# Kruskal-Wallis Test for H0: ranks differ between groups and H1: ranks do not differ

The traditional Kruskal-Wallis Test tests null hypothesis $H_A$: ranks do not differ between groups against the alternative hypothesis $H_B$ in which the ranks differ between groups. However, what if we are interested in rejecting $H_B$ instead of $H_A$. What will the distribution of the test statistic look like?

I'm interested in analytic solutions. Possibly, an analytic solution requires further information and assumptions about the mechanism that generates unequal means. Please, list the assumptions.

My guess is that this is not a simple problem, since I couldn't find any analytic formulas for the calculation of power of the traditional Kruskal-Wallis test.

## 1 Answer

Roughly, if you want to switch roles of point hypothesis and alternative, you are looking for an equivalence test.

Yes, equivalence tests are much more complicated than usual point hypothesis or one-sided hypothesis tests. In particular you need to specify some equivalence margin to be able to find a test. Apart from the mathematically-statistically point, the choice of this margin has to be justified by reasoning about the application's context.

There is a great book about this exciting field by Stefan Wellek: "Testing statistical hypotheses of equivalence" (also obtainable in German). Chapter 7.5 is about "a nonparametric k-samples test for equivalence". This is exactly what you are looking for.

• Thanks for the answer! Any chance, You could pull the relevant equation(s) from the book, or point me to the primary source that Wellek refers to? Both book versions are very rare. To obtain the book in the local library I would need to get it shipped from another library/deposit... – matus Jun 19 '17 at 9:54
• Sorry, same to me, I didn't takte notes about this particular chapter as I had loaned the boot. Nevertheless, it's worthy to read "testing statistical hypotheses of equivalence" or " ... and noninferiority" because the issue of the equivalence margin is discussed there. – Horst Grünbusch Jun 20 '17 at 10:09