Pseudo-Ranks: How to Calculate Them Efficiently in R

Many popular nonparametric inferential methods are based on ranks. Among the most commonly used and most famous tests are for example the Wilcoxon-Mann-Whitney test for two independent samples, and the Kruskal-Wallis test for multiple independent groups. However, recently, it has become clear that the use of ranks may lead to paradoxical results in case of more than two groups. Luckily, these problems can be avoided simply by using pseudo-ranks instead of ranks. These pseudo-ranks, however, suffer from being (a) at first less intuitive and not as straightforward in their interpretation, (b) computationally much more expensive to calculate. The computational cost has been prohibitive, for example, for large-scale simulative evaluations or application of resampling-based pseudorank procedures. In this paper, we provide different algorithms to calculate pseudo-ranks efficiently in order to solve problem (b) and thus render it possible to overcome the current limitations of procedures based on pseudo-ranks.