Fig. 3: Estimation after model averaging.

a Final estimates of causal effects under null, uni-directional, and bi-directional causations in four scenarios (\({s}_{{{{{\mathrm{1,2}}}}},C},\,{s}_{2,C},\,{s}_{{{{{\mathrm{1,2}}}}}}\), and \({s}_{C}\)). When the exposure-specific SNPs exist, the final estimates after model averaging still produced nearly unbiased estimates in simulated scenarios. For null causation (gray) \({\delta }_{12}={\delta }_{21}=0.0\); for uni-directional causation (blue), \({\delta }_{12}=0.1\) and \({\delta }_{21}=0.0\); for bi-directional causation (purple), \({\delta }_{12}=0.1\) 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 of the final \({\delta }_{12}\) and \({\delta }_{21}\) estimates after model averaging in different scenarios. For the zero-effect causal direction, the Type I error rates were well-controlled; for the non-zero-effect causal direction, the presence of the exposure-specific SNPs showed reasonably good power, and the absence of the exposure-specific SNPs showed conservative power of estimation. In the simulations, the mixing proportions of the present component were \(1\times {10}^{-3}\); the pleiotropic effects were correlated (\({\rho }_{C1,C2}=0.1\)); and 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.