Fig. 1. Illumination of Mendelian randomization and G × E.

A Left panel: the path diagram of the MR, where U refers to all confounders. Genetic variants (G) contributing to outcome Y through mediation of exposure X are often selected as the valid genetic instrumental variables (black paths). Genetic variants contributing to Y through both black and red paths independently are horizontal pleiotropic variants. Genetic variants contributing to Y through confounders (U) are invalid instrumental variables and need be blocked (x). Right panel: a scatter plot of effect sizes of genetic instrumental variants for an exposure and an outcome. Each + corresponds to the 95% confidence intervals of the exposure effect size (horizontal line segment) and the outcome effect size (vertical line segment). The horizontal pleiotropic variants (red +) depart from the regression line and can be separated from the variants with no pleiotropic effect (blue +). B Left panel: the $G\times E$ framework, with the goal of testing $G\times E$. Instead of an explicit exposure, we create a pseudo exposure $\tilde{X}$, which can be viewed as a polygenic score for trait Y based on marginal effect sizes. However, our analysis does not require estimating this pseudo exposure. The genetic variants associated with the pseudo exposure $\tilde{X}$ but not through either the environment E or $G\times E$ are valid instrumental variables. The genetic variants interacting with E can be viewed the same as horizontally pleiotropic variants in the MR framework. Genetic variants associated with Y via mediation through E can contribute to both the pseudo exposure $\tilde{X}$ and Y, and thus have similar effects as $G\times E$ and cannot be distinguished from $G\times E$. Thus, testing the combined effect of interaction and mediation is conceptually equivalent with testing the horizontally pleiotropic effect in the MR framework. Right panel: a scatter plot of genetic variants for GWIS main effects and GWAS marginal effects. Each + corresponds to the 95% confidence intervals of the GWIS main effect size (horizontal line segment) and the GWAS marginal effect size (vertical line segment). Like the horizontal pleiotropic variants in the MR framework, $G\times E$ variants (red +) depart from the regression line and can be separated from variants with no $G\times E$ assuming no mediation.