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Exact permutation/randomization tests algorithms. (English) Zbl 1458.62159

Summary: R. A. Fisher [The design of experiments. Edinburgh, London: Oliver & Boyd (1937; JFM 61.0566.03)] described the exact permutation and randomization tests for comparative experiments without assuming normality or any particular probability distribution (see also [R. A. Fisher, Statistical methods, experimental design and scientific inference. A re-issue of ‘Statistical methods for research workers’, ‘The design of experiments’, and ‘Statistical methods and scientific inference’. Ed. by J. H. Bennett. With a foreword by F. Yates. Oxford: Oxford University Press (1990; Zbl 0705.62003)]). While having this as an attractive feature, the computational challenge was a disadvantage at that time but not now with modern computers. This paper introduces a permutation/randomization data algorithm to generate the permutation/randomization distributions under the null hypotheses for calculating the \(P\)-values. The properties of permutation/randomization data matrices developed by algorithms following the proposed mathematical processes are derived. Two illustrative examples demonstrate the usefulness of the proposed computational methods.

MSC:

62K10 Statistical block designs
62G10 Nonparametric hypothesis testing
62-08 Computational methods for problems pertaining to statistics
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[1] Basu, D., Randomization analysis of experimental data: the Fisher randomization test, J Am Statist Assoc, 75, 575-582 (1980) · Zbl 0444.62089
[2] Berry, KJ; Johnson, JE; Mielke, PW, The measurement of association: a permutation statistical approach (2018), Switzerland: Springer, Switzerland
[3] Boik, RJ, The Fisher-Pitman permutation test: a non-robust alternative to the normal theory F Test when variances are Heterogeneous, British J Math Statist Psychol, 40, 26-42 (1987) · Zbl 0616.62056
[4] Box, GEP; Andersen, SL, Permutation theory in the derivation of robust criteria and the study of departures from assumption, J R Stat Society Ser B, 17, 1-34 (1955) · Zbl 0065.12402
[5] Box, GEP; Hunter, WG; Hunter, JS, Statistics for experimenters: an introduction to design, data analysis, and model building (1978), New York: Wiley, New York
[6] Calinski, T.; Kageyama, S., Block designs: a randomization approach. Vol 1: Analysis (2000), New York: Springer, New York · Zbl 0963.62071
[7] Calinski, T.; Kageyama, S., Block designs: a randomization approach. Vol 2: Design (2003), New York: Springer, New York · Zbl 1108.62318
[8] David, HA, The beginnings of randomization tests, Am Stat, 62, 70-72 (2008)
[9] Dugard, P., Randomization tests: a new gold standard?, J Context Behav Sci, 3, 65-68 (2014)
[10] Easterling, RG, Fundamentals of statistical experimental design and analysis (2015), New York: Wiley, New York
[11] Edgington, ES; Onghena, P., Randomization tests (2007), New York: Chapman & Hall, New York
[12] Effron, B.; Hastie, T., Computer age statistical inference : algorithms, evidence, and data Science (2016), New York: Cambridge University Press, New York
[13] Ernst, MD, Permutation methods: a basis for exact inference, Stat Sci, 19, 676-685 (2004) · Zbl 1100.62563
[14] Ferron, JM; Levin, JR; Kratochwill, TR; Levin, JR, Single-case permutation and randomization statistical tests: Present status, promising new developments, Single-case intervention research: statistical and methodological advances, 153-183 (2014), Washington: American Psychological Association, Washington
[15] Fisher, RA, The design of experiments (1935), London: Oliver & Boyd, London
[16] Good, PI, Permutation, parametric and bootstrap tests of hypotheses (2005), New York: Springer, New York
[17] Hoeffding, W., The large-sample power of tests based on permutations of observations, Ann Math Stat, 23, 169-192 (1952) · Zbl 0046.36403
[18] Kempthorne, O., The randomization theory of experimental inference, J Am Stat Assoc, 50, 946-967 (1955)
[19] Kempthorne, O.; Doerfler, TE, The behaviour of some significance tests under experimental randomization, Biometrika, 56, 231-248 (1969) · Zbl 0175.17003
[20] Lehmann, EL, Non-parametrics: statistical methods based on ranks (1975), San Francisco: Holden-Day, San Francisco
[21] Ludbrook, J.; Dudley, H., Why permutation tests are superior to t and F Tests in biomedical research, Am Statist, 52, 127-132 (1998)
[22] Manly, BFJ, Randomization, bootstrap and Monte Carlo methods in biology (2007), Boca Raton: Chapman & Hall/CRC, Boca Raton
[23] Mielke, PW; Berry, KJ, Permutation methods: a distance function approach (2007), New York: Springer, New York
[24] Onghena, P.; Berger, V., Randomization and the randomization test: two sides of the same coin, Randomization, masking, and allocation concealment, 185-207 (2018), Boca Raton: Chapman & Hall/CRC Press, Boca Raton
[25] Pesarin, F.; Salmaso, L., Permutation tests for complex data: theory, applications and software (2010), Chichester: Wiley, Chichester
[26] Pitman, EJG, Significance tests which may be applied to samples from any population, I, J R Stat Soc Ser B, 4, 119-130 (1937) · JFM 63.1095.01
[27] Pitman, EJG, Significance tests which may be applied to samples from any populations, II, the correlation coefficient test, Suppl J R Stat Soc, 4, 225-232 (1937) · Zbl 0019.03503
[28] Pitman, EJG, Significance tests which may be applied to samples from any populations, III, Analf Var Test Biometrika, 29, 322-335 (1938) · Zbl 0018.22601
[29] Rao, CR, Linear statistical inference and its applications (1973), New York: Wiley, New York
[30] Tukey, JW, The future of data analysis, Ann Math Stat, 33, 1-67 (1962) · Zbl 0107.36401
[31] Tukey, JW, Randomization and rerandomization: the wave of the past in the future (1988), Philadelphia: Ciminera Symposium, Philadelphia
[32] Wald, A.; Wolfowitz, J., Statistical tests based on permutations of the observations, Ann Math Stat, 15, 358-372 (1944) · Zbl 0063.08124
[33] Welch, BL, On the z-test in randomized blocks and Latin squares, Biometrika, 29, 21-52 (1937) · Zbl 0017.12602
[34] Yates, F., Sir Ronald Fisher and the design of experiments, Biometrics, 20, 307-321 (1964)
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