Mann–Whitney U test

In statistics, the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric test of the null hypothesis that, for randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X. Nonparametric tests used on two dependent samples are the sign test and the Wilcoxon signed-rank test. Although Mann and Whitney developed the Mann–Whitney U test under the assumption of continuous responses with the alternative hypothesis being that one distribution is stochastically greater than the other, there are many other ways to formulate the null and alternative hypotheses such that the Mann–Whitney U test will give a valid test. A very general formulation is to assume that: All the observations from both groups are independent of each other, The responses are at least ordinal (i.e., one can at least say, of any two observations, which is the greater), Under the null hypothesis H0, the distributions of both populations are identical. The alternative hypothesis H1 is that the distributions are not identical. Under the general formulation, the test is only consistent when the following occurs under H1: The probability of an observation from population X exceeding an observation from population Y is different (larger, or smaller) than the probability of an observation from Y exceeding an observation from X; i.e., P(X > Y) ≠ P(Y > X) or P(X > Y) + 0.5 · P(X = Y) ≠ 0.5. Under more strict assumptions than the general formulation above, e.g., if the responses are assumed to be continuous and the alternative is restricted to a shift in location, i.e., F1(x) = F2(x + δ), we can interpret a significant Mann–Whitney U test as showing a difference in medians. Under this location shift assumption, we can also interpret the Mann–Whitney U test as assessing whether the Hodges–Lehmann estimate of the difference in central tendency between the two populations differs from zero.