Semiparametric Estimation for Causal Mediation Analysis with Multiple Causally Ordered Mediators

Causal mediation analysis concerns the pathways through which a treatment affects an outcome. While most of the mediation literature focuses on settings with a single mediator, a flourishing line of research has considered settings involving multiple causally ordered mediators, under which a set of path-specific effects (PSEs) are often of interest. We consider estimation of PSEs for the general case where the treatment effect operates through $K(\geq1)$ causally ordered, possibly multivariate mediators. We first define a set of PSEs that are identified under Pearl's nonparametric structural equation model. These PSEs are defined as contrasts between the expectations of $2^{K+1}$ potential outcomes, which are identified via what we call the generalized mediation functional (GMF). We introduce an array of regression-imputation, weighting, and "hybrid" estimators, and, in particular, two $K+2$-robust and locally semiparametric efficient estimators for the GMF. The latter estimators are well suited to the use of data-adaptive methods for estimating their nuisance functions. We establish rate conditions required of the nuisance functions for semiparametric efficiency. We also discuss how our framework applies to several causal and noncausal estimands that may be of particular interest in empirical applications. The proposed estimators are illustrated with a simulation study and an empirical example.