Comparing the Effect of Mean- and Median-Centering on the Empirical Type I Error Rates of the Wilcoxon-Mann-Whitney Test

The Wilcoxon-Mann-Whitney (WMW) test checks for the equality of distributions by estimating how likely a random observation from one population is less than a random observation from the other population. The quantity estimated is known as stochastic superiority or relative effect. Even though it is not technically sound in general, in some cases, the WMW test is considered similar to checking for the equality of medians. In his 2003 paper, Donald Zimmerman claimed that the normal approximation of the WMW test is not robust to even slight variance heterogeneity, even if sample sizes are large and equal due to unacceptably high empirical Type 1 error rates. In fact, some of the artificial inflation in the empirical Type 1 error rates in his simulation was caused by shifting the distributions from which random observations were generated to have mean zero (mean-centering) before scaling them to acquire the desired standard deviation ratios. In our adjusted method, we shift them to have median zero (median-centering) instead before scaling. Via simulation, we calculate the empirical Type 1 error rates to both replicate Zimmerman’s results using his original method and to show that median-centering indeed gives less inflated Type 1 error rates. Finally, we provide a brief theoretical justification to illustrate why median-centering improves the Type 1 error rates using the idea of stochastic superiority