Mathematical Ability and Socio-economic Background: IRT Modeling to Estimate Genotype by Environment Interaction: Supplementary material

We have an intercept β0, a main effect of the measured covariate Mij , β1, a main effect of the unmeasured genotypic value Aij , β2, an interaction effect of the measured covariate and the genotypic value, β3, and a residual term with variance e. This model can be extended with a main effect of the shared environment, β4Cij , plus an additional interaction effect, β5CijMij , but for our purpose here it suffices to discuss the model with additive genetic effects alone. The two things that are different from the general ANOVA model is that variable A is unobserved, and that we have data on twin pairs, where phenotypes are correlated. In case of additive genetic effects, the genetic correlation equals 1 for monozygotic twin pairs and 12 for dizygotic twin pairs, that is, for MZ twin pairs we have Cov(Ai1, Ai2) = 1 and for DZ twin pairs we have Cov(Ai1, Ai2) = 1 2 . If we assume that M is a dichotomously scored covariate, the sufficient statistics for our model are the variance of phenotype P, the observed covariance in