$(k - 1)$-Mean Significance Levels of Nonparametric Multiple Comparisons Procedures

We consider the nonparametric pairwise comparisons procedures derived from the Kruskal-Wallis k-sample test and from Friedman's test. For large samples the $(k - 1)$-mean significance level is determined, i.e., the probability of concluding incorrectly that some of the first $k - 1$ samples are unequal. We show that in general this probability may be larger than the simultaneous significance level α. Even when the kth sample is a shift of the other $k - 1$ samples, it may exceed α, if the distributions are very skew. Here skewness is defined with Van Zwet's c-ordering of distribution functions.