Fig. 2. Type I error control of different MR methods in simulations.

Type I error is evaluated by quantile-quantile plots of −log₁₀ P values from different MR methods on testing the causal effect under the null simulations. Compared methods include CAUSE (blue), IVW-R (gold), MRAID (purple), MRMix (black), RAPS (deep pink), Robust (deep sky blue), Weighted median (light salmon), and Weighted mode (green). Four null simulation scenarios are examined. (A) Null simulations in the absence of both correlated and uncorrelated horizontal pleiotropic effects. We simulated 100 instrumental SNPs with their effect sizes drawing from a normal distribution. (B) Null simulations in the absence of both correlated and uncorrelated horizontal pleiotropic effects. We simulated 1000 instrumental SNPs with their effect sizes drawing from a Bayesian sparse linear mixed model (BSLMM) distribution with 1% SNPs having large effects and 99% SNPs having small effects. (C) Null simulations in the absence of correlated horizontal pleiotropic effect but in the presence of uncorrelated horizontal pleiotropic effect (PVE_u = 5%). We simulated 100 instrumental SNPs and set the proportion of instrumental SNPs having uncorrelated horizontal pleiotropy to be 20%. (D) Null simulations in the presence of both correlated (π_c = 5%, $\mathrm{\rho }=\sqrt{0.05}$) and uncorrelated horizontal pleiotropic effects (PVE_u = 5%). We simulated 100 instrumental SNPs and set the proportion of instrumental SNPs having the uncorrelated horizontal pleiotropy effect to be 20%.