In a previous question, I asked about comparing the power of a t test to a Mann Whitney test under different situations. One of the answers pointed out that the worst that the Mann-Whitney can ever perform relative to the t-test is that it would require 1/0.864x as much data to give the same power as the t test so long as the data sets being compared were from the same distribution.
I guess I am confused about how power can be compared between these two tests given that they test different null hypotheses. For example, if I estimate the power for a t-test at 0.9 using effect size equal to a 20% difference between means, then that makes sense for a t test. But a Mann Whitney test does not test for differences between means. If the distributions were the same it would test for differences between medians. Am I right in thinking that the Mann Whitney test would require 1/0.864x as much data to detect a 20% difference in medians with a power of 0.9?