Mediation analysis with unmeasured confounding between parallel mediators and outcome

Mediation analysis extending beyond single mediators has gained significant attention in recent years. However, related methods often assume the absence of unmeasured mediator-outcome confounding. To address this, we develop a mediation analysis framework that accounts for such confounding within a linear structural equation model with parallel mediators. Specifically, we introduce a pseudo proxy variable to capture unmeasured confounding, allowing us to identify causal parameters. Leveraging this proxy, we propose a partially penalized method to identify mediators that significantly affect the outcome. The resultant estimates are consistent, and the estimates of nonzero parameters are asymptotically normal. Motivated by these results, we further introduce a procedure that can consistently select active mediation pathways with large probability. Simulation studies demonstrate the superior performance of the proposed approach. Finally, we apply our approach to genomic data, identifying gene expressions that potentially mediate the impact of a genetic variant on mouse obesity.