Ranks and Pseudo-ranks-Surprising Results of Certain Rank Tests in Unbalanced Designs

Rank-based inference methods are applied in various disciplines, typically when procedures relying on standard normal theory are not justifiable. Various specific rank-based methods have been developed for two and more samples and also for general factorial designs (e.g. Kruskal-Wallis test or Akritas-Arnold-Brunner test). It is the aim of the present paper (1) to demonstrate that traditional rank procedures for several samples or general factorial designs may lead to surprising results in case of unequal sample sizes as compared with equal sample sizes, (2) to explain why this is the case and (3) to provide a way to overcome these disadvantages. Theoretical investigations show that the surprising results can be explained by considering the non-centralities of the test statistics, which may be non-zero for the usual rank-based procedures in case of unequal sample sizes, while they may be equal to 0 in case of equal sample sizes. A simple solution is to consider unweighted relative effects instead of weighted relative effects. The former effects are estimated by means of the so-called pseudo-ranks, while the usual ranks naturally lead to the latter effects. A real data example illustrates the practical meaning of the theoretical discussions.