## Illustrative effect sizes for sex differences

Ingalhalikar et al. (2013) study "sex differences in the structural connectome of the human brain" using a large sample of 949 individuals.

They report "conspicuous and significant sex differences that suggest fundamentally different connectivity patterns in males and females". They claim that their hypothesis that "male brains are optimized for communicating within the hemispheres, whereas female brains are

optimized for interhemispheric communication" was "overwhelmingly supported ... at every level".

The paper contains only t- and p-values, without any estimates of effect size. One can approximately (ignoring covariates) convert t-statistics into Cohen's d effect size estimates using d = t / sqrt(n1*n2 / (n1+n2)), or d = 2 * t / sqrt(df), where df=945 here.

This figure illustrates some effect sizes by plotting a standard normal distribution and a distribution shifted by an amount corresponding to the Cohen's d values. The figure includes the paper's most significant effect (largest absolute t-value reported) and a key interhemispheric difference; these are compared to the effect size for a sex difference in height for illustration (data from Wikipedia).

The substantial overlap of the distributions highlights the danger of assuming that a significant difference from a large sample implies a fundamental/overwhelming difference between the sexes. The optimal (equal error) classification accuracy can be estimated as normcdf(d/2, 0, 1), which for the interhemispheric effect is about 56% (which is statistically significantly -- but not really substantively -- above chance).