Example: Income explains 70% of the variance in expenditure. Variation in driving speed can be explained by drivers' age.

How does one make such statements if the test used is a non-parametric one? I was told before that non-parametric tests such as the Spearman and Kruskal-Wallis does not test for variances, hence it is not right to mention "variance" when interpreting.

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    $\begingroup$ I changed the title to make more salient this interresting aspect about nonparametric tests and the idea of interpreting/stating relations like in terms of ratios of variance but then without the variance. This edit is a very subjective thing (I changed the title because while reading your post I thought your question is much more interresting than what I expected from the title; I expected a lame beginners question about correlation/causation). You can change it back or improve it further if you like. $\endgroup$ – Sextus Empiricus Feb 7 at 8:32
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    $\begingroup$ In logistic regression relationships are often expressed in terms of odds ratio "A increases the odds of B with a factor x". $\endgroup$ – Sextus Empiricus Feb 7 at 8:38

You're right that effect size statistics for nonparametric tests usually can't be expressed in terms of variance explained.

Spearman correlation is equivalent to Pearson correlation on the ranks of the data. I suppose you could convert this, by squaring rho, to "variance of the ranks of y explained by the ranks of x", but I suspect relying the audience to understand rho is a better approach.

For Wilcoxon-Mann-Whitney, an easily understandable effect size statistic is Vargha and Delaney's A, which is simply the probability of an observation in one group being larger than an observation from the other group. This could also be used for pairs of groups in Kruskal-Wallis.

There are other effect size statistics used for nonparametric tests that can be difficult to interpret in words. For example, there is an epsilon-squared used for Kruskal-Wallis. It's a useful statistic, but not easy to interpret in words.

There is also an "eta-squared" used for Kruskal-Wallis, which is supposed to be more related to the usual eta-squared for an anova on the ranks of the y variable. (eta-squared is the same as the r-squared from an anova). But I'm not that enamoured with this statistic.

Likewise there is an "r" commonly used for Wilcoxon-Mann-Whitney, but it doesn't have great properties for an effect size statistic. It's better to use Vargha and Delaney's A, Cliff's delta, or rank biserial correlation.


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