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1920–1939

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A Chronicle of Permutation Statistical Methods

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Abstract

This chapter chronicles the development of permutation statistical methods from 1920 to 1939, when the earliest discussions of permutation methods appeared in the literature. In this period J. Spława-Neyman, R.A. Fisher, R.C. Geary, T. Eden, F. Yates, and E.J.G. Pitman laid the foundations of permutation methods as we know them today. As is evident in this period, permutation methods had their roots in agriculture and, from the beginning, were widely recognized as the gold standard against which conventional methods could be verified and confirmed.

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Notes

  1. 1.

    For an enlightened discussion of the differences and similarities between Neyman and Fisher and their collective impact on the field of statistics, see a 1966 article by Stephen Fienberg and Judith Tanur in International Statistical Review [430] and also E.L. Lehmann’s remarkable last book, published posthumously in 2011, on Fisher, Neyman, and the Creation of Classical Statistics [816].

  2. 2.

    Maxim Gorki (Maksim Gorky) is a pseudonym for Aleksei Maksimovich Peshkov (1868–1936), Russian short-story writer, novelist, and political activist.

  3. 3.

    The Royal Society is a fellowship of the world’s most eminent scientists and is the oldest scientific society in continuous existence. The society was founded on 28 November 1660 when a group of 12 scholars met at Gresham College and decided to found “a Colledge for the Promoting of Physico-Mathematicall Experimentall Learning” and received a Royal Charter on 5 December 1660 from Charles II. The original members included Christopher Wren , Robert Boyle , John Wilkins , Sir Robert Moray , and William, Viscount Brouncker , who subsequently became the first president of the Society [357, 1144, 1351].

  4. 4.

    Fisher and Neyman differed in other ways as well. In general, they differed on the fundamental approach to statistical testing, with Fisher’s ideas on significance testing and inductive inference and Neyman’s views on hypothesis testing and inductive behavior; see an excellent summary in a 2004 article by Hubbard [663] as well as a comprehensive account of the controversy by Gigerenzer , Swijtink , Porter , and Daston published in 1989 [512, pp. 90–106].

  5. 5.

    Those students doing best on the examinations were designated as “Wranglers.” More specifically, the 40 top-scoring students out of the approximately 100 mathematics graduates each year were designated as Wranglers, whereas 400–450 students graduated from the University of Cambridge annually at that time. Wranglers were rank-ordered according to scores on their final mathematics examination, which was a 44-h test spread over 8 days [713, p. 657].

  6. 6.

    The standard biography of R.A. Fisher is that written by his daughter in 1978, Joan Fisher Box [195], but others have provided more specialized biographies, including those by P.C. Mahalanobis [868], F. Yates [1474], F. Yates and K. Mather [1477], M.S. Bartlett [80], S.M. Stigler [1322, 1323], C.R. Rao [1155], W.H. Kruskal [778] , M.J.R. Healy [607] , N.S. Hall [575] , E.L. Lehmann [816] , L.J. Savage [1226] , and G.E.P. Box [191]. The collected papers of R.A. Fisher are posted at http://www.adelaide.edu.au/library/special/digital/fisherj/. In addition, two large volumes of the selected correspondence of R.A Fisher were published in 1983 and 1990 by J.H. Bennett [96, 97].

  7. 7.

    Early on, the probable error was an important concept in statistical analysis and was defined as one-half the interquartile range. In terms of the normal distribution, the probable error is 0.6745 times the standard error. Therefore, as a test of significance a deviation of three times the probable error is effectively equivalent to one of twice the standard error [292, 448, pp. 47–48]. “Probable error” instead of “standard error” was still being used in the English-speaking countries in the 1920s and far into the 1930s; however, “probable error” was rarely used in Scandinavia or in the German-speaking countries [859, p. 214].

  8. 8.

    For comparison, the sex ratio at birth in Germany in 2013 was 1.055.

  9. 9.

    The Irish Poor Law of 1838 was an attempt to ameliorate some of the problems arising out of widespread poverty in the early 1800s in Ireland. Influenced by the Great Reform Act of 1834 in England (q.v. page 11), Ireland was originally divided into 131 poor law unions, each with a workhouse at its center.

  10. 10.

    Mackenzie is sometimes spelled “Mackenzie” [195] and other times “MacKenzie” [191, 576, 720]. In the original article, Mackenzie is all in upper-case letters.

  11. 11.

    The experiment on potatoes had been conducted by Thomas Eden at the Rothamsted Experimental Station, wherein each of twelve varieties of potatoes had been treated with six different combinations of manure [191].

  12. 12.

    Previously, in 1918 in an article on Mendelian inheritance in Eugenics Review, Fisher had coined the term “analysis of variance” [443]; see also a 2012 article by Edwards and Bodmer on this topic [398, p. 29].

  13. 13.

    This 1923 article by Fisher and Mackenzie is often cited as the first randomized trial experiment [484, 517, 893, 925]. However, the first documented publication of a randomized trial experiment was by the American philosopher Charles Sanders Peirce and his colleague at Johns Hopkins University, Joseph Jastrow , in 1885 [1113]; see also, in this regard, discussions by Neuhauser and Diaz [1030, pp. 192–195], Stigler [1321], and an autobiography by Jastrow [682].

  14. 14.

    H.W. Heckstall-Smith, Headmaster, Chippenham Grammar School, had written to Fisher requesting permission to quote from Fisher in an article he was preparing for a medical journal. The article, with M.G. Ellis, was eventually published in the journal Tubercle in December of 1955 under the title “Fun with statistics” [409].

  15. 15.

    Yeoman wheat is a hybrid variety that resists wheat rust. It was developed and released in 1916 by Sir Rowland Biffen, Director of the Plant Breeding Institute at the University of Cambridge School of Agriculture.

  16. 16.

    Because it is possible to hold one block constant and to randomize the remaining blocks with respect to the fixed block, it is only necessary to randomize b − 1 blocks, thereby greatly decreasing the total number of possible arrangements. In this case, (4! )7 = 4, 586, 471, 424 instead of (4! )8 = 110, 075, 314, 176 randomizations.

  17. 17.

    H.A. David has written that the 1933 Eden–Yates paper “may be regarded as introducing randomization [permutation] theory” [326, p. 70].

  18. 18.

    As was customary in scientific societies at the time, these special research papers were printed in advance and circulated to the membership of the society. Then, only a brief introduction was made by the author at the meeting and the remaining time was devoted to discussion. By tradition, the “proposer of the vote of thanks” said what was he thought was good about the paper, and the seconder said what he thought was not so good. Subsequently, there was a general discussion by the Fellows of the Society and often a number of prominent statisticians offered comments, suggestions, or criticisms [192, p. 41]. In this instance the discussants were Arthur Bowley , Leon Isserlis , Joseph Irwin , Julius Wolf , Egon Pearson , Major Greenwood , Harold Jeffreys , Maurice Bartlett , and Jerzy Neyman . As might be evident from the list of names, not all comments were constructive.

  19. 19.

    According to Lehmann [816, p. 48, fn. 1], this statement is in fact not completely true, although very nearly so. See also a 1977 article by Plackett in this regard [1137].

  20. 20.

    As Hitchcock has noted, the variance equals the mean in the archetypical count model of the Poisson, and the normal approximates the Poisson when the mean is large [633, p. 2].

  21. 21.

    It should be mentioned that because Yates was primarily interested in 2 × 2 contingency tables and, therefore, χ 2 was distributed as chi-squared with 1 degree of freedom, he obtained the requisite probability values from tables of the normal distribution since χ 1 2 = z 2.

  22. 22.

    Good has argued that the test should more properly be referred to as the Fisher–Yates–Irwin–Mood test [519, p. 318].

  23. 23.

    Although Metron Rivista Internazionale di Statistica was published in Italy, the article by Irwin was in English.

  24. 24.

    For the early history of Fisher , Yates , Irwin , and the exact analysis of 2 × 2 contingency tables, see articles by Barnard [71] and Good [519–521].

  25. 25.

    Irwin suffered from chronic poor health from early childhood and it is possible that was what delayed publication.

  26. 26.

    Irwin joined the Rothamsted Experimental Station in 1928 and remained there until 1931, which was when Yates joined Rothamsted. Since they were both employed in Fisher’s Statistical Laboratory at Rothamsted and both overlapped as undergraduates at the University of Cambridge, it is likely they were well acquainted.

  27. 27.

    To clarify, Irwin took the positive square root of χ 2, i.e., χ, which with one degree of freedom is a normal deviate, and thus obtained the probability values from a standard unit-normal table of probability values.

  28. 28.

    The controversy as to whether to use the doubling rule or Irwin’s rule to obtain a two-tailed probability value persisted for many years; see for example, articles by Cormack [279, 280] in 1984 and 1986, Cormack and Mantel in 1991 [281], Healy in 1984 [604], Jagger in 1984 [678], Mantel in 1984 and 1990 [884, 885], Yates in 1984 [1476], and Neuhäuser in 2004 [1031].

  29. 29.

    For this section of the book, the authors are indebted to Sir E. John Russell (q.v. page 57) who, in 1942, compiled the early history of the Rothamsted Manor.

  30. 30.

    Originally, Wittewronghele.

  31. 31.

    Sometimes spelled Midleton or Middleton.

  32. 32.

    Today, phosphate-based fertilizers are used throughout the world and there is presently concern that the world will eventually run out of easily accessible sources of phosphate rock [278, 784]. On the other hand, heavy spring rains generate runoff from farmer’s fields into ponds and lakes, spawning growth of toxic blue-green algae, such as Microsystis aeurginosa, which are fed by the phosphorus from the fields [1463].

  33. 33.

    The College of St. Gregory and St. Martin at Wye, more commonly known as Wye College, was an educational institution in the small village of Wye, Kent, about 60 miles east of London.

  34. 34.

    Broadbalk refers to the fields at Rothamsted on which winter wheat was cultivated, not a strain of wheat.

  35. 35.

    Afternoon tea had been a British tradition since one of Queen Victoria’s (1819–1901) ladies-in-waiting, Anna Maria Russell (née Stanhope) (1783–1857), the seventh Duchess of Bedford, introduced it at Belvoir (pronounced Beaver) Castle in the summer of 1840, the idea being a light repast around 4 p.m. would bridge the lengthy gap between luncheon and dinner, which in fashionable circles at that time was not taken until 8 p.m.

  36. 36.

    For a detailed explanation as to why it matters whether the tea or the milk is poured into the teacup first, see a 2012 article by Stephen Senn in Significance [1251].

  37. 37.

    For a biography of Dr. B. Muriel Bristol and a picture, see a 2012 article by Stephen Senn in Significance [1251].

  38. 38.

    In 1956 Fisher published a lengthy discussion of the lady tasting tea experiment titled “Mathematics of a lady tasting tea” in J.R. Newman’s book titled The World of Mathematics [459, pp. 1512–1521].

  39. 39.

    It should be noted that Francis Galton , after much experimentation, always chose to put the milk into the teacup first [1251, p. 32].

  40. 40.

    It is generally understood that the conventional use of the 5 % level of significance as the maximum acceptable probability for determining statistical significance was established by Fisher when he developed his procedures for the analysis of variance in 1925 [292]. Fisher also recommended 0.05 as a level of significance in relation to chi-squared in the first edition of Statistical Methods for Research Workers [448, pp. 79–80]. Today, p = 0. 05 is regarded as sacred by many researchers [1281]. However, Fisher readily acknowledged that other levels of significance could be used [449, p. 504]. In this regard, see discussions by Cowles and Davis [292] and Lehmann [816, pp. 51–53].

  41. 41.

    For a concise summary of the Zea mays experiment, see a discussion by Erich Lehmann in his posthumously published 2011 book on Fisher, Neyman, and the Creation of Classical Statistics [816, pp. 65–66].

  42. 42.

    Authors’ note: actually, 90,548,514,656,103,281,165,404,177,077,484,163,874,504,589,675,413, 336,841,320 ways.

  43. 43.

    The Fisher –Yates shuffle, with little change, became the basis for more sophisticated computer shuffling techniques by Richard Durstenfeld in 1964 [367], Donald Knuth in 1969 [762], and Sandra Sattolo in 1986 [1222]. N. John Castellan [245] and Timothy J. Rolfe [1188] urged caution in choosing a shuffling routine as many widely-used shuffling algorithms are incorrect.

  44. 44.

    Authors’ note: special thanks to Nanci A. Young , College Archivist, William A. Neilson Library at Smith College, Northampton, Massachusetts, for retrieving the information on Margaret Richards Pabst , and to Nancy Lyons , Program Analyst, United States Department of Agriculture, Food and Nutrition Service, for contacting Archivist Nanci Young at Smith College on our behalf. Special thanks also to Sarah Jane Pabst Hogenauer and Dr. Margaret Pabst Battin , Distinguished Professor of Philosophy and Adjunct Professor of Internal Medicine, Division of Medical Ethics at the University of Utah, who are the daughters of Margaret Richards Pabst and who graciously shared details of their mother’s life, including having Muriel Hotelling , Harold Hotelling’s daughter, as a babysitter and, as girls of 10 or 11, having lunch with R.A. Fisher .

  45. 45.

    In 1907, Pearson derived the standard error of Spearman’s rank-order correlation coefficient. Assuming normality, Pearson generated the first four terms of an infinite series to provide an approximate standard error [1109].

  46. 46.

    In the early years of statistics it was common to denote raw scores with upper-case letters, e.g., X and Y, and deviations from the mean scores with lower-case letters, e.g., x and y.

  47. 47.

    This is an area of some controversy. Some researchers hold that, if and only if generalizing from a sample to a population, permutations are equally likely in controlled experimentation, but may not be equally likely in non-experimental research; see for example Zieffler , Harring , and Long [1493, pp. 132–134].

  48. 48.

    A clear and concise explanation of the Friedman analysis of variance for ranks test was given by Lincoln Moses in a 1952 publication on “Non-parametric statistics for psychological research” in Psychological Bulletin [1010].

  49. 49.

    For a concise summary of the 1937 Welch paper, see a 2008 article by H.A. David on “The beginnings of randomization tests” in The American Statistician [326].

  50. 50.

    H.A. David provides a concise summary of the 1937 Pitman paper in his 2008 article in The American Statistician on “The beginnings of randomization tests” [326].

  51. 51.

    Pitman’s use of SS Treatment and SS Error is equivalent to SS Between and SS Within, respectively, as used by others.

  52. 52.

    The method of moments was first proposed by Karl Pearson in 1894 [1105].

  53. 53.

    Today, this approach is termed “data-dependent” analysis.

  54. 54.

    In 1967, the Carnegie Institute of Technology merged with the Mellon Institute of Industrial Research to form Carnegie Mellon University, which abuts the campus of the University of Pittsburgh. The Carnegie Institute of Technology is now the school of engineering at Carnegie Mellon University.

  55. 55.

    There was an error in the denominator of the variance in the 1938 paper. It was first noticed by Scheffé in 1943 [1230] and corrected by Olds in 1949 [1055].

  56. 56.

    While The Advanced Theory of Statistics began as a two-volume work, in 1966 Alan Stuart joined with Maurice Kendall and The Advanced Theory was rewritten in three volumes. Keith Ord joined in the early eighties and a new volume on Bayesian Inference was published in 1994. More recently, Steven Arnold was invited to join with Keith Ord.

  57. 57.

    As Kendall explained in a later publication, the coefficient τ was considered earlier by Greiner [554] and Esscher [414] as a method of estimating correlations in a normal population, and was rediscovered by Kendall [728] who considered it purely as a measure of rank-order correlation [734].

  58. 58.

    A recursive process is one in which items are defined in terms of items of similar kind. Using a recurrence relation, a class of items can be constructed from a few initial values (a base) and a small number of relationships (rules). For example, given the base, F 0 = 0 and \(F_{1} = F_{2} = 1\), the Fibonacci series {0, 1, 1, 2, 3, 5, 8, 13, 21, …} can be constructed by the recursive rule \(F_{n} = F_{n-1} + F_{n-2}\) for n > 2.

  59. 59.

    A correction was proffered by J.A. van der Heiden in 1952 for observers who declined to express a preference between a pair of objects [1390].

  60. 60.

    The coefficient of concordance was independently developed by W. Allen Wallis in 1939, which he termed the “correlation ratio for ranked data” [1411].

  61. 61.

    Also spelled Leibnitz.

  62. 62.

    Also spelled Hariot, Harriott, or Heriot.

  63. 63.

    Actually, the model in the London Science Museum is of Difference Engine Number 2, designed by Babbage between 1846 and 1849 [1290, pp. 290–291].

  64. 64.

    In a wonderful little book on the history of British science during the nineteenth century, Laura Snyder noted that while Lucasian Professor of Mathematics at the University of Cambridge from 1828 to 1839, Charles Babbage never delivered a single lecture [1290, p. 130].

  65. 65.

    The Millionaire calculator was the first commercial calculator that could perform direct multiplication. It was in production from 1893 to 1935.

  66. 66.

    For Fisher’s first major publication in 1921 on “Studies in crop variation, I,” Fisher produced 15 tables [445]. At approximately 1 min for each large multiplication or division problem, it has been estimated that Fisher spent 185 h using the Millionaire to produce each of the 15 tables [618, p. 4].

  67. 67.

    For pictures of the Millionaire calculator and Frank Yates using the Millionaire, see a 2012 article by Gavin Ross in Significance [1196]. Also, there is a YouTube video of a Millionaire calculator calculating the surface of a circle with diameter 3.18311 at http://www.youtube.com/watch?v=r9Nnl-u-Xf8.

  68. 68.

    Iowa Agricultural College and Model Farm was established in 1858 and changed its name to Iowa State University of Science and Technology in 1959, although it is commonly known as Iowa State University.

  69. 69.

    For more interesting stories about Fisher , see a 2012 article in Significance by A.E.W. Edwards and W.F. Bodmer [401].

  70. 70.

    Interestingly, the Statistical Laboratory at Iowa State College initiated four o’clock afternoon tea while Fisher was there in the summer of 1936 [57, 576].

  71. 71.

    Henry A. Wallace served as Secretary of Agriculture from 1933 to 1940. When John Nance Garner broke with then President Franklin Delano Roosevelt in 1940, Roosevelt designated Wallace to run as his Vice-President. Wallace served as Vice President from 1941 to 1945 when Roosevelt jettisoned Wallace in favor of Harry S. Truman , who succeeded Roosevelt upon his death on 12 April 1945 [597]. Finally, Wallace served as Secretary of Commerce from 1945 to 1946.

  72. 72.

    The best accounts of the origins and development of the Iowa State College Statistical Laboratory are Statistics: An Appraisal, edited by H.A. David and H.T. David [327], “Statistics in U.S. universities in 1933 and the establishment of the Statistical Laboratory at Iowa State” by H.A. David [324], “Highlights of some expansion years of the Iowa State Statistical Laboratory, 1947–72” by T.A. Bancroft [58], “Revisiting the past and anticipating the future” by O. Kempthorne [724], “The Iowa State Statistical Laboratory: Antecedents and early years” by H.A. David [322], and “Early statistics at Iowa State University” by J.L. Lush [859].

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  1. Department of Sociology, Colorado State University, Fort Collins, CO, USA

    Kenneth J. Berry

  2. U.S. Government, Alexandria, VA, USA

    Janis E. Johnston

  3. Department of Statistics, Colorado State University, Fort Collins, CO, USA

    Paul W. Mielke Jr.

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Berry, K.J., Johnston, J.E., Mielke, P.W. (2014). 1920–1939. In: A Chronicle of Permutation Statistical Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-02744-9_2

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