Identification, Semiparametric Efficiency, and Quadruply Robust Estimation in Mediation Analysis with Treatment-Induced Confounding

Natural mediation effects are often of interest when the goal is to understand a causal mechanism. However, most existing methods and their identification assumptions preclude treatment-induced confounders often present in practice. To address this fundamental limitation, we provide a set of assumptions that identify the natural direct effect in the presence of treatment-induced confounders. Even when some of those assumptions are violated, the estimand still has an interventional direct effect interpretation. We derive the semiparametric efficiency bound for the estimand, which unlike usual expressions, contains conditional densities that are variational dependent. We consider a reparameterization and propose a quadruply robust estimator that remains consistent under four types of possible misspecification and is also locally semiparametric efficient. We use simulation studies to demonstrate the proposed method and study an application to the 2017 Natality data to investigate the effect of prenatal care on preterm birth mediated by preeclampsia with smoking status during pregnancy being a potential treatment-induced confounder.