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

In experimental and observational studies, there is often interest in understanding the potential mechanism by which an intervention program improves the final outcome. Causal mediation analyses have been developed for this purpose but are primarily restricted to the case of perfect treatment compliance, with a few exceptions that require exclusion restriction. In this article, we establish a semiparametric framework for assessing causal mediation in the presence of treatment noncompliance without 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 behaviour. We derive the efficient influence functions for the principal natural mediation effect estimands, which motivate a set of multiply robust estimators for inference. The semiparametric efficiency theory for the identified estimands is derived, based on which a multiply robust estimator is proposed. The multiply robust estimators remain consistent to the their respective estimands under four types of misspecification of the working models and is quadruply robust. We further describe a nonparametric extension of the proposed estimators by incorporating machine learners to estimate the nuisance parameters. A sensitivity analysis framework has been developed for address key identification assumptions-principal ignorability and ignorability of mediator. We demonstrate the proposed methods via simulations and applications to a real data example.