Kruskal–Wallis non-parametric alternatives for groups with different shaped distributions

My question does't duplicate this question, even if the titles are nearly similar.

Is there an alternative for the Kruskal–Wallis test for groups with different shaped distributions?

• Please do read through a number of highly ranked answers on this site about Kruskal-Wallis or Mann-Whitney tests. The issue of assumptions in these congeneric tests has been extensively discussed. Dec 19, 2014 at 16:01
• Just one of many good threads about it stats.stackexchange.com/q/56649/3277 Dec 19, 2014 at 16:09

I don't think the statement in the quote is accurate.

The Kruskal-Wallis is effectively a test for at least one variable being stochastically larger than at least one other, which doesn't require identity of shape. Indeed, even if it was being used as a test of identical distribution under the null, it would only be necessary for the shape to be identical under the null; if the null is false, there's still no requirement for the shape to be the same then.

If, however, one was looking specifically at say a location shift alternative, in order to use it specifically as say a test of location difference (a test of medians, or of means, or of tenth percentiles or ... against a shift in the same) then the shapes would then be assumed the same under both null and alternative in order that rejection of the null implied that location shift.

• @glen-b Thank you for your answer! What you think, is incorect to use the Kruskal-Wallis test only for two groups comparison? Dec 23, 2014 at 20:57
• When you have two groups you'd normally use a Wilcoxon-Mann-Whitney test rather than K-W (though they should be equivalent for the two tailed case if you use the exact null distribution for both). It can be incorrect to use a Kruskal-Wallis in a variety of situations, but the statement doesn't of itself characterize those situations. It's possible for the Kruskal-Wallis to apply in a fairly broad range of situations. Dec 23, 2014 at 22:35
• @Iurie And when you have >2 groups, you would use the Kruskal-Wallis, and if you wanted to pin down which groups differed from which post hoc (a la the post hoc t tests following rejection of a one-way ANOVA), you would use a something like Dunn's test, or the Conover-Iman test. Jun 26, 2016 at 17:32
• Do you have any citation to back up your comment on the distribution shape regarding KW? Your comment is very helpful, and since I'm writing a paper I need a citation. Thanks Oct 18, 2016 at 21:26
• @Carlos I'm not sure I could find you a reference for a basic statement about how hypothesis tests work. I can prove it easily enough - but I can't imagine any stats journal worth submitting to would let a statistician keep such a statement in a paper. The editor's comment would be "take out all that nanny-state stuff -- you get 5 pages, not 15". ... and if we find a journal where something like that was not seen as obvious, simply getting something into print would be no guarantee of it being true. Some journal outside statistics might publish a paper that says it perhaps. ...ctd Oct 18, 2016 at 23:46