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• When do I use the Kruskal-Wallis test?
• Can I do a 'post hoc' analysis between the groups after a Kruskal-Wallis test?
• Where do I find the Kruskal-Wallis test in SPSS?
• References

The Kruskal-Wallis test is a non-parametric testing procedure for testing hypotheses about the medians of (semi-continuous) data. The Kruskal-Wallis test tests whether the medians of two or more independent groups are equal. Intuitively, the Kruskal-Wallis test is equivalent to a One-way ANOVA, with the data replaced by their rank position.

If you want to test whether the medians of two or more groups are equal to each other, you can use the Kruskal-Wallis test. In practice, the Kruskal-Wallis test is only used when comparing three or more groups. For the comparison of two groups, the Mann-Whitney U test is used. The Kruskal-Wallis test does not assume normally distributed data. It is therefore often used when the assumptions of the One-way ANOVA are not met.

Example of the use of the Kruskal-Wallis test: Suppose you want to test whether the number of children per family is equal in the Netherlands, Belgium, France and Germany.

Table 1. A comparison of family size in different countries
Variable*	Netherlands	Belgium	France	Germany	p-value**
Family size	1.8 (1.1 ; 3.4)	1.6 (0.9 ; 4.1)	2.2 (1.2 ; 4.2)	2.1 (1.5 ; 3.6)	0.12
*Variables are denoted as median (25th percentile; 75th percentile). **Group differences were tested with Kruskal Wallis.

I have three patient groups (polyp size 1-5 mm, 6-9 mm or >10 mm). I want to see whether there is a difference between these groups in absolute blood value (in ngHb/ml). This value, called the absolute FIT value, is not normally distributed. From this I conclude that if I want to know whether there are significant differences between the 3 polyp groups, I need to investigate this with the Kruskal-Wallis test. This gives a p-value of 0.000, so significant. However, I still don't know where the large difference lies — between group 1 and 2, 1 and 3, etc. If I wanted to know this for a normally distributed group, I would do a post-hoc analysis after the one-way ANOVA. Is this also possible after a Kruskal-Wallis test?

Yes, that is indeed possible. You could compare the groups pairwise (1 vs 2, 2 vs 3, 1 vs 3) using separate Mann-Whitney U tests to see exactly where the difference occurs. Do keep an eye on the number of tests and the associated issues with multiple testing. Perhaps you can manage with 2 of the 3 pairwise tests? If needed, you could use a stricter significance level for these 'follow-up tests'.

You will find the Kruskal-Wallis test in SPSS under Analyze -> Nonparametric Tests -> (Legacy Dialogs) -> K Independent Samples, or via the Analyze -> Nonparametric Tests -> Independent samples wizard.

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