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Historical Records of Australian Science, 2004, 15, 223–250

© Australian Academy of Science 2004 0727-3061/04/02022310.1071/HR04005

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CSIRO PUBLISHING

Henry Oliver Lancaster 1913–2001

E. Seneta1 and G.K. Eagleson2

1School of Mathematics and Statistics, F07, University of Sydney, NSW 2006.*2Australian Graduate School of Management, University of New South Wales, NSW 2052.**

1. Some Family History

Henry Oliver Lancaster (HOL) was born inSydney on 1 February 1913 and died there2 December 2001. His parents were EdithHulda (‘Edie’) Smith (1885–1964) andLlewellyn Bentley Lancaster (1871–1921).

The Macleay River region of NewSouth Wales had been the home of theLancasters since 1850, beginning withWilliam Henry Lancaster, who from 1850taught and later served as Postmaster atKempsey and Frederickton. HOL’s grand-parents on his father’s side were JamesHenry Lancaster who died in 1879, prede-ceasing his father William Henry by abouta year; and Jane Norton, who was born atParkmore, Co. Offaly, in Ireland, and cameto Australia at age 15 as part of a familycontingent. Their children were Clara,William, Albert, Stanley, James, Llewellynand Eveline.

Llewellyn became a medical practi-tioner in Kempsey. He had done his train-ing at the University of Sydney. HOL’smother, Edie, was Llewellyn’s second wife.She was a nurse who had completed hergeneral nursing training at KalgoorlieDistrict Hospital, Western Australia.

Edie belonged to a powerful WesternAustralian family that included the For-rests (one first cousin was a Forrest). SirJohn Forrest (1847–1918), the mosteminent of the Forrests, is an outstandingfigure in Australian history. WesternAustralia’s first premier (1890–1901), he

entered the first Federal Parliament in 1901after successful negotiations on WesternAustralia’s behalf. In 1869 he had led asearch expedition for Ludwig Leichhardt;in 1870 an expedition from Perth along theGreat Australian Bight to Adelaide; and in1874 he completed a 4,300 kilometrecrossing of the Australian continent. HOLin his retirement often spoke with pleasureof the Forrest family connection.

Edie, HOL’s mother, was the secondchild and second daughter of MatildaHulda Mattner and Harry Smith, whomshe had married 4 August 1883.

Harry Smith’s real name was ChristianJorgenson or Jensen. As a youth Harry hadshipped out from Moss in Norway as aseaman, and left his ship on arrival inAustralia. Matilda was the eldest of twelvechildren of Pauline Emilie Marks, born inBomst, East Prussia (now Babimost, inPoland) in 1843, and Carl WilhelmMattner, who had reached South Australiain 1857. South Australia’s history andsociety is rich in German names. TheMattners achieved considerable standing;for example, one of South Australia’sSenators to Federal Parliament from the1940s, last elected in 1961, was EdwardWilliam Mattner (1893–1977), and there isa privately-printed book, The Mattners inAustralia 1839–1980. HOL was proud ofthis heritage, and in his retirement spoke toES of a large Mattner family reunionwhich he had attended.

* Eugene Seneta, FAA, succeeded H.O. Lancaster as Professor and Head of Mathematical Statistics at theUniversity of Sydney in 1979. Denoted by ‘ES’ in the sequel.** Geoff. Eagleson was one of Lancaster’s first PhD students in Mathematical Statistics (PhD, Universityof Sydney, 1967).

224 Historical Records of Australian Science, Volume 15 Number 2

Henry Oliver Lancaster 1913–2001 225

2. Early Years. Formation

HOL’s father, Dr Llewellyn BentleyLancaster, had a wide range of interests thatencompassed the entrepreneurial, after hismedical and hospital activities. After hisdecease the local district newspaper spokeof his ‘evincing a deep scientific interest inagriculture and stock raising, entering intopolitical arguments, taking prominentlyactive participation in the establishment ofbutter factories, drainage unions, watersupply schemes and other projects for theadvancement of the district’.

On the medical side, soon after hisreturn to the Kempsey district, an X-rayunit was installed at the local hospital in1902, only a few years after the discoveryof X-rays, through public subscription andhis efforts and that of his senior colleague,Dr Casement. Llewellyn owned one of thefirst motor cars in the district, the ‘noisyapproach’ of which writes his son Richard,HOL’s younger brother and familyhistorian, ‘could be heard from a consider-able distance to the satisfaction of thepatient’s family’ (R.L. Lancaster, 2002).

Llewellyn had a love of horse racingand, as a member of the Australian JockeyClub (AJC) regularly visited Randwick inSydney. He was also president of theMacleay River Jockey Club. Another deepinterest and recreation was chess, which hetook up as a medical student. He played achallenge match against W.S. Viner for theAustralian Chess Championship in 1913,losing with honour by seven games to two.His wife Edie had accompanied him toSydney, and HOL was born there duringthe time Llewellyn was playing in thechampionship.

The marriage of Dr Llewellyn BentleyLancaster and Edith Hulda Smith pro-duced four offspring: Geoffrey Bentley(11/4/1911– ?); Henry Oliver(1/2/1913–2/12/2001); Richard Lonsdale(b. 12/4/1914); and Loaline Lonsdale(14/3/1919–2/2/1977).

HOL showed an exceptional abilitywith mental arithmetic in his earliest years,entertaining his father’s professionalguests with his skill. However, a greatproblem emerged before his fifth birthday.He stuttered. This persisted through hisschooling, and caused him great embar-rassment. It affected his performance inthe viva voce examinations in the earlyyears of his medical studies. As to thecause, he wrote in his autobiography(II.83)*: ‘I have always believed that it waspartly due to dominant personalities.’

Through force of will and evolvingconfidence derived from the strength of hisscholarship he eventually cured himself ofthe stammer, but some after-effects per-sisted throughout his life. He spoke in asomewhat halting, deliberate manner,sometimes coming out with the wrongword, perhaps because he could notquickly enunciate the right one, and then,after a pause, trying to explain what hemeant. This attempt would characteristi-cally begin with: ‘Well, at any rate…’ —aphrase very frequent in his lectures. Diffi-culty in communication coupled with anessential shyness sometimes gave thewrong impression, and led to a reclusive-ness in later life.

During HOL’s early years the familywas prosperous and comfortable, but hisfather, Llewellyn, died a relatively youngman on 8 December 1921 from an attackof pleurisy and pneumonia, the result of aninfluenza epidemic two years earlier.Because of financial difficulties thatbecame apparent after probate, HOL’smother, a strong woman, returned tonursing in Sydney. From November 1923Oliver and his younger brother Richard(‘Rick’) became boarders at St George’s

* HOL’s publications are consecutively numberedwithin each of two separate lists at the conclusionof this biographical memoir. Thus II.83 refers toitem 83 in list II. References in the text to otherworks are in the usual name and date format, andare to be found under ‘References’.


Hostel, a church institution run by the(Anglican) Diocese of Grafton. Here HOLwas to stay for six years. He attended WestKempsey Intermediate High School inwhich he was fortunate in his teachers.

He achieved considerable distinction inhis Leaving Certificate examination inNovember 1929. Throughout his school-ing, in which he excelled, his mathematicalskills flowered. One of his high-schoolclassmates, competitor, and life-longfriend was Alan Heywood Voisey(1911–1995), later Professor of Geology(1954–1965) at the University of NewEngland and then foundation Professor ofGeology and Head of the School of EarthSciences at Macquarie University, Sydney.

HOL was an enthusiastic cricketer atschool and later revelled in Bradman’sprodigious feats and averages. PeterArmitage, whom he met on arrival at theLondon School of Hygiene and TropicalMedicine in 1948, recalls that, havingintroduced himself on his first day, HOLimmediately left to watch the Australiancricket team at Lords. Very much in char-acter, HOL died peacefully in his sleep ona Sunday after watching a game of cricketon television in the company of his young-est son Jon, at Ocean View Nursing Homeat Mona Vale.

HOL’s recreations as a young manencompassed body surfing, billiards (evenplaying for money), bridge, and chess; andin later life, bowls. On weekend visits tohim at his retirement home (‘The Garri-son’, Mosman), HOL would often befound watching on television a RugbyLeague game of his team, the Manly SeaEagles. He had a continuing attachment tothe Manly area of Sydney, going back tohis student days at the University ofSydney.

His interest in chess was another echoof his father’s broad interests, and of thecircumstances of his own birth. In retire-ment in 1996 he showed ES a 1977 photo-graph entitled ‘The Chess Players’ that

pictures HOL and another Sydney gradu-ate, J.W. (later Sir John) Cornforth, NobelLaureate in Chemistry in 1975, playing.Watching is the physicist and radio astron-omer Bernard Mills. The photograph wastaken forty years after a New South WalesChess Championship in which HOLdefeated Cornforth, with Bernie Mills,then a schoolboy, in the audience.

After learning Latin and French atschool, HOL taught himself to readGerman and Russian by reading bilingualBibles. This enabled him to cover widerliterature sources in later life than most ofhis scientific peers.

HOL and his brother Richard remainedclose throughout their lives. Richard editedHOL’s autobiography (II.83), and much ofthe family history of our previous sectioncomes from his unpublished The Family ofthe Lancasters with the Nortons. Hiscompany through regular visits andsupport during HOL’s later retirementyears contributed much to HOL’s positiveview of his past life.

3. 1930–1946. University of Sydney. Hospitals, War Service

On leaving school, HOL wished to studyscience, in particular mathematics; andthere was a sentimental leaning towardsmedicine. His mother opposed the latterchoice, fearing that an associated senti-mental attachment to the Macleay Riverarea, which she regarded as the wilds,would bring him there to practise. It wasdecided that Oliver would become anactuary (possibly because of the potentialfor income), and he began as a trainee inSydney with the MLC (Mutual Life andCitizens) insurance company, starting anevening course in Economics at the Uni-versity of Sydney. After four weeks he wasable to transfer to a full-time Arts course,obtaining amongst other good results aHigh Distinction in Mathematics I, butdismissing on account of bleak careerprospects and nervousness about lecturing


to large classes his desire to become aprofessional mathematician. So, in the hopeof an honourable professional life and aproper reward, he enrolled in first yearMedicine in 1931, with financial supportfrom his mother near whom in Manly heboarded. Clinical studies in Medicine beganin 4th year and continued through the 5thand 6th years of the course, and HOL hadgenerally good memories of these, recol-lecting positively J.C. Windeyer, the Pro-fessor of Obstetrics; Harvey Sutton,Professor of Preventative Medicine; Pro-fessor C.G. Lambie in Medicine; andH.D. Wright and N.E. Goldsworthy in bac-teriology. Professor A.N.St.G. Burkitt washis teacher in Anatomy in 3rd year, asubject HOL did not like although heenjoyed Burkitt’s tales of the great days ofthe Sydney University Anatomy Depart-ment in the 1890s.

Harvey Vincent Sutton (1882–1963)played a role in HOL’s post-war career, andwas one of his heroes.

Having completed the medical degreein 1936, HOL spent 1937 to 1939 atSydney Hospital, first as Resident MedicalOfficer and then as Pathologist and SeniorMedical Officer. The Kanematsu MemorialInstitute of Pathology was located atSydney Hospital, with J.C. (later Sir John)Eccles (1903–1997), Nobel Laureate inMedicine in 1963, as Director, and had astrong research team that included BernardKatz (b. 1911) who in 1970 also became aNobel Laureate, and Stephen W. Kuffler(1913–1980), later a foreign member ofthe Royal Society of London and amember of the US National Academy ofSciences. The afternoon teas, which Ecclesand his staff often attended, were verystimulating to the young graduate. And heclearly made an impression on Eccles, whowrote a reference for him dated 11 Decem-ber 1939, when HOL’s term of three yearsat Sydney Hospital was about to end.HOL, in personal autobiographical notesin his file at the Australian Academy of

Science, says that the two years 1938 and1939 were perhaps the happiest in his life.

In 1940 HOL obtained a Junior Fellow-ship in Medicine at Prince Henry Hospital,Sydney. He writes in his autobiographythat Dr C.J. Walters, the Superintendent ofthe hospital, was not well disposed to theJunior Fellowships scheme, and the short-age of medical officers brought about bywar pressures also helped prevent itsproper development. Consequently HOLleft. After a few further months at RoyalNorth Shore Hospital as resident patholo-gist, on 31 July 1940 he joined the Austra-lian Imperial Forces as Medical Officer,and served until April 1946 with the rankof Captain and later Major.

While at Prince Henry Hospital, he metJoyce Mellon who was training as a nurse.They married on 20 December 1940. Hisson Paul (the first of 5 children, all sons)was born in September 1941.

HOL served as army pathologist at the9th Australian General Hospital outsideAlexandria in Egypt, then moved with it toNazareth. In February 1942 he returned toAustralia and worked in the 117th Austra-lian General Hospital in Townsville, layingthe groundwork for his first publishedpapers (on incidence of eosinophilia,usually caused by hook worm infestation),joint publications with Major T.E. Lowe.These appeared in the Medical Journal ofAustralia in 1944. He also investigated theincidence of intestinal protozoal infec-tions. About some presumably unpublishedanalysis he later wrote (II.83, p.16): ‘Herea statistical problem was to compare inci-dence of infection in the 4 classes oftroops, according to whether or not theyhad been in the Middle East or NewGuinea, a problem in 2 × 2 × 2 tables …’.

He had tried to read in 1939 UdnyYule’s Introduction to the Theory of Statis-tics but the appearance of chi-squared,closely linked to the analysis of suchtables, suggested that he needed to extendhis mathematical studies. In the event, chi-


square analysis of cross-tabulated cate-gorical data was to be one of the dominantthemes of his work in mathematical statis-tics (see Section 10 below).

In August 1944 HOL was seconded tothe Australian New Guinea AdministrativeUnit (ANGAU) as pathologist, attached toheadquarters at hospitals in Port Moresbyand Lae. In the Australian War Memorial’scollection, Canberra, there is a fine pencilsketch (AWM Art 22670) from this periodby the war artist Nora Heysen, entitled‘Pathologist (Major Henry OliverLancaster) 1944’. HOL made a survey ofmore than a thousand native troops andcivil workers from different areas, and in1945 ‘surprised the army Director ofPathology, Colonel E.V. Keogh, by report-ing the results in systematic form withproperly drawn graphs and means andstandard deviations correctly computed’.

Esmond Venner Keogh (1895–1970)was later to play a major role in HOL’ssojourn at the School of Public Health andTropical Medicine in Sydney.

HOL like many returned servicemenalmost never spoke with his family of hisexperiences in the war, and after the earlyyears he rarely attended Anzac Dayservices. At the Ocean View NursingHome where he died, the staff, however,told of his recurring nightmares of firingsoldiers running over the desert towardshim.

4. 1946–1959. School of Public Health And Tropical Medicine, Sydney. Rockefeller Fellow, School of Hygiene and Tropical Medicine, London

With the support of E.V. Keogh, HOL wasgiven a temporary appointment on 8 April1946 as Lecturer in Medical Statistics atthe School of Public Health and TropicalMedicine (SPHTM) at the University ofSydney by Professor Harvey Sutton, whowas the first Director (1930–1947) of theSchool. He was expected to spend his firstyear in acquiring expertise in medical sta-

tistics. In fact, he spent much of his timecompleting his undergraduate mathematicseducation by attending Mathematics III (hewas awarded the BA in 1947), as well asreading the English medical statisticians(Farr, Greenwood, Bradford-Hill) andAmerican statisticians/epidemiologists(Dublin, Lotka, Pearl and Frost). In appliedmathematics he heard ‘very clear lectures’from Professor K.E. Bullen (1906–1976)‘who was a help and encouragement to methen and for many years later’. In puremathematics his lecturer was ProfessorT.G. Room (1902–1986) and the topicprincipally geometry. He later wrote II.83,p.18) ‘I have always been greatlyimpressed by Room’s lectures and hisvirtuosity’. HOL found the elementarystatistical texts that he was reading at thetime to be lacking in precision, but seemsto have found the book of Maurice G.(later Sir Maurice) Kendall (1907–1983)The Advanced Theory of Statistics, Vol.1,first published by Griffin, London in 1943,the most helpful—as had, at about thesame time, his eventual long-time Austra-lian contemporary statistical luminary,P.A.P. Moran (1917–1988) (see Heyde(1992)).

In 1947 HOL did the Pure half of theMathematics IV course, and obtained aRockefeller Fellowship in Medicine thatwas to take him to London for post-graduate training in medical statistics inthe latter part of that year. However, histhird son Llewellyn was born in 1947, andthe trip was postponed for a year.

A. (later Sir Austin) Bradford-Hill(1877–1991) had succeeded MajorGreenwood at the London School ofHygiene and Tropical Medicine (LSHTM)and was Professor of Medical Statisticsthere when HOL, travelling without hisfamily, arrived in August 1948 for a stay ofabout twelve months. He was assigned toshare a room with Peter Armitage, whowas to become an outstanding biostatisti-cian, a President of the Royal Statistical


Society, and a friend for life. HOL wasliving in London House, a hostel forCommonwealth visitors and students inBloomsbury, near LSHTM. He had arrivedwith notes on a variety of topics in medicalstatistics. These included analysis ofamoebic surveys (involving over-dispersedbinomial distributions), control of routineblood counting, partition of the chisquare-statistic in contingency tables in order toidentify particular contrasts (see Section10), the use of what is now known asLancaster’s mid-P test for discrete test sta-tistics (see Seneta (2003)), and the sexratios in families.

In regard to the last subject, in Londonhe obtained 1889 data of A. Geissler ofSaxony, and produced a paper published in1950 in Annals of Eugenics. Armitage(2002) writes that HOL found no evidenceof genetic variability in the probability of amale, although he had four sons and waslater to have a fifth! HOL’s paper wasattacked in a later number of the samejournal by the eminent Italian statisticianCorrado Gini (1884–1965), who hadwritten a dissertation on the subject andwho according to HOL considered itimproper that anyone should criticize thework of German statisticians. Gini’s causewas taken up by A.W.F. Edwards in thesame journal, which caused HOL someundue distress at the time. The work on theover-dispersed binomial, HOL later dis-covered, also overlapped with earlier workof Gini.

Papers on all the topics HOL hadbrought with him were published in 1949and 1950 in prestigious English journals.Of the senior people at LSHTM the majorinfluence on HOL was the mathematicalstatistician J.O. Irwin (1898–1982) whowas particularly interested in HOL’s workleading to his first paper in mathematicalstatistics (II.1), on the partition of chi-square, and who wrote a sequel to it (Irwin,1949). Irwin was something of a recluse.In the early weeks of HOL’s stay Irwin’s

disappointment that it was Bradford-Hilland not himself who had succeeded MajorGreenwood was still much in evidence.Bradford-Hill was an excellent choice inmany respects, pleasant and helpful, but helacked a knowledge of the mathematicalside of medical statistics. HOL had hopedto gain a great deal from Greenwood, whowas then retired but still occupying a room;but Greenwood didn’t seem to recognizeHOL’s talent in epidemiological theory, andHOL’s reticent personality did not permithim to force closer acquaintance.

Under the Rockefeller Fellowshipscheme there was no need to worry aboutstudying for a degree, and HOL attendedas many meetings of the Royal StatisticalSociety as possible and also attendedlectures at the LSHTM and at UniversityCollege nearby. Outside LSHTM he heardlectures from the great names in statisticsand genetics, Hermann Otto Hartley (néHirschfeld) (1912–1980) on analysis ofvariance, Egon S. Pearson (1885–1980) ongeneral statistical theory, J.B.S. Haldane(1892–1964) on genetics, and LionelS. Penrose (1898–1972) on human genet-ics. Penrose encouraged his work on thesex-ratio. (In his correspondence withPeter Armitage in later years, HOL came tothe curious ranking of ‘medical’ statisti-cians: 1. R.A. Fisher; 2. L.S. Penrose, withthe others as ‘also ran’.)

In spite of the friendship of PeterArmitage and his wife Phyllis, who tried todraw him into their circle of friends, HOLin time became increasingly lonely, anddissatisfied with his work environment, thesituation being exacerbated by his separa-tion from his family and the fact that hisfourth son, Andrew James, was due to beborn in Sydney in January 1949. Anotherreason was that before his departure fromSydney, the new director (1947–1968) ofthe SPHTM, Edward (later Sir Edward)Ford (1902–1986), had told him that hiswork would be subsidiary to planned direc-tions of expansion. As a result of HOL’s


concerns about going home to a positioncommensurate with his abilities and pro-ductivity, someone (possibly Bradford-Hill; or Charles Kellaway (1889–1952),former Director of the Walter and ElizaHall Institute, Melbourne) wrote to Ford toask what position HOL might be offeredon his return. The response was non-com-mittal. Terribly upset by this, in late springor early summer in 1949, HOL suddenlywent missing from the LSHTM, and PeterArmitage found him in a depressed state inhis rooms in London House, determined togo to Cambridge to work with the greatmathematical statistician and geneticistR.A. (later Sir Ronald) Fisher(1890–1962). Bradford-Hill arranged thetransfer to Cambridge; but HOL did notfind in Cambridge or in Fisher what he hadhoped, and returned to LSHTM. Bradford-Hill, a kindly man solicitous of HOL’smental welfare, did not take offence.

On his return to Australia in August1949, HOL embarked on the career ofmedical statistician with the Common-wealth Health Department, taking upduties again at the SPHTM at the Univer-sity of Sydney. Salaries at SPHTM werepaid by the Health Department but someofficers held university titles. The univer-sity staff treated HOL as a public servantwhile the Health Department treated himas a raw recruit. His relations with Fordwere strained, and his desire to submit anMD thesis to strengthen his academicstanding was repeatedly not supported.

HOL’s duties were to lecture to post-graduate students on the subject of medicalstatistics within the Diploma of PublicHealth and the Diploma of Tropical Medi-cine. Not wanting to illustrate applicationsof statistics with non-Australian data, heconsidered it his main task to put in orderand publish the medical statistics of Aus-tralia, since they had never been dealt withby a competent medical man. His focus onthe collection and statistical analysis ofAustralian data resulted in some remark-

able successes that will be discussed in ourfollowing section.

In April 1951 HOL wrote toM.G. Kendall that he had been advised toconsider applying for a DSc. Kendall’scautious reply steered him to the PhD,which was awarded in the Faculty ofScience in 1953 for a thesis entitled TheDistribution of X2 in Discrete Distri-butions. Professor Room was appointed ashis supervisor for this until the return ofHarry Mulhall, who was absent inCambridge doing his own PhD underJ. Wishart. It does not appear that Mulhalleventually took over supervision.

By about 1954, when HOL’s positionwas becoming truly anomalous, Ford hadbegun speaking about promotion to Readeror Associate Professor, but nothing hap-pened. HOL’s promotion to Associate Pro-fessor in Medical Statistics finallyoccurred on 2 March 1959, after legaladvice had been sought on both sides.When HOL took up duty as Professor ofMathematical Statistics shortly afterwardson 19 June 1959, the university had nospace for him and the SPHTM had no oneelse to teach statistics to their diplomaclasses. HOL agreed to take the coursesand so kept his room in the depths of theSPHTM. He writes (II.83, p. 30): ‘As Iwalked over to the Professorial Board[meeting] on my first occasion, Ford joinedme and so we entered together. Probablythis was a good thing, as the Board had hadenough of the Room–Bullen debates.’

HOL’s oldest son, Paul, a prominentmedical statistician in his own right whoworked at the SPHTM for many years,once said to ES that in spirit his fathernever left the SPHTM. Indeed, in spite ofthe difficulties in recognition that heencountered at that School, his ambition,drive, and energy resulted in his bestresearch work being done there in bothmedical and mathematical statistics. Hiscorrespondence file of those times containsletters from Joseph Berkson, A.C. Aitken,


M.G. Kendall, A.E. (Alf) Cornish,F.M. Burnet, T.M. Cherry, HughWolfenden, and E.S. Pearson. Cornish’sletter (6 March 1957) is concerned withHOL’s candidature for membership of theInternational Statistical Institute with theproposed support of Cornish from Aus-tralia, J.O. Irwin and M.G. Kendall fromthe U.K., W.G. Cochran and S.S. Wilksfrom the US, and possibly Bradford-Hill.HOL was elected in 1961 and regarded hismembership with great pride. It was thefirst of many international honours. It isclear that two of HOL’s great senior sup-porters in mathematical statistics wereJ.O. Irwin and M.G. Kendall. Of Kendallhe often said, in admiration, that hebelonged to no statistical ‘school’, a flexi-bility to be admired. HOL described Irwinas a lucid thinker and good mathematician.Indeed the times were an era of greatefflorescence for mathematical statistics.

5. Medical Statistics

Initially at SPHTM HOL continued hisinterest in blood counting, analysing someblood counts that he had taken in 1946 inTownsville. He writes (II.83, p. 23): ‘Itwas the blood counting statistical tests thatled me to the theory of chi-squared and hasreally introduced me to a life-long interestin a particular branch of mathematicalstatistics’. We return to the chi-squared(χ2) topic in Section 10. A manifestation ofthis medical-statistics theme is at least oneof the papers published in 1950 in J. Hyg.Camb. (II.3,4); these two papers HOLhimself listed under his publications inMathematical rather than MedicalStatistics.

At about the same time Dr (later Sir)Kempson Maddox (1901–1990) asked himto prepare a statistical survey of diabetes inthe Australian population and in his dia-betic clinic. Four papers resulted. The firstof these (I.3), jointly with Maddox, pub-lished in the Medical Journal of Australia(MJA in the sequel), ‘reassured the then

editor of this journal, Dr M. Archdall, thatmedical statistics had a place in medicineand thereafter he allowed me practically afree hand in submitting papers, indeedabout 50 of them, to his journal’. Amongthe findings of these articles, which initi-ated what HOL described as his MortalitySeries, were that recorded mortality ratesfrom diabetes were high in Australia, andthat under a commonly held genetichypothesis the rates implied that 30 percent of genes were recessive and thatcancer of the pancreas and diabetes werepositively associated.

The Mortality Series was the manifesta-tion of what HOL considered as his maintask at SPHTM, namely to put in order,teach and publish the medical statistics inAustralia. Most of the work in the Seriessubsequent to the diabetes papers was donewithout collaborators.

The most striking finding in his seriesof investigations on the prevalence ofcancer was that melanoma (black molecancer) was associated with latitude inAustralia: that is, malignant melanoma ofthe skin caused higher death rates forpeople living nearer the equator, inQueensland, than in the more southernstates. This appeared in the MJA in 1956under the title ‘Some geographical aspectsof the mortality from melanoma in Euro-peans’ (I.42). HOL was the first to demon-strate this association between latitude andprevalence quantitatively, although therewere qualitative views (for example byV.J. McGovern in MJA in 1952, and DrA.G.S. Cooper of the Queensland RadiumInstitute) that served as motivation. HOL’sstudy was not confined to Australia andthere is analysis by latitude of data fromNew Zealand, the British Isles, variousEuropean countries, the US, Canada andSouth Africa. The list of Acknowledg-ments for the data is extensive, andincludes one of the important names ofstatistics, R.C. Geary (1896–1983), then


Registrar-General of Eire. The danger ofintense ultra-violet radiation has nowpassed into standard knowledge in Aus-tralia, with its discoverer forgotten. Theassociation between ultra-violet radiationand skin cancer is standard doctrine indermatology.

Another paper (I.49) published in 1957in Lancet within the mortality series wason tuberculosis, giving special emphasis togeneration (cohort) analysis, a demo-graphic technique new at the time (I.89).

The most striking of HOL’s discoveriesof this period, however, was outside themortality series, and is increasingly theonly one for which he is medically remem-bered. This was his landmark paper onrubella deafness (I.13). In 1941, theSydney ophthalmologist N.M. (later SirNorman) Gregg (1892–1966) hadobserved that many cases of cataract werethe results of maternal rubella in the firstmonth of pregnancy, due, Gregg and othersthought, to a new, highly virulent form ofrubella in pregnancy in Australia in theyears 1938–1941. Another congenitaldefect, deafness, was described in detail byworkers from South Australia in 1943.HOL’s follow-up was the result of the ideahe had, on passing the old New SouthWales Institution for the Deaf and Dumband the Blind, of examining its well-keptadmission records since its opening in1861. He followed this by a careful exami-nation of similar institutions in otherstates, and of Australian census records for1911, 1921 and 1933, where he found(corresponding) peaks in the age distribu-tion of deaf people that he connected withbirths in 1898 and 1899, a time of knownrubella epidemics in Australia. Theseobservations were aided by knowledge ofthe fact that in small, relatively isolatedpopulations like Australia’s, epidemicstend to die out, so can be observed to comeand go. A passage from his autobiography(II.83, p. 34) is illuminating here:

In England and the large continentalmasses, the females all contracted rubellain the first few years of life. In Australia,however, the population was not largeenough to maintain the rubella epidemicand so it died out. It followed that if thefemales survived up to adult life withouthaving had rubella, [and] the rubella wasintroduced into Australia from outside, …there were epidemics in which women ofchildbearing age were attacked. Childrensubsequently born had congenital defectssuch as congenital cataract.

Thus HOL dispelled the illusion thatwhat had happened in 1940 was a newphenomenon and in effect established acausal connection between ‘ordinary’rubella and congenital deafness. Whenrecollecting his rubella conclusions in con-trast to what had been thought earlier, hequoted, with some satisfaction, Occam’sRazor: Entitia non sunt multiplicanda.

The rubella-deafness study and themelanoma study are both careful quantita-tive and highly original follow-up studiesto clinical observation, and are greatsuccess stories for statistical analysis, inbeing definitive and totally convincing intheir conclusions, and impinging directlyon public health.

HOL gave due credit to the commentaryin the census reports for the censuses of1911, 1921 and 1933 by the Common-wealth Statisticians George Handly Knibbs(1858–1929), Charles Henry Wickens(1872–1939), and Roland Wilson(1904–1996) who reported in 1942. HOLhad followed their lead. He expressed con-tinuing gratitude and admiration forWickens, reading an oration on the centen-ary of his birth, published as II.49.

Knibbs had been first CommonwealthStatistician (from 1906); Wickens hadbeen ‘Compiler’ in the CommonwealthBureau of Census and Statistics underKnibbs and succeeded him in 1922. Someof Wickens’ compilations, especially inrespect of total mortality, saved HOL muchlabour.


6. 1959–1978. Professor of Mathematical Statistics, University of Sydney

The Department of Mathematical Statisticsat the University of Sydney was founded in1959 on the recommendation of theMurray Committee inquiry into tertiaryeducation in Australia. Applications for a(Foundation) Chair of Mathematical Sta-tistics were to close on 9 March 1959.HOL, now Associate Professor of MedicalStatistics, was reluctant to apply, believinghe had left it too late to become a profes-sional mathematician:

I did not wish to remain primarily a medicalstatistician presiding over a group of mathe-matical statisticians. Nevertheless if suc-cessful I would be free of the SPHTM. Mybrother Richard wanted me to apply and sodid Keith Bullen … I was not all that confi-dent about getting the Chair because ofstrong competition but I was not depressedabout the possible outcomes.

In the event, he did apply. In hisapplication under the heading ‘Research’,he said that he was writing a monograph:on the Distribution of X2 in DiscreteDistributions:

As a research program, I propose to con-tinue these investigations into the tests ofsignificance and the mathematical form ofstatistical distributions. Several of themethods introduced in my PhD thesis havenow become standard practice: e.g. the par-tition of X2 by orthogonal matrices, the testsin complex contingency tables and the com-bination of the probabilities from differentexperiments.

His referees were J.O. Irwin, J. Berkson(an American biometrician/medical statis-tician of very considerable eminence instatistics) and M.G. Kendall.

Kendall was to cite seven of HOL’spapers (from 1949 to 1960) in Volume 2:Inference and Relationship (1961), of histhree-volume edition (with A. Stuart) ofThe Advanced Theory of Statistics, and thecontents contain substantial exposition ofLancaster’s work. The section Partitions ofX2: canonical components, on pp. 574–8,

is a particular case in point. At the time ofHOL’s application for the Sydney Chair inearly 1959, this book would have been wellinto preparation and HOL mentioned, insupport and anticipation, his ‘satisfactionof seeing much of this material incor-porated…in…Kendall and Stuart’.

R.A. Fisher’s advice was also sought,and this may have been what tipped thescales. HOL later wrote: ‘Some peoplebelieved that writing on the distribution ofthe sphere, the subject of an importantFisher paper, had cost a rival three Chairs’.That Australian rival, Geoffrey S. Watson,now deceased, went on to important Chairsin Statistics in US universities.

On his return from sabbatical leave,having completed a Cambridge PhD,Harry Mulhall (1915–1995) joined thenew Department of Mathematical Statis-tics, transferring from Applied Mathe-matics where he had kept mathematicalstatistics alive in the old Department ofMathematics (see Phipps and Seneta(1996)). (An account of the teaching ofstatistics at the University of Sydney priorto 1959 is in Seneta (2002c)). HOL (II.83,p. 30) describes Mulhall as ‘a fine mathe-matical scholar and possibly Australia’smost experienced lecturer in statistics…;he continued to play an important role inthe teaching and organisation of thedepartment’.

Mulhall’s teaching helped to attract agalaxy of students who went on to achieveeminence as statisticians in Australian andoverseas universities, the CSIRO, theCommonwealth Bureau of Census and Sta-tistics, and the Australian Department ofHealth and Community Services. Moredetail, with names and photographs may befound in Seneta (2002c). Graduates ofHOL’s Department still form a portion ofthe mathematical statistics component-group of the University’s present School ofMathematics and Statistics, formed in1991.


As regards research, HOL’s activity wastrue to his word as set out in his applica-tion. He later summarized it as focussingon:

the application of orthogonal functions tostatistical problems and included two jointpapers, with M.A. Hamdan and with G.Eagleson [his doctoral students]. For a shorttime it could be said that, with these twoauthors and R.C. Griffiths, a ‘school’ was inexistence. The orthogonal theory has beensummed up in my book The Chi-SquaredDistribution [1969] and in [the surveypaper] ‘Orthogonal models for contin-gency tables [1980]’. In this second mono-

graph, full use is made of the correlationgenerating function, which has very niceproperties in the Meixner classes, namelythe binomial, negative-binomial, Poisson,normal, gamma and inverse hyperbolic dis-tributions. This monograph and the note onthe ‘Development of the notion of statisticaldependence’, reprinted [1977] from theMathematical Chronicle (New Zealand)[1972] were with others part of a generalsurvey which I had hoped to publish. How-ever, statistical dependence proved toolarge a topic to cover in one book.

Lancaster achieved scholarly distinctionin at least four fields: medical and public

HOL in 1978 (year of retirement from the University of Sydney).


health statistics, mathematical statistics,history and biography of medicine and ofstatistics, and statistical bibliography. Thebulk of his creativity during his period asProfessor of Mathematical Statistics wasdevoted to the latter three areas.

Of the topics covered in his seven‘favourite’ papers in mathematical statis-tics, two, ‘Lancaster’s mid-P’ and ‘Nor-mality and independence’, are discussedextensively in Seneta (2002c). In thepresent memoir we have thought it appro-priate to include an expository technicalsection, Section 10, which does not sub-stantially overlap with that account and isrelated to HOL’s beloved topic of the chi-square distribution, his interest in whichhad its origins, as did almost all his work,in his earliest papers of 1949 in mathe-matical statistics.

In the history and biography ofstatistics, we have already mentionedLancaster’s paper (II.49) devoted toWickens. An earlier biographical study, of1962, entitled ‘An early statistician–JohnGraunt (1620–1674)’ (I.65) is meticu-lously researched, clearly written, andwell-documented; sadly, it is so littleknown that it was missed as a source andcitation for the Graunt entry in the collec-tion of sketches edited by Heyde andSeneta (2001).

An article by HOL in 1966, characteris-tically entitled ‘Forerunners of the Pearsonχ2’(II.31) and personal contacts with HOLrelating to one of its protagonists, theFrench statistician I.-J. Bienaymé(1796–1878) led Heyde and Seneta (1972,1977) to a striking historical revelationabout the origin of certain fundamentalresults such as the Criticality Theorem ofbranching processes. (The Bienaymé dis-covery story is told in Seneta (1979).)

HOL’s 1996 autobiography has an inter-esting concluding section (pp. 37–42)entitled ‘Further Thoughts’ in which hespecifically summarizes, from a late per-spective, some memories of his work on

bibliography, biography and history. Hewrites (p. 38):

Before I was elected to the Chair I used totake afternoon tea about once a week withcolleagues in the mathematics departmentin the Physics Building where I used to heartales of Cambridge from some of the grad-uates, usually about G.H. Hardy. So Iresolved to learn about the mathematiciansof other places and I thought a bibliographywould be suitable.

Steps in this direction were alreadyevident in his famous paper on characteri-zation of normality (II.18, p. 375), wherehe says: ‘The history of this theorem beganwith an anonymous review by Herschel inthe Edinburgh Review for 1850’. It maywell be that the preparation of the threevolumes of Bibliography of StatisticalLiterature published in 1962, 1965 and1968, compiled by his mentorM.G. Kendall and Alison Doig (especiallyKendall and Doig (1968)) played a key rolein motivating HOL’s work in history andbibliography.

The compilation of his massive anddetailed card index (in which Statisticsstudents were made to play a part),working from his office in the CarslawBuilding, was a long-term statistical labourof love that resulted in his Bibliography ofStatistical Bibliographies (1968), followedby 21 addenda over the succeeding years.At the same time, apart from professorialresponsibilities, he was running from hisoffice the Australian Journal of Statisticswith the help of G.K. Eagleson and hisdedicated secretaries, Janet Fish and laterElsie Adler. More details on his biblio-graphical activity may be found in Seneta(2002a). Out of this activity came hiscelebrated maxim: ‘Every statisticianshould write his own obituary’.

HOL had an understandable continuinginterest in his predecessor, the first profes-sor of mathematics at the University ofSydney, Morris Birkbeck Pell(1827–1879). HOL published a study ofPell (II.57) in 1977, and this extended to


an extensive study published in 1986entitled ‘The Departments of Mathematicsin the University of Sydney’ (II.74).

There were a number of articles dealingwith the history of the New South WalesBranch of the Statistical Society of Aus-tralia; these are discussed in Seneta(2002c). He wrote several articles andencyclopaedia entries on the history ofmortality surveys in Australia before hisretirement in 1978; but would have hadless time to devote to history and biogra-phy during the tenure of his Chair due tohis other commitments.

Much work of this kind on history andbiography, however, was done after hisretirement, especially in regard to statisticsin medicine, and we continue the storythere (Section 8). However, we shouldmention here his standing in the historyfield, in that he was asked to write the entryentitled ‘History of Statistics’ for theEncyclopaedia of Statistical Sciences;Volume 8 containing this item (II.78) waspublished in 1988.

HOL was a true believer, in his quietway, in the importance of statistics inscience and in the world and was ever itsdefender. At a reception at Old ParliamentHouse in Canberra, he was once intro-duced to HRH Prince Philip as Professorof Mathematical Statistics at the Univer-sity of Sydney. ‘Ah’, said HRH, ‘There are“Lies, damned lies, and statistics” ’,quoting the well-known saying of Disraeliand Mark Twain. Quick as a flash HOLreplied: ‘Figures fool when fools figure’.This must have been a supreme moment inHOL’s eventual mastery of his childhoodstammer.

7. HOL and the Australian Academy of Science

HOL was elected a Fellow of the Austra-lian Academy of Science (FAA) andawarded the Academy’s Thomas RankenLyle Medal for Mathematics and Physics,both in 1961.

His friend and colleague P.A.P. Moran(1917–1988), was Professor of Statistics atthe Institute of Advanced Studies (IAS),Australian National University (ANU),having returned from England in 1952.The ANU had been founded in 1949 as apurely research institution, ‘to secure Aus-tralia a place in international research, andto attract back to Australia some of themany expatriates who had made names forthemselves abroad’ (Heyde (1992), p. 19).One of the functions of the IAS was toprovide supervision of PhD candidateswithin Australia, it having been traditionalfor such candidates to go overseas.Moran’s Department of Statistics become afocus for students who had completed aMaster’s degree in Australia or New Zea-land, and many of them came from Sydney,presumably with HOL’s encouragement.

HOL had known of the ‘English’ stat-istician Moran at Oxford during hissojourn in London, possibly throughM.G. Kendall. It was inevitable that closeacademic contact should develop. The firstvolume of the Journal of the AustralianMathematical Society, 1959–1960, carriedthree articles by HOL including the highlyimportant probabilistic treatment (II.18) ofthe interaction between independence andnormality of distribution in linear andquadratic forms in independent randomvariables, leading to characterization of thenormal. Moran was thus very likely thereferee, being a natural choice since therewere few senior probabilists in Australia atthe time, although E.J.G. Pitman(1897–1993) is also a definite possibility.

The Australian Academy of Science’sfirst set of Fellows, elected in 1954,included the pure mathematicianEric Stephen Barnes (1924–2000), andE.J.G. Pitman as the only mathematicalstatistician. Barnes knew HOL at the Uni-versity of Sydney. HOL’s nomination forFellowship was signed by E.J.G. Pitman,T.G. Room, and E.S. Barnes. The nomina-


tion was considered in 1959 and in 1960,with election in 1961.

HOL’s citation for election reads:

Senior Lecturer in Medical Statistics in theUniversity of Sydney, distinguished for hisstudies of Australian mortality, and for hisrelated statistical studies. The current viewson the epidemiology of deaf-mutism arelargely based on his enquiries in Australiaand elsewhere. He has produced importantevidence that melanoma is caused by sun-light. He has studied the statistics of labora-tory methods, haematological counting,and amoebic surveys, in particular. Thiswork has led him to an extensive enquiryinto the theory and application of Pearson’sX2 test, obtaining new tests of goodness offit and revealing many interrelationsbetween other tests. He started his career asa pathologist and has subsequently appliedmathematical methods in his work toincreasing extent.

The citation was written prior toOctober 1958, but no doubt later paperssuch as II.18, strikingly mathematical andimportant, helped ensure election toSection 1 (Mathematics). HOL in privateconversation with ES did not believe thatmuch attention was paid to his moremedical work in his election, or in his LyleMedal award in 1961, in which Barnes,who had been awarded the medal in 1959,was instrumental as member of the selec-tion committee.

P.A.P. Moran was elected to the Fellow-ship in 1962, one year after HOL, andawarded the Lyle Medal in 1963, jointlywith G.R.A. Ellis. The Convenor of theselection committee for the medal on thisoccasion was HOL.

HOL had a deep pride in his member-ship of the Academy and for many yearsparticipated actively in its affairs. His fileat the Academy contains extensive autobi-ographical and personal notes. Only someof the personal biographical material isencompassed by (II.83) and by the presentmemoir, by his implicit wish. There are inhis file letters to the Academy on variousmatters. Two are discussed in our

Section 9. In the last decade of HOL’s lifehe expressed some dissatisfaction with thechanges being implemented in the Acad-emy’s procedures, which he thoughtdetracted from its elite nature.

8. 1979–2001. Emeritus Professor

On his formal retirement from the Univer-sity of Sydney in 1978, HOL was madeEmeritus Professor and given an office inthe basement of the University’s FisherLibrary, a short walk from his old office inthe Carslaw Building. A grant from theAustralian Research Grants Committee(ARGC) assisted him to complete a projectthat had been in his mind for many years, asurvey of world mortality (II.63). Heworked on this alone, aided only by asecretary, Mrs Philippa Holy. The firstfruits appeared in 1990 as a large-formatbook of 605 pages entitled Expectations ofLife: A Study in the Demography, Statisticsand History of World Mortality (I.92). It isdedicated to Frank J. Fenner and to thememory of Thomas Carlyle Parkinson andDora Lush, bacteriologists. A book ofamazing scholarship and commitment, itsbibliography alone takes up pages505–592, at two columns per page, andcontains an estimated 2,800 items. Thecompilation was done without the aid ofelectronic data bases. The work as a whole,including its meticulously detailed refer-ences, has strong historical colouration, asits subtitle testifies.

HOL was particularly pleased by areview of Expectations in Nature(Vol. 346, 19 July 1990, p. 352) by Profes-sor Roy Porter (1946–2002) who describedthe work as a ‘magisterial survey’. LaterHOL wrote to ES (private communication,1 July 1994), via Philippa Holy: ‘con-cerned that you may not give sufficientweight to his [HOL’s] medical work’, in theanticipation, presumably, that ES would bewriting his obituary and/or biographicalmemoir. He recommended that ES consult


Dr Roy Porter and Professor PeterArmitage for an opinion.

HOL had an enduring regard andindeed filial affection for the University ofSydney. His autobiography, with memora-bilia and archival material, was depositedin its Archives. He contributed some$27,500 to the University of Sydney,largely to support or supplement theARGC grants for the writing of Expecta-tions. He held a strong affection andloyalty also to the Australian Academy ofScience, whose Basser Library also holdshis autobiography in its archives, alongwith his huge collection of bibliographicalcards. His miscellaneous gifts to theAcademy amounted to about $5000.

HOL’s last book, published in 1994, wasentitled Quantitative Methods in Bio-logical and Medical Sciences: A HistoricalEssay (I.94). Its title encapsulates neatlythe various themes of his life-long work,and again it is rich in the breadth of itscoverage. It is dedicated to the biologistErnst Mayr (b. 1904), whose book of 1982,The Growth of Biological Thought: Diver-sity, Evolution and Inheritance, HOL wasoften found to be reading in his laterretirement at ‘The Garrison’. HOLdescribed Mayr as Grandmaster of thehistory of biological science, and claimedto have found what might be called the chi-squared distribution in Mayr’s text. InHOL’s last published papers in medicalstatistics, ‘Semmelweis: a Re-reading ofDie Aetiologie’ (I.95), and ‘Mathematicsin medicine and biology. Genetics beforeMendel, Maupertuis and Réaumur’ (I.96),he was, as ever, concerned with the properattribution of credit and priority and thedismantling of scientific myth. HOL’s ownview on historiography is encapsulated bya comment to ES (22 August 1999):‘Historians should go to primary sources.Reading secondary sources gives thewrong impression of people likeSemmelweis.’

Apart from the two books just men-tioned, HOL’s very productive retirementyears produced some 36 items. On thewhole these were of a general, historical,survey and bibliographical nature, com-mensurate with his senior standing in themedical and statistical areas, but they alsoincluded some technical articles in theAustralian Journal of Statistics. His lasttechnical paper (II.77) in 1987 concludedhis preoccupation over four decades withthe interaction of normality of distributionand independence.

He concluded his own story (II.83,p. 42) thus:

I think I have lived in quite a happy time.Life in an Australian country town whenI was a boy was very pleasant, … I hadopportunities to do the type of workI wished to do, with ample opportunity forsport and in many ways I would not haveliked things to have been changed verymuch. It had been a great satisfaction thatinterest in the histories of science and ofmortality have led me to take a heightenedinterest in the general history of mankind.A continued interest in some general prob-lem of the world at large, related to an indi-vidual’s technical expertise, can be its ownreward.

A favourite memory of ES, so charac-teristic of HOL’s later years and of hisdeeply Australian persona, was a commentthat HOL made in 1996 at ‘The Garrison’,that he liked reading The Australian Ency-clopedia because it ‘keeps me in touchwith what used to be’.

9. Epilogue

HOL was much honoured in his lifetime.His degrees, all from the University ofSydney, were MB,BS (1937); BA (1947);PhD (1953); MD (1967); DSc (1971).

A full account of his honours fromnational and international statisticalsocieties is in Seneta (2002c). Those by theStatistical Society of Australia includedthe Pitman Medal for 1980, given everytwo years for distinction in research instatistical science, and named in honour of


Professor Edwin J.G. Pitman who was itsfirst recipient in 1978. Since 1979 therehas been an annual H.O. Lancaster Lectureat meetings of the New South WalesBranch of the Statistical Society ofAustralia.

As already mentioned, HOL waselected a Fellow of the AustralianAcademy of Science in 1961, and awardedits Lyle Medal in the same year. He wasmade an Officer of the Order of Australia(AO) in 1992 for services to science andeducation. On being congratulated by theAcademy’s then President, ProfessorD.P. Craig, he replied (3 February 1992)still in a firm writing hand:

Dear David,

Please accept my thanks to the Council andFellows of the Academy for their kind con-gratulation on my attaining the rank ofOfficer in the General Division of the Orderof Australia, of which I am very proud. Ithank you also for your kind congratulations.

One of my friends called me a ‘quietachiever’ and I am happy to accept hiswords. There is a good deal to be done thatis not spectacular, even turning up regularlyat meetings of my societies as well as thechores of editing and refereeing, but withthis award I feel very glad to have donethese tasks.

HOL’s children from his marriage toJoyce Mellon were Paul, Peter, Llewellyn,Andrew and Jon. The marriage ended latein HOL’s career, and at age 66 he marriedNancy Gee, but they divorced shortlyafterwards. In his eulogy at his father’sfuneral, Paul said: ‘While Dad rarelyexpressed openly to any of us what hethought about his family’s personal andprofessional achievements, or, indeed,their shortcomings, he was quietly proudof how they etched out their lives.’

HOL was one of the small band ofAustralian statisticians who made Aus-tralia one of the powerhouses of the statis-tical world in the 1950s, 1960s and 1970s.They left a strong legacy. HOL lived forscience, and specifically Australian sci-

ence. Of him it can truly be said, as of theothers: ‘There were giants upon the earthin those days.’

10. Chisquare

Among Lancaster’s many contributions tomathematical statistics there is a majorarea that stands out due to the depth of hisanalysis and the impact his work has hadon the development of further research.The first strand of this area is the decompo-sition of the X2 goodness-of-fit statistic towhich Lancaster was drawn from his workin pathology and epidemiology. Hisresearch in the decomposition of X2 lednaturally into a series of more theoreticalinvestigations that constitute the secondstrand of his major area of interest: thestructure of bivariate distributions.

When Lancaster first started to work inmathematical statistics, the standard wayto assess independence in a contingencytable was to use the X2 goodness-of-fitstatistic which is asymptotically distri-buted, under independence, as a χ2 varia-ble. Contingency tables arise when two ormore categorical variables are measuredon the same object and the joint frequen-cies of the different possible combinationsare displayed in tabular form. The simplestcase, a two-way contingency table, is whenthere are only two variables being consid-ered.

Thus, for example, in order to trial anew vaccine against parasitic infection, agroup of animals can be randomly sepa-rated into a control and a treated group. Foreach animal some outcome variable, suchas whether the animal became infected ornot when exposed to the parasite, can berecorded. The results of this experimentcan be displayed in a 2 × 2 table of whichan example is given in Table 1.

If the vaccine has no protective power,the observed frequencies in the contin-gency table will reflect the hypothesis ofindependence between the administrationof the vaccine and the outcome of the trial.


Lancaster’s interest in the analysis of con-tingency tables is not surprising; in theearly part of his career as a pathologist hewas generating such tables from epidemio-logical data.

The X2 goodness-of-fit statistic for anr × s contingency table is the sum of thesquared standardized deviations of cell fre-quencies from their expected values.Under the hypothesis of independence, it isasymptotically distributed as a χ2 variablewith (r–1)(s–1) degrees of freedom. If wedenote the cell frequencies by {fij, i = 1, …,r; j = 1, …, s} and their expected values by{eij, i = 1, …, r; j = 1, …, s}, then the X2

statistic is calculated as:

(1)

One drawback of using this test on con-tingency tables that are more complex thanthe simple 2 × 2 one is its global nature.A significantly large value of the X2 statisticcalculated under the assumption of inde-pendence provides evidence of a lack ofindependence but no detailed informationas to precisely where the independencehypothesis is likely to have been violated.

In his first mathematical paper (II.1) heshowed how a single, global X2 withmultiple degrees of freedom can bedecomposed into mutually independentcomponents, each one distributedasymptotically as a χ2 with one degree offreedom and each one appropriate fortesting a particular contrast.

He proved this by first showing that thejoint probability function for an r × s table,assuming independence of the frequencies

{fij}, given fixed row and column sums{fi.} and {f.j} (f being the total frequency):

can be written as a product of such proba-bility functions for (2 × 2) tables. Thisprovides asymptotically a decompositionof the associated X2 goodness-of-fit (1)statistic into (r–1)(s–1) asymptoticallyindependent X2 statistics, each one asymp-totically distributed as a χ2 variable withone degree of freedom.

In the same paper, under the influenceof J.O. Irwin (see our Section 4), he gave amethod for partitioning X2 exactly into(r–1)(s–1) components each correspond-ing to a (2 × 2) table.

This was done with the help of a familyof orthogonal matrices, the Helmertmatrices, which were to play a large role inhis future work. Irwin’s (1949) commentaryon the exact decomposition immediatelyfollowed Lancaster’s. Later Lancaster(II .26) reprised this work. The exactdecomposition result is best understood inthe context of finite discrete bivariate proba-bility tables, which can be derived from an(r × s) frequency table by dividing all entriesby the total frequency ƒ. We shall return tothe decomposition result after discussingbivariate probability tables.

The analysis is more complex with con-tingency tables in three or more dimen-sions. However, by using the appropriateHelmert matrices to transform the originaldata, Lancaster (II.7) was able again todecompose the global X2 goodness-of-fitstatistic into components, analogous tothose in an analysis of variance. Thus, in athree-dimensional table he was able tocreate independent contrasts for the threemain effects, the three first-order inter-actions and the second-order interaction.In doing so he showed how, in an analysisof goodness of fit, it is possible to extract

Table 1. Example of a two-way table

Infected Not infected Total

Treated 3 32 35Control 21 10 31Total 24 42 66

X 2 = Σ Σ (ƒi j – ei j)2 /ei j .

r s

i=1 j=1

Π ƒi .! Π ƒ.j !

ƒ!Πi, j ƒi j!

r s

i=1 j=1


detailed information that is both interpret-able and analogous to the decompositionsof the sums of squares in the analysis ofvariance.

In 1953 Lancaster published a paper(II.11) explaining how the classical χ2

goodness-of-fit statistic is related to testsbased on the deviations of samplemoments from their expected values. Heachieved this through a representation ofX 2 based this time not on a Helmertmatrix, but on an orthogonal matrix gener-ated from the orthogonal polynomialsassociated with the marginal distributions.

If pi., i = 1, …, r is a discrete distribu-tion on r real numbers xi, i = 1, …, r, thenthe set of orthogonal polynomials associ-ated with it, {gr (x), r = 0, … (r–1)}, aredefined by the conditions: • g0(x) = 1, • gr(x) is a polynomial in x of degree r,

r >0, and

• pi. = δmn.

An orthogonal matrix, {G = gij}, can beformed from these orthogonal polynomialsby taking:

. , for i, j = 1, …, r.

Given an r × s bivariate probability table:

P = {pij, i = 1, …, r, j = 1, …, s},

it will have marginal distributions definedby:

and

If the marginal distributions are associ-ated with the values:

{xi , i = 1, …, r} and {yj , j = 1, … s},

then we can define two orthogonalmatrices G and H, by using the values ofthe orthogonal polynomials on the mar-ginal distributions as indicated above.

Now consider the matrix Q = {qtk},t = 1, …, r, k = 1, …, s defined by

Then GQHT has every entry in its firstrow and column zero, since polynomials ofdegree greater than 0 are orthogonal to theconstant vector. In general, for fixed (i,j),i = 2, …, r; j = 2, …, s (gi–1(xm), hj–1(yn)),may be regarded as the values taken by apair of random variables (Ui, Vj) withprobability pmn. Since, on account of theorthonormality conditions the randomvariables are already individually standard-ized to mean zero and variance 1, theelements of GQHT are correlations.

Next, when a matrix A is multiplied byan orthogonal matrix O, the sum of squaresof its entries is also preserved:

tr(AT A) = tr(OT OAT A) = tr(OAAT OT )

= tr(OA(OA)T )

= tr((OA)T OA).

Thus

is the sum of the squares of these(r–1)(s–1) correlations. The use of the left-hand side, ϕ2, as a measure of dependencein a bivariate distribution {pij} dates backto the beginnings of Karl Pearson’s work inmathematical statistics and is called thecontingency.

Now let us return to the (r × s) contin-gency table, with underlying probabilitydistribution {pij}, and put pij = fij / f, and

where

gm xi( )gn xi( )i 1=

r

∑

gij gi 1–

xj( ) pj=

�pi .=Σ pij, i=1, …, r�s

j=1

�p.j = Σ pij , j = 1, …, .s� . r

i=1

ptk –

pt.p.k

pt . p.kqtk =

ϕ2 = Σ Σ r s

i=1 j=1

r s

i=1 j=2

( pij –

pi.p.j )2

({GQHT}ij )2

pi . p.j

Σ Σ=

ptk – pt .p.kqtk =pt . p.k

pt .= Σ ptk = ft . / fs

k=1


and similarly for . We know that underthe assumption of independence (pij =pi. p.j) that

has asymptotically (as f →∞) a χ2 distri-bution with (r–1)(s–1) degrees of freedom.On the other hand with the use of suitable(random) orthogonal matrices the(r–1)(s–1) degrees of freedom can beallocated to squares, each multiplied by f,of correlations between orthogonalpolynomials.

Each of these summands in an ‘exact’decomposition will have asymptoticallythe distribution of the square of a standardnormal random variable, i.e. a χ2 distri-bution with one degree of freedom.

This decomposition depends on theorthogonality of G and H. The particularform of G and H allows the individualcomponents to be identified as correlationsbetween orthogonal polynomials. Otherorthogonal matrices can be used to trans-form , each one providing anotherrepresentation of the X 2 statistic. Choosingan orthogonal matrix based on somealternative model will simplify the searchfor how the independence hypothesisbreaks down.

The second strand of Lancaster’s majorarea of interest may be seen to derive fromthe preceding by refocussing on a bivariateprobability distribution table P = {pij},i = 1, …, r, j = 1, …, s, and on thecontingency, ϕ2, which may be written as

(2)

Write in vector notation x = {pi.},y = {p.j}, and put B = {bij} where

so

Note that the symbols x and y just intro-duced have no relation in meaning to thexi’s and yj’s occurring earlier in thisaccount. Since

Z = {zij} = BT B

is (s × s), symmetric and non-negativedefinite, it has real non-negative eigen-values λ1 ≥ λ2 ≥ … ≥ λs ≥ 0. It turns outthat λ1 = 1. Let ki be the (normed) righteigenvector of Z = BT B corresponding tothe eigenvalue λi, i = 1, 2, … s. We maytake k1 = (diag y1/2) 1. Now Z1 = B BT hasthe same non-zero eigenvalues as Z, andcorresponding to the eigenvalue λi >0 thecorresponding eigenvector of Z1 is Bki,with Bk1 = (diag x1/2)1. The number ofstrictly positive eigenvalues is in generalm ≤ min (r,s). Some straightforwardmanipulations give the spectral expansion

(3)

In fact all of the , i = 2, …, m arecorrelations, called canonical correlations,as can be seen from (3) by taking

since uT Pv = . It can be shown for ϕ2

defined by (2):

(4)

so that λ2 = 0 is equivalent to independ-ence: (λ2 = 0 implies that λi = 0, i ≥ 2.)

Notice that if K is the (orthogonal)matrix whose ith column is ki and we writeR = KT BT, C = KT, then

RBCT = KT BT BK = diag λi

so the matrices R and C diagonalize B.Returning to the matrix Q,

we find that the (1, 1) entry of RQCT is 0since λ1 = 1, as are all the other entries of

p .k

ƒ Σ Σ q2tk

r s

t=1 k=1

G, H

Q

ϕ2 = Σ – 1. i,j

p2i j

pi . p.j

bij = pij / pi .p.j

B = (diag x )P(diag y ).12

12

P = xyT� 1 + Σ λi �(diag x ) � (kT (diag y ))�.m

i=2

12

12

Bki

λii

λ i

v = (diag y )ki , u = (diag x ) ,Bki

λi

12

12

λ i

λiϕ2 = Σ i=2

m

Q = B – x (y )T, 12

12


the 1st row and 1st column, because of theorthogonality conditions.

If we now refer to the contingency tablecorresponding to the probability distri-bution {pij} and replace pij everywherewith , we find from (4) that the exactdecomposition of the chi-square statisticis:

,

where is the sample analogue of thecanonical correlation.

To obtain an exact decomposition of X 2

paralleling that obtained from above,we may assume without loss of generalitythat r ≥ s, and plausibly (since is a datamatrix), that m = s. Let be an (r × r)orthogonal matrix whose first s columnsare and let . Then an exactdecomposition of X 2 is obtained from thesquares of the (r –1)(s –1) non-zero clem-ents of .

Eigenfunction expansions such as (3) ofpositive matrices can be used to assess thedependence structure of a contingencytable for departure from independence and,when there is evidence of departure, toestimate the dependence structure, pro-vided that a specific alternative has beenpre-specified. The role of eigenfunctionexpansions of probability matrices in thederivation of an interpretable decomposi-tion of the X2 statistic led Lancaster to hisextensive research on the structure ofbivariate distributions in general.

Eigenfunction expansions can bederived for bivariate continuous distribu-tions (or more precisely for a natural oper-ator defined by them), though in 1953 theonly example known was the expansion ofthe bivariate normal, derived by Mehler in1866. In this case the canonical variablesare the orthogonal polynomials on thenormal distribution, the Hermite-Chebyshev polynomials. That is, when thebivariate normal density function isexpressed as a weighted sum of products of

these polynomials, the expansion is diago-nal; when i is not equal to j, the orthogonalpolynomial in x of ith degree is uncorre-lated with the orthogonal polynomial in yof jth degree.

The Mehler identity provides an elegantway to understand the structure of thedependence between two normal variablesin the case when their joint distribution isbivariate normal. The advantage of writingthe bivariate normal as a sum of cross-product terms is that it reduces a complexdependence relationship into a sum ofsimple constituent parts as well as allow-ing a goodness-of-fit test for the bivariatenormal to be constructed (II.15).

In this seminal 1958 paper Lancasterbegan his study of the structure of bivariatedistributions. In it he showed how theMehler expansion was a special case of amore general result. Any bivariate fre-quency function that satisfies a simpleregularity constraint can be expressed asthe product of its marginal frequency func-tions and a sum of cross-products oforthogonal functions that are complete withrespect to the marginal distributions. WhatLancaster showed was that the orthogonalfunctions can be chosen so that they are‘canonical’: the resulting matrix of coeffi-cients has non-zero terms only along thediagonal. As Lancaster himself noted, thecanonical expansion he derived is nothingmore nor less than the eigenfunction expan-sion of an operator that can be generatedfrom the bivariate distribution.

Let f(x,y) denote a bivariate densityfunction with marginal frequency func-tions, g(x) and h(y). Define a function

.

The regularity constraint then becomesthat

.

pij

X 2 = ƒΣ λ i i=2

m

λ i

GQHT

PRT

BK C = K

RQCT

K(x,y) = ƒ(x,y)

g(x)h(y)

ϕ2 + 1 = ��K2(x,y) dxdy < ∞


K(x,y) can be used to define a condi-tional expectation operator from L2(g) toL2(h) in the following way:

For ∫α2(x)g(x)dx < ∞, define

This operator has an eigenfunctionexpansion that coincides with Lancaster’scanonical expansion.

Lancaster used these canonical expan-sions for two purposes. First he showedhow they could be used to construct newbivariate distributions with given marginaldistributions. He also used them to derive agoodness-of-fit test for the bivariatenormal distribution, associating individualdegrees of freedom with the correlationsbetween the canonical variables.

Knowing that most bivariate densityfunctions can be written in a canonicalexpansion helps understand the structureof probabilistic dependence in general butis not of great value in a particular case.However, when the canonical variables arethe associated orthogonal polynomials theexpansion is particularly useful. Theorthogonal polynomials for a particulardistribution satisfy a recurrence relationand are easily computed. Furthermore, theexpectations of the orthogonal poly-nomials can be determined from and deter-mine the moments of their associateddistribution. As a consequence the condi-tional moments are easy to calculate andthe regression structure of the bivariatedistribution is easy to determine. Onequestion that had concerned many authorswas whether canonical expansions, similarto the Mehler identity, could be found forother bivariate distributions.

Lancaster discovered a paper of aGerman mathematician, Meixner (1934),that turned out to contain the key toanswering the above question, at least in

the case when the bivariate dependencestructure arises from ‘random elements incommon’. Lancaster’s final work on thistheme provided a deep and accessibleunderstanding of the structure of thosebivariate distributions whose marginalscome from the set of distributions studiedby Meixner.

A random variable is said to have adistribution belonging to a Meixner class ifits associated orthogonal polynomials havea particularly simple generating functionk(x;t). More specifically, let X be a centredrandom variable with moment-generatingfunction ϕ and a set of orthogonal poly-nomials, {Pm}, where Pm = xm plus termsof lower degree, m = 1, 2, … .

Then X is said to belong to a Meixnerclass defined by u(t) if the generatingfunction of the orthogonal polynomials hasa specific form, more precisely if:

where ψ(u(t)), is the moment-generatingfunction of the distribution.

There are only six distributions thatform a Meixner class, depending on theform of u(t). They are the normal, thePoisson, the gamma, the positive and nega-tive binomial, and a hypergeometric. Oneadvantage of looking at the Meixnerdistributions is that some properties of theassociated orthogonal polynomials can beproved for all members of the classtogether rather than looking separately atthe individual cases. Thus, for example, thesets of orthogonal polynomials on aMeixner distribution are complete in theassociated L2 space (Eagleson 1964).

The relationship of the Meixner class topolynomial expansions can be derived forthose bivariate distributions that are gener-ated by the random elements-in-commonmodel. If U, V, and W are independentrandom variables, the two variablesdefined by:

β(y) = �K(x,y)α(x) g(x)dx

= �α(x)ƒ(x,y)/ h(y)dx

= E(α(x) / y) h(y).

k(x;t) ≡ 1 + Σ tmPm(x) / m!m

= exp[xu(t)] / ψ(u(t)) ,


X = U + V

Y = V + W

will be correlated, with correlationcoefficient:

The canonical expansion of thebivariate density function of (X,Y) willhave polynomial eigenfunctions if andonly if the U, V and W all belong to thesame Meixner class. This result, togetherwith a simple proof of Meixner’s originalcharacterization of the members of theMeixner class, is given in II.55.

Acknowledgements

ES is indebted to Professor PeterArmitage, Mary Phipps (née Pusey), JonLancaster, and Rosanne Walker for helpspecific to this biographical memoir.

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