New Zealand Statistical Association 2024 Conference

Robin Willink

University of Otago, Wellington

A simple adjustment makes the Wilcoxon rank-sum/Mann-Whitney test uniformly more powerful

The Wilcoxon rank-sum/Mann-Whitney (WMW) test is the standard non-parametric test for a difference between two populations. In its Wilcoxon form, the test can be seen to be a permutation test involving the sum of ranks in one sample. Ranks are equally spaced, and their summation creates artificial ties in the null distribution, which is obtained under all possible permutations. The presence of these ties is unhelpful because they increase the p-value. If we apply a mild non-linear transformation to the ranks before summing, (for example, if we raise each rank to the power of 1.0001), then we can avoid these ties without affecting the ordering in the null distribution in any other way. The result is a valid test procedure that can produce a smaller p-value but not a larger p-value, meaning that the test is uniformly more powerful than the WMW test. The increase in power depends on the sample sizes. When the samples are small and the standard WMW test has a power of 0.50, the procedure can have a power of, say, 0.55. The increase is negligible when the smaller sample size is 20 or more.