Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect

TL;DR

This paper develops a theoretical framework for the nonparametric estimation of the weighted controlled direct effect (WCDE), including conditions for identifiability, influence function derivation, and optimal adjustment set characterization, with applications in fairness and mediation analysis.

Contribution

It introduces necessary and sufficient conditions for WCDE identifiability, derives its influence function, and characterizes the optimal covariate adjustment set for efficient estimation.

Findings

Established conditions for WCDE identifiability

Derived influence function for nonparametric WCDE estimation

Characterized optimal adjustment set minimizing variance

Abstract

The weighted controlled direct effect (WCDE) generalizes the standard controlled direct effect (CDE) by averaging over the mediator distribution, providing a robust estimate when treatment effects vary across mediator levels. This makes the WCDE especially relevant in fairness analysis, where it isolates the direct effect of an exposure on an outcome, independent of mediating pathways. This work establishes three fundamental advances for WCDE in observational studies: First, we establish necessary and sufficient conditions for the unique identifiability of the WCDE, clarifying when it diverges from the CDE. Next, we consider nonparametric estimation of the WCDE and derive its influence function, focusing on the class of regular and asymptotically linear estimators. Lastly, we characterize the optimal covariate adjustment set that minimizes the asymptotic variance, demonstrating how mediator-confounder interactions introduce distinct requirements compared to average treatment effect estimation. Our results offer a principled framework for efficient estimation of direct effects in complex causal systems, with practical applications in fairness and mediation analysis.