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Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect

Ruiyang Lin, Yongyi Guo, Kyra Gan

TL;DR

This work develops a principled framework for nonparametric estimation of the weighted controlled direct effect (WCDE), defined as $\mathrm{WCDE} = \sum_{ extbf{m}'\in\mathcal{M}'} ( \mathbb{E}[Y|\operatorname{do}(a,\mathbf{m}')] - \mathbb{E}[Y|\operatorname{do}(a^*,\mathbf{m}')] ) p(\mathbf{m}')$, where $\mathcal{M}' = \textbf{M}' = \textbf{M} \cap \text{Pa}(Y)$. The authors establish necessary and sufficient conditions for identifiability via valid adjustment sets, derive the influence function for WCDE under nonparametric models, and show that the optimal adjustment set for WCDE generally differs from that for average treatment effects due to mediator–confounder interactions. They prove that the optimal valid adjustment set is $\mathbf{O}(A,Y,\mathcal{G}) = \mathbf{X}_{1\in\text{Pa}(Y)} \cup \mathbf{X}_{3\in\text{Pa}(Y)} \cup \mathbf{X}_{4\in\text{Pa}(Y)}$, and provide an efficient AIPW estimator based on the identified WCDE functional. Theoretical results are corroborated by synthetic experiments and semi-synthetic networks (ASIA, SIGNALING, MILDEW), showing consistent variance reduction and robust performance in finite samples, with practical implications for fairness and mediation analysis in complex causal systems.

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.

Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect

TL;DR

This work develops a principled framework for nonparametric estimation of the weighted controlled direct effect (WCDE), defined as , where . The authors establish necessary and sufficient conditions for identifiability via valid adjustment sets, derive the influence function for WCDE under nonparametric models, and show that the optimal adjustment set for WCDE generally differs from that for average treatment effects due to mediator–confounder interactions. They prove that the optimal valid adjustment set is , and provide an efficient AIPW estimator based on the identified WCDE functional. Theoretical results are corroborated by synthetic experiments and semi-synthetic networks (ASIA, SIGNALING, MILDEW), showing consistent variance reduction and robust performance in finite samples, with practical implications for fairness and mediation analysis in complex causal systems.

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.

Paper Structure

This paper contains 30 sections, 9 theorems, 201 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem Setup and Valid Adjustment Sets for WCDE
  3. RAL Estimators and Influence Function of WCDE
  4. Optimal Valid Adjustment Sets for WCDE
  5. Experiments
  6. Discussion
  7. Definition, Notation and Graph Terminology
  8. Sufficient and Necessary Conditions for CDE identification \ref{['condition:cde']}
  9. Necessary and Sufficient Adjustment Criteria for Identifying do-probability
  10. CDE Adjustment Criteria
  11. Counterexamples to Necessity.
  12. Definitions of Causal Partition Introduced by Maasch et al. [32]
  13. Proofs and Extended Discussions for Sections 2 and 3
  14. Proof of Lemma \ref{['lemma:adjustment_criterion']}
  15. Step 1: Replacing $\mathbf{Z}_2$ with $\mathbf{C}$ Including Non-Mediating Parents of $Y$
  16. ...and 15 more sections

Key Result

Lemma 2.6

Let $\mathbf{Z}$ be any set that satisfies Condition condition:adj for $\text{WCDE}$ with respect to $(A, Y)$ in $\mathcal{G}$. Then $\mathbf{Z}$ is a VAS (Def. def:VAS). Conversely, any adjustment set $\mathbf{Z}$ that identifies the WCDE in Eq. eqution:WCDE must necessarily satisfy Condition condi

Figures (8)

  • Figure 1: The partition visualization follows maasch2024local, with directed squiggly edges indicating the presence of a directed causal path (not necessarily direct parent-child relationships). The dashed edge represents potential parent-child relationships.
  • Figure 2: A DAG illustrating Lemmas \ref{['lemma:1']}, \ref{['lemma:2']}.
  • Figure 3: A DAG illustrating Lemmas \ref{['lemma:3']}, \ref{['lemma:4']}.
  • Figure A.1: Example illustrating Condition \ref{['condition:adj-blocking']} and \ref{['condition:mediator_backdoor']} is not necessary for CDE identification.
  • Figure B.3: A DAG illustrating Condition \ref{['condition:adj']}.
  • ...and 3 more figures

Theorems & Definitions (40)

  • Definition 2.1: CDE, pearl2014interpretation
  • Definition 2.2: WCDE, maasch2025localpearl2000causality
  • Remark 2.3: WCDE definition
  • Definition 2.4: Valid Adjustment Set for WCDE
  • Lemma 2.6: Sufficiency and Necessity of the Adjustment Criterion
  • Definition 3.1: Regular estimator, vanderVaart2000
  • Definition 3.2: Asymptotically linear estimator, vanderVaart2000
  • Remark 3.3: Influence Functions of $\text{WCDE}_{\mathbf{Z}}$
  • Theorem 3.4: IF of $\text{WCDE}_\mathbf{Z}$
  • Definition 4.1: O-set
  • ...and 30 more