since 2001.11.04

final update 2008.08.05

## Mann-Whitney U test and unequal variances

For many of inquiries on this problem, this page was moved to this web site from a private statistical memorandum.

### Beginning

Mann-Whitney U test (i.e., Wilcoxon rank sum test ) is a nonparametric test for comparison of two independent samples, and one of most widely used nonparametric tests. I noticed that there were many of examples where U test was made when variances were unequal.

### Problem

When determining the distribution of test statistic (U) under the null hypothesis, Mann-Whitney U test assumes that two samples are sampled from one identical population. And the distribution of U statistic under the null hypothesis is different between the case of two populations with an identical location but different variations, and the case of an identical population (the table for U test is made for the latter). One population has one variation. So, Mann-Whitney U test assumes the equal variances (homoscedasticity) and the different variations of two populations affect results of the test. It has been noted for a long time in statistical books (for references, see the paper shown in below). However, unfortunately, there are some (not rare) examples where authors wrote that they used Mann-Whitney U test because of unequal variances (!).

I published the following paper. One may feel like "Are nonparametric tests affected by unequal variances?" [Please note this is not a joke, it was an actual question], "Is it true? I do not believe!" or "What should I do?". Please read it.

Kasuya, E.（2001） Mann-Whitney U test when variances are unequal. Animal Behaviour,61:1247-1249. [IngentaConnect] (Hard copies of reprints are exhausted.)

#### Unequal variances

In the case discussed here　(the same as statistical terminology), unequal or equal variances refer to neither difference between the sample variances nor significant difference between the sample variances. They refer to that of population (in statistical sense) variances. It is similar in the case of t-test under unequal variances (Behrens-Fisher problem)

#### Other tests

It is straightforward that the same problem also happens in Kruskal-Wallis test, which is the multi-sample extension of U test.

key words: Mann-Whitney U test, unequal variances, nonparametrics, nonparametric test, homoscedasticity, heteroscedasticity, Wilcoxon rank sum test, type I error rate.