Mann-Whitney test is not just a test of medians: differences in spread can be important

Abstract

The Mann-Whitney (or Wilcoxon-Mann-Whitney) test is sometimes used for comparing the efficacy of two treatments in clinical trials. It is often presented as an alternative to a t test when the data are not normally distributed. Whereas a t test is a test of population means, the Mann-Whitney test is commonly regarded as a test of population medians. This is not strictly true, and treating it as such can lead to inadequate analysis of data. #### Summary points The Mann-Whitney test is used as an alternative to a t test when the data are not normally distributed The test can detect differences in shape and spread as well as just differences in medians Differences in population medians are often accompanied by equally important differences in shape Researchers should describe the clinically important features of data and not just quote a P value The Mann-Whitney test is a test of both location and shape. Given two independent samples, it tests whether one variable tends to have values higher than the other. As Altman states, one form of the test statistic is an estimate of the probability that one variable is less than the other,1 although this statistic is not output by many statistical packages. In the case where the only distributional difference is a shift in location, this can indeed be described as a difference in medians. Hence, for example, the online help facility in Minitab 10.51 states that the Mann-Whitney test is “a two-sample rank test for the difference between two population medians … It assumes that the data are independent random samples from two populations that have the same shape.” Figure 1 shows two distributions for which this is the case. One distribution is shifted 0.75 units …