Twelve British statisticians

Twelve British Statisticians BOSON BOOKS 1 Twelve British Statisticians WARNING THE MATERIALS AND WORKS MADE AVAIL...

Author: Richard H. Williams ... et al. .

153 downloads 898 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Twelve British Statisticians

BOSON BOOKS

1


WARNING THE MATERIALS AND WORKS MADE AVAILABLE BY C&M ONLINE MEDIA INC. THROUGH BOSON BOOKS ARE COPYRIGHTED. YOU ARE PERMITTED TO DOWNLOAD LOCALLY MATERIALS AND WORKS FROM BOSON BOOKS AND TO MAKE ONE (1) HARD COPY OF SUCH MATERIALS AND WORKS FOR YOUR PERSONAL USE. FEDERAL COPYRIGHT LAWS, HOWEVER, PROHIBIT ANY FURTHER COPYING OR REPRODUCTION OF SUCH MATERIALS AND WORKS, OR ANY REPUBLICATION OF ANY KIND. ILLEGAL COPYING OR DISTRIBUTION OF MATERIALS AND WORKS OBTAINED FROM BOSON BOOKS CONSTITUTES COPYRIGHT INFRINGEMENT. ILLEGAL COPIES OF SUCH MATERIAL AND WORKS CAN BE SEIZED AND DESTROYED. FURTHERMORE, MATERIALS AND WORKS CREATED BY YOU OR OTHERS USING COPYRIGHTED MATERIALS OBTAINED FROM BOSON BOOKS WITHOUT THE WRITTEN AUTHORIZATION OF C&M ONLINE MEDIA, INC. ALSO CAN BE SEIZED AND DESTROYED. COPYRIGHT INFRINGEMENT CAN BE INVESTIGATED BY THE FBI. CRIMINAL CONVICTIONS MAY RESULT IN IMPRISONMENT OR A FINE OF UP TO $250,000, OR BOTH. ______________________________

BOSON BOOKS

2


Published by Boson Books 3905 Meadow Field Lane Raleigh, NC 27606 ISBN 1-932482-44-X An imprint of C&M Online Media Inc.

© Copyright 2006 Richard H. Williams, Donald W. Zimmerman, Donald C. Ross, Bruno D. Zumbo All rights reserved For information contact C&M Online Media Inc. 3905 Meadow Field Lane Raleigh, NC 27606 Tel: (919) 233-8164 Fax: (919) 233-8578 e-mail:[email protected] URL: http://www.bosonbooks.com URL: http://www.bosonromances.com

BOSON BOOKS

3


___________________________________

TWELVE BRITISH STATISTICIANS by Richard H. Williams University of Miami Donald W. Zimmerman Carleton University Donald C. Ross New York State Psychiatric Institute Bruno D. Zumbo University of British Columbia _________________________________

BOSON BOOKS

4


TABLE OF CONTENTS ACKNOWLEDGEMENTS INTRODUCTION CHAPTER 1-Karl Pearson: Versatile Statistical Scientist Overview Educational Training and Fellowship Activities The Heart of Karl Pearson’s Versatility The Controversial Karl Pearson Concluding Comments References CHAPTER 2-R. A. Fisher: Contributions to Analysis of Variance and Experimental Design Overview Family Background and Early Vocational Choice Experimental Design Small Sample Theory and the Various Sampling Distributions Population Genetics, Evolution Theory, and Eugenics Controversies with Adversaries Sir Ronald A. Fisher’s Publications and Honorary Awards Concluding Comments References CHAPTER 3-Charles Spearman: Behavioral and Statistical Scientist Overview Educational Experience The Classical Theory of Mental Tests Factor Analysis A Two-Factor Theory of Intelligence Concluding Comments References CHAPTER 4-Florence Nightingale David: Pioneer Feminist Statistician Overview The Four Women Associated with Early History of Statistics David’s Initial Vocational Frustrations David’s Reactions to World Class Statisticians A Sample of David’s Publications On Entering Administration Nan Laird’s Interview of Florence N. David Concluding Comments References CHAPTER 5-George Udny Yule: Premier British Statistician BOSON BOOKS

5


Overview Family Background and Vocational Choice Is Normality Realistic? Professional Organizations Yule’s Textbook Yule’s Regression Approach Classical Mental Test Theory Yule’s Correlation Coefficients Yule’s Desire to Fly Statistical Aspects of Literary Style A Demonstration of Yule’s Research Versatility Concluding Comments References CHAPTER 6-Maurice Kendall: Synthesizer and Commentator of Statistics Overview Early Childhood and Educational Experience Kendall Meets Yule Random Number Generators and Tests for Randomness Administrative Positions and Honors Concluding Comments References CHAPTER 7-George Box: Pioneer in the Forecasting of Time Series Overview Educational Training Box’s Research Contributions Vocational Experiences Two Interviews with George Box Concluding Comments References CHAPTER 8-Student: Inventor of Small Sample Inferential Statistics Overview Student’s Background How Student Learned Statistics Student As A Man Student As A Statistician Concluding Comments References CHAPTER 9- Egon Sharpe Pearson: Contributions to Statistical Inference Overview Early Years and Educational Training Egon Pearson As Teacher and Administrator Reviews of R. A. Fisher’s Statistical Methods for Research Workers BOSON BOOKS

6


Neyman-Pearson Theory of Hypothesis Testing Egon Pearson’s 30 Year Service As Editor of Biometrika Publications In Honor of His Father Concluding Comments References CHAPTER 10- Francis Ysidro Edgeworth: Statistician and Economist Overview Educational Training and Vocational Experiences Francis Bacon and Francis Ysidro Edgeworth Francis Ysidro Edgeworth’s Research and Publications Concluding Comments References CHAPTER 11-Maurice Bartlett: Time Series and Multivariate Statistics Overview Educational Training Vocational Experience Research Contributions Concluding Comments References CHAPTER 12-David Cox: Multivariable Analysis of Failure Rates Overview Educational Background Vocational Experience Journal and Book Publications Prizes, Medals, Awards, Memberships, Fellowships, and Knighthood Concluding Comments References SUMMARY

BOSON BOOKS

7


ACKNOWLEDGMENTS We are grateful to the Regents of the University of California for permission to reproduce the picture of Egon Sharpe Pearson from the frontispiece of The Selected Papers of E. S. Pearson (1966). Thanks to Springer Science and Business Media for permission to reprint three excerpts from Constance Reid’s Neyman…from Life (1982). They appear on pages 24-25, 115, and 133 of her book. We are indebted to John Wiley & Sons, Inc. for permission to reproduce the picture of R. A. Fisher from the set of plates following page 256 in Joan Fisher Box’s R. A. Fisher: The Life of A Scientist (1978). Thanks also go to this publishing house for permission to reprint two quotations from her book on pages 12-13 and 61. Thanks to The Annals of Human Genetics for permission to reprint the picture of William Sealy Gosset from volume 9, part 1 of the frontispiece of the 1939 Annals of Eugenics. The Institute of Mathematical Statistics and Statistical Science were quite helpful in giving permission to reproduce the pictures of George E. P. Box and Florence Nightingale David, the former in volume 2, page 240 (1987), and the latter in volume 4, page 236 (1989) of Statistical Science. Blackwell Publishing gave us permission to reproduce pictures of the following three British statisticians: Sir David Roxbee Cox-His picture appeared as the frontispiece in the Journal of the Royal Statistical Society, Series A, 144(3), 1980. Maurice Stephenson Bartlett-His picture appeared as the frontispiece in the Journal of the Royal Statistical Society, Series A, 130(4), 1967. Francis Ysidro Edgeworth-His picture is from the facing page 238 in Annals of the Royal Statistical Society, 1934-1934, 1934 ESS sup 54-55.

BOSON BOOKS

8


INTRODUCTION This slim volume provides a brief description of the lives and scientific contributions of twelve British Statisticians—Karl Pearson, R. A. Fisher, Charles Spearman, Florence Nightingale David, George Udny Yule, Maurice G. Kendall, George E. P. Box, William Sealy Gosset (“Student”), Egon Sharpe Pearson, Francis Ysidro Edgeworth, Maurice Stevenson Bartlett, and David Cox. The book is written in a non-mathematical manner to make it accessible to a wider reading audience. The Pearson product-moment correlation coefficient is named in honor of Karl Pearson. In addition to studying mathematics and German he took a law degree. He and his wife co-founded the Men and Women Club that was designed to permit free discussion among men and women. His 18 publications in Philosophical Transactions of the Royal Society provided contributions in theoretical as well as substantive areas. Some scientists view Fisher’s work in population genetics and evolutionary theory as being as important as his innovations in statistics. R. A. Fisher introduced pioneering work in the analysis of variance and covariance. These inferential statistical techniques are central tools in much empirical, scientific work. He also laid the groundwork for multivariate statistical analysis. Fisher derived important sampling distributions for correlations and regression coefficients. His publications swept across many substantive disciplines. His work in experimental design, including the idea of randomization, is monumental. It should be mentioned that Fisher was a controversial man and sometimes clashed with fellow scientists. Charles Spearman developed fundamental models in factor analysis, mental test theory, and theories of intelligence. His two-factor theory contained a general factor of intelligence together with certain specific factors. This theory of the intellect is still used as the basis for hierarchical models. His achievements were based on both statistical procedures and empirical psychological studies. Florence Nightingale David’s contributions opened the door to women desiring a career in Statistics, and she was justly rewarded for her achievements. She generated 100 publications. One that is likely to interest both beginners and sophisticates is her 1962 book, Games, Gods, and Gambling. Students flocked to Dr. David’s popular lectures. George Udny Yule conducted early work in time series and contingency table analysis. He is viewed as one of the pioneers of modern statistics. The highlight of his publications is probably the book, An Introduction to the Theory of Statistics, which went through 14 editions and appeared in five languages. Yule also investigated statistical aspects of literary style. Legend has it that he told Spearman the fundamental equation of mental test theory, X=T+E. Maurice G. Kendall made contributions to nonparametric correlational techniques, random walk theory, random number generators and tests for randomness. He published many books, some of them highly respected. A sample of these volumes is Advanced Theory of Statistics (two volumes), Geometrical Probability, and Time Series.

BOSON BOOKS

9


George E. P. Box has published numerous academic and industrial journal articles as well as a good number of books and teaching materials. He holds both Ph.D. and D.Sc. degrees in mathematical statistics from the University of London. Dr. Box has made important research contributions to statistical quality control, Bayesian statistics, and analysis of time series. William Sealy Gosset (“Student”) developed the Student’s t-test and related small sample statistics while working for the brewery of Arthur Guinness Sons & Company of Dublin. This was a major breakthrough as until then the only statistical techniques available were those applicable to rather large sets of data. At the request of the brewery, he published all of his papers, many of which appeared in Biometrika, under the pseudonym “Student.” In his later publications he tried to omit mathematical concepts and he would ask colleagues and friends to view him as a non-mathematician. Egon Sharpe Pearson was a versatile statistical scientist. Since his father was Karl Pearson, he had an impressive tradition to carry on. Egon was a leading statistician. And he did important editorial work as Managing Editor of Biometrika for three decades. His research in developing the Neyman-Pearson theory of hypothesis testing was important to statistical inference. When his father retired he took over as Chairman of the Department of Applied Statistics at University College, London. He was a noted historian of statistics as well as a skilled teacher. Francis Ysidro Edgeworth possessed a mind with great intellect. Although his formal education was in modern and ancient languages, as well as law, he learned a great deal of mathematics, statistics, probability, logic and related disciplines on his own. There is a great deal which is puzzling as to how he did this. Sometimes there were friends or colleagues who stimulated him, such as Francis Galton or Karl Pearson. He never married and might have been lonely, except for the intellectual stimuli that surrounded him. Vocationally he was very stable. He was appointed to the Drummond chair of Political Economy at Oxford and was chosen as a fellow of All Souls in 1891, and retained both until his retirement. Similarly, he became the initial editor of the Economic Journal in the same year and remained in an editorial capacity until retirement. Sometimes he had controversies regarding priority of research findings, especially with Karl Pearson. Maurice Bartlett, like several of the other British statisticians in this book, was very good in mathematics. He lived a long and productive life, excelling in teaching, research, and administration. His most well known statistical finding was the test for homogeneity of variance in the analysis of variance. He was elected to many fellowships, medals, foreign associateships and honorary memberships, and the University of Chicago and the University of Hull awarded him honorary doctorates. His research contributions spanned 55 years. Bartlett’s research interests in statistics and related disciplines covered a wide range of topics including statistical theory and methods, population and epidemic models, stochastic processes, time series analysis, and multivariate analysis. David Cox’s research interests are in inferential theoretical and applied statistics, biostatistics, and applied probability. The applications that he especially focuses on are in the physical, biological, and social sciences. His best-known research contribution is the

BOSON BOOKS

10


proportional hazards model, which has numerous applications in the analysis of survival data. Both the references and the mathematical notation in this book follow the manual of style of the American Psychological Association.

BOSON BOOKS

11


CHAPTER 1 Karl Pearson: Versatile Statistical Scientist Overview: This chapter displays the versatility of Karl Pearson, focusing not only on his contributions to statistics and other quantitative disciplines but also on his research and publications in religion, politics, literary criticism, philosophy of science, Darwinism, biology, history, free thought, evolution, genetics, socialism, anthropology, eugenics, and emancipation of women. Being the chairman of a first class academic department and the managing editor of a major journal, Pearson sometimes used his power to the detriment of other important scientists, such as R. A. Fisher and Jerzy Neyman, and this chapter also brings out this unfortunate characteristic of his personality. Educational Training and Fellowship Activities In the year 2006 Karl Pearson is best remembered for his research in statistics, where he developed the formula for the product-moment correlation coefficient (ρ) and introduced the chi square (χ2) statistic. Occasionally it is recalled that he took a degree in law and wrote a book on the philosophy of science titled The Grammar of Science (1892). The last two items, taken together with his statistical contributions, might give one a glimpse of his intellectual versatility. The main purpose of the present chapter is to display this versatility and elaborate on it in a fully comprehensive way. A secondary purpose is to reveal an unfortunate feature of Pearson’s personality—viz., that he used his professorial and administrative potency to advocate his own research ideas, inhibit those he did not approve of, and insult and intimidate statistical scientists from different philosophical camps. Karl Pearson was born in London on March 27, 1857 and he died at Coldharbour, Surrey on April 27, 1936. His father was a successful Barrister and this may have led him to attain a law degree. Karl was educated at home until he turned nine, and then studied at University College, London for seven years. In his early years he displayed a wide array of interests and a strong intellectual thrust. He frowned upon authority figures, which he viewed as being misdirected, and took delight in argumentation. He graduated Third Wrangler in Mathematics Tripos in 1879 from Kings College, Cambridge and was awarded a flexible fellowship. It extended six years and gave him great freedom regarding scholarly work and travel. Karl Pearson studied law in London and was called to the bar in 1881. In 1882 he took a master’s degree. During his Fellowship years he studied at the Universities of Berlin and Heidelberg, spending a great deal of time in the Black Forest, studying and writing. Germany was always one of his favorite countries. The disciplines he focused on were philosophy, biology, metaphysics, Roman law, mediaeval and renaissance German literature, physics, and Darwinism. After a period of serious study he became fluent in German and well versed in German literature. Karl Pearson was indeed a Germanophile.

BOSON BOOKS

12


Legend has it that he was offered a college post in a German Department, but there is some debate on this point. On returning to England he became engaged in activities that involved the ideas of Karl Marx, Spinoza, and Martin Luther. He also returned to King’s College where he wrote papers on certain mathematical topics. Karl Pearson was appointed Professor of Applied Mathematics and Mechanics at University College, London in 1884 and this college was to be his home base for the rest of his life. He became department head in Applied Mathematics in 1907. For four years he lectured in Geometry. He conducted research in biometry and evolutionary theory from 1891 to 1906 with Rafael Weldon, Professor of Zoology and Comparative Anatomy. As requested by Francis Galton, who was a polymath, Pearson became the first Galton Professor in Eugenics in 1911 and retained that title until his retirement in 1933. Karl Pearson and Maria Sharpe were married in 1890 and they had two daughters, Sigrid and Helga, and one son, Egon. The couple met at the Men and Women Club, which was co-founded by Pearson, and designed to permit free discussion among men and women. Maria died in 1928 and the following year he married Margaret Child, a colleague at University College. The Heart of Karl Pearson’s Intellectual Versatility By perusing Karl Pearson’s academic studies in London and in Germany, one can sense the broad scope in his interest patterns. We now look more closely at his writings, his research, and his publications. He published 18 papers with the title “Mathematical Contributions to the Theory of Evolution,” all of them with different subtitles. They were all published in Philosophical Transactions of the Royal Society. R. A. Fisher also made strong contributions to evolution theory. The foundation of the system of Pearson Curves appears in these publications. Suspicious of the assumption of normality, he explored a number of other possible frequency distributions. Substantively, his publications dealt with problems in genetics, anthropology, biology, eugenics, evolution, etc. Statistically, they provided theoretical contributions to kurtosis, the coefficient of contingency, the correlation ratio, tetrachoric and biserial correlations, product moment correlation (ρ), chi square (χ2), the normal curve and other distributions, multiple correlation and regression, partial correlation, scedasticity, the coefficient of variation, and the standard deviation. He coined this latter term and employed the Greek letter σ to denote its population parameter. His equation for the biserial is the very one that is currently used (1909). Pearson contended that he was the one who named the LaPlace-Gaussian curve the “normal curve,” which he later viewed as unfortunate because it seemed to imply that all other curves were abnormal (Pearson, 1920, p. 25). In Pearson’s first fundamental paper on correlation, “Mathematical Contributions to the Theory of Evolution: III. Regression, Heredity, and Panmixia” (1896), he derived the formula which we now call the “Pearson product-moment correlation coefficient,” although Francis Galton was first to conceptualize it. And he gave the coefficients of the multiple regression equation in terms of the zero order correlation coefficients. Karl Pearson produced over 300 publications in his career. Although he attended especially to studies of heredity and evolution he also conducted research in areas such as

BOSON BOOKS

13


alcoholism, mental deficiency, mental illness, and tuberculosis, as well as meteorology and astronomy. (Lord, 1995) Books edited or authored by Karl Pearson also display his scholarly versatility. In the time period between 1922 and 1934 he edited volumes of the incomplete beta and gamma functions and tables for statisticians and biometricians. Francis Galton was Karl’s mentor and his best friend. When Galton died, he began research on a four volume biography of Galton titled, The Life, Letters, and Labours of Francis Galton which took him many years to write (1914-1930). Karl Pearson also wrote an obituary of Galton which was published in Nature (1911). Pearson produced two volumes, which were quite different from his purely statistical works. They are philosophical or sociological in nature and are titled: The Chances of Death and Other Studies in Evolution (two volumes, 1897) and The Ethic of Free Thought and Other Addresses and Essays (1901). Many of the sections in these two books are from a lecture or an essay or a reprint. The Grammar of Science (1892) is perhaps Karl Pearson’s magnum opus. Vladimir Lenin reacted to this book by saying that it showed integrity and consistency and displayed great precision of expression. (Lenin, 1939) It should be mentioned that Lenin disagreed with some of Pearson’s ideas. In 1915 Jerzy Neyman, who was to become a world-class statistician, read this pilosophy of science book of Karl Pearson’s. He grew very exited about the book and his wife, Olga, said that Karl Pearson was constantly on his mind. He shared the book with fellow students and they too were strongly impressed. Years later Neyman recalled that: “We were a group of young men who had lost our belief in orthodox religion, not from any sort of reasoning, but because of the stupidity of our priests. [But] we were not freed from dogmatism and were prepared in fact to believe in authority, so long as it was not religious. “The reading of The Grammar of Science…was striking because…it attacked in an uncompromising manner all sorts of authorities…At the first reading it was this aspect that struck us. What could it mean? We had been unused to this tone in any scientific book. Was the work ‘de le plague’ [something of a hoax] and the author a ‘canaille’ [scoundrel] on a grand scale…? But our teacher, Bernstein, had recommended the book; we must read it again.” (Reid, 1982, pp. 24-25). It is of interest to note that when the 23year-old Albert Einstein formed a study group called the Olympia Academy with his two younger friends, he recommended that the first book they read was Karl Pearson's The Grammar of Science. George Udny Yule described Karl Pearson as a philosopher, poet, and historian as well as a statistician. (Lord, 1995). Although Yule held a position as demonstrator awarded by Pearson, they differed on certain statistical matters. Years later Neyman would initiate a research project with Karl Pearson’s son, Egon, which would lead to the important Neyman-Pearson Theory of Hypothesis Testing. In October of 1884, Karl Pearson was asked to edit and provide a preface to the incomplete manuscript, The Common Sense of the Exact Sciences. The author, William BOSON BOOKS

14


Kingdon Clifford, had died prematurely and some additional sections had to be added to complete the work. Pearson successfully brought the work to conclusion. It was reissued in 1946 and Bertrand Russell, who praised both Clifford’s original production and Pearson’s editorial work, wrote a new preface. He said the book diminished his bewilderment concerning geometries that differed radically from that of Euclid. Just before the turn of the nineteenth century the Syndics of the University Press requested that Karl Pearson edit and complete the unfinished manuscript, A History of the Theory of Elasticity and of the Strength of Materials from Galilea to Lord Kelvin (two volumes). The deceased author, Isaac Todhunter, had left volume two incomplete, and a great deal of editorial work was needed. Each volume was a thousand pages in length. Pearson successfully completed the task in 1893. Karl Pearson was co-founder, with Francis Galton and Walter Weldon, of the journal Biometrika in 1900. Pearson was its editor from its inception until he died in 1936. His son, Egon Sharpe Pearson, who was a well-known statistician in his own right, took over the editorship at that time and mentored it until almost 1966. This journal publishes studies of the statistical study of Biology. They formed it because the Royal Society refused to publish papers that combined Biology with Mathematics. The reader desiring additional early biographical information on Karl Pearson cannot do better than Egon Sharpe Pearson’s Karl Pearson: An Appreciation of Some Aspects of His Life and Work (1938). Egon’s work, however, is uniformly positive, which, perhaps, is to be expected. The Controversial Karl Pearson As the Head of the Department of Applied Statistics at University College, London and the Managing Editor of the prestigious journal Biometrika, Karl Pearson held a powerful grip on statistical developments in England and elsewhere for many years. His department combined the Biometric Laboratory and the Galton Laboratory. It was not only a matter of his own impressive research developments but also a control of some of the potential research avenues pursued by others, especially by faculty members in his department (Haldane, 1957). When his son, Egon, joined the department in 1921 as a lecturer, Karl did not permit him to teach. Instead he was obliged to attend his father’s lectures. Although Egon produced a great deal of research at that time Karl Pearson chose all of the topics for him. In fact, the elder Pearson dictated who taught what and when for all faculty members in his department and he chose their research pursuits for them. R. A. Fisher was much younger than Karl Pearson. But Pearson recognized Fisher’s talent and tried to hire him. Although the young man was interested in many of the subjects explored under Pearson’s domain, he was aware of the fact that if he accepted the position, Karl Pearson would completely control his teaching load and his research activities, so he turned down the offer. Fisher instead accepted a temporary research position at Rothamsted Experimental Station (Box, 1978, p. 61) Fisher discovered the sampling distribution of the correlation coefficient, which was a very important finding and submitted the paper to Karl Pearson for possible publication in Biometrika. Pearson rejected it and a second paper by Fisher contending that they were controversial. Fisher vowed that he would never again submit a paper to Pearson’s journal. And he never did! (Pearson, 1968)

BOSON BOOKS

15


At one time Karl Pearson had a debate with Jerzy Newman in which Jerzy was correct but Karl refused to back down. Neyman described Pearson as being hostile and skeptical, although Yule said Karl Pearson only became negative when discussing intellectual matters. Major Darwin, a son of Charles Darwin, said that Pearson would never permit a research paper to be published which contained information contrary to his own beliefs. The degree to which Pearson appeared to be a domineering personality was apparent to many, both his students and reputable statisticians, and his combative behaviors appeared even back when he was an undergraduate student. For a further discussion of some of the controversial actions of Karl Pearson, see Theodore M. Porter’s Karl Pearson: The Scientific Life in A Statistical Age (2004, pp. 266-267, 272-273). The odd thing was that when Jerzy Neyman and his wife, Olga, visited the Pearsons at their country home, the guests were delighted with the extreme courtesy, kindness, and politeness with which they were treated. Olga was especially impressed. The elder Pearson joined the younger in welcoming Neyman and his wife. “…Karl Pearson and Egon Pearson and what was the name of Egon’s sister [Sigrid]—oh, they all were wonderful people!” Olga Neyman recalls with enthusiasm. “This Karl Pearson invited us for several days to his country place. That was very shortly after we came. They were extremely nice to us, these Pearsons.’ (In this case it is not difficult to reconcile Mrs. Neyman’s memories of gracious hospitality on the part of the Pearsons with Neyman’s description of him as ‘hostile and skeptical’; for, as George Udny Yule wrote of Karl Pearson, the temper in controversy was the more remarkable because there was no such temper in relation to anything but matters intellectual.) “What was Karl Pearson like,” [Constance Reid] asked Mrs. Neyman. “Oh, he was wonderful! Like some old English duke, you know. Oh, a wonderful face! Just pure blood English face, you know. He was very tall, extremely beautiful old man…Just gentleman, very polite and nice. Oh wonderful, wonderful!” “Was Egon as impressive looking as his father?” “No, no, no, no! But he was awfully nice.” (Reid, 1982. p. 115) Concluding Comments There is no doubt that Karl Pearson’s interests and abilities extended beyond statistics, although his statistical findings such as chi square, the Pearson product-moment correlation coefficient, and the system of Pearson curves and many other quantitative research findings are indeed impressive. Karl’s editorship as founder of two journals and chairman of a pioneering statistics department displayed other skills. The sheer versatility in areas such as philosophy of science, languages, politics, law, physics, Darwinism, biometry, eugenics, evolution, and genetics are quite a display of aptitude, research, motivation, and scholarly output. (Williams, Zumbo, Ross, & Zimmerman, 2003) References Box, J.F. (1978). R. A. Fisher: The life of a scientist. New York: Wiley. BOSON BOOKS

16


Clifford, W.K. (1946). The common sense of the exact sciences, with prefaces by Karl Pearson and Bertrand Russell. New York: Knopf. Eisenhart, C. (1974). Karl Pearson. In C.C. Gillispie (Ed.). Dictionary of Scientific Biography (pp. 447-473). New York: Charles Scribners and Sons. Goodman, L.A. (2002). Contributions to the statistical analysis of contingency tables: Notes on quasi-symmetry, quasi-independence, log-linear models, log-bilinear models, and correspondence analysis models. Annales de la Faculte des Sciences de Toulouse, 11(4), 525-540. Haldane, J.B.S. (1957). Karl Pearson, 1857-1957. A centenary lecture delivered at University College, London. Biometrika, 44, 303-313. Lenin, V.I. (1939). Selected works, Vol. 11, London: Lawrence Wishart. Lord, C.D. (1995). Karl Pearson, English statistician. In E.J. McMurray (Ed.). Notable Twentieth Century Scientists (pp. 1556-1558). New York: Gale Research, Inc. Pearson, E. S. (1938). Karl Pearson: An appreciation of some aspects of his life and works. London: Cambridge University Press. Pearson, E. S. (1968). Some early correspondence between W.S. Gosset, R. A. Fisher, and Karl Pearson, with notes and comments. Biometrika, 55, 445-457. Pearson, K. (1892). The grammar of science. London: J.M. Dent and Company. Pearson, K. (1896). Mathematical contributions to the theory of evolution: III. Regression, heredity, and panmixia. Philosophical Transactions of the Royal Society, 187, 253-318. Pearson, K. (1897). The chances of death and other studies in evolution, 2 vols., New York: Edward Arnold. Pearson, K. (1901). The ethic of free thought and other addresses and essays. London: Adam and Charles Black. Pearson, K. (1909). On a new method of determining a correlation between a measured character of A and a character of B, of which only the percentage of cases wherein B exceeds (or falls short of) intensity is recorded for each grade of A. Biometrika, 7, 96105. Pearson, K. (1911). Francis Galton. Nature, 85, 440-445. Pearson, K. (1914-1930). The life, letters, and labours of Francis Galton, 4 vols., London: Cambridge University Press. Pearson K. (1920). Notes on the history of correlation. Biometrika, 13, 25-45. Porter, T.M. (2004). Karl Pearson: The scientific life in a statistical age. Princeton, NJ: Princeton University Press. Reid, C. (1982). Neyman…from life. New York: Springer-Verlag. Todhunter, I. (1890, 1893). A history of the theory of elasticity and of the strength of materials from Galilei to Lord Kelvin, 2 vols. edited and compiled by Karl Pearson. London: Cambridge University Press. BOSON BOOKS

17


Walker, H. (1968). Karl Pearson. In W.H. Kruskal & J.M. Tanur (Eds.), International Encyclopedia of Statistics (pp. 691-698). New York: Collier and MacMillan. Williams, R.H., Zumbo, B.D., Ross, D., & Zimmerman, D.W. (2003) On the intellectual versatility of Karl Pearson. Human Nature Review, 3, 296-301.

BOSON BOOKS

18


CHAPTER 2 R. A. Fisher: Contributions to Analysis of Variance and Experimental Design Overview: This chapter provides a brief description of Sir Ronald A. Fisher’s scientific contributions and some details of his personal life. His research findings were published in books, monographs, and journal articles. They dealt with work in the substantive disciplines of genetics, biometry, agriculture, evolution and eugenics. Included are his pioneering statistical contributions in the analysis of variance and covariance and multivariate statistics, as well as the following sampling distributions: productmoment correlation coefficient; regression coefficient; partial correlation; multiple correlation. Some other findings were the Fisher Exact Test, maximum likelihood estimation, intraclass correlation coefficient, analysis of cross-classification data, Fisher Information, fiducial probability, multiple comparisons, randomization, replication, genetic findings on natural selection, and the cause of evolutionary change. Fisher provided a unified treatment of many of the important distributions involved in testing hypotheses. His controversies involving other important scientists are displayed in this chapter. The purpose of this chapter is to summarize the scientific contributions of Sir Ronald Aylmer Fisher, focusing especially on his work in statistics and experimental design. Connections between Fisher’s innovative findings in statistics and his personality are made, revealing his many controversies with other luminaries such as Karl Pearson, Jerzy Neyman, and Harold Jeffreys. Family Background and Early Vocational Choice R. A. Fisher was born on February 17, 1890 in London and he died on July 29, 1962 in Adelaide, South Australia. He completed his secondary education at Harrow School in 1909, was awarded a scholarship to the University of Cambridge, and took his B.A. degree in mathematics with honors in 1912. An additional grant was awarded to Fisher in 1913 and he studied the theory of errors, statistical mechanics, and quantum theory. In 1917 he married Ruth Eileen Guinness. They had eight children—six girls and two boys. Joan, who was the second oldest of the girls, married the famous statistician George Box, and she wrote a well-received biography of her father titled R. A. Fisher: The Life of A Scientist (1978). In 1920 R. A. Fisher took the M.A. degree.

BOSON BOOKS

19


Fisher was deferred from military service during World War I because of his deficient eyesight. From 1915 to 1919 he was a public school teacher. During the two decades from 1920 to 1940, he demonstrated remarkable research productivity, publishing three important books, a collection of tables, and over 120 other titles. (Kendall, 1963, pp.1-2). Vocationally, he worked as a statistician at Rothamsted Experimental Station from 1919 to 1933. “Within a few days [of the job offer from Rothamsted] Fisher also received an offer from Professor Pearson at the Galton Laboratory. Fisher’s interests had always been in the very subjects that were of interest at the Galton Laboratory, and for 5 years he had been in communication with Pearson, yet during those years he had been rather consistently snubbed. Now Pearson made him an offer on terms which would constrain him to teach and to publish only what Pearson himself approved. It seems that the lover was at last to be admitted to his lady’s court—on condition that he first submit to castration. Fisher rejected the security and prestige of a post at the Galton Laboratory and took up the temporary job as sole statistician in a small agricultural research station in the country.” (Box, 1978, p. 61) Karl Pearson retired in 1933, and Fisher succeeded him as Galton Professor of Eugenics at University College, London. In 1933 Fisher also became editor of The Annals of Eugenics. In 1943 he moved on to the University of Cambridge, where he was named the Arthur Balfour Professor of Genetics and remained there until 1957. Fisher founded the international journal Heredity in 1947 and co-edited it until his death in 1962. In 1952 the Queen of England knighted him. From 1959 to 1962, R. A. Fisher was Research Fellow, Division of Mathematical Statistics at the University of Adelaide in South Australia. Young Ron Fisher differed from other children in two important ways and both of them could have influenced the way he learned: (1) His eyesight was deficient; (2) He was precocious. The following quotation from Joan Fisher Box’s biography of her father (1978, pp. 1213) displays the precocious behavior of the youngster. “At about [the age of three] when he had been set up in his high chair for breakfast, he asked: ‘What is a half of a half?’ His nurse answered that it was a quarter. After a pause he asked, ‘And what’s a half of a quarter?’ She told him that it was an eighth. There was a longer pause before he asked again, ‘What’s a half of an eighth, Nurse?’ When she had given her reply there was a long silence. Finally, Ronnie looked up, a plump pink and white face framed with red-gold hair, and said slowly, then I suppose that a half of a sixteenth must be a thirty-toof.’” We now turn to the problem of Fisher’s eyesight, which was detected later on. During his adult years, when he was a well-trained sophisticated scientist in solving statistical problems, he made frequent use of viewing n observations in n-dimensional space. That is, he found geometrical solutions to statistical problems more fruitful than algebraic approaches. Some writers link this geometric ability with the fact that Fisher had poor eyesight, almost since birth. If so, it is as though he had transformed a deficit into an advantage. Geometric approaches might be especially advantageous in studying multivariate statistical models. Experimental Design BOSON BOOKS

20


R. A. Fisher’s research in experimental design, which included the notion of randomization, is considered to be among the most important of his contributions to science. As well as independent groups factorial designs, he invented experimental designs where restrictions on randomization are imposed, but random processes are retained. Two examples are the randomized blocks design and the Latin square. These designs ordinarily have greater power or sensitivity than designs with independent groups. In the language that is currently used, randomization supports the internal validity of an experiment, whereas random selection protects against threats to external validity, or generalizability. His work on design is summarized in the book, Design of Experiments (1960). Randomization, replication, and blocking are the fundamental principles of experimental design introduced by R. A. Fisher. Replication is the main source of the estimate of error, while randomization insures that the estimate will be unbiased. Blocking increases precision. Many statistical scientists viewed Fisher’s technique as a revolution. It especially modified research in an agricultural setting. There is controversy regarding the various types of multiple comparisons. Yet many statisticians treat Fisher’s Least Significant Difference (LSD) with respect (Puberty & Morris, 1988). It is a simple procedure. The omnibus F-test is computed first, using a specified α level. If it turns out to be statistically significant, pairwise tests are employed using the same α level. If not, analysis is terminated. Small Sample Theory and the Various Sampling Distributions R. A. Fisher created analysis of variance and analysis of covariance. In fact, George W. Snedecor, who at the time was at Iowa State College, named the F-ratio in honor of Fisher. These two techniques are very widely used in analyzing results in numerous research studies in psychology, sociology biology, agriculture, education, medicine, business, genetics, and many other disciplines. R. A. Fisher helped clarify the vocabulary and notation of some areas of statistical inference. Fisher stressed the idea that one must carefully distinguish between population parameters and sample statistics. He introduced the following features of statistical estimators—sufficiency, consistency, efficiency, and maximum likelihood estimator—in 1925 in his publication titled, “Theory of Statistical Estimation,” which appeared in the Proceedings of the Cambridge Philosophical Society. Also “Student” and Fisher made important inroads into exact small sample distributions, although R. A. Fisher ignored the assumption of normality. (see the summary of reviews of his book, Statistical Methods for Research Workers (1925), which appear in chapter 9 of this book.) Fisher introduced the idea of statistical information. The general notion is that there is a certain amount of information in a sample. And in reducing the data one must minimize the amount of information lost. (Box, 1978) R. A. Fisher created the randomization tests. There are two of them—one for two independent samples and the other for paired observations. These are considered to be nonparametric inferential statistical tests, and, assuming normality, their asymptotic relative efficiencies (ARE), relative to their respective t-tests, are equal to 1.00. Except for very small samples these two tests cannot be computed by hand, and some of the BOSON BOOKS

21


computer packages do not contain routines for these statistics. More generally, he did much original work in nonparametrics, including exact tests in general, tests of runs, various order statistics, the sign test, and normal scores test (Savage, 1976). Fisher corrected the segment of Karl Pearson’s findings regarding degrees of freedom, although Pearson never acknowledged the fact that Fisher’s modification was correct. Fisher introduced other properties of contingency table analysis including the Fisher Exact Test, which yields an exact probability for a 2X2 table. This test is useful because tables of this form are ubiquitous since researchers often deal with dichotomous variables. The user can enter published tables and read out the appropriate probabilities. Leo Goodman (1984) initiated one of his lectures by honoring R. A. Fisher’s contributions to analysis of cross-classification data. Although Karl Pearson originally developed the intraclass correlation, Fisher provided a mathematical link between it and the F-ratio of analysis of variance. Lee Cronbach’s coefficient of generalizability (1972), which is the ratio of universe score variance to expected observed score variance is an intraclass correlation coefficient. And all of the following are intraclass correlations for certain designs: Kuder-Richard Formulas 20 an 21; Cronbach’s coefficient alpha; Hoyt’s formula; and Rulon’s split-half formula. Even the general formula stating reliability as the ratio of true score variance to observed score variance can be regarded as an intraclass correlation coefficient. R. A. Fisher gave a rigorous proof of “Student’s” result for the t-statistic, showing how it could be used to test various statistical hypotheses, and hence gave a unified treatment of many of the important distributions involved in testing (null) hypotheses. He also generalized Student’s result to the case of unequal variances and unequal sample sizes. Further studies by Behrens and others addressed the same topic, which today is referred to as “the Fisher-Behrens problem.” A study by Zimmerman (1996) revealed that the same problem arises in nonparametric statistics. When scores in two groups with unequal variances are combined into a single group and ranked, as done in several nonparametric significance tests, such as the Wilcoxon-Mann-Whitney test, it turns out that the two corresponding groups of ranks also have unequal variances. In producing the sampling distribution of the product-moment correlation, Fisher developed the r to Z transformation, which is in reality the inverse hyperbolic tangent function. One category of application of this transformation is to test hypotheses such as ρ = .80 where it is necessary to deal with a negatively skewed sampling distribution. Another category of application is to test ρ1 = ρ2 = … = ρk. For both of these cases research workers assume the samples are independent. But Fisher also provided the mathematics for the nonindependent case. Sir Ronald A. Fisher proved certain properties of discriminant function analysis, a multivariate statistical technique, and, for the two-group case, he showed a mathematical link between it and multiple regression. He made a number of other contributions to multivariate statistics. For further details see T.W. Anderson’s (1996) “R. A. Fisher and Multivariate Analysis,” which was published in Statistical Sciences. Fisher introduced the idea of fiducial probability. Here one wants an interval rather than a point estimate. Today the theory of confidence intervals of Neyman and Pearson is advocated by most statisticians and Fisher’s fiducial probability method is somewhat BOSON BOOKS

22


neglected. The Neyman-Pearson approach is usually more powerful, which means it produces more narrow confidence intervals. It should be mentioned that some research workers advocate a strategy for testing hypotheses that contains elements of both philosophical camps. Fisher proved certain properties of the maximum likelihood estimator and used them widely in his research. In employing the maximum likelihood estimator, one selects as an estimator of a parameter that value which will maximize the likelihood of the sample that is actually observed to occur. Early on he had no use for Bayesian estimators, but later, partly due to the adversary who became his friend, the luminary Harold Jeffreys, he reversed his position. The reader interested in Fisher’s philosophy of science should consult Nancy BrennerGolomb’s chapter entitled “Fisher’s Philosophical Approach to Inductive Inference,” in Keren and Lewis’s (Ed.) A Handbook for Data Analysis in the Behavioral Sciences: Methodological Issues. (1993) Population Genetics, Evolutionary Theory, and Eugenics Some writers contend that R. A. Fisher’s investigations into Mendelian Genetics are just as important as his contributions to theoretical and applied statistics. Fisher developed a mathematical theory on the basis of extant genetic research to establish the principle of natural selection on a more rigorous basis than Charles Darwin had claimed as the cause of evolutionary change. This work and related matters are presented in his book, The Genetical Theory of Natural Selection (1930). See also Bartlett’s (1968) write up of R. A. Fisher, which was published in the International Encyclopedia of the Social Sciences. Eugenics is the study of possible improvements in a race or breed, often focusing on human beings. Fisher served as honorary secretary of the Eugenics Society and he wrote a great number of reviews for the Eugenics Review. This work was done at the request of Leonard Darwin, one of Charles Darwin’s sons. Controversies With Adversaries Initially Harold Jeffries and R. A. Fisher were adversaries as they presumably represented probabilistic camps at opposite poles. Jeffreys was the most important Bayesian since the Reverend Thomas Bayes. Fisher mounted a frontal attack by uttering a potent negative statement in regard to Jeffreys’ book, Theory of Probability (1939). He said the book contained 395 invalid formulas and that this was due to the fact that page 1 contained Bayes’ postulate. Later they both moved somewhat in the direction of the philosophical position of the other, and then eventually became friends. In fact, one day they both attended a lecture by Arthur Eddington on scientific inference and were so disgusted with the lecture that they shook hands and promised to refrain from insulting each other. Two medical doctors, Raymond Doll and Bradford Hill, published a paper entitled “Smoking and Carcinoma of the Lungs: Preliminary Report” in 1950 in the British Medical Journal. Their findings, which were based on non-experimental (i.e., observational) studies, seemed to imply that smoking causes lung cancer. They attempted to make their field observations resemble an experiment by controlling for certain BOSON BOOKS

23


variables such as gender, age, and general physical health. Fisher attacked them in print and in speeches, indicating that only (true) experimental designs are capable of establishing empirical causality. R. A. Fisher’s arguments on these issues might have been more acceptable to the medical profession if it hadn’t been for the fact that at the time he was serving as a consultant to a tobacco firm. Related to the issue being debated, Jerome Cornfield (1951) introduced the notion of defensible case-control studies. Fisher treated his followers with great respect, but he was negative to the point of outrageous hostility to those who crossed him scientifically or philosophically. His venom for the latter group was reinforced by his eloquent barbed phrases. He was a charming fellow in relation to his disciples but was viciously hostile toward his antagonists, sometimes attacking them in an unfair fashion. (Bartlett, 1968). We next turn to Fisher’s relation with his two mortal enemies, Karl Pearson and Jerzy Neyman. In his book, Statistical Methods in Scientific Inference (1959), attacks on these two and others appear on the following pages: 3 (Pearson); 34-35 (Pearson & Edgeworth); 89 (Neyman and Egon Pearson); 98 (Neyman); 100 (Neyman); 102 (Neyman & Egon Pearson). The names of the persons being attacked appear in the parentheses. It was best to stay out of R. A. Fisher’s path! Now why was Fisher so vicious and vindictive to Karl Pearson and Jerzey Neyman? An interaction was involved. It was partly due to Fisher’s temperament but also, we shall see, both of these other scientists manifested behaviors that triggered Fisher’s ire. Pearson acted aloof to Fisher. He rejected Fisher’s submitted papers intended for Biometrika and used his influence to lead another journal to reject a Fisher paper. When Fisher pointed out that Pearson had misinterpreted his own statistic, chi-square, regarding degrees of freedom, Pearson simply ignored him. Fisher attacked Pearson’s correlation ratio. Jerzy Neyman’s main offence was that he and Egon Pearson had pioneered a method of hypothesis testing that at the time became the dominant theory and overturned Fisher’s approach that used fiducial limits. It should also be mentioned that in March of 1935 Neyman read a paper titled “Statistical Problems in Agricultural Experimentation” at the meeting of the Royal Statistical Society which provided a personal affrontery to Fisher. (Box, 1978). He contended that randomized block and Latin square designs were biased and attempted to correct their deficiencies. Fisher replied that before one attempts to criticize previous work it is necessary to completely understand it. For the hypotheses that Fisher intended to test his statistical models were defensible. The controversy in this case was triggered by the fact that the two men took different approaches to the subject (Box, 1978, pp. 84-88) R. A. Fisher did more than just toss occasional insults to Karl Pearson. He attempted to undermine Pearson’s life work, and imply that he was inadequate as a statistician and as a scientist. And now, retroactively, it is seen that both of them were world-class statisticians. It should be mentioned that R. A. Fisher and Karl Pearson were incapable of self-criticism. Sir Ronald A. Fisher’s Publications and Honorary Awards

BOSON BOOKS

24


A number of present day statisticians would view Fisher as the most important statistician of the 20th century. He published seven books and almost 300 papers throughout his impressive career. In addition to Fisher’s classic on experimental design, and his book on the genetic theory of natural selection he produced Statistical Methods for Research Workers (1925) for which a number of editions appeared subsequently, some of them in German, Japanese, French, Russian, and Spanish. His published journal articles have been packaged in five volumes and are titled Collected Papers of R. A. Fisher (Bennett, 1971). There are 294 papers in all and their publication dates range from 1912 to 1962, spanning a half century. See also Williams, Zumbo, & Zimmerman (2001). The first edition of his statistical tables in research, by Fisher and Yates, was published in 1938. Shewart edited R. A. Fisher’s Contributions to Mathematical Statistics (1950). P.C. Mahalanobis wrote a brief biography of Fisher for this volume. He published it earlier in Sankhya (1938). What is unique about the Shewart volume is that Fisher selected 43 of the articles that he thought were most important. Additionally, he wrote an annotation and described the context in which each paper was written. A number of honorary memberships, foreign associateships, and medals were bestowed on R. A. Fisher. He was also knighted by Queen Elizabeth and received honorary doctorates from nine colleges (Box, 1983, p. 103). Concluding Comments This chapter has displayed the impressive intellectual versatility of Sir Ronald A. Fisher, focusing not only on his contributions to statistics and experimental design, but also on biology, population genetics, evolutionary theory, eugenics, agriculture, and Darwinism. It also should be mentioned that Fisher visited the United States from time to time to give invited lectures and to teach courses in statistics, experimental design, and related topics. In maintaining such contacts he hoped to make his work available to Americans. In judging R. A. Fisher’s work, it is necessary to consider the great wealth of his scientific contributions, but they must be weighed against his bias in controversial matters. References Alexander, T.W. (1996). R. A. Fisher and multivariate analysis. Statistical Sciences, 11, 20-34. Bartlett, M.S. (1968). R. A. Fisher. In D.L. Sills (Ed.). , International Encyclopedia of the Social Sciences (pp. 485-491). New York: Macmillan. Bennett, J.H. (1971). (Ed.). Collected papers of R. A. Fisher. New York: Wiley. Bennett, J.H. (1983). Natural selection, heredity, and eugenics: Including selected correspond of R. A. Fisher with Leonard Darwin and others. Oxford: Clardon Press. Box, J.F. (1978). R. A. Fisher: The life of a scientist. New York: Wiley.

BOSON BOOKS

25


Box, J.F. (1983). Ronald Aylmer Fisher. In S. Kotz & N.L. Johnson (Eds.). Encyclopedia of the Statistical Sciences (pp. 103-111). New York: Wiley. Doll, R., & Hill, A.B. (1950). Smoking and carcinoma of the lungs: Preliminary report. British Medical Journal, 2, 239-248. Efron, B. (1998). R. A. Fisher in the 21st century. Statistical Science, 13, 95-122. Fisher, R. A. (1925). Statistical methods for research workers. New York: Hafner. Fisher, R. A. (1930). The genetical theory of natural selection. Oxford: Oxford University Press. Fisher, R. A. (1949). The theory of inbreeding. London: Oliver & Boyd. Fisher, R.A (1959). Statistical methods and scientific inference. (2nd ed. Revised). New York: Hafner. Fisher, R. A. (1960). The design of experiments (7th ed.). New York: Hafner. Fisher, R. A., & Yates, F. (1938). Statistical tables for biological, agricultural, and medical research. Edinburgh: Oliver & Boyd. Gridgeman, N.T. (1972). Ronald Aylmer Fisher. In C.C. Gillispie (Ed.). Dictionary of Scientific Biography (pp. 7-11). New York: Scribners. Jeffreys, H. (1939). Theory of probability. Oxford: Clarendon Press. Kendall, M.G. (1963). Ronald Alymer Fisher, 1890-1962, Biometrika, 50, 1-15. Mahalanobis, P.C. (1938). Professor Ronald Aylmer Fisher. Sankhya, 4, 265-272. Reid, C. (1982). Neyman…from life. New York: Springer-Verlag. Savage, L.J.H. (1976). On rereading R. A. Fisher (with discussion). Annals of Statistics, 4, 441-500. Shewart, W.A. (1950). (Ed.). Fisher’s contributions to mathematical statistics. New York: Wiley. Williams, R.H., Zumbo, D.W., & Zimmerman, D.W. (2001). The scientific contributions of R. A. Fisher. Starry Night Review, 2, 1-22. Yates, F., & Mather, K. (1963). Ronald Aylmer Fisher. Biographical Memoirs of the Royal Society of London, 9, 91-120. Zimmerman, D.W. (1996). A note on homogeneity of variance of scores and ranks. Journal of Experimental Education, 64, 351-362.

BOSON BOOKS

26


CHAPTER 3 Charles Spearman: Behavioral and Statistical Scientist Overview: This chapter is an outline of the significant achievements of the British Psychologist, Charles Spearman. He was the originator of three important behavioral and statistical domains: (1) Factor analysis; (2) A two-factor theory of intelligence; and (3) The classical theory of mental tests. Spearman utilized both mathematical methods and empirical psychological studies to pioneer research efforts in these three areas. It will be demonstrated that there are strong links between the three. Educational Experience The British psychologist, Charles Edward Spearman, was born the 10th of September of 1863 in London and died in the University College Hospital, London on September 17, 1945. He spent a quarter of his long life of 82 years in the British Army, and as a result didn’t complete his Ph.D. until he was 41 years of age. Wilhelm Wundt, who initiated the first laboratory in Experimental Psychology in Leipzig, Germany, directed his doctoral study. But Spearman was strongly influenced by the work of Francis Galton, who had made a strong case for intelligence testing. Two well-known psychologists who studied under Spearman were Raymond Cattell and David Wechsler, and five others who were influenced by him were Anne Anastasi, J.P. Guilford, Philip Vernon, Cyril Burt, and Arthur Jensen. Charles Spearman occupied chaired professorships at University College, London from 1907-1931, initially assuming a position in Experimental Psychology and then moving to the area of Mind and Logic. Although they were at the same college for many years, Karl Pearson was not impressed by Spearman’s research. The discussion that follows describes Spearman’s contributions to factor analysis, theories of intelligence, and mental test theory, and to provide evidence that he was an innovative force and pioneer in all three areas. Additionally, logical links will be made available to connect the three areas. In other words, the three contributions are not independent of one another. The Classical Theory of Mental Tests Most psychologists, and many other behavioral and social scientists, view Harold Gulliksen’s classic book, Theory of Mental Tests (1950) as the definitive treatment of the classical theory of mental tests. Of course, when it was published in 1950 it was the only comprehensive, mathematical development of psychological test scores. Since then there have been a number of test theory models. See Crocker and Algina’s Introduction to Classical and Modern Test Theory (1986) for a more up to date discussion of the state of psychometrics. In the classical theory a starting point is X = T+E, where X is observed score, T is true score, and E is error score. The investigator would like to know the true measurement, BOSON BOOKS

27


but, unfortunately, errors of measurement obscure the true values. Therefore, one objective of measurement procedures is to minimize errors of measurement. One way to do this is to change test item format from true false to multiple choice to diminish the probability of a successful guess. Another is to lengthen the test. The Spearman-Brown formula shows how test length influences reliability. Two important concepts in test theory are reliability and validity. Reliability has to do with consistency of measurement whereas validity focuses on whether the test or measuring device actually measures what it purports to measure. Incidentally, the term “reliability” was introduced by Spearman. In mental test theory these two features of measuring devices are treated mathematically. For example, the variances of the components of test scores are additive; i.e., Var [X] = Var [T] + Var [E]. Then reliability is defined as the ratio of true score variance to observed score variance or, Var [T]/Var [X]. Similarly, the proportion of observed variance which is due to error is Var [E]/Var [X] (Williams, Zimmerman, Zumbo, & Ross, 2003; Lord & Novick, 1968). On page 1 of Harold Gulliksen’s text (1950), he says that most of the equations of the classical theory of mental tests are attributable to Charles Spearman (1904a, 1904b). One such formula is Spearman’s correction for attenuation. Errors of measurement diminish or attenuate the correlation so this formula gives an estimate of what the correlation would be in the absence of measurement errors. Then there is the Spearman-Brown Prophecy Formula, which was mentioned above. It is even useful to plot curves showing reliability coefficients against test length, prior to constructing a test. In investigating test validity, the test scores to be validated are called X and they are validated against a criterion called Y. For example, X might be Verbal Scores on the Scholastic Aptitude Test and Y could be grade point average in freshman year in college. Then the product moment correlation between X and Y is a validity coefficient. Spearman derived another equation that displays the effect of altering the length of both the test and the criterion on validity. Then the validity of a perfect test for predicting a perfect criterion is given by Spearman’s a generalized form of the correction for attenuation. It yields an estimate of the true score correlations. For further information on Spearman’s contributions to test theory see “Spearman’s Test Score Model” (Ng, 1974) and “Charles Spearman’s Contributions to Test Theory.” (Levy, 1995). It should be mentioned that a number of the equations of the classical theory of mental tests depend, in their mathematical derivation, on error scores being uncorrelated. (Zimmerman, & Williams, 1977) Factor Analysis As a beginning to factor analysis, one obtains a set of n measurements on a set of subjects or examinees. An n x n correlation matrix is then formed, employing productmoment correlations. Some factor analytic technique is then applied to the matrix to find a smaller more basic set of variables that can account for the variance in the original set. This new set of basic variables is called “factors.” The factor matrix is then geometrically rotated to improve interpretation. Several statisticians and behavioral scientists have attributed the pioneering efforts of the development of factor analysis to Charles Spearman. Lovie and Lovie (1993) have carefully analyzed the correspondence between Charles Spearman and Cyril Burt, as both have been presumed to be responsible for the BOSON BOOKS

28


development of factor analysis, and concluded that Spearman is the true pioneer of this methodology. See also Cattell (1968), Harman (1976), and Bartholomew (1995). All three attest to Spearman’s primacy. Factor analysis and other multivariate statistical procedures are difficult to conduct by hand so the eventual presence of prewritten computer packages such as the Statistical Package for the Social Sciences (SPSS) and the Statistical Analysis System (SAS) was welcome. As mentioned above, factor analysis is applied to matrices of intercorrelations among observed score variables. But since it is of greater interest to describe relations among the underlying or latent variables, why not apply Spearman’s correction to the observed score correlations thereby generating a matrix of estimated true score correlations? This would be done prior to the factor analysis proper. It would seem promising because the estimated true score correlations would be larger and hence would account for a greater proportion of the variance. One reason why this isn’t done routinely is because there is some controversy associated with Spearman’s correction. For example, it sometimes yields values greater than 1.00. (Zimmerman & Williams, 1997) Notice that here we have provided a link between mental test theory and factor analysis. Some psychologists, such as J.P Guilford and Raymond Cattell and Philip E. Vernon, and Arthur Jensen have employed factor analysis to assist in the creation of theories of intelligence or personality. Spearman’s paper “General Intelligence Objectively Determined and Measured,” (1904a) was the pioneering contribution to factor analysis. The culmination and highlight of Charles Spearman’s factor analytic research is The Abilities of Man: Their Nature and Measurement. (1927). His final work in factor analysis, which was completed by Wynn Jones after Spearman’s death, was Human Abilities (Spearman & Jones, 1951; Bartholomew, 1995) A Two-Factor Theory of Intelligence Charles Spearman created the first psychometric theory of intelligence, a two-factor theory, and published it in the American Journal of Psychology (1904a). It was based on mathematical notions together with empirical data collected from psychological studies and philosophical concepts. His work in factor analysis paved the way for the application of statistical methods to the study of human intelligence. In analyzing a table of intercorrelations among test scores, Spearman noticed that they could be transformed to produce a hierarchical pattern which revealed that all the tests measured one common factor or, as he termed it, the “general factor (g).” But the various tests exhibited differential correlations with the general factor. Furthermore, each test measured its own specific factor. Hence, he had posited a “two factor theory of intelligence.” (Carroll, 1982) It should be noted that the nature of a theory of intelligence can strongly dictate the way in which intelligence tests are constructed. Some of the well-known intelligence tests are specially designed to measure Spearman’s “g.” It is said that Raven’s Progressive Matrices, which is a non-verbal, cross-cultural test of intelligence, is one of the best measures of Spearman’s “g.” John Raven worked closely with Charles Spearman in constructing this test. Raven was a student of Spearman’s. BOSON BOOKS

29


It is of interest to mention that Charles Spearman developed a rank order correlation coefficient (ρs). It is widely used to find the monotonic relation between two sets of ranks, whereas the usual Pearson product-moment correlation coefficient is used to analyze linear relations between two variables. The latter coefficient is employed in connection with factor analytic studies. The Spearman correlation calculates a Pearson correlation between two sets of observations that consist of integer ranks. One could say that the Pearson formula is general whereas the Spearman formula is special—that is, the general formula applied to a special case. Two other coefficients, phi and point biserial, are also special cases of the Pearson r. Maurice Kendall’s correlation (τ) yields relations between two ordinal variables but it is considered to be a measure of disarray. The Spearman and Kendall statistics are usually described in books on Nonparametric Statistics such as Siegel and Castellan’s Nonparametric Statistics for the Behavioral Sciences (1988). Karl Pearson published an anonymous review of Spearman’s The Abilities of Man in the journal Nature (1927) titled “The Mathematics of Intelligence” in which he attacked Spearman for faulty mathematics and statistics. In this and other publications Pearson criticized the rank correlation and the correction for attenuation, inviting Spearman to produce a paper containing algebraic proofs of his equations to support the unproven assertions. Pearson also pointed put that the correction for attenuation formula sometimes yielded estimates (correlations) greater than one. This problem still exists. Zimmerman and Williams (1997) conducted a simulation study of Spearman’s correction and concluded that the formula should be used only when the reliabilities of both tests are relatively high and sample size is relatively large. Charles Spearman countered Karl Spearman’s criticisms by saying that the GaltonPearson model could no longer be used as a model. On the other hand he criticized psychologists for not becoming familiar with the commendable work being carried out by the Galton-Pearson school (Levy, 1995, p. 223). Dr. Jan Strelau, the Associate Editor of the European Psychologist, was interviewed in the year 2000 and was asked to name the most significant developments in Psychology in the 20th century. He named Charles Spearman’s work in intelligence theory, pointing out that it was catalyst for the development for most subsequent theories of intelligence either in a positive or a negative sense. He also pointed to the significance of Spearman’s “g” as the underlying concept in the construction of a number of intelligence tests. (TeleInterviews, 2000). Charles Spearman utilized the method of tetrad differences in analyzing intelligence test data to support his two-factor theory of intelligence. A tetrad difference is actually the determinant of a 2x2 matrix. If it is zero the data are consistent with the notion of a single general factor. The psychologist L.L. Thurstone was violently opposed to Spearman’s theory and generalized the British psychologist’s criterion of tetrad differences to examination of higher order determinants as a criterion for more than one common factor. Thurstone’s theory of intelligence postulates seven primary mental abilities and he and his coworkers constructed seven tests to correspond to these factors. Others developed hierarchical theories that retained “g” in the structure. One of these was the Canadian psychologist Philip Vernon and another was the American psychologist Raymond Cattell. In Cattell’s theory “g” is at the top of the hierarchy and it becomes BOSON BOOKS

30


partitioned into fluid and crystallized intelligence, with Spearman’s specific ability factors at the bottom of the hierarchy. (Cattell, 1968) Matrix formulations with the advent of high-speed computers have greatly facilitated the development of factor analysis and other multivariate statistical methods. It is now common to distinguish between exploratory and confirmatory factor analysis with the latter more theoretical and hence used to test hypotheses. Concluding Comments Applications linking all three of the domains of research investigated by Charles Spearman are readily conceivable. His intelligence theory could be used to guide construction of a set of ability tests and then the test scores factor analyzed. Mental test theory would be employed to investigate the psychometric properties of the ability tests. Factor analysis itself is a way of studying validity, construct validity, and another approach to validation is to correlate the test scores with an external criterion, such as rank order in class. A method favored by Spearman to estimate reliability was to correlate scores on parallel forms of a test. References Bartholomew, D.J. (1995). Spearman and the origin and development of test theory. British Journal of Mathematical and Statistical Psychology, 48, 211-220. Burt, C. (1909). Experimental tests of general intelligence. British Journal of Psychology, 3, 94-177. Carroll, J.B. (1982). The measurement of intelligence. In R.J. Sternberg (Ed.), Handbook of Human Intelligence (pp. 29-120). Cambridge: Cambridge University Press. Cattell, R.B. (1968). Spearman, C.E. In D.E. Sills (Ed.). International Encyclopedia of the Social Sciences, (pp.108-111). New York: Macmillan. Crocker, L., & Algina, J. (1986). Introduction to classical & modern test theory. New York: Harcourt Brace Jovanovich. Gulliksen, H. (1950). Theory of mental tests. New York: Wiley. Harman, H.H. (1976). Modern factor analysis (3rd ed rev.). Chicago: University of Chicago Press. Joreskog, K.G., & Sorbom, D. (1984). LISREL VI, Analysis of linear structural relationships by maximum likelihood, instrumental variables, and least squares methods. Mooresville, IN: Scientific Software, Inc. Levy, P. (1995). Charles Spearman’s contributions to test theory. British Journal o f Mathematical and Statistical Psychology, 48, 221-235. Lord, F.M., & Novick, M.R. (1968). Statistical theories of mental test scores. Reading, MA: Addison Wesley. Ng, K.T. (1974). Spearman’s test theory model: A restatement. Educational and Psychological Measurement, 34, 487-498. Siegel, S., & Castellan, N.J., Jr. (1988). Nonparametric statistics for the behavioral sciences (2nd ed.). New York: McGraw-Hill. BOSON BOOKS

31


Spearman, C.E. (1904a). “General intelligence” objectively determined and measured. American Journal of Psychology, 5, 201-293. Spearman, C.E. (1904b). Proof and measurement of association between two things. American Journal of psychology, 15, 72-101. Spearman, C.E. (1927). The abilities of man, their nature and measurement. New York: Macmillan. Spearman, C.E., & Jones, L.W. (1951). Human abilities. London: Macmillan. Tele Interviews (2000). Interview of Jan Strelau European Psychologist, 5, 1-4. Williams, R.H., Zimmerman, D.W., Zumbo, B.D., & Ross, D. (2003). Charles Spearman: British behavioral scientist. Human Nature Review, 3, 114-118. Zimmerman, D.W., & Williams, R.H. (1977). The theory of test validity and correlated errors of measurement. Journal of Mathematical Psychology, 16, 135-152. Zimmerman, D.W., & Williams, R.H. (1997). Properties of the Spearman correction for attenuation for normal and realistic non-normal distributions. Applied Psychological Measurement, 21, 253-270

BOSON BOOKS

32


CHAPTER 4 Florence Nightingale David: Pioneer Feminist Statistician Overview: Florence Nightingale David, who preferred to be called simply David, was named in honor of Florence Nightingale, the British nurse who was the founder of trained nursing as a profession. David’s low key and humorous approach to biases against women who chose Statistics as a profession actually opened the door to women preferring that scientific area. Her over 100 publications, which extended from books to monographs to papers in juried scientific journals, were sometimes coauthored by notables such as Karl Pearson, Jerzy Neyman, and Maurice Kendall. She served as a book reviewer for Biometrika and made a number of presentations at the Berkeley Symposium on Mathematical Statistics and Probability. David was awarded for superlative teaching. An award in honor of Florence N. David was approved by the council of Presidents of Statistical Societies and the Caucus of Women in Statistics in 2001. The Four Women Associated With Early History of Statistics Of the many persons who will always be associated with the early history of the statistical sciences, only four are women—Elizabeth Scott, Gertrude Cox, Florence Nightingale, and Florence Nightingale David. Although the present chapter will focus on Florence Nightingale David, a brief summary of the contributions of the other three women will first be presented. Elizabeth Scott received a Ph.D. in Astronomy from the University of California, Berkeley in 1949. She authored many publications in both Astronomy and Statistics and coauthored a number of publications in both disciplines with Jerzy Neyman. It was said that Neyman drew her to Statistics and she drew him to Astronomy. A segment of Scott’s work provided statistical evidence of discrimination regarding advancement of women faculty members and women graduate students. Elizabeth Scott’s excellence as a researcher and a teacher was recognized nationally and internationally. Gertrude Cox took a bachelors degree in Mathematics and Statistics, a master’s degree in Statistics, and took additional graduate studies in Statistics. Her research and writing provided practical applications to horticulture, education, home economics, nutrition, agriculture, industry, and government. She established the Research Triangle Institute and initiated the departments of Statistics and Biostatistics at the University of North Carolina at Chapel Hill. Gertrude Cox was the first woman to achieve full professor and to become chair of the Department of Statistics at North Carolina State College. BOSON BOOKS

33


Florence Nightingale was honored by both the Royal Statistical Society and the American Statistical Association for her creative statistical work. Although she might be best remembered for establishing nursing as a legitimate profession she conducted censuses on sick patients and the conditions under which they were housed and treated before recognized census studies were ever conducted. She gathered large amounts of data that she used to argue her causes before administrators and others who were in a position to make changes in undesirable condition for patients and nurses. Statistical adherents have called her The Passionate Statistician. She was also known to some as The Lady of the Lamp. Her work led to the creation of a statistics department in the army. (Read, 1997). David’s Initial Vocational Frustrations On completing a bachelor’s degree with a major in mathematics from Bedford College for women, which was the first college for women, Florence Nightingale David applied for an actuarial position, using the name F.N. David. Although she was apparently the most well qualified applicant, she was turned down because she was a woman. She wrote to a friend that she experienced many such disappointments early in her career. Her father wasn’t surprised about these disappointments and told her she would be confronted with such problems all of her life and should simply get on with her work. She took this advice, keeping in mind that she remained interested in the actuarial sciences. She soon learned that the famous statistician, Karl Pearson, of the University College in London had some experience in actuarial work and she applied for a job as his research assistant. He immediately hired David and awarded her a scholarship. While working for Pearson, David encountered a number of world-class statisticians. Her favorite among them was William Gosset, a British statistician who worked for the Guinness Breweries in Dublin and published under the pseudonym “Student.” He invented statistical techniques for small samples including the Student t-test. Florence N. David provides us with some insight regarding the statistical luminaries of that time period. David’s Reactions To World-Class Statisticians “I saw the lot of them. Went fly-fishing with Gosset. A nice man. Went to Fisher’s seminars with Cochran and that gang. Endured K.P. Spent three years with Neyman. Then I was on Egon Pearson’s faculty for years. “Fisher was very vague. Karl Pearson was vague. Egon Pearson vague. Neyman vague. Fisher and Neyman fiery. Silly! Egon Pearson was on the outside. “They were all jealous of one another, afraid somebody would get ahead. Gosset didn’t have a jealous bone in his body. He asked the question. Egon Pearson to a certain extent phrased the question that Gosset had asked in statistical parlance. Neyman solved the problem mathematically.” (Reid, 1982, p. 133) David claimed that although R. A. Fisher was atrocious as a lecturer, Karl Pearson was excellent. At times she would sit through Fisher’s talks not understanding what he was talking about, wanting to ask questions. But being a woman she was not allowed to raise questions. She would circumvent this problem by asking two of the well-respected

BOSON BOOKS

34


visiting statisticians, Sam Wilks and Churchill Eisenhart, to ask her questions for her (Utts, 1993, p.1290). F.N. David’s professional career covered six decades and she taught in England and in America. In addition to writing book reviews for Biometrika and making a number of presentations at professional conferences, she published over 100 papers in refereed journals and was the author or editor of nine books. Karl Pearson, Maurice Kendall, and Jerzy Neyman were among her coauthors. (Utts, 1997) When Karl Pearson retired, David stayed on as a lecturer in the Statistics Department at University College, London. His son, Egon, took over as chair of that department and R. A. Fisher became the head of the Department of Eugenics. Karl moved across the college quadrangle and took F.N. Davis with him. There was a great deal of tension in that association because of the negative relation between R. A. Fisher and Karl Spearman. Soon after, Egon hired Jerzy Neyman, which proved beneficial because those two collaborated on research. One of David’s duties was to explain to Neyman’s students what he has talking about. Neyman liked to refer to her as “The Dutchess” because she was stubborn and wouldn’t always comply with his requests. Neyman urged David to complete her Ph.D. in Statistics and she did so, finishing her work in 1938. Research springing from her prior association with Karl Pearson led to her publication of Tables of the Correlation Coefficient (1938). A Sample of David’s Publications After serving in the military during World War II she moved back to her former position in the Department of Applied Statistics. David’s next book publication, Probability Theory and Statistical Methods, appeared in 1949. The main purpose of this work was to state and prove theorems of the calculus of probability found useful for elementary statistics students. Her next two book publications were A Statistical Primer (1953a) and Elementary Statistics Exercises (1953b). These were required as textbooks for a first level course in statistics. A number of her publications in prestigious journals also appeared over those years. Of those, Biometrika was David’s favorite. Almost half of her published journal articles appeared there. One that was notable was her “Dicing and Gaming (A Note On the History of Probability)”, which not only appeared in Biometrika (1955), but also was the lead article in the volume edited by Egon Pearson and Maurice Kendall titled Studies in the History of Statistics and Probability (1970). On Entering Administration In 1968 Florence Nightingale David moved to America where she eventually served as Chairperson of the Department of Biostatistics at the University of California at Riverside. The Riverside years were chaotic, as she had to deal with the development of a new Department of Statistics. There were quarrels among departments as many of them desired to house their own statistics—they frowned upon hearing a central Statistics Department. In 1970 David became Chair of Statistics. Although her work as administrator was filled with obstacles, she succeeded and her department eventually offered a Ph.D. in Applied Statistics. Retiring from Riverside in 1977 she became Emeritus Professor at Berkeley. (Williams, 2003)

BOSON BOOKS

35


Perhaps Professor David’s most popular book publication is Games, Gods, and Gambling (1962). It is probably suitable for both the elementary student and the initiate. In this book she explored the history of probability and included information on many interesting historical figures. Nan Laird’s Interview of Florence N. David Nan Laird, who was a professor of Biostatistics at Harvard University at the time, interviewed Professor David in July of 1988. The interview was conducted in the Jerzy Neyman Conference Room at Berkeley. The entire conversation between the two women was published in 1989 in the journal Statistical Science. In discussing women’s vocational choices in academia, Laird asked David about her view of the present day status. David said that she thought things were better but prejudice still existed. She said that academicians would tend to attack a woman rather than a man. Some female assistants would come to her complaining because David was on the President’s University Committee for Affirmative Action. She would reply in a somewhat negative fashion telling them to show that they could do it. But she would then contact the man who had treated the assistant badly and threaten him. When Laird questioned her on her view of the present day status of the field of Statistics, David cast aspersions on the normal curve and on other theoretical distributions. She urged that people sit down and work with real data, assuming at most randomness. She thought that much of the heavy mathematical formulations in the journals were concealing trivial ideas. And yet David acknowledged that young men and women had to publish to gain promotions and other desirable advances. Laird asked Florence about her early education. When young, David took private lessons up to the age of ten with a British parson. He taught her Greek, Latin, and algebra. While working as Karl Pearson’s assistant she found herself exposed to Pearson’s controversies involving giants in the field of statistics, such as R. A. Fisher and Jerzey Neyman. Karl Pearson didn’t like the hypothesis testing model developed by Neyman and Egon Pearson and he wouldn’t publish their stuff. So they pioneered a journal called Statistical Research Memoirs as a new publication outlet. Egon usually got on well with his father but when the day came that he was obliged to tell his father about the formation of the new journal, it gave him a stomachache. Jerzy thought that was humorous. Two years later, when Karl Pearson died, Egon Pearson assumed the editorship of Biometrika and Statistical Research Memoirs was stopped. Professor David said she got along very well with Jerzy Neyman. He impressed her as being helpful and kind. During the war years David was Senior Statistician for the British Government. The problems she confronted during the six war years gave her ideas that carried over to her subsequent academic work. In June of 1939 she began serving as an experimental officer with the Ordnance Board and a year later went to the Ministry of Home Security analyzing danger zones and safe zones when bombs hit. A special map of London was created to study graphs of where bombs fell and thereby estimate where they came from.

BOSON BOOKS

36


When the war ended on Victory in Europe Day David returned to University College, London. Concluding Comments David won the Academic Senate Distinguished Award for Teaching, although she claimed she disliked teaching, and she was First Recipient of the Elizabeth L. Scott Award. This honor was bestowed for excellence in serving as a role model to women through her professional contributions to teaching, research, multidisciplinary leadership, and statistics education. She was a Fellow of the American Statistical Association and the American Association for the Advancement of Science and Member of the International Statistics Institute and the Institute of Mathematical Statistics In 2001 a Florence N. David Award was established, and Dr. Nan Laird was the first recipient. Also, there is now an F.N. David Library at UCLA. Dr David was a member of the President’s University Committee for Affirmative Action at Riverside, a Member of the University Senate at University College, and Governor of Bedford College for Women. References David, F.N. (1938). Tables of the correlation coefficient. London: Biometric Trust. David, F.N. (1949). Probability theory for statistical methods. New York: Cambridge University Press. David, F.N. (1953a). A statistical primer. London: Griffin. David, F.N. (1953b). Elementary statistical exercises. London: University College. David F.N. (1955). Dicing and gaming (a note on the history of probability). Biometrika, 42, 1-15. David, F.N. (1962). Games, gods, and gambling. London: Griffin. Laird, N.M. (1989). A conversation with F.N. David. Statistical Science, 4(3), 235246. Pearson, E. P., & Kendall, M.G. (1970). (Eds.). Studies in the history of statistics and probability. Griffin: London. Read, C.B. (1997). Florence Nightingale. In N.L. Johnson & S. Kotz (Eds.) Leading Personalities in the Statistical Sciences (pp. 311-314). New York: Wiley. Reid, C. (1982). Neyman…from life. New York: Springer-Verlag. Utts, J. (1993). Florence Nightingale David 1909-1993, Obituary, Biometrics, 49, 12891291. Williams, R.H (2003). Florence Nightingale David: Pioneer feminist statistician. Muse Apprentice Guild, 3,1-10.

BOSON BOOKS

37


CHAPTER 5 George Udny Yule: Premier British Statistician Overview: George Udny Yule was born February 18, 1871 in Beech Hill near Haddington, Scotland and died June 26, 1951 in Cambridge, England. He was a member of an established Scottish family composed of army officers, civil servants, scholars, and administrators. Both his father and a nephew were knighted. At the age of 16 he began the formal study of engineering at University College, London. George gravitated to Bonn, Germany, where he studied under the famous scientist Heinrich Hertz. A great influence in Yule’s academic life was the well-known statistician, Karl Pearson, who lured him back to London, awarding him a directorship. George Yule was prolific in journal and book publications and in activities related to the Royal Statistical Society, the highlight of his publications being perhaps the book, An Introduction to the Theory of Statistics (1911), which went through 14 editions and was published in several languages. Family Background and Vocational Choice In 1899 George Yule married May Winifred. Unfortunately, the marriage was not successful and was annulled in 1912 there having been no children. Yule’s research with Hertz was fruitful as he published four papers on electrical waves. Nevertheless, he turned away from experimental physics and, at the age of 22, in 1893, returned to London, at University College and accepted Karl Pearson’s offer. Pearson was at that time Professor of Applied Mathematics, and Yule thought him to be a good teacher. Is Normality Realistic? Yule’s first publication in Statistics, “On the Correlation of Pauperism with Proportion of Out-Of-Relief,” appeared in Economic Journal (1885). Among other things, the paper developed procedures for applying correlational techniques to two-dimensional contingency tables. It should be mentioned that George Yule was always suspicious of correlational methods that depended on assumptions of normality. He felt that important data in economics and many other fields were unlikely to satisfy the assumption of normality. His view was that the law of normality was invalid for many realistic applications in a number of disciplines. Professional Organizations At the rather young age of 25, Yule was elected a Fellow of the Royal Statistical Society and served with this prestigious organization under several capacities for the next 36 years. He served as Honorary Secretary, which apparently was a somewhat thankless task, was President of the Society, and was awarded the high level Guy Medal in Gold. He was also active in a number of other professional organizations. Yule’s Textbook

BOSON BOOKS

38


Very early in the twentieth century George Udny Yule published a number of papers in correlation and regression. His work paved the way for multivariate developments by R. A. Fisher and Harold Hotelling. And while lecturing in statistics from 1902 to 1909 he produced notes that eventually led to the publication of An Introduction to the Theory of Statistics (1911). This book was readable and comprehensive and widely used—the only textbook of its kind, particularly used in economics and sociology. The book was translated into Czech, Polish, Spanish, and Portuguese. Maurice Kendall coauthored the last three editions. Yule’s Regression Approach George Yule’s major contributions in theoretical statistics dealt with time series, Mendelian Genetics, Epidemiology, and Correlation and Regression, focusing especially on 2x2 contingency tables. He tended to prefer a regression approach to his research problems. It was Yule who first recognized that applications of the chi square statistic to contingency tables are dependent on degrees of freedom (df), where df = (#rows-1)(#cols. –1). He wasn’t able to prove this mathematically but demonstrated it through simulation. Later on R. A. Fisher provided the mathematical proof of Yule’s contention and also corrected Karl Pearson’s misconception of degrees of freedom for chi square tests of goodness of fit. Yule was suspicious of those who spoke of establishing causal relations among variables saying that undetected variables could be responsible for the supposed causality. Many years later George Box made a systematic study of these unmeasured “lurking variables.” A number of modern statisticians have concluded that many of Yule’s research findings have stood the test of time. Although Yule studied under Karl Pearson, the two disagreed on a number of statistical issues, including degrees of freedom and the properties of Spearman’s rank-order correlation. They also had conflicts over correlational techniques applicable to contingency tables and statistical analyses of time series (Porter, 2004). Pearson and Yule were good friends initially, even taking vacations together, but after a while the hard feelings drew them apart. George Yule said that Karl Pearson only became controversial on intellectual or scholarly matters but otherwise was a polite gentleman. Classical Mental Test Theory It is little known that George Udny Yule made some pioneer contributions to the classical theory of mental tests. The fundamental equation in the theory is that X=T+E, where X=observed score, T=true score, and E=error of measurement. T and E cannot be observed directly, but the researcher is able to deal empirically with X. Starting with this equation, one can mathematically derive equations linking means and variances, and expressions for test reliability. Methods designed to reduce E will improve reliability and therefore elevate test validity coefficients. As mentioned earlier, changing the format from true false to multiple choice, which diminishes the probability of a successful guess, is helpful. Legend has it that George Udny Yule communicated the above fundamental equation to Charles Spearman. Spearman then used this equation to develop the basic ideas of the classical theory of mental tests. These concepts were extended by many subsequent investigators and presented systematically by Harold Gulliksen in his wellknown book, Theory of Mental Tests (1950). BOSON BOOKS

39


Errors of measurement tend to depress correlations, including reliability and validity coefficients. Some researchers use Spearman’s correction for attenuation to deal with this problem. For a further discussion of the correction and conditions under which it is applicable see Zimmerman and Williams (1997). Yule’s Correlation Coefficients There are a number of research terms in the literature named after George Yule. Some of them are Yule Process, Yule Distribution, Yule Correlogram, Yule Autoregressive Series, Yule Colligation, and Yule Q. The last named is a measure of association applicable to 2x2 contingency tables. Q can range from –1 to +1 and 0 indicates no correlation. One way in which it can be expressed is Q = (ϕ - 1)/(ϕ + 1), where ϕ = [f(1,1)/f(2,1)]/[f(1,2)/f(2,2)] is the odds ratio; f(i,j) is the observed frequency lying in row I and column j. Yule’s Desire to Fly George Yule retired from academic work at age 60 thinking the mathematics of his discipline had passed him by. He liked to speed in cars so he desired to learn to fly. But at his age he was a poor risk and thus was unable to get insurance. So he purchased his own airplane and got a pilot’s license. It was unfortunate that at that time he was beset with heart problems and became a quasi-invalid for the rest of his life. Ten years later he and Maurice Kendall were observing the planes of a World War II bombing raid and Yule, laughing aloud, said he was still licensed to fly all of the airplanes they were observing but wasn’t familiar with the cockpit controls of any of them (Kendall, 1952). Statistical Aspects of Literary Style As George Yule was about to retire he once again began to study Latin and enjoyed creating Latin verses that related to small sample theory as well as more literary themes. He turned to studying statistical features of literature, such as sentence length. He especially focused on frequency of words, summarizing his work in the book The Statistical Study of Literary Vocabulary. (Yule, 1944) This was his last book. He attempted to provide answers to questions such as the following: Did Shakespeare really write the plays that have been attributed to him? Was the Epistle to the Ephesians written by Saint Paul? What was the order in which Plato’s works were written? A Demonstration of Yule’s Research Versatility We now display the staggering number of substantive variables investigated by Yule in his many years of research, excluding purely statistical studies: pauperism; crop production and price; Mendel’s Laws and their probable relations to intra-racial heredity; duration of life and number of offspring; marriage and birth rates; vaccination statistics; smallpox and vaccination; variation in the number of sepals in anemone nemorosa; sex ratios and births in registration districts; sentence length in prose; harmonic analyzer; Notes and Records of the Royal Society; time series; yield trials; note on the number of Jews in Germany; the growth of population and the factors which control it; sunspot numbers; life table approximations; crop production and price; sampling in Mendelian ratios; anti-typhoid and anti-cholera inoculations; inheritance of quantitative compound BOSON BOOKS

40


characters on the basis of Mendel’s Laws; Census of Production Act; interference phenomena in electrical waves; some bacteriological factors employed in water analysis; a cross between yellow-wrinkled and green-rounded seeded peas; evolution of geographical distribution in plants and animals; statistics of British feeding trials and the starch equivalent theory; exposure to silica dust; occupational mortality; social and economic statistics; distribution of death with age; family size and order of birth; the fall of birth rate; a mathematical theory of evolution; and the statistical theory of accidents. (Williams, 2004; Yates 1952). Concluding Comments George Udny Yule came from a good Scottish stock, noted for its scholarship and administration. He contributed to correlation and regression theory, distribution theory, stochastic processes, statistics of literary vocabulary, and time series. He made continuous and longstanding contributions to the British Royal Statistical Society and produced an impressive textbook, An Introduction to the Theory of Statistics, which went through 14 editions and appeared in five languages. Yule’s contributions along personal lines delighted his friends and colleagues. He was a friendly and gentle man. His intellectual versatility and knowledge of numerous anecdotes made him a fine companion. For many years he kept a notebook replete with negative comments people had to say about statistics. Then, when the nature of such comments became less imaginative, he stopped recording them. (Kendall, 1952) References Gulliksen, H. (1950). Theory of mental tests. New York: Wiley. Johnson, N.L., & Kotzs, S. (Eds.) (1997). George Udny Yule. In Leading Personalities in the Statistical Sciences from the Seventeenth Century to the Present (168-169). New York: Wiley. Kendall, M.G. (1952). Obituary. George Udny Yule. Journal of the Royal Statistical Society, Series A (General), 115(1), 156-161. Kotz, S., & Johnson, N.L. (1988). (Eds.). Yule, George Udny. Encyclopedia of the Statistical Sciences, 9, 664-666. Lumsden, J. (1976). Test Theory. Annual Review of Psychology, 27, 251-280. Palo Alto, CA: Annual Reviews. Porter, T.M. (2004). Karl Pearson: The scientific life in a statistical age. Princeton, NJ: Princeton University Press. Reid, C. (1982). Neyman…from life. New York: Springer-Verlag. Spearman, C. (1910). Correlation calculated from faulty data. British Journal of Psychology, 3, 271-295. Stigler, S.M. (1986). The history of statistics: The measurement of certainty before 1900 (chapter 10-Pearson and Yule-pp. 326-361). Cambridge, MA: Harvard University Press. Stuart, A., & Kendall, M.G. (Ed.). (1971). Statistical papers of George Udny Yule. New York: Hafner. BOSON BOOKS

41


Williams, R.H. (2004). George Udny Yule: Statistical Scientist. Human Nature Review, 4, 31-37. Yates, F. (1952). George Udny Yule. Obituary Notices of the Royal Statistical Society of London, 8, 309-323. Yule, G. U. (1895). On the correlates of total pauperism with proportion of total outrelief. Economics Journal, 5, 477-489. Yule, G. U. (1897). On the theory of correlation. Journal of the Royal Statistical Society, 60, 812-821. Yule, G. U. (1911). An introduction to the theory of statistics. London: Griffin. Yule, G. U. (1944). The statistical study of literary vocabulary. Cambridge: Cambridge University Press. Yule, G. U. (1946). Cumulative sampling: A speculation as to what happens in copying manuscripts. Journal of the Royal Statistics Society, 109, 44-50. Zimmerman, D. W., & Williams, R. H. (1997). Properties of the Spearman correction for attenuation for normal and realistic non-normal distributions. Applied Psychological Measurement, 21, 253-270

BOSON BOOKS

42


CHAPTER 6 Maurice Kendall: Synthesizer and Commentator of Statistics Overview: Sir Maurice Kendall made important contributions to random number generators, statistical tests for randomness, and random walk theory. He was the author of the two-volume work, Advanced Theory of Statistics and authored Rank Correlation Methods. He wrote the humorous “Hiawatha Designs An Experiment.” His main objective was to bridge the gap between sophisticated theory and practical application. Early Childhood and Educational Experience Maurice George Kendall was an only child. He was born September 6, 1907 in Ketttering, Northamtonshire, England. As a youth he contracted potentially fatal cerebral meningitis. At the start of World War I in 1914 his father moved to Derby and there Maurice began his early education. Initially he showed no particular interest or aptitude for mathematics but gravitated toward the study of languages. But then he was awarded a scholarship to St. John’s College, Cambridge. He was named a Mathematics Wrangler in 1929 and entered the British Civil Service in the Ministry of Education in 1930. At Cambridge his gregarious nature led to a circle of friends. He played chess blindfolded with Jacob Bronowski and played cricket for his college. While working in the Civil Service he was introduced to statistical inference. Maurice’s father, John Roughton Kendall, wanted his son to pass up the chance to go to Cambridge and instead train to be an engineer. Fortunately, his mother, Georgina Brewer, supported Maurice’s plan to accept the scholarship. His initial statistical work in the Ministry was of high quality and led to his election as a Fellow of the Royal Statistical Society. One of his first papers demonstrated an application of factor analysis to crop protection. Kendall Meets Yule In 1935 Maurice Kendall met George Udny Yule under rather unusual circumstances. During his holiday vacation Kendall intended to read books on statistics housed in the St. John’s main library. But the library was locked up and Yule had the key, so Kendall had to seek him out. At the time, Yule had authored a successful textbook titled, A n Introduction to the Theory of Statistics, first published in 1911, and in 1935 it was in its 11th edition. The two men became good friends. In fact, Yule turned out to be godfather to Kendall’s second child. Yule requested that Kendall coauthor the next edition of his book and they eventually wrote three editions together. The 14th and last edition of the book appeared in 1950. There is some humor in their writings and interchanges. Kendall wrote a poem called “Hiawatha Designs An Experiment” (1959) which perfectly mimics the famous poem authored by Longfellow, possessing a highly similar rhythm, for example. But Kendall’s version contains a number of allusions to statistics and experimental design, containing BOSON BOOKS

43


parodies on personalities in the field. (Kendall, 1952). Maurice Kendall enjoyed listening to Yule’s anecdotes, which reflected a dry sense of humor. Kendall published The Advanced Theory of Statistics, volume 1 (1943) and volume 2 (1946), which were important advances in the field of statistics. These two books provided a comprehensive treatment of the classical theory of statistical inference. The sixth edition of this work, partitioned into three volumes, authored by Stuart, Ord, O’Hagan, and Kendall appeared in the 1990s under the title Kendall’s Advanced Theory of Statistics with subtitles Distribution Theory, Bayesian Inference, and Classical Inference and the Linear Model. (1994, 1998). The reader can locate a number of volumes authored or coauthored by Maurice Kendall, some of which were published after Kendall’s death, which are still for sale by accessing amazon.com on the Internet. In 1948 Kendall published a monograph titled Rank Correlation Methods. This book is not just a summary of the state of the art of nonparametric correlational techniques, but is a display of methods he invented. For example, it contains the rank order statistics— Kendall’s Tau, Kendall’s Partial Tau, and Kendall’s coefficient of concordance— together with their sampling distributions. The fifth edition of the book appeared in 1990 and the coauthor was Jean Dickinson Gibbons. Gibbons herself took over the task of producing this updated version, although Kendall remains as senior author. A notable feature of the 1990 edition of the monograph is that each chapter ends with a section that outlines the newer developments. Also, a set of realistic applications from substantive areas such as marketing, finance, psychology, education and other fields appears. In his review of this book, Marden (1992) concludes by praising both Kendall’s original work and the changes and updates made by Gibbons. It is interesting to note that although Maurice Kendall died in 1983, his books and extensions of them continue to be published. This must be a sort of immortality. Random Number Generators and Tests for Randomness In the late 1930's Maurice Kendall, together with Bernard Babington Smith (1938, 1939), conducted research on random number generation. Random sequences are widely used, especially in statistics and measurement. They developed mechanical devices designed to generate random digits and a series of tests useful for checking such sequences for randomness. Their findings superceded the methods then in existence and were widely used until Rand Corporation, in 1955, produced A Million Random Digits and 100,000 Normal Deviates. This company used Kendall’s tests for randomness to check out their digits. There were four such tests: frequency test; serial test; poker test; and test for a gap. For some of these analyses chi square (χ2) and the binomial distribution were used. The probability theory concept of random walk, applied to stock market trends, suggests that the price of a stock, or possibly of the overall market, is of no value in predicting the market’s future status or movement. That is, stocks take an unpredictable and random path. Kendall (1953) was a pioneer in random walk theory, applying time series to fluctuations in the stock market. He argued against a cyclical behavior of stock and commodity prices. Administrative Positions and Honors BOSON BOOKS

44


During World War II, Maurice Kendall developed theory on k-statistics and after the war he worked on time series. . In 1949 in took a position with the London School of Economics at the University of London. In the mid-fifties he edited Statistical Sources in the United Kingdom, developed the textbook Multivariate Analysis, and served as editor, with W.R. Buckland, of the Dictionary of Statistical Terms. His aim with the latter was to make the tools and jargon of statistics more available to those in industry and government. In 1961 he left the University of London to take an administrative position with a computer consulting committee that later became known as Scientific Control Systems. That same year he began a two-year term as President of the Royal Society. Although he was scheduled to retire in 1972, he instead became Director of the World Fertility Survey, the largest multinational sample survey ever undertaken. He was knighted in 1974 and received the Peace Medal of the United Nations in 1980 for his work on the World Fertility Survey. He was elected a fellow of numerous British and American scientific and mathematical societies and was Honorary President of the International Statistical Institute at the time of his death in 1983. Concluding Comments Maurice George Kendall was a workaholic, and a brilliant one at that! During the war he carried both a heavy daily workload together with air raid responsibilities at night and yet made significant progress on volume 1 of his Advanced Theory of Statistics. Whether working on his own theoretical developments or that of others, as well as practical administrative tasks, he was an innovative synthesizer and commentator. He avoided personal confrontation and was fair to those in different age groups. He delegated responsibilities in an impartial manner. As mentioned, the importance of his work to statisticians in both academia and industry was rewarded by numerous honors. His scientific prose was lucid, without embellishments, and his writing style was superior to that of most statisticians. In the Inaugural Address given by Kendall at the London School of Economics on January 17, 1950, he stressed the fact that a statistician is not a person who focuses on only numerical facts. Nor is statistics a spin off discipline, but is instead a legitimate field in its own right. He added that the area of statistics yields results that may generalize to other methodologies and substantive areas. As an example, he cited a famous research project on sunspots conducted by Yule. Yule’s methods were later generalized to stock market forecasting, variations in crop yields, standardization in cotton thread, preparation of polished metal surfaces, the designs of gun turrets, and cosmic ray showers. Kendall added that statistics, like mathematics, being a scientific method, is capable of a host of general applications. In 1980 Maurice George Kendall was finally forced to retire because of illness. He died in 1983. His first wife, Sheila, bore him two sons and a daughter and he had a son with his second wife, Ruth. References Barnard, G.A. (1997). Maurice George Kendall. In N.L. Johnson and S. Kotz (Eds). Leading Personalities in the Statistical Sciences, 130-132, New York: Wiley. BOSON BOOKS

45


Kendall, M.G. (1950). The statistical approach. Economica, 17, 127-145. Kendall, M.G. (1952). George Udny Yule. 1871-1951, Journal of the Royal Statistical Society, 115A, 156-161. Kendall, M.G. (1953). The analysis of economic time-series, Part 1. Prices. Journal of the Royal Statistical Society, 96, 11-25. Kendall, M.G. (1959). Hiawatha designs an experiment. American Statistician, 13(3), 363. Kendall, M.G. (1963). Geometrical probability. New York: Hafner. Kendall, M.G. (1971). A dictionary of statistical terms (3rd ed., rev and enl.). New York: Hafner. Kendall, M.G. (1972). A course in multivariate analysis. London: Griffin. Kendall, M.G. (1984). Statistical theory and practice: Selected theory and practice by Sir Maurice Kendall, 1907-1983. London: Griffin. Kendall, M.G., & Doig, A.G. (1981). Bibliography of statistical literature. Great Britain: Arnold Press. Kendall, M.G., & Gibbons, J.D. (1990). Rank correlation methods (5th ed.). New York: Oxford University Press. Kendall, M.G., & Ord, J.K. (1990). Time series (3rd ed.). Great Britain: Arnold Press. Kendall, M.G., & Smith, B.B. (1938). Randomness and random sampling numbers. Journal of the Royal Statistical Society, 101(1), 147-166. Kendall, M.G., & Smith, B.B. (1939). Tables of random sampling numbers. Cambridge: Cambridge University Press. Kendall, M.G., Stuart, A, & Ord, J.K. (1994). Kendall’s advanced theory of statistics: Distribution theory vol. 1, (6th ed.). New York: Wiley. Marden, J.I. (1992). Review of Kendall and Gibbons’ Rank correlation methods (5th ed.). Journal of the American Statistical Association, 87, 249-250. O’Hagan, A., Kendall, M.G., Stuart, A., & Ord, J.K. (1994). Kendall’s advanced theory of statistics: Bayesian inference, vol. 2B (6th ed.). New York: Wiley. Ord, K. (1984). In memoriam: Maurice George Kendall, 1907-1983. American Statistician, 38(1), 36-37. Stuart A. (1984). Sir Maurice Kendall, 1907-1983. Journal of the Royal Statistical Society, Series A (General), 147(1), 120-122. Stuart, A., Kendall, M.G., Ord, J.K., & O’Hagan, A. (1998). Kendall’s advanced theory of statistics: Classical inference and the linear model. Vol. 2A (6th ed.) (rev. and exp.). Great Britain: Arnold Press Yule, G. U. (1911). An introduction to the theory of statistics. London: Griffin.

BOSON BOOKS

46


CHAPTER 7 George Box: Pioneer in the Forecasting of Time Series Overview: George E. P. Box made major contributions to statistical quality control. He received the British Empire Medal for his early experimental work. Later on he became Head of the Statistical Techniques Research Section of the Empirical Chemical Industries. In 1960 he assumed the first Chairmanship of the new Statistics Department of the University of Wisconsin. Box took his B.A., PhD, and D.Sc. degrees in Mathematical Statistics at the University of London and was awarded honorary doctorates from several colleges. He published over 200 academic and industrial papers and a number of books and is the originator of widely used techniques in Response Surface Methodology and Evolutionary Operation. His important contributions include Bayesian Analyses and Analysis of Time Series. Educational Training George E. P. Box initially learned statistics in a manner different from that of most of the other statistical scientists described in this book. Being in situations in industry and in the military he needed to solve problems that required knowledge of statistical methods. With no statisticians in his environment, he had to learn the discipline on his own. Later on he felt it was more meaningful to be confronted with problems in a real setting and then discover or learn the required quantitative method than to first study statistics in academia and then go out and apply the models. It is thought by some that because Box initially had to teach himself statistics, he became a very fine communicator of applied and theoretical statistics. His major field in all three of his degrees was Mathematical Statistics and he studied for all of them at the University of London. Box’s Research Contributions George Box’s research contributions fall into the following categories: Bayesian Inference, Design of Experiments, Time Series, Statistical Quality Control, and Evolutionary Operation. Bayesian Inference is a statistical technique in which probabilities are interpreted as degrees of belief. Those scientists who advocate this type of logic maintain that this method is a formalization of the scientific method. Hence evidence collected tends to support or deny a given hypothesis. The father of BOSON BOOKS

47


experimental design was Sir Ronald A. Fisher. He used such designs to test null hypotheses. The most important concepts in design are randomization, replication, blocking or matching, and use of factorial experiments rather than single factor designs. The design of experiments is closely linked to the inferential statistical method called analysis of variance. Time Series is a sequence of data points, measured at successive times, often spaced apart at uniform time intervals. The analysis of such series is a set of methods designed to understand the theory underlying the data points or the mechanism used to generate the points, and to predict or forecast future events. Quality Control is used to develop and design systems to ensure that industrial products and services satisfy customer requirements or expectations. Evolutionary Operation is a method of increasing industrial productivity. (Box, 1957). Research in the above categories led Dr. Box to produce books, journal articles, monographs, scholarly paper presentations, and published teaching materials. The books deal with Time Series (1976), Model Building and Response Surfaces, (1987), Evolutionary Operation (1998), Experimental Design (1978, 2005), Bayesian Inference (1992), and Process Improvement (2006). In all, George Box has published nine books and over 200 research articles. Perhaps his two most widely cited publications are those by Box and Cox (1964) on transformations and Box and Jenkins (1970) on time series. Vocational Experiences George Box was raised in England and was a chemist in the British Army of Engineers in World War II. By reading books and other materials designed to help one understand design and statistics he acquired a sophistication that led to his work being recognized, and in 1946 he received the British Empire Medal. This would be the first of many honors bestowed on him. Later he served as a consultant and lecturer to the government, the academic, and industry. Over the years his clients included Ford Motor Company, Hewlett Packard, R.J. Reynolds, The Federal Reserve Board, World Bank, and Boeing Aircraft. His post-war vocation was as an applied statistician at Imperial Chemical Industries (ICI). He stayed there for eight years and then went on to Princeton, New Jersey to become Director of the Research Group. While at ICI Box completed his Ph.D. in Mathematical Statistics in 1952 from the University of London. In 1960 he accepted a post at the University of Wisconsin with the challenge to initiate a Department of Statistics. It wasn’t too long before that department became recognized as one of the best statistics departments in the world (Pena, 2001). George Box constantly demonstrated that he was a good administrator as well as a good researcher and a good teacher. The University of Wisconsin awarded the Smith-Reynolds Teaching Award to Box. In 1971 he was appointed Ronald Aylmer Fisher Professor of Statistics and then, in 1980, he became Vilas Research Professor, which is the highest award given to faculty at that institution. George E. P. Box has had bestowed on him many fellowships, medals, prizes, and awards by both America and the United Kingdom, as well as three honorary doctorates. The recognition he prizes the most is becoming a Fellow of the Royal Society of London. Two Interviews With George Box BOSON BOOKS

48


M.H. DeGroot interviewed George E. P. Box in Chicago in August of 1986 and the result was published in Statistical Science (1987). There are brief but interesting comments in this conversation showing links between Box and R. A. Fisher. Early in Box’s career, when he was trying to learn statistics on his own, he attempted to read Fisher’s Statistical Methods for Research Workers (1925), not having much success. It is interesting that Egon Pearson and “Student” and many others had difficulty with this book. Box was given a reading list that was replete with Fisher’s publications including Design of Experiments (1960) and Fisher and Yates’ Statistical Tables for Biological, Agricultural, and Medical Research (1938). George Box was reading at night to help him decide what to do the next day on his job. He was eventually given the task to go see R. A. Fisher. So he went to Cambridge and spent an entire day with the great man. Fisher welcomed him and gave some useful advice. Later Box went to interview Karl Pearson at University College. Not knowing there was a feud between Fisher and Pearson, Box told Pearson that Fisher was a very fine fellow. Pearson didn’t seem to mind and was very friendly. George Box spoke favorably about R. A. Fisher throughout most of the interview. He even named him as one of his main influences. He said that Fisher seemed to be the pioneer in almost every aspect of statistics and experimental design. Box said that one day Fisher came to him and said he was electing him for the International Statistical Institute. One time, while bordering on an argument with Fisher, Box decided that they approached inference with different presuppositions and that there was no point in futher discussions. R. A. Fisher’s daughter, Joan, was serving as secretary at Princeton University and she and George E. P. Box met there and eventually married. Hence Fisher turned out to be Box’s father-in-law. Furthermore, Joan Fisher Box wrote a biography of her father titled R. A. Fisher: The Life of a Scientist (1978), which received favorable reviews. In the conversation with DeGroot, Box mentioned that it was John Tukey who was enthusiastic about bringing him to Princeton. It was Tukey, too, who said that in statistical analysis, such as analysis of variance, the data analyzer had two choices. One was to apply a suitable transformation to the data, such as reciprocal, square root, logarithmic, arc sine square root of a proportion, etc.; the other was to seek a new method of analysis, such as Kruskal-Wallis. It would usually be more difficult to find a new technique. Box spoke of “robustness,” which was of great interest to him. Nonparametric statistics are sometimes referred to as “distribution free statistics.” George Box frowned upon this nomenclature, saying that it is doubtful that any methods exist that are completely free of underlying assumptions. Although such methods may be free of an assumption of homogeneity of variance or normality, they could still be vulnerable to violation of assumptions such as symmetry. He went so far as to speak of nonparametric statistics as a “swindle.” [It should be noted that Box is referring to tests such as the Wilcoxon-Matched-Pairs-Signed Ranks Statistic. There are other nonparametric tests, like the Sign Test and the Median Test that do not assume symmetry. Some statisticians have suggested that these tests be referred to as “assumption freer tests.”] George Box made major contributions to the concept of “robustness,” even coining the word. Box often spoke of the difficulty of dealing with nonlinear situations, telling DeGroot that R. A. Fisher was the first scientist to deal with nonlinear designs. One problem is BOSON BOOKS

49


that there is only one unique way for data to be linear but numerous ways to experience non- linearity. When DeGroot asked George Box to list the statisticians who had influenced him in his thinking, Box mentioned R. A. Fisher, George Bernard, and Maurice S. Bartlett. Daniel Pena of the Universidad Carlos III of Madrid Spain conducted the second interview with Box. It took place in Chicago in October of 1999, the day after a party to celebrate Box’s 80th birthday, and was published in 2001 in the International Journal of Forecasting. George Box told Daniel Pena that in 1986 he created, with Bill Hunter, the Center for Quality and Productivity Improvement that has had a pioneering role in the Quality Movement in the USA and elsewhere. Pena asked Box about the origin of Box’s work in time series and forecasting. His reply was that after the war he completed a course on time series, but it was very mathematical and didn’t seem to have any practical applications. But while working at Imperial Chemical Industries (ICI) he analyzed their monthly sales and compared the forecasts to what actually happened. He devised methods that gave better predictions. Later on, after moving to the United States, he did consulting for chemical industries, and entertained the possibility of curvilinear relations. Pena asked Box about a paper on canonical correlation that Box had coauthored with George Tiao. George Box said that his own encounter with multivariate statistics occurred because he continued to study statistics after he left University College. While at ICI he turned in work beyond the call of duty so his boss permitted him to attend lectures given by M.S. Bartlett. The course given at that time was multivariate statistics and canonical analysis was included. Box enjoyed Bartlett’s lectures, finding them extremely clear and he liked Bartlett’s use of n-dimensional geometry. Pena said that after time series Box had become interested in quality and inquired as to the factors that led Box to quality. Box replied that the best way of looking at quality control charts is as a plot of time series. Additionally, good forecasts are needed to run a quality company. The better the methods employed for forecasting, the better the decisions that will be made and hence the better the quality. Daniel Pena asked about the best kind of advice Box might give to young persons just starting out in research and are trying to discover innovations in statistics. Box claimed that undergraduates are merely a receptacle for knowledge and then, when they graduate and go on to graduate school, they are all at once expected to come up with totally originally ideas. Undergraduate studies should give students more chances to employ their creativity. Apart from work, what do you like to do? Box said, I find it pleasant to go for walks in the woods. To me, getting back to the environment, to the hills, the mountains, the fields, and the sea is a fundamental need. Concluding Comments We have seen in this chapter that George E. P. Box is versatile. And he is strong along several dimensions: researcher; consultant; administrator; and teacher. It can be added that he is also a pleasant gentleman. Some of his work as a teacher is somewhat invisible even though he was given an award in teaching for excellence. BOSON BOOKS

50


Going to amazon.com on the computer reveals not only that his many books are for sale—put them in your cart, but there are also displays for teaching in areas such as Designing Industrial Experiments (S10), Time Series Modeling (S14), Expert Time Series Forecasting (S16), General Statistical Analysis (S17), Quality Improvement and Design of Experiments (S18), and Published Papers by George E. P. Box (Q67). More specifically, S10 consists of 5 books plus 6 videotapes and sells for $2500.00. S14, S16, S17, and S18 are in the form of computer software and their prices range from $195.00 to $1195.00. If you desire his published papers you must call in to get the price. But George Box is not doing this just to make money. The fact is, he is an impressive teacher. He is also a builder of models, although he is known to make statements such as the following from time to time: “All models are wrong, but some are useful.” Box’s unusual ability to communicate rather complex notions in a simple manner, coupling his teaching with humor and striking examples, is appealing. His deep knowledge of industrial settings and of statistical inference and experimental designs make this possible. The well-known Box-Jenkins model (1976) together with the Box and Cox transformations (1964) are now a part of the standard curriculum learned by students in Economics, Psychology, Engineering, Chemistry, History, Biology, and Sociology. It seems that George E. P. Box excels at whatever he attempts. But there is one exception! By his own admission, whenever he and his wife were working in the garden he was relegated to “assistant gardner.” References Box, G.E. P. (1957). Evolutionary operation: A method of increasing industrial evolution. Applied Statistics, 6, 81-101. Box, G.E. P. (Ed.). (2006). Box on process improvement, quality, and discovery. (rev. ed.). New York: Wiley-Interscience. Box, G.E. P., & Cox, D.R. (1964). An analysis of transformations. Journal of the Royal Statistical Society B, 26, 211-252. Box, G.E. P., & Draper, N.R. (1987). Empirical model building and response surfaces. New York: Wiley. Box, G.E. P., & Draper, N.R. (1998). Evolutionary operation: A statistical method for process improvement. New York: Wiley-Interscience. Box, G.E. P., Hunter, J.S, & Hunter, W.G. (2005). Statistics for experimenters: Design, innovation and discovery. (2nd ed.). Wiley: Interscience. Box, G.E. P., Hunter, W.G., & Hunter, J.S. (1978). Statistics for experimenters: An introduction to design, data analysis, and model building. New York: Wiley-Interscience. Box, G.E. P., & Jenkins, G. (1976). Time series analysis: Forecasting and control. New York: Holden-Day. Box, G.E. P., & Tiao, G.C. (1984). (Eds.). The collected works of George Box, vol. II, New York: Chapman and Hall/CRC.

BOSON BOOKS

51


Box, G.E. P., & Tiao, G.C. (1992). Bayesian inference in statistical analysis. New York: Wiley-Interscience. Box, J.F. (1978). R. A. Fisher: The life of a scientist. New York: Wiley. DeGroot, M.H. (1987). A conversation with George Box. Statistical Science, 2, 239258. Pena, D. (2001). George Box: An interview with the International Journal of Forecasting. International Journal of Forecasting, 17 (1), 1-9.

BOSON BOOKS

52


CHAPTER 8 Student: Inventor of Small Sample Inferential Statistics Overview: William Sealy Gosset was educated as a chemist and mathematician at New College, Oxford. His statistical contributions were published under the pseudonym “Student.” He made important contributions to 20th century statistics by introducing small sample inferential methods, including the well-known Student's t-test. He worked for the Guinness Brewery of Dublin and then of London, applying his new methods to their breweries and farms. Student’s Background William Sealy Gosset was born in 1876 in Canterbury, England, and was the oldest of five children. He died in Beaconsfield, England in 1937. Although he planned to follow his father’s path vocationally, becoming a member of the Royal Engineers, faulty eyesight made him change his goals. Being an excellent student, he won a scholarship at Winchester College and then another at New College, Oxford. In the latter he excelled in mathematics and graduated with honors in chemistry in 1899. By chance, Guinness Brewery had at that time decided to introduce more scientific methods into their brewing and were interested in employing young men with high level science degrees from Oxford and Cambridge. So they employed Gosset as a brewer at their Dublin brewery. His entire vocational career would be with Arthur Guinness Sons and Company. It is interesting to note that he was hired as a chemist, not as a statistician. Legend has it that a researcher at Guinness had published a manuscript containing Guinness trade secrets. Because of this the firm forbade publications bearing the names of their employees. Gosset therefore used the pseudonym “Student” for all of his publications. How Student Learned Statistics When Student began work at Guinness he had no knowledge of inferential statistics but possessed a good mathematics background. He learned the theory of errors and common statistical methods after encountering problems in the brewery requiring statistics, by reading Airy’s Theory of Errors of Observation (1879) and Merriman’s Method of Least Squares (1884), and through trial and error. By 1904 he had gained enough understanding of these and related topics to construct an internal report titled, The Application of the Law of Error to the Work of the Brewery.

BOSON BOOKS

53


Luckily, he met Karl Pearson, who in an initial meeting helped him. As a result, Guinness assigned Student to Pearson’s Biometric Laboratory at University College, London for the academic year 1906-1907. Student’s coworkers at Guinness began to recognize his skill and turned to him for problems of data analysis and experimental design. It soon became apparent that the brewery designs must contain small samples, unlike the large sample analyses that Karl Pearson conducted in his Biometric Laboratory (Box, 1987). Student had to be concerned with the rather large errors of estimate that occurred in estimating population means and variances from sample values, whereas in Pearson’s research the sample statistics were essentially the same as the population parameters. The Guinness Brewery would ask very practical questions such as: How can the best flavor be produced? What types of hops should be grown? How can the life of a beer be lengthened, considering that it must be exported? How expensive is it to support the various growth conditions? (Brown, 1980) Student As A Man Student was a modest man. It seems as though all of his coworkers and friends liked him. When complimented on one of his pioneer findings in small sample statistics he might say, “Fisher would have discovered it all anyway.” Although R. A. Fisher and Karl Pearson despised each other, each having a very large ego, Student managed to be a friend to both. Pearson was not appreciative of small sample statistical endeavors, although he was willing to help Student in his statistical work. Fisher, on the other hand, felt that Student had initiated a logical revolution. One of Fisher’s research thrusts was close to Student’s work. In spite of their mutual admiration, in the late 1930's Student and Karl Pearson's son, Egon, were involved in a controversy with Fisher over the utility of nonrandom assignment schemes for experimental designs (Ross, 1988). Fisher argued that balanced assignment schemes designed to make treatment groups as similar as possible necessarily reducing power while Student and Egon Pearson argued persuasively that this was not always true. A number of Student’s close friends admired him for his personal traits. He was thought to be tolerant, helpful, kind, gentle, and yet unconventional. He played golf for a while, using clubs that were of turn of the century vintage. Student also enjoyed walking, cycling, sailing, fishing, skating, gardening, and shooting. He was a good carpenter and built several boats, one of which had rudders at both ends for better control of direction and speed. (McMullen, 1970) But then he experienced a serious automobile accident, overturning his old, high twoseater Model-T Ford, which he called “The Flying Bedstead.” He took several months to heal and retained a limp for the rest of his life. Student enjoyed Gilbert and Sullivan operas, played the penny whistle, and sang songs to his children. He listened to good music, being especially appreciative of Beethoven, who was his favorite composer. It may be thought that Student was well organized in his work, including arithmetical calculations. But in reality Student’s scribbling manifested occasional errors and he tended to conduct calculation on small bits of paper, on stray envelopes, and he even BOSON BOOKS

54


worked on trains or while in the hospital. It is unlikely that his method of work would be suitable for others. It has often been asked whether he was viewed as an advisor or consultant to Guinness or was he a brewer allocating a portion of his time to statistics? The truth apparently is some combination of the two. Why would a person of his genius desire to work in industry? Why did he not gravitate to academia? From time to time he was made offers by colleges and always turned them down, perhaps because his salary was better than that of a college instructor. Student was immersed in his vocation. He corresponded with a number of agricultural and experimental experts around the world and sometimes visited some of them. His letters were lucid, especially on more difficult points, and sometimes displayed the message more clearly than in his published papers. He even managed to run comparative studies with some of his correspondents. In certain of the analyses he conducted he used a special extension of Student’s t-test that yielded results similar to the F test. (Box, 1987) Karl Pearson always pointed out that a statistician, serving as a consultant to persons in the various substantive fields, must learn something about their disciplines, but must refrain from acting like an expert in the other’s area. In a similar way, those whose subject is not statistics and are seeking consultative advice must be willing to learn some of the vocabulary and concepts of statistics. Student avoided these communication problems, partly because of his likable personality and partly because he was enthusiastic about learning the other person’s subject. (McMullen, 1970) Student As A Statistician Although Student’s statistical work might be thought of as extremely complex, in reality it was based on simple elements—the mean, the standard deviation, the variance, and the correlation coefficient. The variance of a sum or a difference also appeared often in his work and he stressed the fact that in this latter equation, where differences were involved, it was more fruitful to increase the correlation between the two variables than to attempt to reduce the variance of X or of Y. (Pearson, 1939) One can get a glimpse of this by comparing the sensitivity of the t-test for paired observations with the t-test for two independent samples. Nine years after Guinness hired Student, he changed the countenance of inferential statistics by publishing two papers in Karl Pearson’s journal, Biometrika. One was titled “The Probable Error of A Mean” (1908a) and the other the “Probable Error of A Correlation Coefficient.” (1908b) With large samples, confidence intervals for population means can be constructed by knowing the sample mean and standard deviation together with sample size, and normal curve properties. In the early history of applied statistics there existed no corresponding result for small samples. Student discovered that when the observations are normal, confidence intervals for the population mean can be developed in a manner analogous to that for large samples. He also showed that exact significance tests and confidence intervals could be derived quite often. The multiplicative values in the small sample case come from Student’s t-distribution rather than from the normal curve. The t-table actually displays a family of curves, each with its own degrees of freedom. In a similar way, the F BOSON BOOKS

55


and χ2 tables are families of curves, F indexed with two sets of degrees of freedom and χ2 with one. The t-distribution with infinite degrees of freedom is the normal curve. The tdistribution and the normal curve are unimodal symmetrical whereas F and χ 2 are positively skewed. E.L. Lehmann (1999, p. 418) has delineated three stages in the establishment of small sample theory. These began in 1908 with the publication of Student’s two papers in Biometrika dealing with the standard errors of the mean and the correlation coefficient, and extended through 1933: (1) sampling distributions under normality; (2) robustness in the presence of nonnormality; and (3) How to choose the best statistic. Lehmann describes Student’s contributions to all three stages as follows Student expressed interest in robustness by considering parent populations that are triangular or rectangular. The third stage mentioned above alludes to choice of statistic and hence the Neyman-Pearson method of testing hypotheses. Here again Student was contributing pioneer efforts. In a letter to Egon Pearson he said that the only valid reason for rejecting the null hypothesis is because some alternative hypothesis is more likely to be true. R. A. Fisher never did invoke the notion of an alternative hypothesis. Of course once one includes an alternative hypothesis in the decision making scheme, ideas such as Type I errors and Type II errors and their probabilities (α and β) come to mind, with the former probability being the significance level and 1-β being “statistical power,” and power is tied up with robustness. Student’s 1908 papers deal with Lehmann’s stage #1. Concluding Comments In his later publications Student tried to omit or at least diminish mathematical concepts, and he would ask colleagues and friends to view him as a non-mathematician. As an example, when he sent Karl Pearson a copy of his Lanarkshire Milk experiment he accompanied it with a note such as the following: I hope you will like this study. To me, its main virtue is that it is essentially devoid of mathematics. (Student, 1931, p. 398) Perhaps this feature will not impress you. And then again when writing to Egon Pearson he said that there was little of Karl Pearson’s famous χ2 paper that he could understand. In 1935 he left Dublin to become the Head Brewer of Guinness Brewers and Sons in London. When he died of a heart attack in October of 1937, he was survived by a grandson, his children, his wife, and both of his parents. References Airy, G.B. (1979). On the algebraic and numerical theory of errors of observations and the combinations of observations. (3rd ed.). London: Macmillan. Box, J.F. (1987). Guinness, Gosset, Fisher, and small samples. Statistical Science, 2(1), 45-52. Brown, J.F. (1980). Guinness and hops. Arthur Guinness Son and Co. (Central Park), Ltd., London. Lehmann, E.L. (1999). “Student” and small sample theory. Statistical Science, 14(4), 418-426.

BOSON BOOKS

56


McMullen, L. (1970). “Student” as a man. In E. S. Pearson and M.G. Kendall (Eds.). Studies in the history of statistics and probability. (355-360). London: Griffin. Merriman, M. (1884). A textbook on the method of least squares. New York: Wiley. Pearson, E. S. (1938). “Student” as a statistician. Biometrika, 30, 210-250. Pearson, E. S. (1990). Student: A statistical biography of William Sealy Gosset. Oxford: Claredon Press (Edited and augmented by R.L. Plackett with the assistance of G.A. Barnard). Pearson, E. S., & Kendall, M.G. (1970). Studies in the history of statistics and probability. London: Griffin. Read, C. B. (1988). Studentization. In S. Kotz and N.L. Johnson (Eds.-in-Chief). Encyclopedia of Statistical Sciences. (35-36). New York: Wiley. Ross, D.C., & Klein, D.F. (1988). Group matching: Is this a research technique to be avoided? Educational and Psychological Measurement, 48, 281-295. Student (W.S. Gosset). (1908a). The probable error of a mean. Biometrika, 6, 1-25. Student (W.S. Gosset) (1908b). Probable error of a correlation coefficient. Biometrika, 6, 302-310. Student (W.S. Gosset). (1926). R. A. Fisher: Statistical methods for research workers. Eugenics Review, 18, 148-150. (book review) Student (W.S. Gosset). (1931). Lanarkshire milk experiment. Biometrika, 23, 398-406.

BOSON BOOKS

57


CHAPTER 9 Egon Sharpe Pearson: Contributions to Statistical Inference Overview: Some say that Egon Sharpe Pearson’s greatest contribution was as a teacher and advisor. But he was also noted as a researcher, an historian of statistics, an editor, and an administrator. He served ably as Managing Editor of Biometrika for three decades. He and Jerzy Neyman developed the Neyman-Pearson theory of hypothesis testing. Early Years and Educational Training Egon Sharpe Pearson was born in August of 1895 in London and died in June 1980 in Midhurst, Sussex. He was the only son of the British statistician, Karl Pearson. Egon had two sisters, Sigrid and Helga. He initially took his education at Dragon School, Oxford and then attended Winchester College where he graduated in 1914. Although he was accepted for study at Trinity College, Cambridge, he had health problems including a heart murmur that kept him from enlisting in the military and a siege of influenza that interrupted his studies at Trinity College. Since he was a tall man, over the years he had back problems. When Egon returned to studies he didn’t focus directly on statistics but rather on physics and astronomy. He attended lectures by George Yule, F.L. Engledow, A.S. Edington, and F.J.M. Stratton. After World War I he took a job in his father’s department, the Department of Applied Statistics at University College, London. Karl Pearson didn’t permit him to teach initially and assigned to him the research projects to be conducted. Egon stayed in that capacity from 1921 until 1926. Egon Pearson As Teacher and Administrator When Karl Pearson began to experience health problems he asked Egon to begin teaching. Perhaps because the son didn’t undergo the rigorous mathematical training that Karl had expected, Egon seemed to have good communication skills as a teacher and he was a good advisor too. He showed patience in thrashing out problems. Furthermore, this lucidity carried over to his scientific writing. Egon Pearson was somewhat overwhelmed by his father’s reputation as a famous statistician and for many years thought his father could do no wrong. Egon had been especially impressed with Karl’s pioneer research on frequency curves. But then R. A. Fisher challenged Karl Pearson on the degrees of freedom for the chi square statistic and a number of other issues, so that a vicious rivalry occurred. Egon then realized that his father was fallible, although he was angry with Fisher for insulting his father and BOSON BOOKS

58


undermining his reputation. On the other hand, he respected Fisher for his knowledge even though he couldn’t understand all of the mathematics in Fisher’s equations. Egon Pearson came to know that he needed to get out from under his father’s protection. He needed to develop his own philosophy of mathematical statistics. It turned out that the three persons who would help him to do this were: Student (W.S. Gosset), R. A. Fisher, and Jerzy Neyman. Karl Pearson retired in 1933 and the department he governed with an iron hand was split into two areas: Department of Eugenics, which was headed by R. A. Fisher and Department of Statistics, headed up by Egon Pearson. By chance Fisher took over Karl Pearson’s office and immediately did away with the scientific embellishments on the office walls, such as the array of skulls, that Karl had enjoyed. A year later Egon hired Jerzy Neyman. This was designed in part to speed up the research project that they had previously been obliged to conduct through mail correspondence. R. A. Fisher’s fiducial limits approach had been the primary model for hypothesis tests but they would be overthrowing it before long. Egon Pearson and Student were on friendly terms. Student also communicated with R. A. Fisher and Karl Pearson. Student was always on the lookout for statistical techniques applicable to small samples. At Guinness Brewery the experimental designs they employed implied small sample sizes in the cells. But just as Student was seeking useful information from these other statisticians, he was bestowing his wisdom on them. One idea he gave to Egon that helped to further the Neyman-Pearson research was to include an alternative hypothesis. In a letter he said to Egon that the only reason for rejecting the null hypothesis is because some alternative hypothesis is more likely to be true. Once Egon and Jerzy Neyman included this alternative hypothesis, which is the idea Fisher overlooked, concepts such as type 1 error and type 2 error and their respective probabilities were suggested in a natural way. Another idea Student gave to Egon Pearson was that it was necessary to carry out studies of robustness. He was interested in seeing studies conducted with parent populations that are rectangular, skewed, and triangular. In Student’s vocation, with the studies involving small samples, one can easily see why research into robustness would be of great interest. The presence of non-normal populations would be likely to have a greater impact on sampling distributions when sample sizes are small. His studies dealt with chemical analysis or barley breeding. Karl Pearson, like most other scientists, liked Student but wasn’t interested in small sample studies. In his biometric research the samples were always large so there was almost no error in estimating the parameters from the statistics. He once jokingly exclaimed to Student: “Only naughty brewers deal in small samples.” Reviews of R. A. Fisher’s Statistical Methods for Research Workers The published reactions to the first edition of R. A. Fisher’s Statistical Methods for Research Workers (SMRW) (1925) are structured as follows: First comes Egon Pearson’s review (1926) and it is followed by Fisher’s reply (1927). Reviews of the second edition of the same book (1928) began with Egon Pearson’s review (1929a), followed by Student’s letter (1929), which is in the form of a review, followed by

BOSON BOOKS

59


Fisher’s (1929) reply, followed by Egon Pearson’s rejoinder (1929b). There were also a great number of private letters that were never published. Egon contended that the book was too mathematical for the intended reading audience and that it was unlikely that the reader of SMRW could subsequently deal with problems different from those included in the book. In other words there would be little if any transfer of training. He went on to say that the book contained short cuts without thorough grounding. To really understand it the reader would have to read some of Fisher’s published papers. Yet two other criticisms were that the proofs based on degrees of freedom were not convincing and the treatment of the correlation ratio was inadequate. It appeared that in Egon Pearson’s last two comments he was defending his father. Fisher’s main reply was to speak against the correlation ratio at length, demeaning it. Pearson’s review of the second edition of SMRW indicates that since Fisher’s book gives a thorough treatment of small sample techniques, but without attending a great deal to the assumption of normality, a greater focus should have been on normality. In other words, the important normal curve assumption was neglected. Egon continued by pointing out that the book was too difficult for the intended reading audience. And, once again, the reader would need to peruse some of Fisher’s published papers in order to understand SMRW. Another criticism was that although Fisher speaks of small sample distributions as being “exact,” he fails to discuss how far the the population distribution drifts from normality before they become “inexact.” Finally, Egon points out that Fisher has given the impression that the book will solve all of the researcher’s statistical problems, and that is misleading. And Student’s reaction to the second edition of SMRW is that the book implies that the small sample techniques will be exact even in the presence of non-normality. And he asks how Fisher would modify his statistical tables if the population shape were neither symmetrical nor mesokurtic. R. A. Fisher’s rejoinder to these two statisticians is that he disagreed with the idea of modifying his tables and gave three reasons why such modified tables would be exposed to criticism. In doing so he threw a criticism at Karl Pearson’s system of distributions. He said there were examples in Student’s writing and his own involving frequency or order data that were free from all taint of normality but lacked the sensitivity to be useful. And Fisher added that in his own research in biology he never encountered non-normality nor had it been reported in the published literature. Egon Pearson’s rejoinder was that the study of non-normal distributions for sample statistics must be a part of the study of biology students for Fisher’s book to be comprehensive. And he said the research work conducted using normal theory proceeded without knowing the shape of the population distribution or having any way to check it, as well as its impact. Neyman-Pearson Theory of Hypothesis Testing One cannot speak of the Neyman-Pearson approach to the testing of statistical hypotheses without also describing R. A. Fisher’s procedure. Historically, there was a great rivalry between Neyman and Fisher. Student played a role in this same rivalry. Before beginning it should be said that there are approaches other than these two that we BOSON BOOKS

60


will name but not provide detailed discussion. One is the construction of confidence intervals, which is closely related to hypothesis testing. Another is Bayesian estimation. Fisher introduced a formal conception of testing the null hypothesis (H0). Neyman and Pearson brought the alternative hypothesis (H1) into the paradigm. (Pearson, 1966). Often the researcher is thinking, I don’t believe in H0 , therefore I do believe in H1. Where does this setup come from? The concept of disbelief in H0 rests on the notion of falsification introduced by the famous philosopher of science, Karl Popper (1902-1994). Popper maintained that we cannot conclusively affirm an hypothesis but we can conclusively negate it. At the height of their powers, back in the 1930s, Jerzy Neyman and Egon Pearson overthrew R. A. Fisher’s fiducial approach and, afterwards, their view of hypothesis testing reigned supreme. It is more likely that modern day statistical inference favors a blend of the two approaches. In the Neyman-Pearson method there are two hypotheses, whereas Fisher’s system advocated the test of significance of just the null hypothesis. Fisher introduced a procedure that led the data itself to produce a significance level where Neyman and Pearson favored a procedure where the α level, where α is the probability of a Type 1 error, is set before any data are gathered. A Type 1 error is rejecting a true H0 and a type 2 error is retaining or “accepting” H0 when it is false. Also, β is the probability of committing a type 2 error and 1-β is statistical power. Power is important in the Neyman-Pearson approach and it is tied up with robustness. George E. P. Box coined the term “robustness.” Recall that Student was pushing hard for studies of robustness, especially for the small data sets that he worked with in the brewery. Concepts such as the two types of decision-making errors and their respective probabilities and power were not a part of Fisher’s plan for testing the null hypothesis. It should be mentioned that later on Fisher suggested using .05 for a “significant” result and .01 for a “highly significant” result. This can be viewed as a slight concession to the Neyman-Pearson theory of hypothesis testing. Here is a scheme that displays Fisher’s test of significance and we follow it with the analogous set up for Neyman-Pearson. We show these as lists to facilitate comparisons. The paradigm of R. A. Fisher is: 1. State H0 2. Chose the test statistic, assuming H0 is true. 3. Compute test statistic. 4. Assuming H0 true calculate p-value for test statistic. The p-value is the area under the curve beyond the test statistic. If p is sufficiently small, reject H0, if not a decision cannot be made. The Neyman-Johnson procedure is: 1. Identify H1, which is sometimes called the research hypothesis. It can be directional (one-tailed) or non-directional (two-tailed). State H0. 2. Choose test statistic assuming H0 is true. 3. Chose a significance level (α) and, assuming H0 is true, calculate the critical value. BOSON BOOKS

61


4. Calculate the test statistic. If p< α, reject H0 and support H1. It is interesting to note that R. A. Fisher didn’t consider the possibility of robustness in his hypothesis testing, just as he refused to discuss the possibility of non-normality in his book, Statistical Methods for Research Workers. Neyman and Pearson discarded Fisher’s idea that the null hypothesis alone is to be tested. For the α and β are long run values, with α being dependent on H0 being true and beta dependent on it being false. Furthermore, the alternative or research hypothesis permits multiple explanations for H0 being false and the one chosen is the very one that maximizes power. Thus their method includes a choice of the best statistical mechanism. Fisher, like Popper, viewed the [null] hypothesis as a tentative explanation designed to give the data a chance to nullify it. Much of the language of hypothesis testing still resides with Fisher’s original formulations. Egon Pearson’s 30-Year Service as Editor of Biometrika Egon Sharpe Pearson assumed the editorship of Biometrika when his father, Karl, died in 1936. Egon then served as Managing Editor until late 1965. Thus he was editing this prestigious scientific journal for almost three decades, the journal his father had cofounded with Francis Galton and Walter Weldon in 1900. Egon and Karl had therefore edited the journal for almost two-thirds of a century. When Egon stepped down as Managing Editor, the Biometrika Trustees awarded him with a volume titled The Selected Papers of E. S. Pearson (1966). This book contains 21 of his published papers. And there is a bibliographic appendix listing 112 of his published scientific papers. The papers he coauthored with Professor Jerzy Neyman were displayed in a separate book. The Trustees spoke of his editorial work as being skillful and with enthusiastic zeal, being responsible for Biometrika’s status in the world of scientific journals. The journal’s financial stability at that time was attributed to Egon’s wise leadership. Publications In Honor of His Father With the collaboration of L.J. Comrie and H.O. Hartley, Egon Pearson revised Karl Pearson’s Tables for Statisticians and Biometricians, a two-volume set. He purposely added new tables to Karl’s work including additional calculations that Egon himself conducted. Most importantly, he edited The History of Statistics in the Seventeenth and Eighteenth Centuries. New York; Macmillan. (Lectures by Karl Pearson given at University College, London, during the academic sessions 1921-1933.). Egon published this work in 1979, the year before he died. In completing this project, he was keeping a promise to his father. This volume contains rich editorial embellishments by Egon that makes the work more attractive. Egon Pearson also wrote a biography of Karl Pearson (1938) and a summary of his father’s early statistical papers (1956). He published three papers in Biometrika which relate to late nineteenth century and early twentieth century history of statistics. This series of journal articles and the books mentioned above reflect Egon Sharpe’s impressive contributions to the history of his discipline. One dealt with the correspondence among BOSON BOOKS

62


his father, “Student,” and Fisher (1968), the second surveyed the early history of biometry, and statistics (1965), and the third focused on continuity in the development of mathematical statistics (1967). Egon and Maurice Kendall edited a collection of papers describing the history of statistics and probability (1970). Concluding Comments Egon Sharpe Pearson initially seemed shy, and was perhaps overwhelmed by his father’s reputation as a renowned statistician. It was Karl Pearson’s initiation of a Statistics Department that is the reason that Statistics Departments now exist in many colleges in Great Britain and America. During the years Egon held a lectureship in his father’s Departments he was still living at home, and most of his friends were relatives. The stimuli that enabled him to move away from his father’s domination and to form his own philosophy regarding mathematical statistics were the personal interactions and the letters with Jerzy Neyman, R. A. Fisher, and Student. Of course, he had mixed feelings about Fisher. On the one hand he respected Fisher’s research. But it angered him that Fisher attacked his father in such a brutal manner. And yet he could see that Fisher’s position on certain issues was more correct. Of course, after a while Neyman and Fisher became enemies. Student was always the neutral friendly fellow and everybody liked him. Egon Sharpe Pearson produced an impressive publication record, part of it coauthored by Jerzy Newman. He was awarded the Weldon Medal in 1935. In 1946 a C.B.E. was bestowed on him for his service in World War I. In 1955 he was awarded the Gold Medal of the Royal Statistical Society and he was their President in 1955-1956. In 1966 he became a Fellow of the Royal Statistical Society, although some felt he should have been given that honor at an earlier time. In speaking and writing ability Egon Sharpe had a very simple and clear style, showing restraint, supported by a deep knowledge. Some adjectives that were sometimes associated with Egon Sharpe Pearson’s communication abilities are: clear; fond of simplicity; precise; restrained. He was a man who would avoid drawing conclusions that go beyond the evidence at hand. He was an exceptional model for his students and colleagues. References Biometrika Trustees (1966). The selected papers of E. S. Pearson. Berkeley, CA: University of California Press. David, F.N. (1983). Egon Sharpe Pearson. In S. Kotz & N.L. Johnson (Editors-inChief). Encyclopedia of the Statistical Sciences (pp. 650-653). New York: Wiley. Fisher, R. A. (1927). Statistical methods for research workers. Science Progress, 21, 340-341. (Reply) Fisher, R. A. (1929). Statistical methods for research workers. Nature, August 17th, 266-267. (Reply) Pearson, E. S. (1926). [Review of the book Statistical methods for research workers], by R. A. Fisher]. Eugenics Review, 20, 733-734. BOSON BOOKS

63


Pearson, E. S. (1929a). [Review of the book Statistical methods for research workers, by R. A. Fisher]. Nature, June 8, 866-867. Pearson, E.S (1929b). Statistical methods for research workers. Nature, October 19th, 615. (Rejoinder) Pearson, E. S. (1938). Karl Pearson: An appreciation of some aspects of his life and work. Cambridge: Cambridge University Press. Pearson, E. S. (1939). “Student” as a statistician. Biometrika, 30, 205-250. Pearson, E. S. (Ed.). (1956). Karl Pearson’s early statistical papers. Cambridge: Cambridge University Press. Pearson, E. S. (1965). Some incidents in the early history of biometry and statistics, 1890-1894. Biometrika, 52, 3-18. Pearson, E. S. (1966). The Neyman-Pearson story: 1926-1934. Historical sidelights on an episode in Anglo-Polish collaboration. Festschrift for Jerzy Neyman. New York: Wiley. Pearson, E. S. (1967). Some reflexions on continuity in the development of mathematical statistics, 1885-1920. Biometrika, 54, 341-355. Pearson, E. S. (1968). Some early correspondence between W.S. Gosset, R. A. Fisher, and Karl Pearson, with notes and comments. Biometrika, 55, 450-457. Pearson, E. S. (Ed.). (1979). The history of statistics in the seventeenth and eighteenth centuries. (Lectures by Karl Pearson given at University College and University of York) New York: Macmillan. Pearson, E. S. (1990) “Student”: A statistical biography of William Sealy Gosset. Oxford: Clarendon Press (edited and augmented by R.L. Plackett with the assistance of G.A. Barnard). Pearson, E. S., & Kendall, M.G. (Eds.). (1970). Studies in the history of statistics and probability. London: Griffin. Pearson, E. S., & Wishart, J. (1942). “Student’s” collected papers. University College, London: Biometrika Office. Student (W.S. Gosset). (1929). [Review of the book Statistical methods for research workers]. Nature, July 20th, 866-867.

BOSON BOOKS

64


CHAPTER 10 Francis Ysidro Edgeworth: Statistician and Economist O v e r v i e w : Francis Ysidro Edgeworth was initially named Ysidro Francis Edgeworth but he decided to transpose his first two names. Until the age to enter the university, he did not attend a formal school but was schooled by private tutors in his own home. At the age of 17 he matriculated at Trinity College, Dublin studying French, Spanish, German, and Italian. He went on to Oxford and graduated in 1869 with a First in L i t e r a e Humaniores. There is some mystery as to what he did after leaving Oxford, but it is known that he had little financial support and lived in London. Apparently, he studied law, for he was called to the Bar by the Inner Temple in 1877. Somewhere in his unusual educational career, perhaps when he was immersed in legal studies or at Oxford, he learned a great deal about mathematics, statistics, physics, probability, and related disciplines. It appears that he was self-taught in these fields. Being a distant cousin of Sir Francis Galton, his knowledge of statistics strengthened markedly through their frequent conversations, although they did not conduct research together. Edgeworth’s great intellectual versatility labeled him by many as a polymath or a Renaissance man. Educational Training and Vocational Experiences Francis Ysidro Edgeworth was born into a large, commercially sound, creative family in Edgeworthstown, Ireland in 1845 (Keynes, 1933). He died in Oxford, United Kingdom in 1926. He was the fifth of six sons and all of the other heirs died before him, so that he inherited Edgeworthtown in 1911. A number of F. Y. Edgeworth’s forebears had displayed talent in the field of literature. Francis was educated by tutors at his family home, never engaging in formal schooling until 17 years of age, at which time he attended Trinity College, Dublin specializing in ancient and modern languages. While at Trinity Edgeworth achieved prizes in Greek Prose and Verse Composition and was thought by some to be the best scholar in his class (Stigler, 1986, p. 289). From there he matriculated at Oxford on a scholarship, beginning in 1867, where he achieved a First Class in classics. In the study of his early years he never showed interest in mathematics or statistics. Edgeworth was an Irish and British economist and statistician of Irish, Spanish, and Huguenot descent. There is some question about his whereabouts and activities in the years following his stay at Oxford, although he must have studied law, for he was called to the Bar in 1877. He never did practice law but in 1880 took a position as Lecturer in Logic at King’s College in London. He was promoted to Professor in Political Economics BOSON BOOKS

65


in 1888 and was appointed to the Tooke Chair in Economic Science in 1890. In 1891 he left London and assumed the chair of Drummond Professor of Political Economy and fellow of All Souls at Oxford, retaining those positions until retirement in 1922, at the age of 77. Francis Edgeworth lectured on topics such as Greek, English theater, logic, and moral sciences. Galton held Edgeworth in high esteem and invited him to accept a Professorship at the Royal Institution in London, but Edgeworth had already agreed to the Oxford appointment (Stigler, 1986, p. 291). Syllabi for Edgeworth’s 1885 Lectures at King’s College and his 1892 Newmarch Lectures at University College, London can be seen in Appendices A and B of Stephen Stigler’s The History of Statistics: The Measurement of Uncertainty before 1900 (1986, pp. 367- 369). Francis became the very first editor of the Economic Journal in March of 1891 and directed the journal with professional care until 1911, at which time John Maynard Keynes assumed the post. Edgeworth returned as coeditor in 1919 and was active in this editorial role until his death in 1926. F. Y. Edgeworth invited Karl Pearson to submit a paper to his new journal but was unsuccessful (Stigler, 1986, pp. 327-328). To see two letters from Edgeworth to Karl Pearson along this line, together with the speech Pearson made at the February 1926 Galton dinner in honor of Edgeworth, consult Kendall (1970, pp. 261-262). Pearson organized a series of such annual biometric dinners in honor of Francis Galton. The first was in 1920 and they extended to 1933 when Karl Pearson retired (Porter, 2004, p. 275). Pearson gave eloquent speeches at these occasions. It is difficult to locate information of a personal nature on F. Y. Edgeworth. It is well known, however, that he never married and in his will he required that his handwritten manuscripts be destroyed (Kendall, 1968). Also, he was very well read and enjoyed bicycle riding, walking, mountaineering, golf, and boating. Accurate records of Edgeworth’s studies in mathematics and statistics are not available. Apparently he self-studied mathematics during the periods he was at Oxford and subsequently studying law, achieving the equivalent of a university curriculum in that discipline. It is thought that Francis Edgeworth mastered algebra and the calculus and was conversant with the calculus of variations by the age of 30. Soon after he was called to the bar he published New and Old Methods of Ethics (1877), which displayed his talent in mathematical physics, psychophysics, mathematical economics, physics, and the theory of probability (Stigler, 1986, p. 290). By 38 years of age, he produced his first paper on probability, one that dealt with the law of error. From the measurement of utility in economics, he moved to the measurement of belief, and subsequently became a probabilist and a statistician (Kendall, 1970, pp. 258-259). His earliest publications disclosed a mathematical talent (Stigler, 1986, p. 305). Edgeworth defended and employed inverse probability or Bayesian probability in several of his publications. (Dale, 1999, pp. 439-447). A marriage of his view of probability with his statistical investigations appears in an article titled “Probability” which was published in the eleventh edition of the Encyclopedia Britannica in 1911. Egon Pearson published a paper titled “Some Reflexions on Continuity in the Development of Mathematical Statistics, 1885-1920.” (Egon, 1970, pp. 339-353). It is

BOSON BOOKS

66


helpful in linking the scientific contributions of Galton, Edgeworth, Karl Pearson, Gossett, and Fisher. Francis Bacon and Francis Ysidro Edgeworth Francis Galton was co-founder with Walter Weldon and Karl Pearson of the prestigious journal Biometrika (Reid, 1982, p. 55). It was designed to publish papers in which the methodology of statistics or mathematics was applied to substantive biological research. Pearson became its first editor in 1901 and held the post until he retired in 1933. The Royal Society had declined such papers, and so Biometrika filled a gap in the literature. Galton was a half-cousin to Charles Darwin. Darwin’s The Origin of The Species was published in 1859 and it led Galton to study human intelligence (Fisher, 1978, p. 24). R. A. Fisher followed Galton along the same path in being aware of the new opportunities evolutionary theory provided. Galton communicated with younger men, especially with Walter Weldon and Karl Pearson about biometric problems in his research (Fisher, 1978, p. 65). Pearson was stimulated by both Weldon’s interest and the publication of Francis Galton’s Natural Inheritance (1889), and he enthusiastically moved in the direction of biometrics. It should be mentioned that Karl Pearson sometimes failed to give credit to Edgeworth when credit may have been due. Further, Pearson claimed that Galton and Darwin committed many errors but nevertheless their research led to progress (Reid, 1982, p. 172). And Pearson failed to give credit to Galton for discovery of the correlation coefficient. Apparently Edgeworth’s conversations with Galton in the early 1880s strengthened the former’s interest in statistics, for Edgeworth went on a spree publishing 40 notes and articles on probability and statistics, a slim volume titled Metreike (1887), and numerous book reviews during the decade from 1883 to 1893 (Stigler, 1886, p. 307). Galton was a eugenicist and an anthropologist and he conducted original research in a number of other disciplines. He was one of the first to recognize the importance of Darwin’s theory of evolution. Francis Galton coined the word “eugenics” and defined it as the study of planned human improvement through selective mating. Galton published important research on applied statistics, biometrics, human psychology, and geography, as well as on criminology and meteorology (Stevens, 2000, p. 627). He relied heavily on the normal curve in his research. By 1885 Francis Galton was using linear regression for the main statistical technique in studying heredity (Porter, 2004, pp. 257-258). It should be mentioned that Galton preferred to work independently and never took on students. Sir Francis Galton attended medical school but did not complete his training, and experienced a mental breakdown in the early 1840s. He was widely travelled and received a gold medal from the Royal Geographical Society. Galton was the first to formulate a measure of correlation and it was subsequently formalized mathematically by Karl Pearson. Galton established the first mental test center in London, where measurements could be taken of sensory, reaction time, and physical variables. (Eysenck, 1983, p. 254). Galton became president of the Eugenics Society of London in 1907. The statistical methods developed by Galton and Pearson were employed in studies conducted at the Galton Laboratory. When Francis Galton died in 1911, funds were bequested from his

BOSON BOOKS

67


will that led to Karl Pearson’s appointment as the first Galton Professor of Eugenics (Porter, 2004). Francis Ysidro Edgeworth’s Research and Publications Edgeworth’s most famous publication was Mathematical Psychics: An Essay on the Applications of Mathematics to the Moral Sciences (1881), and his last publication, Papers Relating to Political Economy (1925), appeared the year before his death. Edgeworth did pioneering research in mathematical economics, statistics, general equilibrium theory, index numbers, probability, regression, and correlation (Oppenheimer, 1983). Alfred Marshall, who was John Maynard Keynes’s teacher, and W. Stanley Jevons (Jevons, 1881) both gave favorable reviews to Edgeworth’s Mathematical Psychics, although Jevons found the book difficult to read. Marshall detected clear signs of genius in the book and Jevons found it very remarkable and saw it as containing unquestionable power and originality (Stigler, 1986, p. 306). As a result of reading the book together with Jevons’s review, Galton wrote a letter to Edgeworth praising the book highly, and saying that he disagreed with Jevons regarding the difficulty in Edgeworth’s writing style and said that the topics themselves were inherently difficult (Stigler, 1986, pp. 306-307). By coincidence, Edgeworth and Galton both gave important papers on the 12th of September 1885 meeting of the British Association for the Advancement of Science at Aberdeen (Stigler, 1986, p. 313). Galton’s address, which was more noteworthy, was titled “Regression Toward Mediocrity in Heredity Stature.” Although it is well known that Galton introduced the word “regression” to the world in his speech, it is seldom recalled that two days later Edgeworth demonstrated the analysis of variance for a two-dimensional design (Stigler, 1986, pp. 300, 313). Edgeworth was never elected to the Royal Society. Yet Francis Galton attempted to gain election for him on three occasions. The problem, apparently, was that one could not indicate a single essay of Edgeworth’s that possessed both power and originality. And yet, Francis Edgeworth’s influence was great—both in the application and the theory of statistics. Stephen Stigler felt that Edgeworth’s influence on statistical theory was appreciable and hasn’t been duly acknowledged. His development of the theory of estimation, study of families of curves (a precursor to Karl Pearson’s work), investigation of multiple correlation surfaces, work with goodness of fit tests, and advocacy of inverse probability all strike favorably. It should be mentioned that from 1912 to 1914 Edgeworth served as President of the Royal Statistical Society (Stigler, 1986, pp. 291, 310). Following Stigler’s 1986 paper is a Discussion section. Those who participated, some statisticians and some economists, were: R. L. Plackett, D. M. G. Wishart, Maurice G. Kendall, T. Lewis, Churchill Eisenhart, P. R. Fisk, and P. R. Freeman.These scientists evaluated both Edgeworth’s scientific career and Stigler’s paper. In general, all of these reviewers had something good to say about at least portions of Stigler’s publication. And they all admired Edgeworth’s work, but they weren’t as positive as Stigler. Although much has been said about Francis Edgeworth’s difficult prose and even more difficult mathematical communication skills, Galton and some others have contended that Edgeworth was often quite lucid. For evidence of the latter, see, for example his explanation of the additive effects statistical model (Stigler, 1978, pp. 314-315). In a brief

BOSON BOOKS

68


paragraph, without employing mathematical symbolism, Edgeworth clearly conveys the meaning more lucidly than some other texts needing pages of algebraic symbolism. There is no doubt that Francis Edgeworth’s writing and content of speeches require great effort to require a complete understanding. And such effort may lead one to become addicted to his combining poetry, pedantry, science, art, wit, and learning (Keynes, 1933). Even so, more modern commentators may accuse the work of being obscure. An even greater problem readers can have with Edgeworth’s works lies in his mathematical style. One source of the difficulty is organization and another is scarcity of detail. In a sequence of publications in Philosophical Magazine and Mind during the period from October 1883 through the year 1884, Edgeworth applied the statistical methods of the theory of errors to the quantification of uncertainty in the social, and especially in the economic, sciences. In these papers he dealt with a priori and physical probability, data reduction, the philosophy of chance, the law of error, and the method of least squares (Stigler, 1986, p. 307-308). By 1885 Edgeworth’s knowledge of the literature of probability and the method of least squares probably excelled all others in England. Yet another contribution by Edgeworth was the earliest rendition, regardless of form, of the Student t-distribution. (Stigler, 1986, p. 294). Edgeworth showed great variation in his research tastes, some additional examples being bimetallism, examination grades, anthropometry, the attendance of club dinners, the meter of Virgil’s poetry, and statistics of bees. He also concluded that the proportion of Liberal votes to (Liberal + Conservative) in the British Constituencies was approximately normally distributed (Stigler, 1986, p. 297). In his book Metretike he devised a method of measuring probability and utility, where probability is viewed as the basis of inductive reasoning. He had much to say about maxima minima in physics as well as in psychics. Francis Edgeworth invented indifference-curves and utility surfaces, both of which are now accepted concepts in economic theory. One reason his writing is difficult to understand is that he tended to sprinkle his work with both differential equations and terms from ancient languages, possibly on the same page. The use of multisyllable words in his writing carried over to his speech. From the measurement of pleasure, Edgeworth moved to the measurement of utility, and therefore became an economist. Once he became editor of the Economic Journal, he tended to introduce mathematical symbolism into that journal. He would sometimes apologize for such transgressions saying that, regarding such symbols, one can only practice temperance, not abstinence (Kendall, 1970, p. 259-261). Although Edgeworth was often kind on flattering other scientists regarding their discoveries, sometimes they did not reciprocate. For example, he always referred to Karl Pearson’s work with high praise. In particular, he wrote to Pearson saying that the χ2 statistic was a beautiful contribution to mathematical statistics. Sometimes others were given credit for his findings. It has been said that Edgeworth anticipated R. A. Fisher’s pioneering research in maximum likelihood estimation, although Fisher failed to give him any credit (Pratt, 1976). Galton or Weldon should not have been given credit for the development of theory that leads to a system of skew curves. The originator was Francis Ysidro Edgeworth. Alfred Marshall wrote a review of Edgeworth’s Mathematical Psychics (1881) in which he said that Edgeworth could not be considered to be a serious

BOSON BOOKS

69


economist. And this may have driven Edgeworth from writing books to instead producing journal articles. Francis Edgeworth published statistical papers dealing with topics such as probable errors, the law of error, correlation, correlated averages, least squares analysis, theory of errors, rates of birth, deaths, and marriages, elimination of chance, the mathematical theory of banking, relations among social phenomena, symmetrical and asymmetrical frequency distributions, and the factor of chance in competitive examinations. So once again we see his remarkable versatility. Concluding Comments A variety of words or expressions have been attached to Francis Ysidro Edgeworth’s name. Here is a sampling of them: brilliant, difficult, dense, forceful, original, penetrating, lack of orderly presentation of ideas, polymath, Renaissance Man, interesting, versatile, use of too many multisyllabic words, both in written prose and speech, and impressive scholarship. Unlike the other eleven statisticians in this book, Edgeworth’s formal education was not in science, but was in the classics and law. Exactly how he learned mathematics and statistics is likely to be forever a mystery. F. Y. Edgeworth’s knowledge was not as narrow in scope as the others in this volume. It could be that his apparent versatility made him rather thin in both economics and statistics so that his place in posterity will not enable him to reside with the great in either of these two disciplines. For sheer multidimensional knowledge, however, he may stand alone. References Bowley, A. L. (1928). F. Y. Edgeworth’s contributions to mathematical statistics. London: Royal Statistical Society. Dale, A. I. (1999). A history of inverse probability from Thomas Bayes to Karl Pearson (2nd ed.). New York: Springer-Verlag. Edgeworth, F. Y. (1877). New and old methods of ethics. London: James Parker. Edgeworth, F. Y. (1881). Mathematical psychics: An essay on the applications to mathematical statistics. London: Kegan Paul. Edgeworth, F. Y. (1887). Metretike: Or, the method of measuring probability and utility. London: Temple. Edgeworth, F. Y. (1926). Papers relating to political economy, 3 volumes. London: Macmillan. Eysenck, M. (1983). Francis Galton. In A. Bullock & R. B. Woodings (Eds.). Twentieth century culture: A biographical companion (pp. 254-257). New York: Harper & Row Fisher, J. A. (1978). R. A. Fisher: The life of a scientist. New York: Wiley. Galton, F. S. (1889). Natural inheritance. London: Macmillan. Jevons, W. S. (1881). [Review of the book Mathematical Psychics, by F. Y. Edgeworth]. Mind, 6, 581-583.

BOSON BOOKS

70


Kendall, M. G. (1970). Francis Ysidro Edgeworth 1845-1926. In E. S. Pearson & M. G. Kendall. (Eds.). Studies in the history of statistics and probability (257-263). London: Griffin. Keynes, J. M. (1933). F. Y. Edgeworth. In Essays in biography. New York: Harcourt, Brace, & Company. London: Macmillan. Marshall, A. (1881). [Review of the book Mathematical psychics, by F.Y. Edgeworth], Academy, June 18, 1881, p. 457. Pearson, E. S. (1970a). Some incidents in the early history of biometry and statistics, 1890-1894. In E. S. Pearson & M. G. Kendall (Eds.). Studies in the history of statistics and probability (pp. 323-338). London: Griffin. Pearson, E. S. (1970b). Some reflexions on continuity in the development of mathematical statistics, 1885-1920. In E. S. Pearson & M. G. Kendall (Eds.). Studies in the history of statistics and probability. (pp. 339-353). London: Griffin. Porter, T. M. (2004). Karl Pearson. Princeton, NJ: Princeton University Press. Pratt, J. W. (1976). F. Y. Edgeworth and R. A. Fisher: On the efficiency of maximum likelihood estimation. Annals of Statistics, 4, 501-514. Reid, C. (1982). Neyman…from life. New York: Springer-Verlag. Stevens, M. A. (2000). Francis Galton. In Merriam-Webster’s Collegiate Encyclopedia. Springfield, MA (p. 627). Stigler, S. M. (1978). Francis Ysidro Edgeworth, statistician (with discussion). Journal of the Royal Statistical Society (A), 141, 287-322. Stigler, S. M. (1986). The History of Statistics: The measurement of uncertainty before 1900. Cambridge, MA: The Belknap Press of Harvard University Press.

BOSON BOOKS

71


CHAPTER 11 Maurice Bartlett: Time Series and Multivariate Statistics Overview: Maurice Bartlett, like most of the other eleven British statisticians discussed in this book, was good at mathematics, being a graduate with the rank of wrangler from Queen’s College, Cambridge. Two assumptions underlying defensible construction of the F-ratio of the analysis of variance are normality and homogeneity of variance. The Fstatistic is, however, robust with respect to violations of normality but not with those of equal variances. And Maurice Bartlett’s most well-known research contribution is perhaps his test for homogeneity of variance. Most books that deal with analysis of variance include a discussion of his test. It is of interest to examine the relation between R. A. Fisher, who is viewed by many statisticians as the most important statistical scientist of the twentieth century, and Maurice Bartlett. Bartlett was always impressed with Fisher’s research, especially his philosophy of the application of statistics, not just viewing statistics as yet another academic discipline. He also admired Fisher’s geometric approach to many statistical problems. Bartlett learned that Fisher was kindly to those who advocated his work but could be nasty to those who opposed him in any way. Several times Bartlett found errors in R. A. Fisher’s work or discovered procedures that were adverse to the great man’s findings. And the result was that Fisher became very negative. Bartlett married Sheila Chapman in 1957 at a rather late age. They got on very well. She died in 1998. Their daughter, Penny, lives in Devon, United Kingdom. Educational Training Maurice Bartlett was born June 18, 1910 in Chiswick, London. He was raised in a family of modest means. But when he completed elementary school he won a scholarship to the famous Latymer Upper school where he received his secondary education. Maurice then received a scholarship to Queen’s College, Cambridge. There he read in mathematics and graduated with the rank of Wrangler. While at Cambridge he was on the rowing team, and he designed the cover of the college journal. After graduation he remained at Cambridge for a fourth year studying as John Wishart’s first postgraduate mathematics student. In one of the classes John Wishart suggested that the Wishart distribution could be derived using characteristic functions. The very next day Barlett came in and gave him the proof. They coauthored two papers. In addition to attending lectures in statistics given by Wishart, he sat in on the following classes just for fun: Dirac’s lectures on “Quantum Mechanics”: Eddington’s class on “Relativity”; Colin Clark’s BOSON BOOKS

72


“Statistical Sources”: and George Udny Yule’s “Vital Statistics.” Bartlett’s initial interest in probability was sparked at Latymer by reading a chapter in Hall and Knight’s book, Algebra, and was extended further through his studies at Cambridge (Bartlett, 1982, Gani, 2006). Vocational Experience In 1933 Maurice Bartlett was recruited by Egon Pearson to teach statistics at University College, London. This was his first vocational position. The environment there was intellectually stimulating but anxiety provoking. The carryover of the feud between Karl Pearson and R. A. Fisher was still there. Egon Pearson and Jersey Neyman were conducting their research on hypothesis testing, R. A. Fisher had recently been appointed as Galton Professor, and J. B. S. Haldane often visited the Galton Laboratory. In 1934 Bartlett joined Imperial Chemical Industries (ICI) agricultural research station as statistician at Jealott’s Hill. Years later, when he examined his various vocational experiences retroactively, he said that he was happier at Jealott’s Hill than anywhere else. Bartlett became lecturer in mathematics at Cambridge in 1938. With the advent of World War II, he entered into rocket research. After the war he became Chair of Mathematical Statistics at Victoria University of Manchester, and then, in 1960, he succeeded Egon Pearson as Chair of Statistics back at University College, London. His final regular position was as professor of biomathematics at the University of Oxford. He commented that the biology professors complained that the biomathematicians were too mathematical. Bartlett began there in 1967 and left in 1975, when he retired. Maurice Bartlett remained active in statistics while in retirement, visiting the Institute of Advanced Studies, visiting the Australian National University several times. He published on random walks, spatial patterns, and catastrophe models. Although Bartlett was large physically and austere in appearance, he always showed a kindness to students and colleagues. They remembered him with great affection. He was quiet and reserved—a man of great sensitivity (Gani, J., 2006, Olkin, 1989). He was a talented draftsman with a great deal of artistic talent. Research Contributions Maurice Stephenson Bartlett’s most well known statistical finding was his test for the assumption of homogeneity of variance. It appears in almost any textbook that deals with analysis of variance (ANOVA). His test is used prior to the computation of the F-ratio. The other assumption of ANOVA is normality, and it must be satisfied in order for Bartlett’s test to be valid. In simple analysis of variance there is an independent variable, which is the one manipulated by the investigator. And there is a dependent or outcome variable whose values depend on the experimental manipulations. Both assumptions, as well as the F-test itself are computed from the values that comprise the dependent variable. Two examples of independent variables are different teaching methods or administration of different drugs. The corresponding dependent variables might be scores on an achievement test or measures of reaction time. Monte Carlo studies have shown that Bartlett’s test is more powerful than its competitors, which are those of Cochran, Hartley, Cadwell, Box, and the jackknife procedure. Some statisticians claim that preliminary tests of assumptions prior to the F-test of ANOVA are unnecessary. Bartlett’s statistic is closely related to one discovered by Jerzy Neyman and Egon Pearson. BOSON BOOKS

73


While at Imperial Chemical Industries Bartlett was very productive, publishing in a great variety of areas, including: intelligence; inbreeding; robustness of the t-distribution; contingency table analysis; sampling errors; mental factors; genetic factors influencing ability; cotton experiments; nutritive value of milk; and the above test of homogeneity of variance. He published two dozen papers in all! (Gani, 2006). On returning to Cambridge after the war he focused on research in time series and diffusion processes. Bartlett was in the Statistical Laboratory of the Department of Mathematics at the University of Manchester for thirteen years. While there he defined modern courses in mathematical statistics and formulated a Diploma in Statistics. In 1955 he published what is viewed to be his most important book, An Introduction to Stochastic Processes. David Cox called this book Bartlett’s masterpiece, although he said it was a difficult read. In 1960 his second book, Stochastic Population Models in Ecology and Epidemiology, appeared. On succeeding Egon Pearson as Chair of Statistics at University College, London, Maurice Bartlett extended his research on multivariate statistics, statistical physics, stochastic phenomena, and epidemic models (Gani, 2006). Barlett began communicating with Godfrey Thomson about various methods of estimating factors. At the time Thomson was at the University of Edinburgh and was considered to be one of the leading experts in factor analysis in Great Britain, especially for applications of assessing students’ mental abilities. Thomson’s method involved regression, whereas Bartlett’s indicated that estimating the factors for an individual pupil by taking more tests will increase the sample size and thereby yield more information for that individual. The two approaches differed only by a simple mathematical transformation. After writing to Thomson, Bartlett became even more interested in factor analysis. He indicated that the main point that led to the controversy in factor analysis was how real the factors were perceived. Most psychologists, according to Bartlett, viewed these factors as being something real in themselves, whereas the attitude of both Thomson and Barlett was that they were merely useful statistical constructs (Olkin, 1989). According to Bartlett, a statistician pursuing a general statistical philosophy, the first specifies a model, and secondly tests the data at hand against the model. If both physical and mental measurements were taken, and the relation between the two sets of variables were of interest, then the multivariate statistical technique introduced by Harold Hotelling would be applicable. For a more complete treatment of factor analysis by Maurice Bartlett, see his essay, “Factor Analysis in Psychology As A Statistician Sees It” (Bartlett, 1962, pp. 37-48). In September of 1952, Maurice Bartlett delivered a paper titled “Stochastic Processes or the Statistics of Change” at the Joint Conference of the Royal and Manchester Statistical Societies held in Manchester. A stochastic process is sometimes viewed as an area of probability theory and is widely studied. The statistical study of stock-market fluctuations and subsequent measures of rainfall or temperature are examples. The theory of stochastic processes deals with the statistics of change. Some of the older techniques, according to Bartlett, were the statistical analysis of time series, the practice of industrial sampling—where sequential methods were finally developed, so that sampling is continued until an adequate amount of information is acquired for a specified decision— and the manner in which a discrete distribution arises (example: the occurrence of BOSON BOOKS

74


accidents to a single individual). Bartlett gives examples in the following disciplines in his speech: Physics and Communication Engineering; Industry; Biology; and Medicine (Bartlett, 1962, pp. 49-69). Maurice Bartlett delivered the first “R. A. Fisher Memorial Lecture” in Chicago on December 29, 1964. It was titled “R. A. Fisher and the Last Fifty Years of Statistical Methodology” and was published in the Journal of the American Statistical Association (1965). The meeting took place at a joint session of the American Statistical Association, the Institute of Mathematical Statistics, and the Biometrics Society. Bartlett initially spoke of Fisher’s three lines of research—sampling distribution problems, more general principles of statistical inference, and experimental design and analysis. Among other things, he stressed the power and generality of Fisher’s geometric approach. R. A. Fisher’s contributions to exact small-sample theory were mentioned as well as the credit Fisher gave to Student. Bartlett declined to discuss Fisher’s research in genetics, which is given equal importance to his work in statistics and design. He gave the book The Genetical Theory of Natural Selection (1930) as a suitable reference. Until the exact small-sample distributions and their significance tests appeared, only large sample procedures would yield accurate results. Fisher introduced the concept of “amount of information” in estimation theory. In analyzing data one could possibly lose information but never gain any. In his speech Bartlett spent some time on certain controversies in statistics, one prominent example being the rudeness of Fisher regarding Neyman and Pearson’s theory of hypothesis testing. And another being the Bayesian versus non-Bayesian argument. On these and some other debates Fisher did not argue in a fair manner. On the other hand, R. A. Fisher introduced value concepts such as sufficiency, likelihood, and information that even Bayesian or inverse probability advocates could appreciate. Fisher clung to his idea of fiducial inference, which was perhaps his greatest error, far too long. Bayesian statistics, however, are still with us and going strong, thanks to the research efforts of Jeffreys, Savage, Good, Lindley, and others. One of Fisher’s greatest advances was in research design configurations and their statistical analyses. Concepts such as randomization and blocking are key to many types of experiments. In addition to Fisher’s discovery of the analysis of variance, in which the F stands for Fisher, he made contributions to the analysis of covariance, where an experiment can be analyzed, adjusting for an initial variable or covariate. Bartlett concluded by saying that as great a man of science as Fisher was, he was not infallible. And his greatest weakness was the inability to admit when he was wrong. Similar statements could be made about Karl Pearson. Concluding Comments Maurice Stevenson Bartlett lived to be 91, and 55 of those years were spent in making important contributions to statistical science. He was successful as a researcher, teacher, administrator, and constructor of statistical curricula. He published five books and almost 200 research papers, and made numerous conference presentations. Bartlett was elected to a Fellowship of the Royal Statistical Society in 1961 and was the president of that organization in 1966-1967. In 1980 he became an Honorary Member of the International Statistical Institute and Oxford University awarded him the Weldon Medal in 1971. He BOSON BOOKS

75


was President of the Manchester Statistical Society, 1959-60, and was awarded the Guy Medal in Gold of the Royal Statistical Society in 1969. Finally, Maurice Bartlett received Honorary Doctorates from the University of Chicago in 1966 and the University of Hull in 1986. Bartlett’s research specialties were multivariate analysis, time series, stochastic processes, population and epidemic models, spatial statistics, and statistical theory and methods. References Bartlett, M. S. (1935). Contingency table interactions. Journal of the Royal Statistical Society Supplement, 2, 248-252. Bartlett, M. S. (1942). The use of transformations. Biometrics, 3, 39-52. Bartlett, M. S. (1952). The statistical significance of odd bits of information. Biometrika, 39, 228-237. Bartlett, M. S. (1955). An introduction to stochastic processes, (2nd ed., 1966). Cambridge England: Cambridge University Press. Bartlett, M. S. (1960). Statistical population models in ecology and epidemiology. London: Methuen. Bartlett, M. S. (1962). Essays on probability and statistics. New York: Wiley. Bartlett, M. S. (1965). R. A. Fisher and the last fifty years of statistical methodology. Journal of the American Statistical Association, 60, 395-409. Bartlett, M. S. (1982). Chance and change. In J. Gani (Ed.). The making of statisticians. (pp. 42-60). New York: Springer-Verlag. Fisher, R. A. (1930). The genetical theory of natural selection. Oxford, England: Oxford University Press. Gani, J. (2006). Maurice Stevenson Bartlett. In S. Kotz (Founder and Editor-in- Chief). Encyclopedia of Statistical Sciences (2nd ed.)(pp. 369-371). New York: Wiley. Glaser, R. E. (1983). Bartlett’s test of homogeneity of variance. In S. Kotz & N. L. Johnson (Editors-in-Chief). Encyclopedia of Statistical Sciences (pp. 189-191). New York: Wiley. Olkin, I. (1989). A conversation with Maurice Bartlett. Statistical Science, 4, 151- 163.

BOSON BOOKS

76


CHAPTER 12 David Cox: Multivariable Analysis of Failure Rates Overview: Sir David Cox is one of the most well known statisticians of the twentieth and twenty-first centuries. He writes lucidly and yet economically, as is best evidenced by his book Planning of Experiments (1958). He feels that maximum conciseness also maximizes clarity. Cox’s test for randomness is useful in determining whether a sequence of events measured over time is random. The alternative would be that the rate of occurrence shows a pronounced trend. David Cox’s paper on regression models and life tables (Cox, 1972), which is highly useful in medical research, is used to measure the dependence of survival time on a set of independent variables or covariates. Independent variables that might be included are the state of the disease, the time period in which symptoms were manifest prior to the onset of the experimental treatment, and a dummy coded variable depicting the experimental vs. the control conditions. The model can be generalized to other substantive areas, such as those in industry. Educational Background David Cox was born in Birmingham in 1924 and he attended Handsworth Grammar School there. He studied mathematics at St. John’s College, University of Cambridge where he received the Master of Arts Degree. He took his PhD in 1949 at the University of Leeds. When Cox was a student at Cambridge, he attended a short course given by one of the luminaries of Bayesian Probability, Harold Jeffreys, who authored Theory of Probability (1961). He said that Jeffreys was rather incomprehensible and yet intriguing. He also sat in on a class given by J.O. Irwin on the characteristics of a number of wellknown distributions (Reid, 1994, p. 440). Vocational Experience As part of his war commitment, he worked for the Royal Aircraft Establishment (RAE) from 1944 to 1946. He then went on to the Wool Industries Research Association (WIRA) in Leeds in 1946 and remained there until 1950 at which time he accepted an academic position as Assistant Lecturer at the University of Cambridge, where he worked in the statistical laboratory. Cox spoke highly of both students and colleagues at Cambridge, but said that his salary was inadequate. While at RAE Cox was assigned to the department of structural and mechanical engineering. They tested components of BOSON BOOKS

77


aircraft, especially strengths of materials. They also focused on the rates of aircraft accidents. The work at WIRA was funded by government and the purpose was to conduct basic research on problems in industry (Reid, 1994, pp. 439-441). In 1947 David Cox married Joyce Drummond and they have four children and two grandchildren. In the early years of their marriage they couldn’t afford to own a car or to even heat their house. From 1956 to 1966 he was First Reader and then eventually Professor at Birkbeck College in London. Beginning in 1966 he became full Professor of Statistics at Imperial College, London and remained in that post until 1988. He was a member of the Department of Statistics at Oxford University as well as a Warden of Nuffield College beginning in 1988 and remained there until 1994, when he retired. In 1994 he was made a Fellow of Imperial College. Journal and Book Publications Sir David Cox has written or co-authored over 300 research papers and 15 books. The books, like his lectures, are concise and lucid and sprinkled with wit. The reader perusing the most well known of his books, Planning of Experiments (1958), will find that there is a sparsity of equations coupled with fine communication skills. Cox said that there is a certain advantage to writing in a nonmathematical way. It obliges the writer to explain the crucial topics that the student must learn in a qualitative manner. He added that in some ways experimental design is the most interesting feature of statistics. It should be mentioned that his lecture notes are more mathematical than his books and he doesn’t require textbooks. Dr. Nancy Reid, who co-authored The Theory of The Design of Experiments with Cox, interviewed him in October of 1993. The result, which was titled “A Conversation with Sir David Cox,” was published in 1994 in Statistical Science. This book that they coauthored was published in 2006. It is designed for a general audience and sweeps across a number of disciplines. The mathematics level is low and data analysis is minimized, so that the focus in clearly on design. Nevertheless, the book contains appendices dealing with review of algebra and of statistical analysis. The applications are in the fields of medicine, agriculture, physical sciences, and industry. Most related books tend to stress data analysis and minimize design and in this sense the publication by Cox and Reid fill a gap in the literature. David Cox’s research interests are in both applied and theoretical statistical inference and in applied probability. His applied interests are in biostatistics and medicine as well as in the social, physical, and biological sciences and industry. His proportional hazards model (Cox, 1972) is known worldwide in the biostatistical sciences. It is a multivariable analytic technique that makes it possible to identify the differential influence of a set of observed predictor variables, as well as treatment variables, on rate of failure. All variables in the model are taken into consideration including covariates. Adjustments for interactions and time are included. A research paper by Roodnat et. al. (2004) titled “The Cox Proportional Hazards Analysis in Words: Examples in the Renal Transplantation Field,” gives non-mathematical examples in the area of medical transplantations. Another research contribution of David Cox that is often cited in the literature is the Cox test of randomness (Cox, 1955). It is useful for determining whether a sequence of

BOSON BOOKS

78


events over time is random. The alternative hypothesis would be that there exists a trend in the rate of occurrence. Prizes, Medals, Awards, Memberships, Fellowships, and Knighthood Karl Pearson was one of the founders and the initial editor of the journal Biometrika. When he retired his son, Egon Pearson, became editor. And when Egon stepped down David Cox assumed the role of editor of Biometrika, and retained it until 1991. The journal completed a century of successful and prestigious existence in the year 2001. At that time Titterenton, the present editor, and Cox published a book titled Biometrika One Hundred Years (2001) and then Stephen Senn reviewed it in 2002. David Cox was awarded his first honorary doctorate in 1982 from the University of Reading and his most recent from the Federal University of Rio de Janiero in 2000. In between these two events he received 16 additional honorary doctorates. Cox only came to America once, during 1955-1956, where he was a Visiting Professor at the University of North Carolina. He was the President of three organizations, viz., the prestigious Royal Statistical Society (1980-1982), the Bernoulli Society (1979-1981), and the International Statistical Institute (1995-1997). Queen Elizabeth II knighted David Cox in 1985. Among his academic prizes were the Guy Medal in Silver (1961) and in Gold (1973) of the Royal Statistical Society. Some others were the Weldon Memorial Prize (1984), the Kettering Prize and Gold Medal for Cancer Research (1990), and the Max Planck Forschungspreise (1992). In 1998 Harvard University recognized him with the Marvin Zelen Leadership Award. His Fellowships and Honorary Memberships were appointed by Imperial College, Birkbeck College, the Royal Society, American Academy of Arts and Sciences, Institute of Actuaries, Nuffield College, International Biometric Society, National Academy of Sciences, and the America Philosophical Society. “So how does Sir David Roxbee Cox feel about all of these honors that have been bestowed on him?” asked Dr. Nancy Reid in their Statistical Science interview (1994). He replied that when he was young he was pretty much ignored, but now he was being showered with “bizarre excess.” Concluding Comments Unlike many statistical scientists, David Cox has good communication skills, both as a lecturer and a writer of books. He is able to write a book such as Planning of Experiments that is almost devoid of equations. The influence he has had on graduate students, colleagues, and other researchers is also notable. He has been bombarded with prizes and awards of various types, even outstriping those bestowed on the great R. A. Fisher. His reaction to these honors is that they are a superfluity of awards. He would have liked to have some more attention paid to him when he was young and unknown. The research publication he is most proud of is the one that introduced the proportional hazards model (1972) and that is also the one that has impressed other scientists the most. The Guy Medal in Gold of the Royal Statistical Society came the very next year. And BOSON BOOKS

79


because many of this model’s applications are in the field of medicine, this contribution may have been a strong factor in his being awarded the Kettering Prize and Gold Medal for Cancer Research. References Cox, D. R. (1955). Some statistical methods connected with series of events (with discussion). Journal of the Royal Statistical Society. Series B, 17, 129-164. Cox, D. R. (1958). Planning of experiments. New York: Wiley. Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society. Series B (Methodological), 34(2), 187-220. Cox, D. R., & Reid, N. (2006). The theory of the design of experiments. London: CRC Press. Davison, A., Dodge, Y, & Wermuth, N. (Eds.). (2005). Celebrating statistics: Papers in honor of Sir David Cox on his 80th birthday. Oxford: Oxford University Press. Jeffreys, H. (1961). Theory of probability (3rd ed.). Oxford: Clarendon Press. Reid, N. (1994). A conversation with Sir David Cox. Statistical Science, 9(3), 439-455. Roodnat, J. I., Mulder, P. G., Tielens, E. T., Van Riemsdick, I. C., Van Gelder, T., & Weimar, W. (2004). The Cox proportional hazards analysis in words: Examples in the renal transplantation field. Transplantation, 77(4), 483-488. Senn, S. (2002). [Review of the book Biometrika One Hundred Years]. International Journal of Epidemiology, 31, 269-271. Titterenton, D. M., & Cox, D. R. (2001). Biometrika One Hundred Years. Oxford: Oxford University Press.

BOSON BOOKS

80


SUMMARY Perhaps it has come as a surprise to many of the Behavioral, Social, Biological, and Educational Researchers reading this book that so many of the statistical models presented seem to have been discovered long ago by these twelve world-class British statisticians. Although some of these methods were initiated long ago, the pioneers did not always finalize them. Picasso once said that you do something, and then somebody else comes along and does it pretty. Considering the fact that most of these statistical scientists didn’t have access to high speed computers or the associated statistical packages, such as the Statistical Analysis System (SAS), and the Statistical Package for the Social Sciences (SPSS), one has to wonder how they got so much accomplished. If one were to summarize each of their main accomplishments in just a few words, the results might be: Karl Pearson- product-moment correlation; The Grammar of Science; 18 publications in Philosophical Transactions of the Royal Society. R. A. Fisher- analysis of variance and covariance; experimental design; derived several sampling distributions. Charles Spearman- factor analysis; theory of intelligence; mental test theory Florence Nightingale David-women’s vocational rights in statistics; very fine teacher. George Udny Yule-time series; An Introduction to the Theory of Statistics; contingency table analysis. Maurice G. Kendall-Advanced Theory of Statistics; generation of and tests for random numbers; time series. George E. P. Box-fine teacher; statistical quality control; analysis of time series. William Sealy Gosset (“Student”)-developed small sample statistical techniques; made important contributions to Neyman-Pearson theory. Egon Sharpe Pearson-Neyman-Pearson Theory; history of statistics; good editor, administrator and, teacher. Francis Ysidro Edgeworth-expertise in many disciplines; a polymath; Mathematical Psychics; editor of the Economics Journal; self-taught in mathematics and statistics. Maurice S. Bartlett-His research spanned fifty-five years and covered a wide scope within statistics and related disciplines including stochastic processes, epidemiology, and time series. David Cox-His multivariable model can be used to determine the influence of many variables, including covariates and treatment variables, on survival rate. The recent investigations by George Box solved some of the problems that confronted the earlier statisticians. “Robustness,” is a term that he coined. His research in this area, including his treatment of nonlinear transformations, clears up many of the concerns BOSON BOOKS

81


statisticians like Student and Egon Pearson and others had. The present day work of Leo Goodman on multidimensional contingency tables, where analysis of variance-like models are applied to frequency data, far supersedes the classical approaches used by Karl Pearson and George Yule. It is now apparent that Lord and Novick’s (1968) investigations of mental test theory extends the classical measurement model developed by George Yule and Charles Spearman and explores other more modern mental test theory paradigms. And it is now realized that the F ratios of analysis of variance and covariance proposed by Fisher are sensitive to sample size and so a more comprehensive analytic technique would be to support them with appropriate correlational methods. Joreskog’s (1984) factor analytic mode easily supersedes the classical model proposed by Charles Spearman. The depth and width of intellect displayed by Francis Ysidro Edgeworth are not likely to be encountered again. Maurice Bartlett’s research longevity and scope across sub-disciplines within statistics are impressive. David Roxbee Cox’s most widely cited contributions are the proportional hazards model, the test of randomness, and his book Planning of Experiments. END OF BOOK

BOSON BOOKS

82

Twelve British statisticians

Twelve

Twelve

Twelve

Twelve

Twelve

Numerical analysis for statisticians

Numerical analysis for statisticians

Twelve

Twelve

Probability for Statisticians

Stereology for Statisticians

Numerical Analysis For Statisticians

Probability for Statisticians

Statistics for Non-Statisticians

Probability for Statisticians

Numerical Analysis for Statisticians

Stereology for Statisticians

A matrix handbook for statisticians

Twelve Days

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Across

Twelve Sharp

Twelve Across

Twelve British statisticians

Twelve

Twelve

Twelve

Twelve

Twelve

Numerical analysis for statisticians

Numerical analysis for statisticians

Twelve

Twelve

Probability for Statisticians

Stereology for Statisticians

Numerical Analysis For Statisticians

Probability for Statisticians

Statistics for Non-Statisticians

Probability for Statisticians

Numerical Analysis for Statisticians

Stereology for Statisticians

A matrix handbook for statisticians

Twelve Days

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Sharp

Twelve Across

Twelve Sharp

Twelve Across

Recommend Documents