Title: Optimal Adjustment Sets for Nonparametric Estimation of Weighted Controlled Direct Effect
Venue: NeurIPS 2025
Accepted: yes
Keywords: Optimal adjustment sets

causal inference

weighted controlled direct effect

nonparametric estimation
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. In this work, we first establish necessary and sufficient conditions for the unique identifiability of the WCDE, clarifying when it diverges from the CDE. Next, we derive the efficient influence function for the WCDE and consider the class of regular and asymptotically linear estimators. We characterize the optimal covariate adjustment set that minimizes 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.


Peer reviews:
comments: SUMMARY:
The paper makes three substantial contributions: conditions for identifiability, derivation of the influence function, and characterization of optimal adjustment sets for WCDE. These are theoretically significant and extend existing work on average treatment effect (ATE) adjustment.

STRENGTHS AND WEAKNESSES:
Strengths

The introduction effectively motivates the Weighted Controlled Direct Effect (WCDE) as a generalization of the Controlled Direct Effect (CDE), with clear links to fairness analysis and policy relevance.

The formal criteria for Valid Adjustment Sets (VAS) are well-defined and justified.

The contrast between CDE and WCDE is articulated well, especially regarding the limitations of fixing mediator levels arbitrarily.

Weaknesses

Even though the paper is theoretical, a small simulation study could help validate the efficiency gains from the optimal VAS, enhancing accessibility for applied researchers.

The identifiability and efficiency results hinge on having the true DAG and the absence of unmeasured confounding. Without access to the true DAG, the proposed method may no longer be valid.

QUESTIONS:
n/a

LIMITATIONS:
Since fairness is a key motivation, consider sketching a concrete fairness scenario where the WCDE provides insights unattainable through the CDE or ATE. Including a real-world case would make the example even more compelling.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-22

comments: SUMMARY:
This paper presents both necessary and sufficient conditions under which weighted controlled direct effect are identifiable. These necessary and sufficient conditions are proved to yield unique weighted controlled direct effect. Later, the paper also proposes a way to identify and verify optimal adjustment set that yields asymptotically lower variance of the estimated weighted controlled direct effect.

STRENGTHS AND WEAKNESSES:
Strengths:

1. The paper is written well and easy to understand.
2. This paper nicely extends the theoretical results from literature on effect identifiability and optimal adjustment set identification to the case of weighted controlled direct effect which has applications in domains such as fairness.

Weaknesses:

1. A major limitation of this paper is lack of experimental results. While the theoretical results are interesting and useful, experimental results verifying the theoretical results further significantly improve the paper.
2. Considering the complexity of the paper, it benefits more from several illustrative figures explaining various aspects of the paper. For instance, more causal diagram based examples are needed to understand condition 2.3 (4).
3. Minor: In L133, the phrase "let $M'\subseteq M$ be parents of $Y$" needs a little refinement. Currently it reads like $M'=Pa(Y)$. Similarly in L145, clearly explain what counterfactual independence conditions are.

QUESTIONS:
See the weaknesses section above and provide your responses for those comments.

LIMITATIONS:
Limitations are not explicitly presented but future directions are discussed.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-29

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies nonparametric estimation of the Weighted Controlled Direct Effect (WCDE), which is an extension of the standard controlled direct effect (CDE) that averages over the mediator distribution, providing a robust estimate when treatment effects vary across mediator levels.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n- This paper studies nonparametric estimation of the Weighted Controlled Direct Effect (WCDE), which is an interesting and important extension of the standard controlled direct effect (CDE).\n- This paper defines a Valid Adjustment Set and establishes that Condition 2.3 is both necessary and sufficient for identifying a unique WCDE (Lemma 2.4).\n- This paper presents the identification theory and formulas for WCDE via Valid Adjustment Set.\n- This paper proposes three fundamental advances for WCDE in observational studies:\n  - Establishing necessary and sufficient conditions for the unique identifiability of WCDE, clarifying when it diverges from CDE.\n  - Deriving the influence function for nonparametric estimation of WCDE, considering the class of regular and asymptotically linear estimators.\n  - Characterizing the optimal covariate adjustment set that minimizes asymptotic variance, demonstrating how mediator-confounder interactions introduce distinct requirements compared to average treatment effect estimation.\n\n**Weaknesses:**\n\n- While the paper provides a strong theoretical foundation, the practical implementation of determining the \"O-set\" (optimal adjustment set) in real-world scenarios might be complex.\nThe entire framework, including the definition of VASs and the optimal adjustment set, assumes a known - - Directed Acyclic Graph (DAG) encoding the underlying causal structure. In many real-world applications, the true DAG is unknown and must be learned from data.\n- The paper's notion of \"optimal\" is based solely on minimizing asymptotic variance. While this is a critical statistical property, other practical considerations, such as bias in finite samples, robustness to model misspecification (beyond what's implied by RAL estimators), or computational feasibility for very high-dimensional data, are not discussed in detail.\n- The absence of experimental validation limits the immediate practical applicability and empirical understanding of the theoretical advancements presented.\n\nQUESTIONS:\n- Could the authors include simulation studies to demonstrate the practical advantages of  the proposed methods. \n- How robust are the results (identifiability, influence function, and optimal adjustment set) to misspecification or uncertainty in the underlying DAG? Could the authors discuss the implications if the true DAG is unknown and has to be learned from data?\n- Could the authors elaborate on how practitioners can leverage the derived influence function (Theorem 3.4) to build efficient RAL estimators for WCDE?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the estimation of \"Weighted Controlled Direct Effect (WCDE)\". CDE measures the effect of a treatment holding a mediator to a fixed level. WCDE averages over mediator distribution. The paper provides necessary and sufficient conditions for valid adjustment sets for unique identifiability of WCDE (condition 2.3), gives influence function for WCDE (Theorem 3.4), and shows that the optimal adjustment set that minimizes asymptotic variance is different from the ATE adjustment set.\n\nSTRENGTHS AND WEAKNESSES:\n** Strength: **\n\nS1. The paper is in general well written. \n\nS2. I have not checked the proofs in the appendix, but proof sketches are provided in the main paper. One contribution is condition 2.3 (4), which uses interesting ideas for sufficiency and necessity. An example 2.5 is provided to illustrate the condition. \n\nS3. Similarly the proof idea of finding the optimal valid adjustment set is interesting and a proof sketch for 4 steps is given. I have not checked the proof in the appendix.  \n\n** Weakness: **\n\nPresentation: In some places a better clarity would be helpful.\n\n(a) in the introduction, better explanation can be provided for \"unique identifiability\" and the optimal adjustment set that minimizes variance, suggesting that there are multiple adjustment sets. \n\n(b) In condition 2.3, line 2.3, M should be defined since A and Y are not defined as treatment and outcome but arbitrary disjoint nodes. \n\n(c) In condition 2.3, backdoor and mediator paths are used for the first time. They should be defined for the sake of completeness. \n\n\nMinor typo in the 3rd condition in 2.3 -- \"are block\"\n\nQUESTIONS:\nPlease explain unique identifiability. \n\nPlease also explain limitations of the assumptions and method in detail. \n\nSatisfactory explanations to these points might increase my score to Accept.\n\nLIMITATIONS:\nNot adequate. The Q/A at the end says that they discuss future work, but I only find a summarization of the contributions and not much future work or limitation. Following the solicitation a clear specification and discussion of limitations and future work will be useful. For instance, the methods seem to be fully dependent on the correctness of causal DAG and correct specification of mediator variables, which may not hold in practice as is largely understood. Studying robustness when the causal DAG is not correct will be a future work.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}