Mann-Whitney U Test (Wilcoxon Rank-Sum Test)

The Wilcoxon Rank-Sum Test was developed by Wilcoxon in 1945, and it is useful when comparing the location of two independent samples. A slightly different version of the test was later introduced by Mann and Whitney in 1947. Therefore, it is some times referred to as the Wilcoxon Mann-Whitney test.

The underlying assumptions of the Wilcoxon Rank-Sum Test are that the scores are independent and come from a continuous probability distribution. The null hypothesis is H0: fi(x) = gi(x), or the two samples come from identical distributions. The alternative hypotheses are Ha: fi(x) ≠ gi(x) (two-tail), Ha: fi(x) < gi(x) (lower-tail), or Ha: fi(x) > gi(x) (upper-tail).

The null hypothesis suggests that the Wilcoxon Rank-Sum is a test of general differences. Even though the Wilcoxon procedure is a test of stochastic ordering, it is particularly powerful in detecting differences between group means. As a rank-based procedure, it is not useful in testing for differences in scale (variance).

The Wilcoxon test is nonparametric. This means that it preserves the Type I error rate (i.e., false positive rate) to nominal alpha regardless of the population shape. This is a fundamental advantage over its parametric counterpart, the Student's t test, which relies on the normality distribution assumption.

When sampling from nonnormal distributions, the Wilcoxon Rank-Sum Test is often more powerful than the t test when the hypothesis being tested is a shift in location parameter. This was suggested by the large sample property known as the asymptotic relative efficiency (ARE). The ARE of the Wilcoxon (Mann-Whitney) relative to the t test under population normality is 0.955. However, under population nonnormality, the ARE of the Wilcoxon Rank-Sum Test can be as high as ∞.

Small-sample Monte Carlo studies confirmed the comparative statistical power advantage of the Wilcoxon Rank-Sum Test over the t test for departures from nonnormality. It is often three to four times more powerful for sample sizes and treatment effect sizes common in education and psychology.

Because of the relationship between statistical power and sample size, research studies may be designed with considerably fewer participants when using the Wilcoxon Rank-Sum Test instead of the t test. It provides a considerable efficiency advantage in terms of cost, time, and effort in conducting an experiment.

In order to compute the Wilcoxon Rank-Sum Test, combine the two samples, order the scores from lowest to highest, and keep track of the score's group membership. The ordered scores are assigned ranks. If there are tied values, the average of the ranks is assigned to each of the tied scores. The Wilcoxon formula is

where Ri are the ranks and Sn is the sum of the ranks for a sample of size n.

The rank-sum statistic can be converted to a Mann-Whitney U in order to use commonly available tabled critical values for the U statistic. The formula for the conversion is

If a table of critical values is not available, the Wilcoxon Rank-Sum Test can be evaluated with a large-sample approximation using the following formula:

where m and n are the sample sizes of the two groups and Sn is the rank sum for sample n. The large-sample approximation for the critical value to test the obtained Mann-Whitney U statistic

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