Guillaume Rousselet explains how and when group comparisons with Cohen’s d fail, and what robust statistics one could use instead:

basic statistics

When I was an undergrad, I was told that beyond a certain sample size (n=30 if I recall correctly), t-tests and ANOVAs are fine. This was a lie. I wished I had been taught robust methods and that t-tests and ANOVAs on means are only a few options among many alternatives. Indeed, t-tests and ANOVAs on means are not robust to outliers, skewness, heavy-tails, and for independent groups, differences in skewness, variance (heteroscedasticity) and combinations of these factors (Wilcox & Keselman, 2003; Wilcox, 2012). The main consequence is a lack of statistical power. For this reason, it is often advised to report a measure of effect size to determine, for instance, if a non-significant effect (based on some arbitrary p value threshold) could be due to lack of power, or reflect a genuine lack of effect. The rationale is that an effect could be associated with a sufficiently large effect…

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