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A Chronicle of Permutation Statistical Methods

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Abstract

This chapter chronicles the period from 1980 to 2000. Progress on the development of permutation methods continued unabated during this period, paralleling advancements in high-speed computing and the subsequent wide-spread availability of both university mainframes and, later in the period, personal desktop computers. Also, a number of books were published that introduced permutation methods to a wide variety of audiences and there was a decided shift in the literature away from computer science journals into discipline journals. These progressions were accompanied by an increasing emphasis on statistical applications of permutation methods, both exact and resampling, since efficient permutation generators were readily available.

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Notes

  1. 1.

    Continued by Journal of the Royal Statistical Society, Series C.

  2. 2.

    Continued by Plant Ecology.

  3. 3.

    On 8 August 2013 a retired school psychologist from Sacramento, California, sold one of the few remaining original Apple I computers (Serial Number 01-0025) that had been assembled by hand by Steve Wozniak in Steve Jobs parents’ garage. It fetched $387,750 at Christie’s Auction House; original price in 1976: $666.66. When the Apple II personal computer was introduced in 1977, customers were allowed to trade in their Apple I computers, making surviving Apple I computers very rare.

  4. 4.

    The 386 SLC was known inside IBM at the Super Little Chip for its initials.

  5. 5.

    Stata is a portmanteau of the words “statistics” and “data.”

  6. 6.

    On this topic, see a wonderful little book published in 2011 by Simon Garfield titled Just My Type: A Book About Fonts [495].

  7. 7.

    For an excellent bibliography on contingency table analysis from 1900 to 1974, see a 1976 article by Killion and Zahn in International Statistical Review [754].

  8. 8.

    When Goodman and Kruskal’s γ statistic is restricted to 2 × 2 contingency tables, it reduces to Yule’s Q statistic, introduced in 1912 [1480, 1481].

  9. 9.

    Authors’ note: the actual number is 30,414,093,201,713,378,043,612,608,166,064,768,844,377, 641,568,960,512,000,000,000,000 contingency tables.

  10. 10.

    In 1941 Edwin B. Wilson published a short note on “The controlled experiment and the four-fold table” in Science in which he explored the use of the chi-squared test statistic, chi-squared with Yates’ correction, and Fisher’s exact probability test. The analyses were based on data on the potency of two viruses injected into a small sample of six mice.

  11. 11.

    Fisher, quoted in Yates [1476, p. 444].

  12. 12.

    Emphasis in the original.

  13. 13.

    This was in contrast to Starmer , Grizzle , and Sen who stated in 1974 “[t]here seems to be no good reason to use the exact test as the standard of comparison for competing tests” of association, instead suggesting as a gold standard a randomized version of the exact test [1315, p. 377].

  14. 14.

    Loosely, “for lack of an alternative.”

  15. 15.

    Hirji and Johnson showed in 1996 that stage-wise processing is a very memory-intensive approach [630, p. 420].

  16. 16.

    See in this regard, a 1994 paper by Kulinskaya on “Large sample results for permutation tests of association” published in Communications in Statistics—Theory and Methods [780].

  17. 17.

    For a rigorous proof of the exact contingency formula, see a 1969 article by John Halton in Mathematical Proceedings of the Cambridge Philosophical Society [578].

  18. 18.

    Nathan Mantel long held to Irwin’s rule, e.g., [883, p. 379], but recanted in 1990 to support doubling the observed one-tail probability value [885, p. 369].

  19. 19.

    The Pearson type III distribution was one of four distributions introduced by Karl Pearson in 1895 [1106], although the type III distribution had previously been presented without discussion by Pearson in 1893 [1104, p. 331]. The type V distribution introduced by Pearson in 1895 was simply the normal distribution and the Pearson type I distribution was a generalized beta distribution.

  20. 20.

    Mielke , Berry , and Brier were, of course, not the first to use the Pearson type III distribution to approximate a discrete permutation distribution. B.L. Welch utilized the Pearson type III distribution in a paper on the specification of rules for rejecting too variable a product [1427, p. 47] and used it again in a paper on testing the significance of differences between the means of two independent samples when the population variances were unequal [1430, p. 352].

  21. 21.

    Authors’ note: special thanks to Charles Greifenstein , Manuscript Librarian at the Library of the American Philosophical Society in Philadelphia, for retrieving this manuscript from their extensive collection of papers, articles, and books by John W. Tukey , which was donated to the American Philosophical Society by the estate of John Wilder Tukey in 2002.

  22. 22.

    Emphasis in the original.

  23. 23.

    As noted by John W. Whitfield in 1950 [1444], Spearman argued that the process of squaring the deviations, i.e., v = 2, increased the error of the correlation measure and repeated this argument in various publications; see especially three articles by Pitman in 1904, 1906, and 1910 [1300–1302].

  24. 24.

    “Raters” are variously termed “judges” or “observers” in the agreement literature.

  25. 25.

    Emphasis in the original.

  26. 26.

    Wilhelm Maximilian Wundt is known as “the founding father of experimental psychology” and many psychologists trace their academic legacy to Wundt , perhaps because he produced 186 Ph.D. students during his long career at the University of Leipzig. [755, 1497, pp. 141–143]

  27. 27.

    Spearman’s test statistic, R, is based on the absolute differences between x i and y i , i = 1, …, n. Thus, the absolute distance between two rank vectors is often referred to as the “footrule distance.” When the variables are quantitative, the absolute distance is known as a “city-block metric” or “Manhattan distance.”

  28. 28.

    It is not generally recognized that under special conditions Spearman’s rank-order correlation coefficient is also a chance-corrected measure of agreement. When x and y consist of ranks from 1 to n with no ties, or x includes tied ranks and y is a permutation of x, then Spearman’s rank-order correlation coefficient is both a measure of correlation and a chance-corrected measure of agreement [772, p. 144].

  29. 29.

    This criticism is largely moot today with fast permutation generators and the selection of 1,000,000 or more random permutations of the observed data being quite common.

  30. 30.

    Walters reported an exact probability value of 0.066, but the correct value should be 0.067 as the exact probability value is actually 0.066628.

  31. 31.

    Resampling was required in this case as an exact test would have required generating \({(t!)}^{b} = {(4!)}^{8} = 110,075,314,176\) F values.

  32. 32.

    For a lucid and cogent description of the Pagano–Tritchler algorithm, see a 1998 article by Gebhard and Schmitz in Statistical Papers [503].

  33. 33.

    Edgington and Khuller cite (n)k!  possible arrangements of the data, but this is obviously incorrect.

  34. 34.

    In 1996 π had been calculated to 3 × 231 = 6, 442, 450, 938 decimal digits. On 22 October 2011 Alexander Yee and Shigeru Kondo announced that π had been calculated to 10 trillion digits on a dedicated desktop computer; the execution time was 371 days.

  35. 35.

    Sergey Brin married Anne Wojcicki, Susan’s younger sister, in May 2007.

  36. 36.

    Winsorizing, or Winsorization, is named for the physiologist-turned-biostatistician Charles P. Winsor [1380, p. 18]. It was Charles Winsor who convinced John Tukey to convert from mathematics to statistics while both were at Princeton University’s Fire Control Research Office in the 1940s [814, p. 194]. As Tukey noted in his foreword to Volume VI of The Collected Works of John W. Tukey, “[i]t was Charlie, and the experience of working on the analysis of real data, that converted me to statistics. By the end of late 1945, I was a statistician rather than a topologist …” [871, p. xlviii].

  37. 37.

    Trimming has long been advocated by the psychologist Rand Wilcox . His extensive writings on the subject have provided a modern impetus to the procedure of trimming and, to a lesser extent, Winsorizing [1448–1452].

  38. 38.

    This reiterated the position held, for example, by Altman and Bland [15], Bradbury [200], Edgington [389], Feinstein [421], LaFleur and Greevy [789], Ludbrook [850], Ludbrook and Dudley [856], and Still and White [1324], that assuming a random sample from an infinite population was untenable in many disciplines.

  39. 39.

    John Ludbrook is Professional Research Fellow in the University of Melbourne, Department of Surgery, Royal Melbourne Hospital and Hugh Dudley is Professor Emeritus at the University of London, Department of Surgery, St. Mary’s Hospital Medical School.

  40. 40.

    It should be noted that the second paper was written in response to a stinging criticism of the 1915 paper by H.E. Soper , A.W. Young , B.M. Cave , A. Lee , and K. Pearson that had appeared in Biometrika in 1916 and was titled “On the distribution of the correlation coefficient in small samples” [1297].

  41. 41.

    As a testament to computing power in 2000, the authors computed three different confidence intervals at three confidence levels on four values of the population parameter 1,000,000 times with four different sample sizes of 10, 20, 40, and 80; a feat that was inconceivable just a decade earlier.

  42. 42.

    Authors’ note: one of the authors often advises his students regarding the Fisher Z transform:

    1. 1.

      Do not use the Fisher Z transformation.

    2. 2.

      If you do use it, don’t believe it.

    3. 3.

      If you do believe it, don’t publish it.

    4. 4.

      If you do publish it, don’t be the first author.

    Adapted from a description of a tiltmeter in Volcano Cowboys by Dick Thompson [1357, p. 258].

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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). 1980–2000. In: A Chronicle of Permutation Statistical Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-02744-9_5

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