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 $\alpha$. Even when the $k$th sample is a shift of the other $k - 1$ samples, it may exceed $\alpha$, if the distributions are very skew. Here skewness is defined with Van Zwet's $c$-ordering of distribution functions.
"$(k - 1)$-Mean Significance Levels of Nonparametric Multiple Comparisons Procedures." Ann. Statist. 8 (1) 75 - 86, January, 1980. https://doi.org/10.1214/aos/1176344892