Test statistic for follow up procedure (stepwise step down) of Kruskal Wallis

When conducting a Kruskall Wallis test, it is possible to follow up the procedure with a post hoc test, looking for homogeneous subsets. My question is which statistic is used to perform this procedure (e.g., is it Chi-Square, z, or the original H)?

• You will want to look at Dunn test (1964) as a post hoc to K-W. I don't understand your question about which statistic is used. Aug 16 '18 at 17:08

You are looking for Dunn's test, or the less well known, but more powerful Conover-Iman test. Both post hoc tests (a) perform pairwise comparisons using the same rankings used in the Kruskal-Wallis test (as opposed to just performing a bog-standard rank sum test for each pairwise comparison), and (b) use a pooled variance estimate implied by the Kruskal-Wallis test's null hypothesis. Dunn's test is based on an asymptotic z distribution, while the Conover-Iman test is based on an asymptotic t distribution.

I am unsure what you mean by "stepwise" or "step down", but implementations of both tests for R (dunn.test and conover.test) and for Stata (dunntest and conovertest) include step up and step down family-wise error rate and false discovery rate control for these tests.

References
Conover, W. J. and Iman, R. L. (1979). On multiple-comparisons procedures. Technical Report LA-7677-MS, Los Alamos Scientific Laboratory.

Dunn, O. J. (1964). Multiple comparisons using rank sums. Technometrics, 6(3):241–252.

• Thank very much, really this is really helpful. To be more precise: I'm using SPSS which offers a stepdown stepwise multiple comparisons procedure to follow up a KW test. See here. This page doesn't really mention which statistic is used to perform the hypothesis tests. Concretely, how would you report this procedure using a statistical formula(e.g., t (5) = 4.32, p < .1). What would take the place of the t when reporting this procedure? Thanks! Aug 21 '18 at 9:41
• @MartijnGoudbeek Eh... I don't use SPSS, and tend to recommend R (it's free, and well documented). Aug 21 '18 at 13:55
• I know all about R and use it all the time. Great piece of software. But in this case, for all kinds or reasons, SPSS is what I need to use. Aug 21 '18 at 14:12
• @MartijnGoudbeek SPSS is using methods described in Campbell, G. and Skellings, J. H. (1985). Nonparametric stepwise multiple comparison procedures. Journal of the American Statistical Association, 80(392):998–1003. Aug 21 '18 at 14:28
• Thanks, that paper is really helpful. From that paper, it seems they're using the same statistic (H) as in the "omnibus" test. Per the paper: Step 1. The statistic T(k) tests the equality of all k treatments [...] Step 2. The statistic T(k-1) is applied to all k subsets of k - 1 [..] Etc. Does this make sense? Aug 22 '18 at 8:44