Figure 2.

Type I error control and power for testing the causal effects under various simulation scenarios

(A and B) Quantile-quantile plot of −log10 p values for testing the causal effects either in the absence or in the presence of horizontal pleiotropic effects under null simulations. Null simulations are performed under different horizontal pleiotropic effect sizes: (A) $\mathbf{\gamma }={\left(\mathrm{0,0,0,0}\right)}^T$; (B) ${\gamma }_j$ randomly selected from (0, 1 × 10⁻⁴, 5 × 10⁻⁴, 1 × 10⁻³, 2 × 10⁻³), j = 1,2,3,4. p values from moPMR-Egger are on the expected diagonal line across a range of horizontal pleiotropic effect sizes.

(C–F) Power (y axis) at a Bonferroni adjusted threshold to detect the causal effects is plotted against different causal effect sizes characterized by PVE_zy for the first trait (x axis) in the heterogeneous causal effect settings, where the PVE_zy for the remaining traits are 15%, 85%, 50% of the PVE_zy for the first trait. Compared methods include moPMR-Egger (magenta), PMR-Egger (blue), PrediXcan (green), and TWAS (orange). Different line symbols represent whether the four traits are correlated or not in (C) and the direction of causal effects in (D)–(F). Simulations are performed under different number of affected traits being from one to four (C–F) in the absence of horizontal pleiotropic effect.