Mann-Whitney U-Test

The Mann-Whitney U-Test can be used to test whether there is a difference between two samples (groups), and the data need not be normally distributed.

To determine if there is a difference between two samples, the rank sums of the two samples are used rather than the means as in the t-test for independent samples.

The Mann-Whitney U test is thus the non-parametric counterpart to the t-test for independent samples; it is subject to less stringent assumptions than the t-test. Therefore, the Mann-Whitney U test is always used when the requirement of normal distribution for the t-test is not met.

Assumptions Mann-Whitney U-Tests

To compute a Mann-Whitney U test, only two independent samples with at least ordinal scaled characteristics need to be available. The variables do not have to satisfy any distribution curve.

If the data are available in pairs, the Wilcoxon test must be used instead of the Mann-Whitney U test.

Hypotheses Mann-Whitney U-Tests

The hypotheses of the Mann-Whitney U-test are very similar to the hypotheses of the independent t-test. The difference, however, is that in the case of the Mann-Whitney U test, the test is based on a difference in the central tendency, whereas in the case of the t-test, the test is based on a difference in the mean values. Thus, the Mann-Whitney U test results in:

• Null hypothesis: There is no difference (in terms of central tendency) between the two groups in the population.
• Alternative hypothesis: There is a difference (with respect to the central tendency) between the two groups in the population.

Calculate Mann-Whitney U-Test

To calculate the Mann-Whitney U test for two independent samples, the rankings of the individual values must first be determined (An example with tied ranks follows below).

These rankings are then added up for the two groups. In the example above, the rank sum T₁ of the women is 37 and the rank sum of the men T₂ is 29. The average value of the ranks is thus R̄₁= 6.17 for women and R̄₁= 5.80 for men. The difference between R̄₁ and R̄₂ now shows whether there are possible differences between the reaction times. In the next step, the U-values are calculated from the rank sums T₁ and T₂.

where n₁, n₂ are the number of elements in the first and second group respectively. If both groups are from the same population, i.e., the groups do not differ, then the value of both U values is the expected value of U. After the mean and dispersion have been estimated, z can be calculated. For the Mann-Whitney U value, the smaller value of U₁ and U₂ is used.

Depending on how large the sample is, the p-value for the Mann-Whitney U-test is calculated in a different way. For up to 25 cases, the exact values are used, which can be read from a table. For larger samples, the normal distribution can be used as an approximation.

Note: In this example, we would actually use the exact value, but we will nevertheless use the normal distribution. To do this, the z-value was simply inserted into the z-value to p-value calculator of DATAtab.

If the calculated z-value is larger than the critical z-value, the two groups differ.

Calculate Mann-Whitney U test with tied ranks

If several people share a rank, connected ranks are present. In this case, there is a change in the calculation of the rank sums and the standard deviation of the U value. We will now go through both using an example.

In the example it can be seen that the...

• ...reaction times 34 occur twice and share the ranks 2 and 3
• ...reaction times 39 occur three times and share the ranks 6, 7 and 8.

To account for these connected ranks, the mean values of the joined ranks are calculated in each case. In the first case, this results in a "new" rank of 2.5 and in the second case in a "new" rank of 7. Now the rank sums T can be calculated.

Since the rank ties are clearly visible in the upper table, a term is calculated here that is needed for the later calculation of the u-value in the presence of rank ties.

Now all values are available to calculate the z-value considering connected ranks.

Again, noting that you actually need about 20 cases to assume normal distribution of U values.

Example with DATAtab

A Mann-Whitney U-Test can be easily calculated with DATAtab. Simply copy the table below or your own data into the statistics calculator and click on Hypothesis tests. Then click on the two variables and select Non-Parametric Test.

DATAtab then gives you the following table for the Mann-Whitney U-Test:

The Mann-Whitney U-Test works with ranks, so the result will first show the middle ranks and the rank sum. The reaction time of women has a slightly lower value than that of men.

DATAtab gives you the asymptotic significance and the exact significance. The significance used depends on the sample size. As a rule:

• n₁ + n₂ < 30 → exact significance
• n₁ + n₂ > 30 → asymptotic significance

Therefore the exact significance is used for this example. The significance (2-tailed) is .931 and thus above the significance level of 0.05. Therefore, no difference between the reaction time of men and women can be determined with these data.

Interpret Mann-Whitney U-Test

The reaction time female group had the same high values (Mdn= 39) as the reaction time male group (Mdn= 39). A Mann-Whitney U-Test showed that this difference was not statistically significant, U=14, p=.931, r=0.06.

Mann-Whitney U-Test effect size

In order to make a statement about the effect size in the Mann-Whitney U-Test, you need the Standardised test statistic z and the number of pairs n, with this you can then calculate the effect size with the equation below

In this case, an effect size r of 0.06. In general, one can say about the effect strength:

• effect size r less than 0.3 → small effect
• effect size r between 0.3 and 0.5 → medium effect
• effect size r greater than 0.5 → large effect

In this case, the effect size of 0.06 is therefore a small effect.