# Non-parametric test for unequal samples with subsequent post-hoc analysis?

Is it okay to perform a Kruskal-Wallis on four unequal samples? Further to this, is there a subsequent pair-wise post-hoc suitable for two unequal sample sizes?

Yes and Yes. Kruskal-Wallis analysis does not require equal sample size. In latest SPSS versions (from 18, if I remember correctly) there is a new nonparametrics procedure that performs pairwise comparisons with sig. adjustment, as well as step-down post-hoc method. Alternative would be to use very nice macro by Marta Garcia-Granero http://gjyp.nl/marta/

• @ttnphs Thanks so much! What is the new procedure in SPSS then? – Platypezid Jan 30 '12 at 19:12
• Starting from version 18 you may notice new nonparametric menus (= NPTESTS command) and legacy nonparametric menus (= NONPAR TESTS command) under Nonparametric Tests. The new menus are ugly but there is an option for post-hoc which was absent in NONPAR TESTS. – ttnphns Jan 30 '12 at 19:25

According to the formula for the Kruskal-Wallis test statistic, each group can have a different number of observations, so "yes".

Whether this is the best test or not I'm not sure - if you're still in doubt you'd need to post more details. But perhaps this is all you needed to know. Good luck!

• Post-hoc analyses after this rely on "pair-wise" comparison, so are Mann-Whitney tests okay to look at where the significances are? – Platypezid Jan 30 '12 at 18:53

In case of heavy unbalanced groups the Kruskal-Wallis-test may be far off and you should not use it. There is a recent paper published on arXiv by Brunner et al. 2018 "Ranks and Pseudo-Ranks - Paradoxical Results of Rank Tests" in which the authors show that under certain conditions Kruskal-Wallis-test for more than two groups with unequal sample sizes may lead to intransitive decisions and false rejection of the null hypothesis.

This means that for the same set of distributions $$F_1 ,..., F_d$$ and unequal sample sizes the p-value of the test may be arbitrary small if $$N$$ is large enough.

A possible solution is provided by the concept of pseudo-ranks. Apparently, the authors have implemented solutions (which I have not tested yet) for this in R and SAS. See this reference for more information:

Brunner, E., Bathke, A.C., and Konietschke, F. (2018). Rank- and Pseudo-Rank Procedures for Independent Observations in Factorial Designs - Using R and SAS. Springer Series in Statistics, Springer, Heidelberg.

Yes, Mann-Whitney tests are the normal post-hoc test to use for a Kruskal-Wallis test.