Effect of within-sample dependence on the Mann–Whitney–Wilcoxon statistic

Motivated by a problem involving correlation due to genetic relationships within a sample, we investigate the effect of within-sample dependence and between-sample independence on the variance of the Mann–Whitney–Wilcoxon statistic, given that the marginal distributions are identical in each sample. The dependence increases the variance of the statistic by a factor equal in order to the combined sample size when three specified joint marginal distributions are chosen. The results for the two-sample statistic are extended to the mean of the Kruskal–Wallis statistic, when between-sample independence and within-sample dependence are assumed. The effect is similar in this case in that the mean of the statistic is increased by a factor of the same order as the sample size. A numerical example is given.