Aust. N. Z. J. Stat. 44(4), 2002, 385–400
IN MEMORIAM EMERITUS PROFESSOR HENRY OLIVER LANCASTER, AO FAA 1 February 1913 – 2 December 2001
Eugene Seneta1 University of Sydney Summary The death, in Sydney, of Oliver Lancaster marks the end of an era in the histories of the Statistical Society of Australia, which (in its previous existence as the Statistical Society of New South Wales) he helped found in 1947, and of the Australian Journal of Statistics of which he was founding editor (1959–1971). Oliver Lancaster was Foundation Professor of Mathematical Statistics at the University of Sydney (1959–1978), where he spent his life as student and academic. During his academic career, he achieved scholarly distinction in at least four fields: mathematical statistics, medical and public health statistics, the history of medicine and of statistics, and statistical bibliography. With E.J.G. Pitman (1897–1993), M.H. Belz (1897–1975), E.A. Cornish (1909–1973) and P.A.P. Moran (1917–1988) he was part of a cohort of renowned Australian mathematical statisticians who laid the foundation of the glory days of Australian mathematical statistics. This obituary and tribute focuses on some of these aspects, within a broader historical picture. Key words: Australian Journal of Statistics; characterization; chi-squared; history of statistics; Lancaster’s mid-P; mathematical statistics; medical statistics; normal distribution; Statistical Society of Australia; University of Sydney.
1. History to 1946 Henry Oliver Lancaster (H.O.L.) was born the second son of Dr Llewellyn Bentley Lancaster, a medical graduate of Sydney University, and his wife Edith Hulda Smith, a nurse, in Sydney, and spent his early life in Kempsey, NSW, where his father practised. His father died just before H.O.L.’s 9th birthday, and he and his brother Richard (‘Rick’), two years younger, to whom he was to remain close throughout his life, boarded in Kempsey through his years at primary school and West Kempsey Intermediate High School. H.O.L. showed an extraordinary talent for mental arithmetic, from his childhood years. After a year in Economics and then Arts at Sydney University, he enrolled in Medicine I in 1931, graduating MB BS in 1937. In 1938 and 1939 he was Pathologist and Senior Medical Officer at Sydney Hospital, where he had contact with the future Nobel laureates J.E. Eccles and B. Katz. From 1940 H.O.L. worked as Medical Officer in the A.I.F., first in the Middle East, and then from early 1942 in the 117 Australian General Hospital in Townsville, this laying the groundwork for his first (joint) papers which appeared in the Medical Journal of Australia in 1944. Service in New Guinea followed. In the Australian War Memorial’s collection, Canberra, there is a fine pencil sketch (AWMArt 22670) by the war artist Nora Heysen, ‘Pathologist 1 School
of Mathematics and Statistics FO7, University of Sydney, NSW 2006, Australia. e-mail:
[email protected]
c Australian Statistical Publishing Association Inc. 2002. Published by Blackwell Publishing Ltd.
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H.O. Lancaster (1978)
Nora Heysen’s pencil portrait (1944) of Lancaster, from the Australian War Memorial’s collection (AWM Art 22670) c Australian Statistical Publishing Association Inc. 2002
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(Major Henry Oliver Lancaster) 1944’, from this time (see below). It was reproduced in his obituary in the Sydney Morning Herald , Thursday, March 14, 2002, p . 38. In 1945, while still in New Guinea, he enrolled as an external student in second-year honours Mathematics at Sydney University. He wrote of this time (Lancaster, 1982): I surprised the army director of Pathology, Colonel E.V. Keogh, by reporting the results in systematic form with properly drawn graphs, and means and standard deviations correctly computed.
It was E.V. Keogh (to become Director General of Health) who in the words of another famous recently deceased Australian-born statistician, Geoffrey S. Watson (email to the author, 1997), ‘pushed us both towards . . . Stats’as a postwar career. Although H.O.L’s papers of 1944 do not contain any quantitative statistical inference, they tabulate and graph counting data, and draw qualitative conclusions from this. In relation to the time in Townsville in 1942, presumably of some unpublished analysis, Lancaster (1982) wrote: Here a statistical problem was to compare incidences of infection in the 4 classes of troops according to whether or not they had been in the Middle East or New Guinea, a problem in 2 × 2 × 2 tables, . . .
H.O.L. had been interested in blood-counting before and during his wartime service. One can see in all this the seeds of his later work in the mathematical statistics of counting data (categorical data analysis) in general, and chi-squared in particular. 2. 1946–1959: Two directions of statistics 2.1. Medical statistics Given a temporary appointment in 1946 as Lecturer in Medical Statistics at the School of Public Health and Tropical Medicine (SPHTM at Sydney University) by Professor Harvey Sutton with Keogh’s support, H.O.L. spent much of his time completing his undergraduate mathematics education and reading the English medical statisticians (Farr, Greenwood, Bradford Hill) and American statisticians/epidemiologists (Dublin, Lotka, Pearl and Frost). Herbert A. David (personal communication) recollects that H.O.L., . . . still in the uniform of major, joined the Honours Math class in 1946. I was most impressed by his being one of the very few who responded to Professor T.G. Room’s plaintive appeals to the class to hand in geometry homework.
Austin Bradford Hill had succeeded Major Greenwood as Professor of Statistics at the London School of Hygiene when H.O.L. arrived there in 1948 for about 12 months as Rockefeller Fellow in Medicine. In London he shared an office with Peter Armitage, who was to become an outstanding biostatistician, a President of the Royal Statistical Society, and a friend for life. The major influence on H.O.L. at the School of Hygiene, however, was the mathematical statistician J.O. Irwin (1898–1982) who encouraged H.O.L. with his first paper in mathematical statistics, published in 1949, and in continuing work on partitions of chisquared statistics. This advice H.O.L. followed, with a PhD from Sydney University in 1953 and a series of publications highlighted by his book The Chi-squared Distribution (1969). In later years, H.O.L. also expressed admiration for M.G. Kendall, whose Advanced Theory of Statistics had been an early guide to the theory, who had given him good advice on his future career, and who belonged to no ‘school’. On H.O.L.’s return to Australia, there was a period of intense activity in medical statistics. Among the 50 or so articles which he was eventually to publish in the Medical Journal of Australia alone, were his striking findings — in publications of which Lancaster (1956) c Australian Statistical Publishing Association Inc. 2002
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is representative — that death rate from melanoma (black mole cancer) was associated with intensity of sunlight, by studying effects in different Australian states (variation with latitude). The danger of intense ultra-violet radiation has now passed into standard knowledge in Australia, with its discoverer forgotten. The landmark paper on rubella deafness was published as Lancaster (1951). It was motivated by the association noted by N.M. (later Sir Norman) Gregg and others between congenital deformities and what was thought to be a virulent form of rubella in pregnancy in Australia in the years 1938–1941. H.O.L’s follow-up was the result of careful thought following on from the idea he had, on passing the old New South Wales Institution for the Deaf and Dumb and the Blind, of examining its well-kept admission records since its opening in 1861. He followed this by a careful examination of the records of similar institutions in other states, and of Australian census records for 1911, 1921 and 1933, where he found (corresponding) peaks in the age distribution of deaf people, which he connected with births in 1898 and 1899, a time of known rubella epidemics in Australia. This knowledge was aided by the fact that, in small relatively isolated populations, epidemics tend to die out, and so can be observed to come and go over time. It is illuminating to read a passage from late in his life in Some Recollections (Lancaster, 1996 page 34): . . . in England and the large continental masses, the females all contracted rubella in the first few years of life. In Australia, however, the population was not large enough to maintain the rubella epidemic and so it died out. It followed that if the females survived up to adult life without having had rubella, [and] the rubella was introduced into Australia from outside, . . . there were epidemics in which women of childbearing age were attacked. Children subsequently born had congenital defects such as congenital cataract, which was described by Norman Gregg in 1940, or deafness which was described in detail by workers from South Australia.
Thus H.O.L. dispelled the illusion that what had happened in 1940 was a new phenomenon, and in effect established a causal connection between ‘ordinary’rubella and congenital deafness. We can see here the groundwork for his strong disagreement with R.A. Fisher’s denial of smoking as a cause of lung cancer, during Fisher’s visit to Sydney in 1959. In Lancaster (1996), when recollecting his rubella conclusions in contrast to what had been thought earlier, he quotes, with some satisfaction, Occam’s Razor: ‘Entitia non sunt multiplicanda’. Lancaster (1951) is additionally interesting from an Australian standpoint in that it cites reports by the first two Commonwealth Statisticians, George Handley Knibbs (1858–1929), and Charles Henry Wickens (1872–1939) who had been ‘Compiler’ in the Commonwealth Bureau of Census and Statistics under Knibbs since its creation in 1906. Wickens succeeded Knibbs in 1922; and H.O.L. expressed continuing gratitude and admiration for Wickens, reading an oration on the centenary of his birth. This period of H.O.L.’s activity included statistical correspondence with Joseph Berkson in 1952, and Sir Frank Macfarlane Burnet in 1957. Berkson was to be one of the referees, along with J.O. Irwin and M.G. Kendall in support of his application for the Chair of Mathematical Statistics at Sydney University. R.A. Fisher’s opinion was also sought. 2.2. The Statistical Society of New South Wales There was, after World War 2, a clear need for an umbrella organization for statisticians in Sydney, who were spread over a number of departments and institutions. The driving forces behind the founding public meeting of the Statistical Society of New South Wales, held at the University of Sydney on 25 September 1947, were H.O.L., Helen Newton-Turner (who c Australian Statistical Publishing Association Inc. 2002
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became first President), and R.S.G. Rutherford (later Professor of Economic Statistics). This initial meeting formed a steering committee which included G.D. Bradshaw, D.B. Duncan, W.A. McNair, H. Mulhall, A.H. Pollard and A.W. Prescott. On his return from London, H.O.L. was largely responsible for publication of the Bulletin of the Statistical Society of New South Wales which had begun publication in March 1949, until September 1958 when the last number appeared. It was replaced in 1959 by the Australian Journal of Statistics with H.O.L. as founding editor. He served continuously till 1971, running the journal from his office in the Carslaw Building, and compiling his famous bibliographic card index, in which mathematical statistics students were made to play a part. The name for the new journal anticipated there soon being a Statistical Society of Australia, whose nucleus was the amalgamation of the NSW and Canberra Statistical Societies in 1962. With the formation of the Statistical Society of Australia, the Statistical Society of New South Wales became the NSW Branch, and responsibility for the journal was assumed by the central organization. H.O.L.’s conception of the journal inclined more to the theoretical than to applied statistics articles, for sound reasons. This policy was continued by the successor editors (C.R. Heathcote, C.C. Heyde, E.J. Williams, J.S. Maritz, C.A. McGilchrist and I.R. James) of the Australian Journal of Statistics. The change of name and of support base (to include the New Zealand Statistical Association) occurred in 1998. Up to the last number (No. 3) of Volume 18, 1976, the journal continued to be printed by the Australasian Medical Publishing Co. Ltd, located in what is now the extended main campus of Sydney University, in a format established by H.O.L., due to his medical connections. The NSW Branch of the Statistical Society commemorated the 50th anniversary of its founding in the Norman Gregg Lecture Theatre of the School of Public Health and Community Medicine (the most recent name for the SPHTM) on 25 September 1997 (Eyland, 1997). A keynote speech was given by one of H.O.L.’s earliest colleagues in the Society, Emeritus Professor A.H. Pollard of Macquarie University. Greetings from H.O.L. were presented, as was a paper (Lancaster, 1997) written for the occasion. Lancaster (1988) gives a detailed account of the progress and meetings of the Statistical Society of New South Wales, of its Bulletin and of its successor, the Australian Journal of Statistics. The account contains names additional to those already mentioned which many Australian readers will know, such as M.H. Belz, W.D. Borrie, E.G. Bowen, C.G. Clark, H.M. Finucan, E.J. Hannan, F.G. Garrett, M.A. Hertzberg, F.B. Horner, H.S. Konijn, D.W. Maitland, Norma McArthur, G.A. McIntyre, J.E. Moyal, R.W. Rutledge, J.H. Weiler and P.C. Wickens. Presidents of the Society’s successor, the NSW Branch, have included many people closely associated as academics or students with H.O.L.’s Department of Mathematical Statistics: Les Balaam (1968–1969), John Pollard (1974–1975), Doug Shaw (1985–1986), Ron Sandland (1988–1989), John Robinson (1990–1991), Nick Fisher (1992–1993), Ann Eyland (1994–1995), David Griffiths (1996–1997), and President for 2002–2003, John Rayner. H.O.L. was also involved in the founding of the Australian Mathematical Society in 1956–1957, and later (1967–1968) served as its President. At the founding of its organ, the Journal of the Australian Mathematical Society , whose Volume 1 contained three of H.O.L.’s papers (including probably his most important one on the characterization of the normal distribution), the first editor, T.G. Room, FRS, FAA, Professor of Pure Mathematics at Sydney University, declared that ‘the standard of our Journal can be taken as equivalent to the Annals of Mathematical Statistics’. Although professedly surprised at the time by this choice c Australian Statistical Publishing Association Inc. 2002
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Sydney University Department of Statistics, 1960. Background (left to right): Harry Mulhall, Henry Konijn, H.O. Lancaster, Janet Fish (secretary), Chris Heyde, David Hamdan, Bro. Bennett, R.M. Armson, Les Balaam. Foreground (left to right): Eve Bofinger, Comptometrist. Absent: Murray Aitkin.
of model, H.O.L. aspired to such standards in ‘his’ own journal, and the policy was successful for both journals. 3. 1959–1978: The Department of Mathematical Statistics There was no Department of Mathematical Statistics at Sydney University till H.O.L.’s appointment to the Chair of Mathematical Statistics in 1959. Until his retirement in 1945, D.T. Sawkins offered a statistics course of two lectures a week, and D.B. Duncan had been appointed in 1938 to the Faculty of Agriculture to teach Biometry. For the last year (1945) of Sawkins’ class, Herbert A. David (personal communication to Eugene Seneta, 22 January 1998) writes: As a text we used Aitken’s Statistical Mathematics and one searched round for other texts. Yule and Kendall was the most helpful.
In 1946 H. Mulhall was teaching probability out of Uspensky’s book in third-year Honours mathematics, and through him mathematical statistics maintained a presence within the applied mathematics program. By 1952 there were Honours programs in economic statistics and in biometry, but no full Honours program in mathematical statistics till 1960. By then H. Mulhall had joined H.O.L. in the newly formed Department of Mathematical Statistics, and between them the two men laid the foundations for its development (Eyland, 1979). H.O.L. in his retirement frequently recalled that the relationship was always professional and amicable. After H.O.L.’s appointment in 1959, the first 4th year students that he taught were E. Fackerell (later Associate Professor in Applied Mathematics at Sydney University) and Ann Eyland (n´ee Wight), in recent years an elected member of the Sydney University Senate. The course was based on Feller, Volume 1, of which the second edition had appeared in 1957. The first 4th year Honours class in mathematical statistics was in 1960, with M. Aitkin, R.B. Armson, Bro. Bennett, C.C. Heyde, and M.A. (David) Hamdan. Most of this group went on to distinguished academic careers in mathematical statistics, as did G.K. Eagleson, who came from 4th year pure mathematics to do a PhD under H.O.L., and is presently Professor at the c Australian Statistical Publishing Association Inc. 2002
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Sydney University Department of Statistics, 1963. Left to right: Tim Brown; David Hamdan; Eddie Oliver; K.Y. Chan; Vic Bofinger; Geoff Eagleson; Harry Mulhall; Janet Fish (secretary); Comptometrist; H.O.L.; Ailsa Ferrier; Mary Pusey (now Phipps); John Pollard; Murray Aitkin; Comptometrist.
Australian Graduate School of Management (AGSM). Professor C.C. Heyde FAA, who after an MSc supervised by H.O.L. went on to the Australian National University (ANU) for PhD, is now Professor of Statistics at both ANU and Columbia University in New York. The photograph on page 390 currently hangs adjacent to H.O.L.’s old office in the Carslaw Building, Sydney University, in the room used by his secretary, Mrs Elsie Adler, before his retirement, and shows the members of the Department as it was in that first year (1960) on the steps of its then location, the Transient Building, still the home of the biometricians. Apart from the 4th year Honours students, it includes H.O.L., Harry Mulhall, Henry Konijn, Les Balaam, Eve Bofinger and Janet Fish. Many of the students which the Department of Mathematical Statistics (DMS) produced over the years still work as statisticians in the Sydney area, including John Robinson, Mary C. Phipps (n´ee Pusey), Howard D’Abrera and Neville C. Weber in the mathematical statistics component-group of the University’s present School of Mathematics and Statistics. Beyond New South Wales, Canberra has a special place as destination — in particular the ANU, the Commonwealth Bureau of Census and Statistics (as it then was) and Health and Community Services. The ANU–Sydney University link is especially relevant. Professor P.A.P. Moran (1917–1988) was Professor of Statistics at the Institute of Advanced Studies (IAS), ANU. He had returned from England in 1952. One of the functions of the IAS was to provide for the supervision of PhD candidates within Australia. It had been traditional for such candidates to go overseas. For example, D.B. Duncan had gone to Iowa State University, Harry Mulhall to Cambridge (to work under J. Wishart), and Herbert A. David to University College London (to work under H.O. Hartley). Consequently, Moran’s Department of Statistics at IAS became a focus for students who had completed a research Master’s in Australia or New Zealand, and many of them came from Sydney, presumably with H.O.L’s encouragement. The relationship between Lancaster and Moran appears to have been amicable and supportive; both had Australian origins, and they knew the same people in England. Both were building up something very strongly Australian, on the basis largely of native talent. The times were right: the Murray Report had empowered Australian universities financially; the subject of mathematical statistics was young and flourishing world-wide, and its importance was recognized at home and abroad. c Australian Statistical Publishing Association Inc. 2002
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4. Retirement After retirement, H.O.L. wrote in his autobiographical sketch (Lancaster, 1982): . . . my chief interest is in the survey of world mortality. A grant from the Australian Research Grants Committee is assisting me to complete a project which I have had in mind for many years; indeed from the time I was working on Australian mortality; further, in my sabbatical leaves of 1969 and 1973, I took advantage of the London libraries to prepare for this survey. The survey needs to bring together the official statistics, demography, some mathematical ideas on epidemiology and laboratory epidemiology . . .
In the event, he worked away at this survey project in a basement office in the stacks of the Fisher Library on the main Sydney University campus, aided only by his secretary Mrs Philippa Holy, and the first fruits appeared as a large format book of 605 pages entitled Expectations of Life: A Study in the Demography Statistics and History of World Mortality. This is a book of amazing scholarship and commitment. The Bibliography alone takes up pages 505–592, at two columns per page, and contains (at an estimate) about 2800 items. This compilation was done without the aid of electronic databases. The whole work, including its meticulously detailed references, has a strong historical colouration, to which its subtitle already testifies. The history of statistics and probability permeates almost all of H.O.L.’s oeuvre, and towards the study of such history he influenced substantially both C.C. Heyde and the author of this obituary. H.O.L’s last book (1994) was a kind of complement to the preceding one. Entitled Quantitative Methods in Biological and Medical Sciences. A Historical Essay, it was dedicated to the biologist Ernst Mayr (b. 1904), whose book of 1982, The Growth of Biological Thought: Diversity, Evolution and Inheritance, I would often find H.O.L. reading in his retirement. In Lancaster (1996 p . 39) he says: ‘I found what might be called the chi-squared distribution in Mayr’s text’. There was another, less manifest, feature of H.O.L.’s retirement: his continuing preoccupation with the interaction of normality and independence, particularly the characterization of the normal distribution by the independence of linear forms of independent random variables without assuming finiteness of moments (see Section 5.2 below). H.O.L’s last technical research paper, published in the Australian Journal of Statistics in 1987, discussed aspects of this general topic, and his last seminar to his old Department of Mathematical Statistics in 1990 was, appropriately, on the same topic. 5. Mathematical statistics: some highlights Lancaster’s own ‘favourite’ papers in mathematical statistics (personal communication to Eugene Seneta, dated 21 January 1993), with himself as sole author, are as follows. They are an accurate representation of his work of this kind. The comments are his own. (1949). The derivation and partition of χ 2 in certain discrete distributions. Biometrika 36, 117–129. Corrig. 37, 452. (‘Sentimental! The beginnings of it all.’) (1954). Traces and cumulants of quadratic forms in normal variables. J. Roy. Statist. Soc. Ser. B 16, 247–254. (1958). The structure of bivariate distributions. Ann. Math. Statist. 29, 719–736. Correction, 35 (1964) 1388. (1960). The characterization of the normal distribution. J. Austral. Math. Soc. 1, 368– 383. (‘Here I was in competition with many masters, who had made unnecessary assumptions before 1960, only excepting Zinger.’) c Australian Statistical Publishing Association Inc. 2002
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(1961). Significance tests in discrete distributions. J. Amer. Statist. Assoc. 56, 223– 234. (‘This is humbler mathematics, but there are some shrewd maneuvers. Most authors cite E.S. Pearson: it looks more erudite.’) (1975). Joint probability distributions in the Meixner classes. J. Roy. Statist. Soc. Ser. B 37, 434–443. (‘Thought highly of by the editors!’) (1980). Orthogonal models for contingency tables. In Developments in Statistics Vol. 3, ed. P.R. Krishnaiah, pp . 99–157. New York: Academic Press. For this obituary, in the professional journal which H.O.L. can be credited with founding, we single out two areas from a number into which H.O.L.’s work can be partitioned. 5.1. Lancaster’s mid-P The finished form of Lancaster’s mid-P is in the 1961 paper above, although the idea has its origins in H.O.L.’s second published paper, written while he was still Rockefeller Fellow in London (Lancaster, 1949b in References, below). If U is a random variable which is uniformly distributed in (0, 1), then Y = −2 ln U is distributed exponentially with density f (y) = 21 e−y/2 (y > 0), and so has a χ22 distribution, with E(Y ) = 2. Thus if Y1 , Y2 , . . . , Yn are independent and identically distributed (iid) with d 2 , using the additive property of chi-squared. this distribution, then ni=1 Yi = χ2n For a test statistic continuously distributed under the null hypothesis, the P-value is uniformly distributed on (0, 1). Hence if n hypotheses are being tested, using independent test statistics, the sum of the corresponding −2 ln P can be used as an overall test criterion. Lancaster (1949b) shows that biases are introduced when a test statistic is discrete under the null hypothesis, Y taking s successive values yi = −2 ln pi
(i = 0, 1, 2, . . . , s − 1) ,
where the pi are the realizable values of the P-value P . His focus is the quadrivariate table. Here one is testing for equality of success probabilities θ1 , θ2 in a 2 × 2 table of frequencies, Success
Failure
a c
b d
Sample 1 Sample 2
emanating from two independent and not generally large samples, using the usual (discrete) chi-squared test statistic χ2 =
n(ad − bc)2 , (a + c)(b + d)(a + b)(c + d)
where n = a + b + c + d ,
the hypothesis H0 : θ1 = θ2 being rejected if the value of the test statistic is too large. With the transformation Y = −2 ln P of its P-value, however, E(Y ) = 2, so a bias occurs in using χ22 as the distribution of each Y, when cumulating Y values for repeated experiments. The evolution of his thinking is as follows. (1) Under H0 , E(χ 2 ) = 1, precisely, which is consistent with the mean of a χ12 random variable. Using Yates’ correction to produce a χc2 statistic without or with the log transformation makes matters worse, since E(χc2 ) and E(Yc ) are considerably below 1 and 2 respectively. c Australian Statistical Publishing Association Inc. 2002
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(2) If one insists on proceeding with a log transformation in general for an upper tail test with test statistic W, one should use quantities whose expectation is closer to 2 than Y = −2 ln P where the observed value of P is Pr(W ≥ w), where w is the observed value of W. (3) Lancaster (1949b) proposes two such quantities; the first has expectation precisely 2 for a general W. The second is simpler in structure; its expectation is not quite 2, but very close. He calls the second the median value χ 2 , and writes its observed value as: χm 2 = −2 ln
1
2 Pr(W
≥ w) + 21 Pr(W > w) ,
= 2 − 2 ln Pr(W ≥ w),
if Pr(W > w) > 0 ; if Pr(W > w) = 0 .
The above ideas crystallize in general format in Lancaster (1961), in which he changes both terminology and position, and proposes ‘the median probability or perhaps the midprobability’ as the test function for a single experiment, where this test function is defined by Pm (w) = 21 Pr(W ≥ w) + 21 Pr(W > w) with a rule of rejection (in a single experiment): Pm (w) ≤ α , where α is the prespecified (nominal) size of test. Thus he is proposing to use it in the manner of a P-value for a continuously distributed test statistic. He is of course aware (and dismissive) of the Neyman–Pearson randomization proposal to give exact significance level α. In support of the use of Pm (w) in this way, it was subsequently noticed by various authors that n 1 1 3 E Pm (W ) = 2 , 1− var Pm (W ) = pr , 12
r=0
where pr = Pr(W = r), which is very much in line with the properties of U. The rediscovered relevance and use of this Lancaster’s mid-P significance measure has been discussed in recent years by Plackett (1984), Barnard (1989), Agresti (1992), Routledge (1992, 1994), Berry & Armitage (1995), Seneta, Berry & Macaskill (1999), Phipps (2000, 2002) and Seneta & Phipps (2001), amongst others. At July 2002, there were 58 articles listed on the electronic database Science Citation Index Expanded (beginning with the year 1985) which referred to topic ‘mid-P’; and 44 on the database Medline. 5.2. Normality and independence H.O.L.’s work on normality and independence originates from his interest in the partition of chi-squared statistics into independently distributed components. The first paper in a more general setting, Lancaster (1954) — which is in the list of his favourites — is a matrix treatment of necessary and sufficient conditions for statistical independence of quadratic forms of normal variables. There are no citations in it of his own earlier papers. The underlying tools are that if A is an (n×n) symmetric matrix, then all its eigenvalues ρ1 , ρ2 , . . . , ρn , are real, and there exists an orthogonal matrix T such that T TAT = diag(ρi ). The rank of A, r(A) is thus the number of non-zero eigenvalues. If A is symmetric, and idempotent (A2 = A), its eigenvalues are all 0 or 1, so r(A) = tr(A). c Australian Statistical Publishing Association Inc. 2002
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If Z = (Z1 , Z2 , . . . , Zn )T is a column vector of iid N(0, 1) random variables, and A is symmetric, then use of the orthogonal transformation permits a straightforward derivation (going back to Cayley in 1869, as H.O.L. was fond of pointing out) of the moment generating function (MGF) of the quadratic form Q = Z TAZ, T
MQ (t) = E(exp tZ AZ) = det(I − 2tA)
−1/2
=
n i=1
(1 − 2tρi )−1/2 ,
for t in a neighbourhood of 0, since I − 2tA is then symmetric and positive definite. Since (1−2t)−1/2 is the MGF of a χ12 random variable, if A is also idempotent and r(A) = r (≤ n), d then Z TAZ = χr2 . The converse is also true; and thus there is an equivalence between the chi-squared distribution and that of a quadratic form in iid N(0, 1) random variables if A is symmetric and idempotent. If A and B are symmetric matrices, the joint MGF of the quadratic forms Y1 = Z TAZ and Y2 = Z TBZ is −1/2 MY1 ,Y2 (t1 , t2 ) = E exp Z T (t1 A + t2 B)Z = det(I − 2t1 A − 2t2 B) . Thus a necessary and sufficient condition for Y1 , Y2 to be independent is (Cochran, 1934) det(I − 2t1 A − 2t2 B) = det(I − 2t1 A) det(I − 2t2 B) . Since det((I − 2t1 A)(I − 2t2 B)) = det(I − 2t1 A − 2t2 B) if AB = 0, from the multiplicative property of determinants, it follows that Y1 and Y2 are independently distributed if AB = 0. The condition AB = 0 is also necessary for independence (Craig, 1943), and thus provides a matrix characterization of independence of quadratic forms in iid N(0, 1) random variables. It is this result of Craig’s that seems to have ignited H.O.L.’s preoccupation with characterization, although his focus (in Lancaster, 1960) was to shift from characterization of independence given normality, to characterization of normality by independence. The necessity for independence of Craig’s condition AB = 0 had been the subject of controversy. Aitken (1950) proved it definitively. Lancaster (1954 p . 248) states ‘My own proof may also be regarded as an abbreviated form of Aitken’s (1950) . . . ’, and points out (p.253) the essence of both proofs: It is not sufficiently stressed in the statistical literature that the relation AB = 0, A and B being symmetric matrices, is the condition for a simultaneous reduction [by an orthogonal matrix H ] of A and B to disjunct diagonal form.
The presentation of H.O.L.’s proof in his Section 2 is followed by a number of then-recent theorems as simple corollaries of his methodology. The simultaneous reduction is also used in his Section 3, which is entitled ‘The Independence of χ 2 ’ to give a simple proof of a matrix version of Cochran’s (1934) theorem, which can be stated as: k A = If Ai , i = 1, 2, . . . , k, are symmetric matrices, ni = r(Ai ), i = 1, 2, . . . , k and i=1 i In , then any one of the following three properties implies the other two: (i) ni=1 ni = n; (ii) Aj2 = Aj , j = 1, 2, . . . , k; (iii) Ai Aj = 0, i = j. The χ 2 distribution of component quadratic forms Z TAj Z arises through (ii); and their independence through (iii). c Australian Statistical Publishing Association Inc. 2002
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A presentation of matrix normal theory in the above style appears as Hogg & Craig (1965 Chapter 13), presumably under the influence, in part, of Lancaster (1954), specifically cited in relation to distribution of quadratic forms on page 365, and given in its References (its Appendix A), along with most of his own references. The first edition of Hogg & Craig (1959; reprinted 1962) lacks such matrix presentation and citations. In Section 4, ‘Characterizations of the Normal Distribution’, Lancaster (1954) assembles some of the results on this topic available at the time, such as Geary’s characterization of the normal by independence of sample mean and variance, but is clearly hampered by the need to assume a priori the existence of all moments of the parent distribution. His next paper — also among his favourites — in the same area, Lancaster (1960), consequently contains the following generalization of Craig’s theorem. Let X = (Xi , X2 , . . . , Xn )T denote a vector of iid random variables and A = [aij ] and B = [bij ] denote non-negative definite symmetric matrices. Then if XTAX and X TBX are independently distributed and ajj bjj is not zero for at least one j, then the Xi s have finite moments of all orders. Further, if the Xi s are also symmetrically distributed and AB = 0, the common distribution of the Xi s is normal. The most important result in Lancaster (1960) is preliminary to the preceding, showing that independence implies existence of moments. We state first the results which motivated him. Bernstein’s Theorem (1941). If X1 and X2 are independent random variables, with E(X12 ) < ∞, E(X22 ) < ∞, and Y1 = X1 + X2 ,
Y2 = X1 − X2
are independently distributed, then each of X1 and X2 is normally distributed. Skitovich–Darmois Theorem (circa 1953). If X1 , X2 , . . . , Xn are independent random variables, with E(Xj2 ) < ∞, j = 1, 2, . . . , n, and Y1 =
n j =1
Xj ,
Y2 =
n j =1
bj Xj ,
where bj = 0, j = 1, . . . , n ,
are independently distributed, then each Xj is normally distributed, j = 1, 2, . . . , n. Bernstein and Skitovich wrote in Russian, and H.O.L. had taught himself to read this language at a time when translations were unavailable. In Lancaster (1960) he cites their paper and other Russian-language papers. The second theorem is a generalization of the first, but we have stated it here since it was to the n = 2 case (which has an essentially simpler structure) that he was to return, after years of preoccupation with this problem, in his last published research paper in the Australian Journal of Statistics, Lancaster (1987). His fundamental result is Lancaster (1960 Lemma 5) wherein he shows that the independence of Y1 and Y2 alone is sufficient to guarantee the existence of all moments of the Xj , j = 1, 2, . . . , n, so the preliminary assumption E(Xj2 ) < ∞, j = 1, 2, . . . , n, could be dropped from both theorems. In fact he had been anticipated in this result by the Soviet mathematicians B.V. Gnedenko (in 1948 for the n = 2 case) and A.A. Zinger (in the 1950s). H.O.L. actually cited Zinger’s papers, but in conversation with me in later years admitted that he had not fully understood them at the time. He continued to acknowledge Zinger’s priority till the end of his days. It c Australian Statistical Publishing Association Inc. 2002
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is fitting, nevertheless, that this paper earned him the great respect of the Soviet ‘masters’, as evinced during visits by Gnedenko and Yu.V. Linnik to Australia in the 1960s. An exploration of the technology involved in the above problem, with particular reference to Lancaster (1960) is contained in Quine (1993) and Quine & Seneta (1999). It seems appropriate to complete this subsection with a proof of a result which arose out of his continuing work to elucidate the power of the assumption of independence. Its name will be familiar to H.O.L.’s Honours students; it appears in print with only a laconic proof in Lancaster (1987). Doubling Lemma. If X1 and X2 are independent, Y1 = aX1 + bX2 ,
Y2 = cX1 + dX2 ,
where ac = 0 ,
and Y1 , Y2 are independent, with E(|Y1 Y2 |k ) < ∞ for some k > 0, then E(|X1 |2k ) < ∞. Proof. If Gi (x) denotes the cumulative distribution function of Xi , then for some M < ∞, M > E(|Y1 Y2 |k ) =
∞ ∞
−∞ −∞
|ax1 + bx2 |k |cx1 + dx2 |k dG1 (x1 ) dG2 (x2 ) .
If for constants C, K > 0, |x2 | < C, |x1 | > K C, then |ax1 + bx2 | ≥ |a||x1 | − |b||x2 | ≥ |a||x1 | − |b|C ≥ |a||x1 | − |b||x1 |/K = |x1 |(|a| − |b|/K) ; and similarly
|cx1 + dx2 | ≥ |x1 |(|c| − |d|/K) .
Now select D, 0 < D < min(|a|, |c|), and K sufficiently large so that |a| − |b|/K > D and |c| − |d|/K > D. Select C so that Pr(|X2 | < C) > 0. Then k |x1 |2k D 2k dG1 (x1 ) dG2 (x2 ) M > E(|Y1 Y2 | ) ≥ |x1 |>KC |x2 |KC
Thus E(|X1 |2k ) < ∞. The name ‘Doubling’ arises because (by an analogous argument, say) E(|Y1 |k ) < ∞ ⇒ E(|X1 |k ) < ∞. 6. Conclusion H.O.L. was much honoured in his lifetime, beginning with his election to the Australian Academy of Science (FAA) and the award of its Lyle Medal for Mathematics and Physics, both in 1961. He was made Officer of the Order of Australia (AO) in 1992 for services to Science and Education. Honours by the Statistical Society of Australia constituted: • President, Statistical Society of New South Wales (1952–1953), • President, Statistical Society of Australia (1965–1966), c Australian Statistical Publishing Association Inc. 2002
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• Honorary Life Member, 1972, • No. 3 (November) Issue of the Australian Journal of Statistics 21, 1979: ‘In Honour of H.O. Lancaster, Founding Editor’, • E.J.G. Pitman Medal (1980), • Annual ‘H.O. Lancaster Lecture’, commencing in 1979 at a meeting of the NSW Branch of the Statistical Society of Australia. The November issue, 1979, contains papers by V.M. Zolotarev, E. Seneta, G.K. Eagleson, R.C. Griffiths, R.K. Milne & R. Wood, C.D. Lai & D. Vere-Jones, P. Armitage, C.C. Heyde, P.A.P. Moran, J. Robinson, M.A. Hamdan and J.H. Pollard. H.O.L. was President (1966–1967) of the Australian Mathematical Society, and made Honorary Life Member in 1981. Other honours included: • • • • • •
International Statistical Institute (ISI): Elected Member, 1961; Royal Statistical Society: Honorary Fellow, 1975; American Statistical Association: Honorary Fellow, 1974; Institute of Mathematical Statistics (USA): Honorary Fellow, 1975; Emeritus Professor, University of Sydney, 1979; American Association for the Advancement of Science: Honorary Fellow, 1987.
His degrees, all from the University of Sydney, were MB BS (1937); BA (1947), PhD (1953), MD (1967); DSc (1971). A full account of his career (Lancaster, 1996; edited by his brother Richard) is in the Sydney University Archives with other memorabilia and archived material. It subsumes an earlier more widely available account (Lancaster, 1982). An ISI-specific perspective is Seneta (2002). An early bibliography of his writings is on pages 188–192 of the November 1979 issue of the Australian Journal of Statistics 21, and is continued in Lancaster (1996). The reference list below includes, as Lancaster items, non-technical writings which pertain specifically to the Statistical Society of Australia. In addition to the three books mentioned in the preceding account, a long-term statistical labour of love — one of the products of his bibliographic work — which must be mentioned for completeness, was Bibliography of Statistical Bibliographies (1968) and its 21 addenda over the succeeding years. H.O.L. died peacefully in his sleep at a Mona Vale nursing home, the quintessential quiet Australian achiever, on a Sunday after watching a game of cricket on TV in the company of his youngest son Jon. His other children from his marriage to Joyce Mellon are Paul, Peter, Llewellyn and Andrew. The author succeeded Professor Lancaster in the Chair of Mathematical Statistics in 1979, and delivered the H.O. Lancaster Lecture, on which this obituary is based, at the University of Sydney, to the NSW Branch of the Statistical Society of Australia, Inc., on 26 March 2002. References The list below includes H.O.L.’s seven favourite mathematical statistics papers (see Section 5). Not all items are specifically cited in the text. Agresti, A. (1992). A survey of exact inference for contingency tables. Statist. Sci. 7, 131–177. Aitken, A.C. (1950). On the statistical independence of quadratic forms in normal variables. Biometrika 37, 93–96. Anonymous (1972). Honorary Life Member (Professor H.O. Lancaster), Austral. J. Statist. 14, 90–91. c Australian Statistical Publishing Association Inc. 2002
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Anonymous (2002). Lancaster, Henry Oliver. In Who’s Who in Australia 2002, p . 1097. Melbourne: Crown Content.
Barnard, G.A. (1989). On alleged gains in power from lower P-values. Statistics in Medicine 8, 1469–1477. Berry, G. & Armitage, P. (1995). Mid-P confidence intervals: a brief review. The Statistician 44, 417–423. Cochran, W.G. (1934). The distribution of quadratic forms in a normal system, with applications in the analysis of covariance. Proc. Camb. Phil. Soc. 30, 178–181.
Craig, A.T. (1943). Note on the independence of certain quadratic forms. Ann. Math. Statist. 14, 195–197. Darroch, J.N. (1974). Multiplicative and additive interaction in contingency tables. Biometrika 61, 207–214. Eagleson, G.K. (1964). Polynomial expansions of bivariate distributions. Ann. Math. Statist. 35, 1208–1215. Eyland, Ann (1979). Mathematical Statistics at Sydney University. Statistical Society of Australia Newsletter, No.8, Sept. 1979, pp . 1–2.
Eyland, Ann (1997). Reflections on Statistics 1947 to 1997 to . . . Statistical Society of Australia Incorporated Newsletter, No. 81, 30 November 1997, pp . 1–4.
Feller (1957). An Introduction to Probability Theory and its Applications, Vol. 1, 2nd edn. New York: Wiley. Heyde, C.C. & Seneta, E. (1972). The simple branching process, a turning point test and a fundamental inequality. A historical note on I.J. Bienaym´e. Biometrika 59, 680–683.
Hogg, R.V. & Craig, A.T. (1965). Introduction to Mathematical Statistics, 2nd edn. London–New York: Collier–Macmillan.
Koudou, A.E. (1996). Probabilit´es de Lancaster. Exposition. Math. 14, 247–275. Koudou, A.E. & Pommeret, D. (2000). A construction of Lancaster probabilities with margins in the multidimensional Meixner class. Austral. and New Zealand J. Statist. 42, 59–66.
Lancaster, H.O. (1949a). The derivation and partition of χ 2 in certain discrete distributions. Biometrika 36, 117–129. Corrig. 37, 452.
Lancaster, H.O. (1949b). The combination of probabilities arising from data in discrete distributions. Biometrika 36, 370–382. Corrig. 37, 452.
Lancaster, H.O. (1951). Deafness as an epidemic disease in Australia: A note on census and institutional data. British Medical Journal 2, 1429–1432.
Lancaster, H.O. (1954). Traces and cumulants of quadratic forms in normal variables. J. Roy. Statist. Soc. Ser. B 16, 247–254.
Lancaster, H.O. (1956). Some geographical aspects of the mortality from melanoma in Europeans. Medical Journal of Australia 1, 1082–1087.
Lancaster, H.O. (1958). The structure of bivariate distributions. Ann. Math. Statist. 29, 719–736. Correction, 35 (1964) 1388.
Lancaster, H.O. (1960). The characterization of the normal distribution. J. Austral. Math. Soc. 1, 368–383. Lancaster, H.O. (1961). Significance tests in discrete distributions. J. Amer. Statist. Assoc. 56, 223–234. Lancaster, H.O. (1968). Bibliography of Statistical Bibliographies. Edinburgh: Oliver and Boyd. Lancaster, H.O. (1969). The Chi-squared Distribution. New York: Wiley. Lancaster, H.O. (1975). Joint probability distributions in the Meixner classes. J. Roy. Statist. Soc. Ser. B 37, 434–443.
Lancaster, H.O. (1980). Orthogonal models for contingency tables. In Developments in Statistics Vol. 3, ed. P.R. Krishnaiah, pp . 99–157. New York: Academic Press.
Lancaster, H.O. (1982). From medicine, through medical to mathematical statistics: Some autobiographical notes. In The Making of Statisticians, ed. J. Gani, pp . 236–252. New York: Springer.
Lancaster, H.O. (1987). Finiteness of the variances in characterization of the normal distribution. Austral. J. Statist. 29, 101–106.
Lancaster, H.O. (1988). Statistical Society of New South Wales. Austral. J. Statist. 39(B), 99–109. [Bicentennial History Issue, eds C.C. Heyde & E. Seneta].
Lancaster, H.O. (1990). Expectations of Life: A Study in the Demography Statistics and History of World Mortality. New York: Springer.
Lancaster, H.O. (1994). Quantitative Methods in Biological and Medical sciences. A Historical Essay. New York: Springer.
Lancaster, H.O. (1996). Some Recollections of Henry Oliver Lancaster, ed. R.L. Lancaster. (Privately printed, for University of Sydney Archives.) Sydney, xxiii + 42 pp.
Lancaster, H.O. (1997). The Statistical Society of New South Wales and the Statistical Society of Australia. Statistical Society of Australia Incorporated, Newsletter, No. 80, 31 August 1997, pp . 1–2. c Australian Statistical Publishing Association Inc. 2002
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Lancaster, H.O. & Seneta E. (1998). Chi-square-distribution. In Encyclopedia of Biostatistics, Vol. 1, eds P. Armitage & T. Colton, p . 608. Chichester: Wiley.
Mayr, E. (1982). The Growth of Biological Thought: Diversity, Evolution and Inheritance. Cambridge, Mass.: Belknap Press.
Phipps, M.C. (2000). Exact tests and the mid-P. In VIII International Scientific Kravchuk Conference, Institute of National Academy of Science of Ukraine, Kyiv, pp . 471–475.
Phipps, M.C. (2002). Inequalities between hypergeometric tails. Journal of Applied Mathematics and Decision Science (to appear).
Plackett, R.L. (1984). Discussion of paper by Yates. J. Roy. Statist. Soc. Ser. A 147, 426–463. Quine, M.P. (1993). On three characterizations of the normal distribution. Probab. Math. Statist. 14, 257–263. Quine, M.P. & Seneta, E. (1999). The generalization of the Kac–Bernstein theorem. Probability and Mathematical Statistics 19, 441–452.
Routledge, R.D. (1992). Resolving the conflict over Fisher’s exact test. Canad. J. Statist. 20, 201–209. Routledge, R.D. (1994). Practicing safe statistics with the mid-P. Canad. J. Statist. 22, 103–110. Seneta, E. (1979). Round the historical work on Bienaym´e. Austral. J. Statist. 21, 209–220. [No. 3. Issue in Honour of H.0. Lancaster, Founding Editor.]
Seneta, E. (2002). Henry Oliver Lancaster (1 February 1913–2 December 2001) Newsletter. International Statistical Institute 26, No. 1 (76), p . 7.
Seneta, E. & Phipps, M.C. (2001). On the comparison of two observed frequencies. Biometrical J. 43, 23–43. Seneta, E., Berry, G. & Macaskill, P. (1999). Adjustment to Lancaster’s Mid-P. Methodol. Comput. Appl. Probab. 1, 229–240.
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