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.