The derivation of Eq. (3) assumes that G, U, EX and EY are pairwise uncorrelated, which is unlikely to hold in general. Although by definition U, EX and EY are uncorrelated, G may be correlated with any of those variables through LD or gene-environment correlation. However, we might expect that across all SNPs in a GWAS, any systematic such correlation will be negligible. Equation (3) also assumes no statistical interaction between G and U in their effects on X, and between G, U and X in their effects on Y. Again, and in view of the low number of detectable interactions in GWAS compared with main effects36, we may safely assume that any systematic interactions are negligible in comparison with the main effects.
