Identification and multiply robust estimation in causal mediation analysis with treatment noncompliance

In experimental and observational studies, there is often interest in understanding the mechanism through which an intervention program improves the final outcome. Causal mediation analyses have been developed for this purpose but are primarily considered for the case of perfect treatment compliance, with a few exceptions that require the exclusion restriction assumption. In this article, we consider a semiparametric framework for assessing causal mediation in the presence of treatment noncompliance without the exclusion restriction. We propose a set of assumptions to identify the natural mediation effects for the entire study population and further, for the principal natural mediation effects within subpopulations characterized by the potential compliance behavior. We derive the efficient influence functions for the principal natural mediation effect estimands and motivate a set of multiply robust estimators for inference. The multiply robust estimators remain consistent to their respective estimands under four types of misspecification of the working models and are efficient when all nuisance models are correctly specified. We further introduce a nonparametric extension of the proposed estimators by incorporating machine learners to estimate the nuisance functions. Sensitivity analysis methods are also discussed for addressing key identification assumptions. We demonstrate the proposed methods via simulations and an application to a real data example.