Idea

[krzysiektr]

Kruskal-Wallis test:

This non-parametric test, robust to non normal distributions, has the same null and alternative hypotheses, and the same interpretations than the ANOVA.

Note that the Kruskal-Wallis test does not require the assumptions of normality nor homoscedasticity of the variances.

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Note 1

The Kruskal–Wallis one-way analysis of variance by ranks (named after William Kruskal and W. Allen Wallis) is a non-parametric method for testing whether samples originate from the same distribution. It is used for comparing two or more samples that are independent, and that may have different sample sizes, and extends the Mann–Whitney U test to more than two groups. The parametric equivalent of the Kruskal-Wallis test is the one-way analysis of variance (ANOVA). When rejecting the null hypothesis of the Kruskal-Wallis test, then at least one sample stochastically dominates at least one other sample. The test does not identify where this stochastic dominance occurs or for how many pairs of groups stochastic dominance obtains. Dunn's test would help analyze the specific sample pairs for stochastic dominance. Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal distribution of the residuals, unlike the analogous one-way analysis of variance. If the researcher can make the more stringent assumptions of an identically shaped and scaled distribution for all groups, except for any difference in medians, then the null hypothesis is that the medians of all groups are equal, and the alternative hypothesis is that at least one population median of one group is different from the population median of at least one other group.

source: Cortés, Omar. (2015). Re: Which statistical test to use?. Retrieved from: https://www.researchgate.net/post/Which_statistical_test_to_use4/55edbeb260614b7ac18b458d/citation/download.

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Note 2

A problem with the Kruskal-Wallis test is that, while it does not assume normality for groups, it still assumes homoscedasticity (i.e. the groups have the same distributional shape). As a solution Brunner et al. (1997) proposed a heteroscedastic version of the Kruskal-Wallis test which utilizes the F-distribution. Along with being robust to non-normality and heteroscedasticity, calculations of exact P-values using the Brunner-Dette-Munk method are not made more complex by tied values. This is another obvious advantage over the traditional Kruskal-Wallis approach.

source: asbio

[AntoineSoetewey]

Dear @krzysiektr,

Thanks again for your suggestions!

I have included Note 1 in the article (see the assumptions).

Regarding Note 2: if Kruskal-Wallis also requires homoscedasticity, which test can be used when both the normality and the homogeneity are not met?

[AntoineSoetewey]

Ok, this article helped me to clarify it:

• As long as you use the Kruskal-Wallis test to, in fine, compare groups, homoscedasticity is not required
• if you wish to compare medians, the Kruskal-Wallis test requires homoscedasticity

I added a footnote with this remark.

Thanks again for your feedback!