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Causal mediation analysis concerns the pathways through which a treatmentaffects an outcome. While most of the mediation literature focuses on settingswith a single mediator, a flourishing line of research has examined settingsinvolving multiple mediators, under which path-specific effects (PSEs) areoften of interest. We consider estimation of PSEs when the treatment effectoperates through K(\geq1) causally ordered, possibly multivariate mediators. Inthis setting, the PSEs for many causal paths are not nonparametricallyidentified, and we focus on a set of PSEs that are identified under Pearl'snonparametric structural equation model. These PSEs are defined as contrastsbetween the expectations of 2^{K+1} potential outcomes and identified via whatwe call the generalized mediation functional (GMF). We introduce an array ofregression-imputation, weighting, and "hybrid" estimators, and, in particular,two K+2-robust and locally semiparametric efficient estimators for the GMF. Thelatter estimators are well suited to the use of data-adaptive methods forestimating their nuisance functions. We establish the rate conditions requiredof the nuisance functions for semiparametric efficiency. We also discuss howour framework applies to several estimands that may be of particular interestin empirical applications. The proposed estimators are illustrated with asimulation study and an empirical example.

Semiparametric Estimation for Causal Mediation Analysis with Multiple Causally Ordered Mediators