On semiparametric estimation of a path-specific effect in the presence of mediator-outcome confounding

Path-specific effects constitute a broad class of mediated effects from an exposure to an outcome via one or more causal pathways along a set of intermediate variables. Most of the literature concerning estimation of mediated effects has focused on parametric models, with stringent assumptions regarding unmeasured confounding. We consider semiparametric inference of a path-specific effect when these assumptions are relaxed. In particular, we develop a suite of semiparametric estimators for the effect along a pathway through a mediator, but not through an exposure-induced confounder of that mediator. These estimators have different robustness properties, as each depends on different parts of the likelihood of the observed data. One estimator is locally semiparametric efficient and multiply robust. The latter property implies that machine learning can be used to estimate nuisance functions. We demonstrate these properties, as well as finite-sample properties of all the estimators, in a simulation study. We apply our method to an HIV study, in which we estimate the effect comparing two drug treatments on a patient's average log CD4 count mediated by the patient's level of adherence, but not by previous experience of toxicity, which is clearly affected by which treatment the patient is assigned to and may confound the effect of the patient's level of adherence on their virologic outcome.