. In progressing Wells:Youjustmentionedthatwhenteaching lin- toward the all-important result about a quadratic ear models that set you off to start thinking about form in normal variables having a chi-squared dis- doinga book.Tell me aboutwritingyour firstbook. tribution, Graybill (1961) had nineteen preparatory Searle: Federer was on sabbatic leave 1962–1963, theorems!Thatstruckmeasjusttoomuch.Tohigh- and in his absence Professor D. S. Robson chaired light the differences between each theorem and the theBiometrics Unit.InJanuary1963hetoldmethe next I summarized the nineteen in one line each. secretaries were shortof work, and he asked, “Don’t That immediately showed most of those differences you have some notes on matrices they could type?” to bevery small; for example, normalvariables with I protested that although I’d written the notes for zero mean in one theorem had nonzero mean in the teaching a 1957 summer course when I was a grad- next. Among my biometry colleagues was a Ph.D. uate student, they needed plenty of work to make graduate of Graybill’s who explained that was what them worthy of a typist’s time. Robson’s reaction Graybill wanted his students to learn and so be able was, “Why don’t you write a book?”So I did. I sent itin1964tofourpublishers:twoturneditdown,one in exams to regurgitate theorems and their proofs. never replied and Wiley & Sons accepted it. Months Not for me, I decided. I wanted students to know later they had a change of editors and turned it where they could read the importantly useful theo- down. But luckily one of their senior editors, Ms. rems(whichtheymightneedtouseinpractice),and Beatrice Shube, saw the manuscript and promoted to thoroughly understand them. So I concentrated its publication. Thus was born my firstbook, Searle on the overriding theorem in this topic, namely the (1966)whichsoldmorethan10,000copiesbeforego- conditionsunderwhichaquadraticformofnonzero- ingoutofprint.Itspawnedamildlyplagiarized ver- mean normal variables has a noncentral chi-squared sionintheformofSearleandHausman(1970)which distribution. Armed with that, many of Graybill’s throughgetting little ornopromotion frombusiness nineteen theorems became just special cases. This academia sold barely 5,000 copies. Nevertheless in appealedtomeasamathematicallytidywayofhan- 1974 it did have reprint editions in Taiwan (in En- dling things. Thus there was only one theorem, but glish)andinRussia(inRussian),bothofwhichwere avital one, thatstudents neededto know andin do- initially denied by their respective publisher. The ing so needing to know that they understood it and successor to both the 1966 and 1970 books is Ma- knew how to use it. This set me to thinking about trix Algebra Useful for Statistics. It (Searle, 1982) doing a book. hassoldmorethan10,000 copies—thanks toGeorge So then, armed with matrix algebra and the gen- Styan for the “Useful.” Prior to that helpful word, eralized inverse, and motivated by unbalanced data, reviewers of the manuscript had strongly disliked I went on to describe in detail the various sums of the title. I started, in 1965, and for thirty years taught a squaresandtheircorrespondinghypothesesthatcan Matrix Algebra course at Cornell; it never had less be derived from unbalanced data in the analysis of than twenty students and up to seventy one year. variance context. Not much of this was dealt with They were from as many as 8–12 departments in byGraybilloranyotherbook.Noneofitwaspretty, agriculture most years, despite the 8:00 am starting but it was only the use of matrices that made it at time three days a week. Because of that early hour all feasible. As well as fixed effects models, Linear I never admonished anyone for being late; to be late Models also(initslastthreechapters)dealswithap- was better than not coming. Thirty and more years plying to unbalanced data the analysis of variance ago teaching matrix algebra because of its practical method of estimating variance components, namely use in statistics was never doubted as being useful equating observed sumsof squares to their expected (in contrast to some of the concepts of linear alge- values. Nearly all of that has now been relegated bra).But nowadays, because, I suppose,of thecom- to history by the widespread application of max- puting software for doing the algebra, the teaching imum likelihood (starting with Hartley and Rao, of the algebra seems to have become somewhat of 1967) and other methods, and the amazing growth just an add-on, if that. What a pity; matrix alge- of computability. bra is fun. My initial intrigue at being able to have 10 M. T. WELLS ton University in Washington, D.C. and in Berlin, Germany; and to lecture in such various locales as Budapest,Sydney,Auckland,andFreiburgamBreis- gau and a raft of conferences and seminars in the U.S.A. and elsewhere. This included in each of 1985 and 1986 a 4-month stay in the mathematics de- partment of the University of Augsburg in Bavaria, funded as a U.S. Senior Scientist by the Alexander von Humboldt Foundation of Bonn, Germany. As well as having a thoroughly enjoyable time in the historical city of Augsburg, I finalized a number of papers, gave some seminars and made a good start on Linear Models for Unbalanced Data publishedby Wiley in 1987. Towhateverextent allthisrepresentedsuccessfor Linear Models it motivated me to more books, five more actually, the most recent being Generalized, Linear, and Mixed Models co-authored with C. E. McCulloch, published by Wiley & Sons in 2001. An important feature of this book is its distinct em- phasis on mixed models, a topic which is very much in evidence in today’s statistical research. A con- tributing reason for this is that today’s computing facilities can handle the arithmetic that is needed for coping with random effects when modeling un- Fig. 4. Shayle at the blackboard, Hoehenheim University, balanced data. July, 1977. 7. SOME PERSPECTIVE AB =0 without having A=0 or B =0 has never Wells: Looking back over your career, do you see left me. a recurring theme? Wells: How did your classic Linear Models book Searle: I find it hard to believe that through my come about? Searle: In 1968 I was invited by H. O. Hartley to activities with animal breeding data it was more than fifty years ago when I was first trying to deal take my sabbatic at Texas A&M University and it withrandomeffects andvariancecomponentsinun- was therethat I started my Ph.D. level Linear Mod- balanced data. After all, half the genetic contribu- els book, published by Wiley & Sons in 1971. It has tion to a cow’s milk production comes from its sire had sales of more than 15,000 and another 1,800 in but it is only a random half—and thus we have a the paperback Wiley Classic Edition which started random effect when including the effect of sire in in 1997. It is, I believe, a book which did make an a linear model for its daughter’s milk production impression on the understanding of linear models, record.Andthisinturngivesrisetoavariancecom- especially of the complications emanating from un- balanced data. This was, and still is, especially im- ponentfortherandomeffect. Thishasbeenastatis- portant for using the high-powered computing soft- tical interest of mineever sincethe C.R. Henderson ware designed for doing linear model and analy- (1953)paper“Estimationofvariancecomponents...” sis of variance calculations of such data—software in Biometrics. My contributions motivated by that suchas SAS, SPSS, STATAandmany others.Their paper are in the Annals of Mathematical Statistics early output descriptions and labels were often not in 1956, 1958 and 1961. My most recent effort on a model of clarity, sothat knowing the mathematics this topic is the 1992 Wiley book Variance Compo- of the calculations was important. nents with G. Casella and C. E. McCulloch. The book led to many interesting and enjoyable Wells: Where do you see basic statistical research invitationstogiveshortcoursesforGeorgeWashing- heading?