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Kruskal–Wallis, Multiple Comparisons and Efron Dice

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  • B.M. Brown
  • T.P. Hettmansperger

Abstract

The Kruskal–Wallis test is a rank–based one way ANOVA. Its test statistic is shown here to be a quadratic form among the Mann–Whitney or Kendall tau concordance measures between pairs of treatments. But the full set of such concordance measures has more degrees of freedom than the Kruskal–Wallis test uses, and the independent surplus is attributable to circularity, or non–transitive effects. The meaning of circularity is well illustrated by Efron dice. The cases of k = 3, 4 treatments are analysed thoroughly in this paper, which also shows how the full sum of squares among all concordance measures can be decomposed into uncorrelated transitive and non–transitive circularity effects. A multiple comparisons procedure based on patterns of transitive orderings among treatments is implemented. The testing of circularities involves non–standard asymptotic distributions. The asymptotic theory is deferred, but Monte Carlo permutation tests are easy to implement.

Suggested Citation

  • B.M. Brown & T.P. Hettmansperger, 2002. "Kruskal–Wallis, Multiple Comparisons and Efron Dice," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 44(4), pages 427-438, December.
    • Handle: RePEc:bla:anzsta:v:44:y:2002:i:4:p:427-438
    DOI: 10.1111/1467-842X.00244
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    Cited by:

    1. Dennis Dobler & Sarah Friedrich & Markus Pauly, 2020. "Nonparametric MANOVA in meaningful effects," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(4), pages 997-1022, August.
    2. Edgar Brunner & Frank Konietschke & Markus Pauly & Madan L. Puri, 2017. "Rank-based procedures in factorial designs: hypotheses about non-parametric treatment effects," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1463-1485, November.
    3. Daniel Fischer & Hannu Oja & Johanna Schleutker & Pranab K. Sen & Tiina Wahlfors, 2014. "Generalized Mann–Whitney Type Tests for Microarray Experiments," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(3), pages 672-692, September.
    4. Lumley, Thomas & Gillen, Daniel L., 2016. "Characterising transitive two-sample tests," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 118-123.
    5. Umlauft, Maria & Placzek, Marius & Konietschke, Frank & Pauly, Markus, 2019. "Wild bootstrapping rank-based procedures: Multiple testing in nonparametric factorial repeated measures designs," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 176-192.

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