Extended Data Fig. 4. Simulation study assessing sex-specific heritability and genetic correlation divergence.

Simulation of environmental effect that reconciles sex-difference in heritability with the similarity of the SNP effect sizes. a, Frequency density distributions of the liabilities for different models. Blue line, base model, $\phi =X\beta +\epsilon$, as assumed to be present in males with h² = 0.1395, X and β as determined by GWAS, $\epsilon ~N\left(0,1\right)$, and a disease threshold in keeping with the male RLS prevalence of 0.06 (shaded area under the curve). Black line, model with non-interacting binary environmental effect, $\phi =X\beta +\tau E+\epsilon$, with $X,\beta ,\epsilon$ and threshold as in the base model plus an additional binary effect $E~Bernoulli\left(p=0.21\right)$, representing childlessness with a weight vector τ such that that prevalence is 0.13 as in females. Red line, analogous G×E model, $\phi =X\beta +X\eta \circ E+\epsilon$, but where the environmental effect now interacts with the genetic effects and the corresponding weight vector η is chosen in accordance with the female prevalence. b, c, Optimization of the model $\phi =X\beta +X\eta \circ E+\tau E+\epsilon$ with $X,\beta ,E,\epsilon$ and threshold as above, where the additional degree of freedom is covered by also considering the mean effect size ratio rb observed in the GWAS. Heatmap and contour plot for logistic regression-based liability scaled LDSC h² (b) and effect size ratio rb (c) as functions of $Var\left(\tau E\right)$ and $Var\left(X\eta \circ E\right)$. Optimal values for $Var\left(\tau E\right)$ and $Var\left(X\eta \circ E\right)$, that is, for τ and η, respectively, comply with female prevalence, female heritability, and observed effect size ratio as well. The optimal τ turns out to be close to zero so that the environmental factor acts mostly via genetic interaction.