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It looks like Kruskal-Wallis is the standard nonparametric test for more than two groups. The problem is that it does not tell which groups are different, except that whether there exists significant difference among the groups.

Is there another nonparametric test for more than two groups? Or better, is there one that would tell you which groups are different?

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    $\begingroup$ ANOVA - or indeed any omnibus test - doesn't tell you which groups are different either. In each case you can use post-hoc testing. $\endgroup$
    – Glen_b
    Commented Oct 29, 2014 at 17:15
  • $\begingroup$ @Glen_b, but ANOVA is not a nonparametric test? If we can't assume the normality of the data, then how can we use ANOVA? $\endgroup$
    – MLister
    Commented Oct 30, 2014 at 3:23
  • $\begingroup$ As I said, what matters is how it performs (i.e. the type I error rate should preferably be close to what you want and the power should not be very poor compared to a competing test); particular kinds of violations of the assumption of normality make relatively little difference to level and power holds up well. For example, if I knew the data were from a symmetric beta distribution (with unspecified parameter), I would much prefer ANOVA to Kruskal Wallis. $\endgroup$
    – Glen_b
    Commented Oct 30, 2014 at 4:06

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The generalization of the Kruskal-Wallis test is the proportional odds ordinal logistic model. Such a model can provide the multiple degree of freedom overall test as you get with K-W but also can provide general contrasts (on the log odds ratio scale) including pairwise comparisons.

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  • $\begingroup$ How would log-odds ratio make sense when the predicted value is not a probability? $\endgroup$
    – Dave
    Commented May 19, 2021 at 20:15
  • $\begingroup$ You can state the predicted values in many ways including median, mean, exceedance probabilities, and odds of exceedance. Underlying all this is odds ratios which are ratios of odds of exceedance, or more accurately odds that $Y \geq y$. Or you can state the model in terms of $Y \leq y$. See BBR. $\endgroup$ Commented May 20, 2021 at 11:36
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You can still run the Kruskal-Wallis, all you need to do is run subsequent pair-wise tests comparing each group to the other groups.

After running a Kruskal-Wallis test and determining that there is a significant difference, you could run additional post hoc tests, for example a Dunn's test, to compare each individual group and determine which are significantly different from each other.

For a reference to the rationale for a Dunn`s test vs. Wilcoxon Rank-sum: Post-hoc tests after Kruskal-Wallis: Dunn's test or Bonferroni corrected Mann-Whitney tests?

For the original paper describing the test: Dunn OJ. Multiple comparisons using rank sums. Technometrics 1964; 6(3):241-52. http://dx.doi.org/10.1080/00401706.1964.10490181

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    $\begingroup$ No this has been warned against at other places on this site. Pairwise Wilcoxon tests (using the mean rank for group A in the context of groups A and not A) is not consistent with a K-W test that distinguishes B and C among the not-A observations. $\endgroup$ Commented Oct 30, 2014 at 19:28
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    $\begingroup$ Thanks for pointing that out! Answer revised to reflect this. $\endgroup$
    – Red_Star
    Commented Oct 31, 2014 at 1:50

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