Fig. 1: Schematic diagram of MRCI.

a \({Y}_{1}\) and \({Y}_{2}\) represent a pair of phenotypes. Genotypes can be divided into four components: \({Y}_{1}\)-specific causal SNPs (\({G}_{1}\)), \({Y}_{2}\)-specific causal SNPs (\({G}_{2}\)), pleiotropic causal SNPs (\({G}_{{C}}\)) and null SNPs (\({G}_{0}\), not shown). Lines connecting these genotypes represent the LD correlation between SNPs. Arrow lines from genotype to phenotype represent the direct effect of corresponding SNPs (\({\gamma }_{1}\), \({\gamma }_{2}\), \({\gamma }_{{C1}}\) and \({\gamma }_{{C2}}\)) on the phenotypes. Covariance (\({\rho }_{{C1,C2}}\)) between \({\gamma }_{{C1}}\) and \({\gamma }_{{C2}}\) is allowed. Arrow lines between the two phenotypes represent the reciprocal causal paths (\({\delta }_{12}\) and \({\delta }_{21}\)). Non-additive genetic effects on phenotypes are represented by \({e}_{1}\) and \({e}_{2}\). In real situations, one or two components in the model could be absent and our method could handle these sub-model scenarios in a robust way. b Illustration of several representative simulation scenarios. simIDs are LoS1, LoS3, LoS7 (upper panel from left to right), LoS2, LoS4, LoS8 (lower panel from left to right) (see Supplementary Table 1). \(x\)-axis and \(y\)-axis show the standardized effect size estimates for GWAS \({Y}_{1}\) and \({Y}_{2}\), respectively. Red, orange, green, and gray points represent \({Y}_{1}\)-specific, \({Y}_{2}\)-specific, pleiotropic, and null SNPs in the simulation respectively. For null causation, \({\delta }_{12}={\delta }_{21}=0.0\); for uni-directional causation, \({\delta }_{12}=0.1\), \({\delta }_{21}=0.0\); for bi-directional causation, \({\delta }_{12}=0.1\), \({\delta }_{21}=0.05\). For independent pleiotropy, \({\rho }_{{C1,C2}}=0.0\); for correlated pleiotropy, \({\rho }_{{C1,C2}}=0.1\). In these plots, the mixing proportions for non-null components were set as \({\pi }_{1}={\pi }_{2}={\pi }_{{c}}=1\times {10}^{-4}\), and the heritabilities contributed by \({Y}_{1}\)-specific, \({Y}_{2}\)-specific, and pleiotropic SNPs were set as 0.3, 0.3, and 0.1, respectively (see Supplementary Fig. 1 for other simulated scenarios).