Mann-Whitney test

Also known as Wilcoxon-Mann-Whitney test, for reasons that we explain here.

Definition of the Mann-Whitney test

The Mann-Whitney test is used to test the null hypothesis H₀ according to which 2 independent samples were drawn from the same population (or identical populations).

The observations must be on a numeric or ordinal scale (not just categorical). The samples do not need to have the same number of observations.

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It is occasionally read that the null hypothesis bears only on the equality of the medians of the distributions, leaving aside the issue of the shapes of the distributions. It is important to understand that the null hypothesis demands that the two distributions, although arbitrary, have the same shape, as in this illustration :

The alternative hypothesis are :

    * H₁ : H₀ does not hold  (two-sided test).

    * H₁ : the central tendency of D₁ is larger (resp. smaller) than that of D₂    (one-sided tests).

Mann-Whitney test and t test

There is a clear similarity between the t test on independent samples and the Mann-Whitney test as both are testing the identity of two independent populations.

    * The t test on independent samples is the archetype of the parametric test : the assumptions that the two populations are normal with identical variances is essential (so, in fact, the t test is testing the identity of two populations).

    * The Mann-Whitney test, on the other hand, is non parametric : it does not rest on any assumption concerning the underlying distributions. It is therefore more widely applicable than the t-test.

The price to pay for this generality is that the Mann-Whitney test is less powerful than the t test, because it first converts the values of the observations into ranks, and some information is lost in the process. Yet :

    * Even when the assumptions of the t test are justified, it can be shown that the Mann-Whitney is hardly less powerful that the t test for large samples.

    * The t test is sensitive to departures from normality, and to the difference between the two variances, especially when the sample sizes are different.

So whenever there exist some doubts about the validity of these assumptions, the Mann-Whitney test is an excellent alternative.

Other non parametric identity tests

The Mann-Whitney test is fullfiling the same need as two other classical non parametric identity tests :

    * The Chi-square test of identity,

    * The Kolmogorov-Smirnov test.

Mann-Whitney and Kruskal-Wallis tests

The t test cannot be straightforwardly extended to more than two populations : another, more complicated test, has to be designed for the purpose of testing the equality of the means of more than two normal populations (ANOVA).

The same happens with the Mann-Whitney test, and the non parametric test for testing the identity of more than two independent distributions is the Kruskal-Wallis test.

The two statistics : Wilcoxon and Mann-Whitney

The test was first introduced by Wilcoxon, and independently by Mann and Whitney shortly thereafter. The two presentations do not use the same test statistic, but we show that these two statistics are equivalent : one can be calculated from the other.

We will show how the probability distributions of these statistics are calculated.

Large samples

For large samples, the distribution of the test statistic is normal, with known mean and variance. Specific tables of critical values are therefore needed only for small samples.

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We here describe the (Wilcoxon)-Mann-Whitney test.

    * We first describe the process of merging the two samples and ranking the assignment.

    * We then go over the rationale of the Wilcoxon statistic, which is somewhat simpler than the Mann-Whitney statistic. We spend some time explaining how the distribution of the statistic is calculated when the null hypothesis is true. Although this information is of little value to the practitioner, it sheds some light on the concept of "distribution free" test. In this case, the calculation relies on very simple probabilistic, and somewhat intricate combinatorial arguments.

    * We then go over the Mann-Whitney statistic, which we show to be equivalent to the Wilcoxon statistic.

    * We state (without proof) that the limit distributions of these statistics are normal.

    * We finally give numerical examples of the test both for small and large samples. In the latter case, we also conduct a t test and compare the result with that of the Mann-Whitney test.

THE (WILCOXON-) MANN-WHITNEY TEST

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