A SAS R macro for the multivariate extension of the Kruskal-Wallis test including multiple comparisons: Randomization and ϰ2 criteria

Abstract

In a multi-group experimental design where interest is in a univariate response, the nonparametric Kruskal-Wallis test [Kruskal and Wallis (1952)] provides a potentially more powerful alternative to the parametric one-way analysis of variance when the assumptions of normality are in question. For multivariate response, Puri and Sen (1966) proposed a generalization of the Kruskal-Wallis test that is a nonparametric analog of the one-way multivariate analysis of variance and can be used when multivariate response data are measured on at least an ordinal scale. Large sample theory suggests that the multivariate Kruskal-Wallis statistic is approximately distributed as chi-square, but in small samples the behavior of the statistic warrants conditioning on the observed data and using randomization theory to tabulate the exact distribution. We have written a SAS [SAS Institute, Inc. (1985)] macro that computes the probability values and tabulates the exact distribution for the univariate and multivariate Kruskal-Wallis test that will be useful to applied statisticians.