Fig. 4: Estimation comparison with CAUSE and MRMix.

a Estimates from MRCI (green), CAUSE (orange), and MRMix (blue) under null, uni-directional, and bi-directional causations in four scenarios (\({s}_{{{{{\mathrm{1,2}}}}},C},\,{s}_{2,C},\,{s}_{{{{{\mathrm{1,2}}}}}}\), and \({s}_{C}\)). CAUSE and MRMix produced severely over-estimated causal effects when exposure-specific SNPs were absent in the sub-model \({s}_{2,C}\) and \({s}_{C}\). For null causation, \({\delta }_{12}={\delta }_{21}=0.0\); for uni-directional causation, \({\delta }_{12}=0.1\) and \({\delta }_{21}=0.0\); for bi-directional causation, \({\delta }_{12}=0.1\) and \({\delta }_{21}=0.05\). The true causation values of \({\delta }_{12}\) and \({\delta }_{21}\) are indicated by up- and down-pointing triangles, respectively. b Rejection rates of the null hypothesis for \({\delta }_{12}\) and \({\delta }_{21}\) estimates from MRCI (green), CAUSE (orange), and MRMix (blue) in different simulated scenarios. CAUSE and MRMix produced inflated Type I error rates for the causal direction where exposure-specific SNPs were absent (\({s}_{2,C}\) and \({s}_{C}\)). c Estimates of causal effects and rejection rates of null hypothesis from MRCI (green), CAUSE (orange) and MRMix (blue) under null, uni-directional, and bi-directional causations in \({s}_{{{{{\mathrm{1,2}}}}},C}\) scenarios with small sample sizes (20,000 individuals). Decreased GWAS power led to over-estimates for CAUSE and larger estimation variance for MRMix. MRCI produced nearly unbiased estimates and correct Type I error rates. In these results, the estimates of MRCI came from the final estimates after model averaging; p-value thresholds for CAUSE and MRMix were \(1\times 1{0}^{-3}\) and \(5\times 1{0}^{-8}\), respectively. In the simulations, the mixing proportions of the present component were \(1\times {10}^{-3}\); the pleiotropic effects were correlated (\({\rho }_{C1,C2}=0.1\)); the heritabilities contributed by \({Y}_{1}\)-specific, \({Y}_{2}\)-specific and pleiotropic SNPs (if present in the sub-model scenario) were 0.3, 0.3, and 0.1, respectively.