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.

翻译：加权控制直接效应（WCDE）通过对中介变量分布进行加权平均，推广了标准的控制直接效应（CDE），从而在治疗效果随中介变量水平变化时提供更稳健的估计。这使得WCDE在公平性分析中尤为重要，因为它能够分离暴露对结果的直接影响，独立于中介路径。本研究在观测性研究中为WCDE建立了三个基础性进展：首先，我们建立了WCDE唯一可识别性的充分必要条件，阐明了其与CDE产生差异的情形。其次，我们考虑WCDE的非参数估计并推导其影响函数，重点关注正则且渐近线性估计量类。最后，我们刻画了能够最小化渐近方差的最优协变量调整集，证明了与平均处理效应估计相比，中介变量-混杂因子交互作用如何引入不同的估计要求。我们的研究结果为复杂因果系统中直接效应的高效估计提供了原则性框架，在公平性与中介分析中具有实际应用价值。