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1940–1959

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Book cover A Chronicle of Permutation Statistical Methods

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Abstract

This chapter chronicles the development of permutation statistical methods from 1940 to 1959. This period may be considered a bridge between the early years of 1920–1939 where permutation tests were first conceptualized and the next period, 1960–1979, in which gains in computer technology provided the necessary tools to successfully employ permutation tests. The recognition of permutation methods as the gold standard against which conventional statistical methods were to be evaluated, while often implicit in the 1920s and 1930s, is manifest in many of the publications on permutation methods that appeared between 1940 and 1959. Also, a number of researchers turned their attention during this time period to rank tests, which simplified the calculation of exact probability values; other researchers continued work on calculating exact probability values, creating tables for small samples; and still others continued the theoretical work begun in the 1920s.

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Notes

  1. 1.

    A comprehensive overview of statistics in the 1950s is provided by Tertius de Wet in his presidential address to the South African Statistical Association in 2003 [335].

  2. 2.

    Unfortunately, these two important articles by Samuel Silvey went largely unnoticed, published as they were in Proceedings of the Glasgow Mathematical Association, a journal that was not widely distributed at the time.

  3. 3.

    Atanasoff’s proposal for construction of the computer was funded by Iowa State College, (now, Iowa State University) which granted Atanasoff $5,000 to complete his computing machine.

  4. 4.

    Technically, the computer was originally called the Aiken–IBM Automatic Sequence Controlled Calculator (ASCC) and was renamed the Mark I by Harvard University when it was acquired from IBM on 7 August 1944.

  5. 5.

    For an interesting biography of John von Neumann , as related to computers and computing science, see Chap. 4 in the 2012 book Turing’s Cathedral by George Dyson [370, Chap. 4].

  6. 6.

    This statement paints a rosy picture of the relationship between Shockley , on the one hand, and Brattain and Bardeen , on the other hand, that was nothing but congenial. For a more detailed account, see a 2010 book by Sam Kean titled The Disappearing Spoon [712, pp. 41–43].

  7. 7.

    In fact, it took 70 years for the Government Communications Headquarters (GCHQ) to release two papers written by Alan Turing between April 1941 and April 1942 while he was head of wartime code-breaking at Bletchley Park. The two papers on “Paper on the statistics of repetitions” and “Other applications of probability to cryptography” were finally released in April of 2012 [25].

  8. 8.

    For a history of the development of fortran, see the recollection by John Backus in the special issue of ACM SIGPLAN Notices on the history of programming [44].

  9. 9.

    This statement is from Kurt Beyer , Grace Hopper and the Invention of the Information Age. Actually, the first woman to earn a Ph.D. in mathematics from Yale University was Charlotte Cynthia Barnum (1860–1934) who received her Ph.D. in mathematics in 1895 [1018].

  10. 10.

    The maximum number of inversions required to reverse the order of ranks follows an irregular series. For n = 2, the maximum number of inversions, I, is 1; for n = 3, I = 3; for n = 4, I = 6; and so on. Thus, the sequence is 1, 3, 6, 10, 15, 21, and so on. The sequence is a component of Pascal’s triangle; see Column 3 in Table 3.11 of this chapter, page 186. Any successive number can be obtained by n(n + 1)∕2. Thus, for n = 9 objects the maximum number of inversions is 9(9 + 1)∕2 = 45. It stretches the imagination that a subject made more than 70 paired comparisons to rank order only nine objects.

  11. 11.

    This is the reason that the Academy of Motion Picture Arts and Sciences places a maximum limit of 10 nominations for the Academy Awards (Oscars), as it is too difficult for the judges to rank order a larger number of nominations.

  12. 12.

    These results were later proved by Patrick Moran in a brief article on “The method of paired comparisons” in Biometrika in 1947 [1003].

  13. 13.

    This was a typical approach for the time. Because a permutation test generally did not generate a value of the statistic that coincided exactly with α (e.g., 0.05 or 0.01) of the permutation distribution, a value of the permuted statistic, \(C_{\alpha }^{2}\), was defined as the smallest value of statistic C 2 for which \(P({C}^{2} \geq C_{\alpha }^{2}) \leq \alpha\).

  14. 14.

    The National Bureau of Standards was founded in 1901 as a non-regulatory agency of the United States Department of Commerce in Gaithersburg, Maryland. The NBS was renamed The National Institute of Standards and Technology (NIST) in 1988.

  15. 15.

    In a 1943 article in The Annals of Mathematical Statistics, Wolfowitz commented that the choice of U as a test statistic was somewhat arbitrary and that other reasonable tests could certainly be devised [1465, p. 284], such as that proposed by Wilfrid Dixon in 1940, also in The Annals of Mathematical Statistics [353].

  16. 16.

    A clear exposition of the Wald–Wolfowitz runs test was given in an article by Lincoln Moses on “Non-parametric statistics for psychological research” published in Psychological Bulletin in 1952 [1010].

  17. 17.

    Tests based on permutations of observations require that, under the null hypothesis, the probability distribution is symmetric under all permutations of the observations. This symmetry can be assured by randomly assigning treatments to the experimental units. As a result, these tests are often referred to as “randomization tests” in the literature [254, p. 729].

  18. 18.

    A summary in English of the Rodrigues 1839 article is available in Mathematics and Social Utopias in France: Olinde Rodrigues and His Times [39, pp. 110–112].

  19. 19.

    This paper was cited by Moran in [1005, p. 162] as “Rank correlation and a paper by H.G. Haden, ” but apparently the title was changed at some point to “Rank correlation and permutation distributions” when it was published in Proceedings of the Cambridge Philosophical Society in 1948.

  20. 20.

    Technically, Fig. 3.1 is a permutation graph of a family of line segments that connect two parallel lines in the Euclidean plane. Given a permutation {4, 2, 3, 1, 5} of the positive integers {1, 2, 3, 4, 5}, there exists a vertex for each number {1, 2, 3, 4, 5} and an edge between two numbers where the segments cross in the permutation diagram.

  21. 21.

    See also a discussion about the relationship between Turing and Church by George Dyson in a 2012 book titled Turing’s Cathedral [370, pp. 249–250].

  22. 22.

    This was actually Barnard’s first, of many, published papers. It was published in Nature while Barnard was employed at the Ministry of Supply and is only one-half page in length.

  23. 23.

    In 1984 Barnard revealed the meaning behind labeling the statistic CSM, recalling “there was a private pun in my labelling the suggested procedure CSM—it referred …to the Company Sergeant Major in my Home Guard unit at the time, my relations with whom were not altogether cordial. I still feel that the test, like the man, is best forgotten” [70, p. 450].

  24. 24.

    The Barnard test will not die and from time to time the test is resurrected and advocated; see for example, articles by McDonald, Davis, and Milliken [913] in 1977; Barnard [72], Hill [621], and Rice [1167, 1168] in 1988; Dupont [365] and Martín Andrés and Luna del Castillo [900] in 1989; and Campbell [239] in 2007.

  25. 25.

    The Gordon Research Conferences on Statistics in Chemistry and Chemical Engineering began in 1951 and continued through the summer of 2005.

  26. 26.

    The Wilcoxon two-sample rank-sum test statistic is conventionally expressed as W in textbooks, but Wilcoxon actually designated his test statistic as T. Also, many textbooks describe the Wilcoxon test as a “difference between group medians” test, when it is clearly a test for the difference between mean ranks; see for example, an article by Bergmann , Ludbrook , and Spooren in 2000 [100] and an article by Conroy in 2012 [274].

  27. 27.

    A clear and concise exposition of the Wilcoxon unpaired and paired sample rank tests is given in an article by Lincoln Moses on “Non-parametric statistics for psychological research” published in Psychological Bulletin in 1952 [1010].

  28. 28.

    For a brief history of the factorial symbol, see a 1921 article in Isis by Florian Cajori on the “History of symbols for \(\underline{\text{n}} =\mathrm{ factorial}\)” [237].

  29. 29.

    Wilcoxon’s use of the term “partitions” here is a little misleading. These are actually sums of T = 20, each sum consisting of five integer values between 1 and 2q = 10 with no integer value repeated e.g., {1, 2, 3, 4, 10} = 20 which consists of five non-repeating integer values, but not {5, 7, 8} = 20 which consists of only three integer values, nor {1, 3, 3, 5, 8} = 20 which contains multiple values of 3.

  30. 30.

    MacMahon’s monumental two-volume work on Combinatory Analysis, published in 1916, contained a section in Volume II, Chap. III, on “Ramanujan’s Identities” in which MacMahon demonstrated the relationship between the number of q-part unequal partitions without repetitions with no part greater than 2q and the number of partitions with repetitions with no part greater than q [865, pp. 33–48].

  31. 31.

    Several sources list Festinger earning his Ph.D. in 1942 from Iowa State University, not the University of Iowa. Since his dissertation advisor was Kurt Lewin, who was at the University of Iowa from 1935 to 1944, the University of Iowa appears correct.

  32. 32.

    The decomposition \(\binom{n}{r} = \binom{n - 1}{r} + \binom{n - 1}{r - 1}\) has been well known since the publication of Blaise Pascal’s Traité du triangle arithmétique in 1665, 3 years after his death [1088]. Thus, considering any one of n objects, \(\binom{n - 1}{r}\) gives the number of combinations that exclude it and \(\binom{n - 1}{r - 1}\) the number of combinations that include it.

  33. 33.

    A particularly clear exposition of the Mann–Whitney U test is given in a 1952 paper by Lincoln Moses on “Non-parametric statistics for psychological research” published in Psychological Bulletin [1010].

  34. 34.

    Here, the decomposition is identical to Festinger’s , as given in [427].

  35. 35.

    Whitfield’s article was followed immediately in the same issue of Biometrika with a comment by M.G. Kendall noting that “Mr Whitfield has correctly surmised the variance [of τ] when one ranking contains ties, and the other is a dichotomy” [733, p. 297].

  36. 36.

    In 1968 Charles R. Kraft and Constance van Eeden showed how Kendall’s τ can be computed as a sum of Wilcoxon W statistics [768, pp. 180–181].

  37. 37.

    Whitfield lists the date of the Kendall article as 1946, but Kendall’s article was actually published in Biometrika in 1945.

  38. 38.

    It was common at this time to assume continuous variables as this ensured no tied values, cf. articles by Sun and Sherman in 1996 [1335, p. 90] and Gebhard and Schmitz in 1998 [502, p. 76].

  39. 39.

    Cedric Smith, Roland Brooks, Arthur Stone , and William Tutte met at Trinity College, University of Cambridge, and were known as the Trinity Four. Together they published mathematical papers under the pseudonym Blanche Descartes , much in the tradition of the putative Peter Ørno , John Rainwater, and Nicolas Bourbaki .

  40. 40.

    Equation (3.5) is incorrect in Haldane and Smith [573, p. 117] and is corrected here.

  41. 41.

    It should be noted that while the decomposition in Eq. (3.8) is different from that employed by Mann and Whitney in Eq. (3.4) [880], it is similar to the decomposition used by Festinger [427], although there is no indication that Haldane and Smith were familiar with the work of Festinger .

  42. 42.

    Phenylketonuria (PKU) is a autosomal recessive metabolic genetic disorder that can lead to mental retardation, seizures, behavioral problems, and autism. Dr. Asbjørn Følling , a Norwegian biochemist and physician, was the first to publish a description of phenylketonuria as a cause of mental retardation in 1934 [475].

  43. 43.

    The Fisher–Yates test of significance for 2 × 2 contingency tables was independently developed by R.A. Fisher in 1935 [452], F. Yates in 1934 [1472], and J.O. Irwin in 1935 [674] (qq.v. pages 25, 37, and 48).

  44. 44.

    Unfortunately, Finney recommended doubling the obtained one-tailed probability value when using a two-tailed test [434, p. 146]. This was destined to become a procedure of considerable controversy in the mid-1980s (q.v. page 51).

  45. 45.

    It should be noted that Kendall neglected to mention the two-sample rank-sum test developed by Festinger 2 years prior, perhaps because it was published in the psychology journal, Psychometrika, which was not commonly read by statisticians.

  46. 46.

    Southern Rhodesia was shortened to Rhodesia in 1965 and renamed the Republic of Zimbabwe in 1980.

  47. 47.

    For a brief history of the development of the two-sample rank-sum test, see a 2012 article by Berry , Mielke , and Johnston in Computational Statistics [160].

  48. 48.

    Authors’ note: in deciphering the article by van der Reyden we were often reminded of a comment by Nathaniel Bowditch . In the memoir prefixed to the fourth volume of Bowditch’s translation of Laplace’s Mécanique Céleste, page 62, Bowditch wrote: “[w]henever I meet in La Place with the words ‘Thus it plainly appears’ I am sure that hours, and perhaps days of hard study will alone enable me to discover how it plainly appears.” (Bowditch, quoted in Todhunter [1363, p. 478]; emphasis in the original).

  49. 49.

    Recall that the two-sample rank-sum method proposed by Wilcoxon in 1945 considered only equal sample sizes [1453] and Festinger , in 1946, was the first to develop a two-sample rank-sum procedure that could accommodate different sample sizes [427].

  50. 50.

    For brief biographical sketches of Evelyn Fix and Joseph L. Hodges , see Lehmann’s wonderful little book titled Reminiscences of a Statistician: The Company I Kept, published in 2008 [814, pp. 27–35].

  51. 51.

    In this regard, see an article by John Ludbrook on “The Wilcoxon–Mann–Whitney test condemned” in British Journal of Surgery [851] as well as a rejoinder by G.D. Murray [1017]. See also an exact permutation computer program for the Wilcoxon–Mann–Whitney test by Berry and Mielke in 2000 [155].

  52. 52.

    It should be explained that “computer” was a common term that referred to the person who was responsible for calculations, usually a woman or group of women. In this case the computer was Miss Joan Ayling of the National Institute for Social and Economic Research who was given due credit by the authors “for her customary patience and accuracy” [328, p. 131]. See also a discussion by George Dyson in a 2012 book titled Turing’s Cathedral [370, p. 59].

  53. 53.

    A rigorous derivation of the exact contingency formula was given by John Halton in Mathematical Proceedings of the Cambridge Philosophical Society in 1969 [578].

  54. 54.

    In many applications, “panels” are sometimes referred to as “slices” or “levels.”

  55. 55.

    As noted by Edwards [399, p. x], it was Pierre Raymond de Montmort who, in 1708, first attached the name of Pascal to the combinatorial triangle; however, he changed the form to a staggered version [334]. Then, in his Miscellanea Analytica of 1730, Abraham de Moivre christened Pascal’s original triangle “Triangulum Arithmeticum PASCALIANUM.”

  56. 56.

    As Box and Andersen noted in 1955, although there are (m + n)! possible arrangements of a sample, there are only (m + n)! ∕(m! n! ) arrangements that result in possibly different mean differences [193, p. 7].

  57. 57.

    Presently, resampling permutation routines, which are essentially sampling without replacement routines, generate hundreds of thousands of permutations per second when powered by an efficient uniform pseudorandom number generator (PRNG) such as the Mersenne Twister (MT) or the SIMD-oriented Fast Mersenne Twister (SFMT) on high-speed work stations [905, 1214].

  58. 58.

    Emphasis in the original.

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Authors and Affiliations

  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). 1940–1959. In: A Chronicle of Permutation Statistical Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-02744-9_3

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