More generally, let si=1 denote the event that individual i is selected into our study, and let Ci denote a vector of covariates describing individual i (which may include the phenotype of individual i). Then we can represent an arbitrary biased sampling scheme by specifying the selection probabilities f(Ci):=P[si=1|Ci] (note that case/control ascertainment is the special case where Ci=yi). Suppose that phenotypes are generated following the model from Section 1.1 of the Supplementary Note, but that our sample is selected following the biased sampling scheme f. Let aij denote the additive genetic component for phenotype j in inidividual i. If there is no direct ascertainment on genotype (i.e., if Ci does not include genotypes), then the proof of Proposition 1 in the Supplementary Note goes through, except that ρ is replaced with E[yi1yi2|si=1] and ρg is replaced with E[ai1ai2|si=1].
