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 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 (ATE) estimation. Using synthetic and real-world data, we validate our theory numerically, showing that the proposed optimal valid adjustment set yields the lowest variance at practical sample sizes. Our results offer a principled framework for efficient estimation of direct effects in complex causal systems, with practical applications in fairness and mediation analysis.

To formally define valid adjustment sets for the WCDE, we first introduce notation for graphical models. We then present the inferential framework and associated notation required to specify the estimation problem. Lastly, we classify variable types with respect to the exposure and outcome and provide a formal definition of VASs.

An SCM is represented by a DAG, G = (V, E), a graph with directed edges and no directed cycles, where vertices V = X 1 , . . . , X d represent random variables and edges E encode direct causal relationships. An SCM is equipped with a probability distribution P over V. The parent set of a vertex X j , denoted Pa(X j ), consists of all vertices X i for which X i → X j ∈ E, representing direct causes. The children Ch(X i ) are vertices X j with X i → X j ∈ E, representing direct effects. Ancestors An(X j ) comprise all vertices connected to X j via directed paths (Def. A.2), while descendants De(X i ) are all vertices reachable from X i via directed paths.

The d-separation (Def. A.1) criterion formally characterizes conditional independence relationships implied by the graph structure. Under the assumption of causal sufficiency (no unmeasured confounding), d-separation perfectly captures the conditional independencies in the joint distribution through the Markov property:

distributions that are Markov with respect to G. This Markov property implies two key consequences. First, the joint distribution factorizes as:

where each component represents the conditional distribution of a variable given its direct causes. Second, it yields the local Markov property that each variable is conditionally independent of its non-descendant non-parents given its parents:

where ND(X j ) denotes the non-descendants of X j . These properties form the foundation for deriving identifiability conditions for causal effects in the presence of mediator-confounder relationships.

Weighted Controlled Direct Effect Consider a DAG G = (V, E) representing a structural causal model. We focus on estimating WCDE of a binary treatment variable A ∈ V with values a, a * (treatment vs. control) on an outcome Y ∈ V.

Let M ⊂ V be the set of all observed mediators between A and Y , defined through ancestral relations as M := De(A)∩An(Y ), Y . The CDE measures the expected change in outcome as the exposure changes when mediators M are uniformly fixed to a constant value m through intervention:

We provide the identifiability condition of CDE in Def. A.3. When mediator-exposure interactions are present, CDE becomes mediator-dependent, taking different values across M [35]. To obtain an unique population-level measure that admits valid adjustment, we define WCDE using the subset of mediators that are also direct parents of Y , ensuring that all mediator paths (Def. A.2) are blocked: Definition 2.2 (WCDE, [31,34]). Given a DAG G, let M ′ := M ∩ Pa(Y ) and let M ′ be the set of all possible joint values of the mediators in M ′ . The WCDE can be expressed using do-probabilities as:

2.1) Definition 2.2 builds upon prior work by Pearl [34, 35], 1 who established that fixing only the mediators that are parents of Y is sufficient for identifying direct effects. 2 As M ′ is uniquely determined given G, for any given SCM, Eq. (2.1) admits an unique value assuming CDE's identifiability. Remark 2.3 (WCDE definition). Let C be the set of confounders between A and Y . In defining WCDE, we preserve the conceptual distinction between mediators and confounders, rather than treating mediators as additional confounders through joint conditioning. We weight the marginal distribution P (m ′ )