Solved – How to interpret Mann-Whitney’s statistical significance if median is equal

Testing the difference between the observations of two groups by using Mann-Whitney Test has given the following output (from minitab):

                N  Median positives  137  1.0000 negatives  892  1.0000 Point estimate for η1 - η2 is 0.0000 99.0 Percent CI for η1 - η2 is (0.0001,-0.0001) W = 56899.5 Test of η1 = η2 vs η1 ≠ η2 is significant at 0.0000 The test is significant at 0.0000 (adjusted for ties) 

When I was testing other pairs of groups, I concluded that a group tends to have larger values than another group based on their medians, if the test shows a statistically significant difference. However, the example above shows that I may not be interpreting the results correctly.

Here is some additional descriptive statistics for the example above:

Variable   |  Mean | StDev  | Minimum | Q1 |Median| Q3 | Maximum positives  |  4.13 | 13.17  |  1.00   |1.00| 1.00 |1.00| 116.00 negatives  | 6.851 | 20.503 |  0.000  |1.00| 1.00 |5.00| 434.000 
  • What have I done wrong?
  • How can I figure out the direction of the difference? Can I conclude that the negatives tends to have larger values based on their larger Q3, for example?

The Mann-Whitney is not a test of medians. At best, the Mann-Whitney test can only be claimed to a be a test of differences in mean-rank between two populations' pooled ranking.

You can easily calculate medians empirically and perform a basic Wald test if you need a test of medians.

The Mann-Whitney test happens to be a reasonably powerful test of medians only when the underlying distributions are symmetric, an assumption that is clearly violated in these data. However, if a distribution is symmetric, the median also happens to be the mean (when variance is finite). This means the Mann-Whitney and the t-test are testing the same hypothesis in symmetric distributions.

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