# Multiple comparison method after Kruskal Wallis test

In my study, I have 4 quantiles and to ascertain the statistically significant difference in systolic blood pressure across the quantiles, I did a Kruskal Wallis test in Stata (Medians; 1116, 119, 120 and 127, p<0.001). What is the best multiple comparison method for me to use in order to determine which groups were statistically significant?
Thanks for a great job from this group, I have really benefited a lot as it relates to the appropriate application of statistical methods.

Information-losing transformations of variables is seldom a good idea. Quantile groups are piecewise flat transformations of inherently continuous variables. I recommend modeling Y=SBP using raw data for X.

The model that contains the K-W test as a special case is the proportional odds ordinal logistic semiparametric model. Whatever you decide to do with X, the PO model will handle Y without making a distributional assumption for Y given X=x.

Thank you for the advice, but is there a posthoc test for Kruskal Wallis test?

You must use the proportional odds model to get rational post hoc tests. The model likelihood ratio \chi^2 test from the PO model provides the overall test for k groups, and then do any contrasts of interest using the estimated regression coefficients and their standard errors and covariances. This is laid out in https://hbiostat.org/doc/bbr.pdf in the Nonparametrics chapter. The score \chi^2 test from the PO model is the K-W test.

The reviewer for my work is insisting that I should use a multiple comparison method. so, I was wondering if I can use Dunn’s test in place of PO model. I have seen a paper that has used Dunn’s test after a significant Kruskal Wallis test. What is your comment about that?

Unlike embedding this in a proper model that preserves the transitivity of the groups (A > B \cap B > C \implies A > C), Dunn’s test just deals with p-value corrections and does not get the contrasts right. That is because with Dunn’s test you re-rank each of the two groups being compared as if the third group doesn’t exist. The proportional odds model does one overall ranking.

Thanks, prof, so when is it appropriate to use Dunn’s test?

It’s not appropriate in the context of the Kruskal-Wallis test. I hesitate to do multiplicity adjustments in general, except for the multiple degree of freedom overall chunk test, e.g., overall ANOVA F or \chi^2.

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