Sorry, I planned to get back to you on this yesterday, but I was injured on a bus trip and lost pretty much a whole day; it wasn't until about 4 this morning that I felt in a state to read about the Wilcoxon-Mann-Whitney. Having now had a quick look at Welleck on that, he works with the obvious choice of unit: P(Xᵢ >Yⱼ ) (or its complement, it doesn't matter which)

This is readily estimated from the sample U. That doesn't remove the issue of it not being nonparametric under the alternative.

What he does is work with an asymptotic result. He calls it "asymptotically nonparametric" which is technically true but that's perhaps a little misleading; by the same token a z-statistic is "asymptotically nonparametric" as well (as long as the variance of the numerator is finite).

If you're happy to call a z-test asymptotically nonparametric then that's fine. Anyway, the asymptotic result is (unsurprisingly) Gaussian, since U/mn is a kind of average (though not of independent terms, the calculation of the dependence is an old result -- though not one I had seen before)

@Nick Hi

Sorry to hear about your accident.

Thanks. The bus went around a hairpin bend, and (due to it being full) I was seated near the back. I wasn't watching and the bus took it too fast, I slid out of the seat and the back of my neck landed pretty hard on a step just in front of the rear bench seat.

I think I'll be fine in a few days.

@Alexis If a specific size of P(Xᵢ >Yⱼ ) - 1/2 is of interest for equivalence in the WMW (I guess there would be some occasions where that's quite suitable) then this is all fine, it will do what is required in sufficiently large samples. It's a bit hard to say how large that might need to be but I'd expect it's fairly robust so the procedure will probably work reasonably across a wide range of cases as long as that form of equivalence statement is suitable.

This does make me wonder if there might indeed be a similarly simple - but potentially meaningful in a practical sense of 'equivalent' - population quantity for the Shapiro-Wilk test (or if not, at least for the Shapiro Francia) that might be suitable in some cases and for which an asymptotically normal result under a collection of alternatives would be feasible.

That is, it may be feasible to get somewhere with the Shapiro-Wilk; I hadn't really thought about just trying for large sample results.

I don't hold high hopes but maybe there's some way to get there.

Yipes, Glen_b! Wishing you best recovery. Yes, large sample is definitely where I was coming from with TOST using z.

I haven't, however, formulated TOST for the rank sum test using an equivalence range about the 1/2 in the null... that'd be pretty sweet. Hmm...

Holler if you have ideas about the Shapiro-Wilk/Shapiro-Francia asymptotic tests... those seem a natural fit for the motivation of equivalence tests to me, since the motivation for their application is often to show that some variable is not different than normal.