Wilcoxon-Mann-Whitney test for mean difference under a location shift assumption

Question

If we make an assumption that the difference in the two groups is simply a shift in location, we can say that the test is a test of the difference in medians. However, if the groups have the same distribution, then a shift in location will move medians and means by the same amount and so the difference in medians is the same as the difference in means. Thus I think the Wilcoxon-Mann-Whitney test should also be a test for the difference in means.

It is right? How about a confidence interval for mean difference, comparing with a Hodges-Lehmann estimate.

Answer 1

If you assume a pure location-shift alternative and the population mean is finite, this is so; it's as much a test of a shift in mean as it is a shift in median, 10th percentile, 25% trimmed mean and so on through all manner of location statistics.

(Indeed a number of posts on site discuss this already)

As for an estimate (and an interval) for that population location shift, the usual one based on the Hodges-Lehmann statistic will be suitable (as long as that assumption is reasonable).

If the pure-shift assumption were doubtful, it would be necessary to think more carefully about what it was you wanted to measure, and the best way to go about testing it.