Power comparisons of significance tests of location using scores, ranks, and modular ranks

The Type I error probability and the power of the independent samples t test, performed directly on the ranks of scores in combined samples in place of the original scores, are known to be the same as those of the non-parametric Wilcoxon-Mann-Whitney (WMW) test. In the present study, simulations revealed that these probabilities remain essentially unchanged when the number of ranks is reduced by assigning the same rank to multiple ordered scores. For example, if 200 ranks are reduced to as few as 20, or 10, or 5 ranks by replacing sequences of consecutive ranks by a single number, the Type I error probability and power stay about the same. Significance tests performed on these modular ranks consistently reproduce familiar findings about the comparative power of the t test and the WMW tests for normal and various non-normal distributions. Similar results are obtained for modular ranks used in comparing the one-sample t test and the Wilcoxon signed ranks test.