Fig. 2. Simulation performance of T_{MR_GXE} and the two-step procedure.

A–D No medication was present. The simulation details were described in “Methods”. A Box plots of $\widehat{\theta }$ in simulations under different environments in GWAs data. The top and bottom edges of the box plots represent the 25th and 75th percentiles of $\widehat{\theta }$, and the horizontal middle line represents the 50th percentile. The vertical bars extend from the 25th (or 75th) percentile of $\widehat{\theta }$ to the minimum (or maximum) value of simulated data. $E\left(\widehat{\theta }\right)$ is close to 1 as expected. B Box plots of the direct estimate of ${\beta }_3$ in GWIS (top panel) and by $\left(\widehat{\alpha }-{\widehat{\beta }}_1\widehat{\theta }\right)/{\mu }_e$ through MR- $G\times E$ analysis (bottom panel). The box plots are interpreted the same as in (A) accordingly. Both the estimates of ${\beta }_3$ and that by $\left(\widehat{\alpha }-{\widehat{\beta }}_1\widehat{\theta }\right)/{\mu }_e$ are unbiased. Here s = −1 refers to the scenario when the main effect and interaction effect have opposite effect directions; s = 0 refers to no main effect; and s = 1 refers to the scenario when the main effect and interaction effect have the same effect direction. C Type I error rate comparison between $T_{MR\_GxE}$ and the direct test for different main and interaction effect directions. Both $T_{MR\_GxE}$ and the direct test maintain the type I error rate well. D Power comparison between $T_{MR\_GxE}$ and the direct test for different main and interaction effect directions. E, F 20 variants were tested when mediation was present or not. The simulation details were described in ”Methods”. E Type I error comparison for $T_{Direct}$, $T_{MR\_GxE}$ and two-step procedure. The dash lines represent the 95% confidence interval. F Power comparison for $T_{Direct}$, $T_{MR\_GxE}$ and two-step procedure.