Single replicate of the simulated scenarioUndGX-DAGY, where an undirected graph encodes the dependency pattern within X and a DAG represents the dependency relations within Y along with the simulated causal effects from the exposures to the outcomes, resulting in an overall partially oriented DAG. In this scenario, the strength of correlation between consecutive X is set at rX=0.6 and then decreases exponentially for non-consecutive exposures, and the average level of the mediation parameters within Y is set at mY=1. Individual-level outcomes Y=(Y1, Y5) and exposures X=(X1, X15) as well as genetically predicted outcomes Y*=(Y1*, Y5*) and exposures X*=(X1*, X15*) are represented with orange and blue nodes, respectively. Directed edges indicate dependency relations, while undirected edges denote partial correlation. Dashed lines depict the true (unconfounded by U) and estimated dependency structure within the exposures and the outcomes, while solid lines indicate true and estimated causal effects between them. Red color denotes false positives, either falsely detected effects (regardless of the directionality) or wrong directionality of the edges. Besides the proposed model, alternative methods considered Mendelian randomization with Bayesian model averaging (MR-BMA),1 multi-response Mendelian randomization (MR2),2 Mendelian randomization with PC algorithm (MRPC),51 and partition-DAG (ParDAG).44 We report the results of MR-BMA and MR2 obtained by thresholding the marginal posterior probability of inclusion (mPPI)>0.5, which correspond to the median models.52 No threshold is applied to MrDAG posterior probability of edge inclusion (PPEI). MRPC partially directed acyclic graphs (PDAGs) are obtained by specifying the type I error rate for the conditional independence test at a=0.01. ParDAG results are the solutions of causal effects estimation with Lasso penalization set at l=0.9.
