Two-Stage Nuisance Function Estimation for Causal Mediation Analysis (2404.00735v2)

Abstract: 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 propose a two-stage estimation strategy for certain nuisance functions in the influence function of the mediation functional that are based on these three mechanisms. This strategy is guided by the role those nuisance functions play in the bias structure of the influence function-based estimator for the mediation functional. In the first stage, we estimate two primary nuisance functions, namely the inverse treatment mechanism and the outcome mean mechanism. In the second stage, we leverage these primary functions to estimate two additional nuisance functions that encapsulate the needed information about the mediator mechanism. We propose a nonparametric weighted balancing estimation approach to design the estimator for one of the nuisance functions in Stage 1 and one of in Stage 2, where the weights are designed directly based on the bias of the final estimator for the mediation functional. The remaining two nuisance functions are estimated using standard parametric or nonparametric regression methods. Once all four nuisance functions are obtained, they are incorporated into the influence function-based estimator. We provide a robustness analysis of the proposed method and establish sufficient conditions for consistency and asymptotic normality of our estimator for the mediation functional.