effects [Goetgeluk, Vansteelandt and Goetghebeur (2008)]. An important advantage of a double robust method is that it carefully combines both of the aforementioned dimension reduction strategies for confounding adjustment, to produce an estimator of the causal effect that remains consistent and asymptotically normal, provided at least one of the two strategies is correct, without necessarily knowing which strategy is indeed correct [van der Laan and Robins (2003)]. Unfortunately, similar methods for making semiparametric inferences about marginal natural direct and indirect effects are currently lacking. Thus, this paper develops a general semiparametric framework for obtaining inferences about marginal natural direct and indirect effects on the mean of an outcome, while appropriately accounting for a large number of confounding factors for the exposure and the mediator variables.
Our semiparametric framework is particularly appealing, as it gives new insight on issues of efficiency and robustness in the context of mediation analysis. Specifically, in Section 2, we adopt the sequential ignorability assumption of Imai, Keele and Tingley (2010) under which, in conjunction with the standard consistency and positivity assumptions, we derive the efficient influence function and thus obtain the semiparametric efficiency bound for the natural direct and natural indirect marginal mean causal effects, in the nonparametric model $M_{nonpar}$ in which the observed data likelihood is left unrestricted. We further show that in order to conduct mediation inferences in $M_{nonpar}$, one must estimate at least a subset of the following quantities:
(i) the conditional expectation of the outcome given the mediator, exposure and confounding factors;
(ii) the density of the mediator given the exposure and the confounders;
(iii) the density of the exposure given the confounders.
Ideally, to minimize the possibility of modeling bias, one may wish to estimate each of these quantities nonparametrically; however, as previously argued, when as we assume throughout, we wish to account for numerous confounders, such nonparametric estimates will likely perform poorly in finite samples. Thus, in Section 2.3 we develop an alternative multiply robust strategy. To do so, we propose to model (i), (ii) and (iii) parametrically (or semiparametrically), but rather than obtaining mediation inferences that rely on the correct specification of a specific subset of these models, instead we carefully combine these three models to produce estimators of the marginal mean direct and indirect effects that remain consistent and asymptotically normal (CAN) in a union model, where at least one but not necessarily all of the following conditions hold:
(a) the parametric or semi-parametric models for the conditional expectation of the outcome (i) and for the conditional density of the mediator (ii) are correctly specified;