# Mann-Whitney test W statistics

Below is Mann-Whitney test result from Minitab. At the end of result, there is no p-value but "Cannot reject since W is < 363.0". What does it mean? I cannot reject H0? But the median looks significantly different here.

========================================================================

N  Median

Point estimate for ETA1-ETA2 is -5.000
95.1 Percent CI for ETA1-ETA2 is (-9.000,-0.001)
W = 315.5
Test of ETA1 = ETA2 vs ETA1 > ETA2

Cannot reject since W is < 363.0


The Mann-Whitney U-test isn't actually a test of the median, although that is often the way it is explained. (If you literally wanted a test of the equality of the median, you could do that by bootstrapping or using a permutation test.)

In truth, MW is a test for stochastic dominance. That is, it is a test of whether, if you drew one value at random from each group, is the value from one group typically higher than the value from the other? If the shapes of the distributions for the two groups differ, it is quite possible that the medians could differ without one stochastically dominating the other significantly, and it is very possible for one to significantly stochastically dominate the other even if the medians are identical.

I have never used Minitab, but it appears to me that you may have run a one-tailed test and selected the wrong tail by mistake. I notice the output says:

...
... ETA1-ETA2 is -5.000
...
Test of ETA1 = ETA2 vs ETA1 > ETA2
...


If you tested Test of ETA1 = ETA2 vs ETA1 < ETA2 instead, you might get a significant result.

• Nice answer; you can also use a M-W to test for differences in the median if you add sufficient assumptions to make it one (e.g. assuming that the distributions will be identical under the null and that the medians will differ under the alternative, which - while it narrows the scope of what a M-W can test - is usually the kind of assumption people have in mind, at least implicitly, when they use it). I agree, it looks like a one-tailed test was probably done; it's a pity that sufficient information was not supplied to be sure – Glen_b Apr 9 '16 at 1:43
• @Glen_b; for information, Minitab seems to make those assumptions. Its help text says "An assumption for the Mann-Whitney test is that the data are independent random samples from two populations that have the same shape and a scale that is continuous or ordinal (possesses natural ordering) if discrete." – user20637 May 3 '16 at 15:42