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)?
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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. – Sal Mangiafico Aug 16 '18 at 17:08
1 Answers
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.

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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](https://www.ibm.com/support/knowledgecenter/SSLVMB_22.0.0/com.ibm.spss.statistics.algorithms/alg_nonparametric_multiple-comparisons_stepwise.htm). 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! – Martijn Goudbeek Aug 21 '18 at 09:41
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@MartijnGoudbeek Eh... I don't use SPSS, and tend to recommend [R](https://cran.r-project.org/) (it's free, and well documented). – Alexis Aug 21 '18 at 13:55
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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. – Martijn Goudbeek Aug 21 '18 at 14:12
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@MartijnGoudbeek SPSS is using methods described in Campbell, G. and Skellings, J. H. (1985). [Nonparametric stepwise multiple comparison procedures](https://sci-hub.tw/). *Journal of the American Statistical Association*, 80(392):998–1003. – Alexis Aug 21 '18 at 14:28
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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? – Martijn Goudbeek Aug 22 '18 at 08:44
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$H$ is just the Kruskal-Wallis omnibus test statistic (the nonparametric equivalence of the oneway ANOVA), **and there is no multiple comparisons adjustment for a single omnibus test.** The multiple comparisons procedures enter in for the *post hoc* pairwise tests following a rejection of the Kruskal-Wallis null hypothesis. – Alexis Aug 22 '18 at 15:23
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Exactly, so my question still is what the test statistic is that is used in the pairwise tests. I am not trying to be obnoxious here, but I'm still in the dark. For an ANOVA, you have the F-test, which can be follow by t-test. So what is the equivalent of that t-test here? Thanks again for all your efforts. – Martijn Goudbeek Aug 26 '18 at 19:41
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@MartijnGoudbeek I am unfamiliar with the methodology cited in the SPSS documentation. Dunn's test is, I believe, the most common appropriate *post hoc* test. – Alexis Aug 27 '18 at 07:09