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

Causal mediation analysis (Pearl, 2001; VanderWeele, 2015; Imai et al., 2010a) is widely applied in experimental and observational studies to investigate the mechanism underlying a treatment-outcome relationship. Causal mediation methods have been developed under the potential outcomes framework with a primary objective to decompose the total treatment effect into an indirect effect that works through a specified mediator and a direct effect that works around the mediator. While alternative definitions exist, the natural indirect and direct effects have been considered as the most relevant for studying causal mechanisms (Nguyen et al., 2021). The natural indirect effect compares potential outcomes by switching the mediator from the value it would have taken under the control condition to the value it would have taken under the treated condition, while fixing the assignment to the treated condition. The natural direct effect compares potential outcomes by switching the assignment from the control condition to the treated condition, while fixing the mediator to the value it would have taken under the control condition. Parametric regressions (e.g., Valeri and VanderWeele, 2013; Cheng et al., 2021, 2023), semiparametric methods (e.g., Tchetgen Tchetgen and Shpitser, 2012), and nonparametric methods (e.g., Kim et al., 2017) have been proposed for estimating natural mediation effects, typically assuming that all study units perfectly comply with their treatment assignments. Experimental and observational studies are often subject to treatment noncompliance, where the actual treatment received for each unit may differ from the treatment assignment (Angrist et al., 1996). The intention-to-treat (ITT) effect (Lee et al., 1991) and the principal causal effect (PCE) (Frangakis and Rubin, 2002) represent two typical estimands to quantify the impact of intervention under noncompliance. To elaborate, the ITT estimand quantifies the'pragmatic effectiveness' of the treatment under real-world conditions, by measureing the effect of treatment assignment on the outcome among the study population regardless of the actual treatment receipt.