| Interpreting the Mann-Whitney test
How the Mann-Whitney test works
The Mann-Whitney test, also called the rank sum test, is a nonparametric test that compares two unpaired groups. To perform the Mann-Whitney test, Prism first ranks all the values from low to high, paying no attention to which group each value belongs. If two values are the same, then they both get the average of the two ranks for which they tie. The smallest number gets a rank of 1. The largest number gets a rank of N, where N is the total number of values in the two groups. Prism then sums the ranks in each group, and reports the two sums. If the sums of the ranks are very different, the P value will be small.
The P value answers this question: If the populations really have the same median, what is the chance that random sampling would result in a sum of ranks as far apart (or more so) as observed in this experiment?
If your samples are small, and there are no ties, Prism calculates an exact P value. If your samples are large, or if there are ties, it approximates the P value from a Gaussian approximation. Here, the term Gaussian has to do with the distribution of sum of ranks, and does not imply that your data need to follow a Gaussian distribution. The approximation is quite accurate with large samples, and is standard (used by all statistics programs).
How to think about the results of a Mann-Whitney test
The Mann-Whitney test is a nonparametric test to compare two unpaired groups. The key result is a P value that answers this question: If the populations really have the same median, what is the chance that random sampling would result in medians as far apart (or more so) as observed in this experiment?
If the P value is small, you can reject the idea that the difference is a coincidence, and conclude instead that the populations have different medians.
If the P value is large, the data do not give you any reason to conclude that the overall medians differ. This is not the same as saying that the medians are the same. You just have no compelling evidence that they differ. If you have small samples, the Mann-Whitney test has little power. In fact, if the total sample size is seven or less, the Mann-Whitney test will always give a P value greater than 0.05 no matter how much the groups differ.
Checklist. Is the Mann-Whitney test the right test for these data?
Before interpreting the results of any statistical test, first think carefully about whether you have chosen an appropriate test. Before accepting results from a Mann-Whitney test, ask yourself these questions (Prism cannot help you answer them):
| Question |
Discussion |
| Are the "errors" independent? |
The term "error" refers to the difference between each value and the group median. The results of a Mann-Whitney test only make sense when the scatter is random - that whatever factor caused a value to be too high or too low affects only that one value. Prism cannot test this assumption. You must think about the experimental design. For example, the errors are not independent if you have six values in each group, but these were obtained from two animals in each group (in triplicate). In this case, some factor may cause all triplicates from one animal to be high or low. See The need for independent samples.
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Are the data unpaired?
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The Mann-Whitney test works by ranking all the values from low to high, and comparing the mean rank in the two groups. If the data are paired or matched, then you should choose a Wilcoxon matched pairs test instead.
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Are you comparing exactly two groups?
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Use the Mann-Whitney test only to compare two groups. To compare three or more groups, use the Kruskal-Wallis test followed by post tests. It is not appropriate to perform several Mann-Whitney (or t) tests, comparing two groups at a time.
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Are the shapes of the two distributions identical?
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The Mann-Whitney test does not assume that the populations follow Gaussian distributions. But it does assume that the shape of the two distributions is identical. The medians may differ — that is what you are testing for — but the test assumes that the shape of the two distributions is identical. If two groups have very different distributions, transforming the data may make the distributions more similar. |
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Do you really want to compare medians?
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The Mann-Whitney test compares the medians of two groups. It is possible to have a tiny P value - clear evidence that the population medians are different — even if the two distributions overlap considerably.
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If you chose a one-tail P value, did you predict correctly?
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If you chose a one-tail P value, you should have predicted which group would have the larger median before collecting any data. Prism does not ask you to record this prediction, but assumes that it is correct. If your prediction was wrong, then ignore the P value reported by Prism and state that P>0.50. See One- vs. two-tail P values.
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Are the data sampled from non-Gaussian populations?
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By selecting a nonparametric test, you have avoided assuming that the data were sampled from Gaussian distributions. But there are drawbacks to using a nonparametric test. If the populations really are Gaussian, the nonparametric tests have less power (are less likely to give you a small P value), especially with small sample sizes. Furthermore, Prism (along with most other programs) does not calculate confidence intervals when calculating nonparametric tests. If the distribution is clearly not bell-shaped, consider transforming the values to create a Gaussian distribution and then using a t test. |
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