# 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 Dunnâ€™s test in place of PO model. I have seen a paper that has used Dunnâ€™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), Dunnâ€™s test just deals with p-value corrections and does not get the contrasts right. That is because with Dunnâ€™s test you re-rank each of the two groups being compared as if the third group doesnâ€™t exist. The proportional odds model does one overall ranking.

Thanks, prof, so when is it appropriate to use Dunnâ€™s test?

Itâ€™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.

1 Like