What can we say when the Wilcoxon signed-rank paired test shows significance but the means and the medians have different orderings?

Question

In Wilcoxon signed-rank paired test, the hypotheses are

$H_0$: difference between the pairs follows a symmetric distribution around zero
$H_1$: difference between the pairs does not follow a symmetric distribution around zero

So 0 is both the mean and the median. Now suppose I have two groups of measurements and perform a Wilcoxon signed-rank paired test on them. The result shows significance so we can reject $H_0$. However, we also have $\mathrm{Mean}_1 > \mathrm{Mean}_2$ and $\mathrm{Median}_1 < \mathrm{Median}_2$. So what is the conclusion in this case (i.e., which group is "better")? Or is it indicating that Wilcoxon signed-rank paired test is not strong enough in this situation, so I cannot have conclusions here?

Answer 1

Everything in your question is perfectly consistent.

1. You have assumed symmetry of pair-differences under the null, but you rejected the null, so it seems you are not in that situation

2. Even if you had symmetry in the population, the sample mean and median may not be equal and you can easily get a reversal of direction from one to the other.

3. The test itself doesn't compare means or medians; when the assumption of symmetry under the null is true the population values - the thing the test looks at, versus the difference in medians and the difference in means (assuming means exist) - will all be the same: $0$, but again, you rejected, so there's no good reason to think that even the population differences in mean and median would necessarily need to be in the same direction.

   What the test is really "measuring" (in the sense of how far the pair-differences are from 0) is the median of pairwise averages of the pair-differences. This is usually close to the median if the distribution of differences is nearly symmetric, but doesn't have to be in general. So if you want to talk about the direction of difference your test detected, report that. Many stats packages will give you a confidence interval for the difference.