The test of Prentice [1] is a non-parametric statistical test for the two-way analysis of variance using ranks. The null distribution of this test typically is approximated using the Chi-square distribution. However, the exact null distribution deviates from the Chi-square approximation in certain cases commonly found in applications of the test, motivating adjustments to the distribution. This manuscript presents adjustments to this null distribution correcting for continuity, multivariate skewness, and multivariate kurtosis. The effects of alternative scoring methods as non-polynomial functions of rank sums are also presented as a broader application of the approximation.
References
[1]
Prentice, M.J. (1979) On the Problem of m Incomplete Rankings. Biometrika, 66, 167-170. https://doi.org/10.2307/2335259
[2]
Kruskal, W.H. and Wallis, W.A. (1952) Use of Ranks in One-Criterion Variance Analysis. Journal of the American Statistical Association, 47, 583-621. https://doi.org/10.1080/01621459.1952.10483441
[3]
Friedman, M. (1937) The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance. Journal of the American Statistical Association, 32, 675-701. https://doi.org/10.1080/01621459.1937.10503522
[4]
Cressie, N. and Read, T.R. (1984) Multinomial Goodness-of-Fit Tests. Journal of the Royal Statistical Society Series B: Statistical Methodology, 46, 440-464. https://doi.org/10.1111/j.2517-6161.1984.tb01318.x
[5]
Jensen, D. (1977) On Approximating the Distributions of Friedman’s χ2 r and Related Statistics. Metrika, 24, 75-85. https://doi.org/10.1007/BF01893394
[6]
Gaunt, R.E., Pickett, A.M. and Reinert, G. (2017) Chi-Square Approximation by Stein’s Method with Application to Pearson’s Statistic. Annals of Applied Probability, 27, 720-756. https://doi.org/10.1214/16-AAP1213
[7]
Gaunt, R.E. and Reinert, G. (2023) Bounds for the Chi-Square Approximation of Friedman’s Statistic by Stein’s Method. Bernoulli, 29, 2008-2034. https://doi.org/10.3150/22-BEJ1530
[8]
Iman, R.L. and Davenport, J.M. (1980) Approximations of the Critical Region of the Friedman Statistic. Communications in Statistics-Theory and Methods, 9, 571-595. https://doi.org/10.1080/03610928008827904
[9]
Iman, R.L. and Davenport, J.M. (1976) New Approximations to the Exact Distribution of the Kruskal-Wallis Test Statistic. Communications in Statistics Theory and Methods, 5, 1335-1348. https://doi.org/10.1080/03610927608827446
[10]
Yarnold, J.K. (1972) Asymptotic Approximations for the Probability That a Sum of Lattice Random Vectors Lies in a Convex Set. The Annals of Mathematical Statistics, 43, 1566-1580. https://doi.org/10.1214/aoms/1177692389
[11]
Esseen, C. (1945) Fourier Analysis of Distribution Functions. A Mathematical Study of the Laplace-Gaussian Law. Acta Mathematica, 77, 1-125. https://doi.org/10.1007/BF02392223
[12]
De Leeuw, J. (2012) Multivariate Cumulants in R. https://escholarship.org/content/qt1fw1h53c/qt1fw1h53c_noSplash_8cb15933a039988ef5b788a1b4ef1b38.pdf?t=mkq6eg
[13]
Kolassa, J. (2020) An Introduction to Nonparametric Statistics. Chapman and Hall/ CRC, New York. https://doi.org/10.1201/9780429202759
[14]
Minitab, L.L.C. (2023) Example of Friedman Test. https://support.minitab.com/en-us/minitab/21/help-and-how-to/statistics/nonparametrics/how-to/friedman-test/before-you-start/example/
