Fig. 2: Comparison of estimates for the reciprocal causal effect by our method and instrumental variable (IV)-based MR methods in simulations with correlated pleiotropy.

a Estimates in null causation scenarios (\({\delta }_{12}={\delta }_{21}=0.0\), LoS2). b Estimates in uni-directional causation scenarios (\({\delta }_{12}=0.1\) and \({\delta }_{21}=0.0\), LoS4). c Estimates in bi-directional causation scenarios (\({\delta }_{12}=0.1\) and \({\delta }_{21}=0.05\), LoS8). d Rejection rates of the null hypothesis in bi-directional, uni-directional, and null scenarios. In plots a–c, our method took genome-scale SNPs for estimation and produced nearly unbiased estimates in different scenarios; the true values of \({\delta }_{12}\) and \({\delta }_{21}\) are indicated by up- and down-pointing triangles, respectively. For MR methods, IVs were selected in three ways: (1) use the exposure-specific true causal SNPs in the simulation as IVs; (2) use exposure-associated SNPs (p-value < \(5\times 1{0}^{-8}\)) after clumping but exclude potential outcome-associated SNPs (defined as p-value < \(5\times 1{0}^{-5}\) with the outcome); (3) use significant exposure-associated SNPs after clumping regardless of their association with outcome. In plot d, the exclusion criteria were applied to IV-based MR methods and our method shows well-controlled Type I error rates and adequate power. In these plots, \({\rho }_{{C1,C2}}\) for the correlated pleiotropy is 0.1; the mixing proportions for non-null components were \({\pi }_{1}={\pi }_{2}={\pi }_{{c}}=1\times {10}^{-4}\); the heritabilities contributed by \({Y}_{1}\)-specific, \({Y}_{2}\)-specific, and pleiotropic SNPs were 0.3, 0.3, and 0.1, respectively (note: The selection of exposure-specific true causal SNPs was not applied to MRMix due to its assumption of a normal-mixture distribution).