Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect

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.

💡 Research Summary

This paper tackles the problem of estimating the Weighted Controlled Direct Effect (WCDE) in observational studies, providing a complete theoretical framework for identification, non‑parametric estimation, and efficiency optimization. The WCDE extends the standard Controlled Direct Effect (CDE) by averaging the CDE over the natural distribution of the mediators, thereby yielding a single population‑level measure that remains robust when treatment effects vary across mediator levels. This property makes WCDE especially attractive for fairness analyses, where one wishes to isolate the direct effect of a protected attribute on an outcome, independent of mediating pathways.

The authors first formalize the causal setting using a Directed Acyclic Graph (DAG) and define the set of mediators (M) as the descendants of the treatment (A) that are also ancestors of the outcome (Y). The WCDE is then defined with respect to the subset (M’ = M \cap \text{Pa}(Y)), i.e., those mediators that are direct parents of (Y). The effect is expressed as a weighted sum of CDEs: \