Two-Stage Nuisance Function Estimation for Causal Mediation Analysis

Tchetgen Tchetgen and Shpitser (2012) introduced an efficient, debiased, and robust influence function-based estimator for the mediation functional, which is the key component in mediation analysis. This estimator relies on the treatment, mediator, and outcome mean mechanisms. However, treating these three mechanisms as nuisance functions and fitting them as accurately as possible may not be the most effective approach. Instead, it is essential to identify the specific functionals and aspects of these mechanisms that impact the estimation of the mediation functional. In this work, we consider a specific reparametrization of the likelihood function that requires four nuisance functions. To estimate them, we propose a two-stage estimation strategy guided by the role of the nuisance functions in the bias structure of the influence function-based estimator. In particular, two of the functions are estimated using a novel nonparametric weighted balancing approach that directly targets the bias of the final mediation functional estimator. We show that the resulting estimator is consistent and asymptotically normal under certain conditions and attains multiple robustness against misspecifications of the nuisance functions. In simulations, our estimator demonstrated better stability and reduced bias and mean squared error compared to the original influence function-based estimator and a naive estimator. In an application to NHANES 2013-2014 data, our approach suggests that obesity results in a 60% higher odds of coronary heart disease, with roughly 35% higher odds attributable to the direct pathway independent of Glycohemoglobin.