Model-Robust Direct Effect Under Confounder-Mediator Ambiguity

1. Problem Statement

In causal inference, direct effect analysis typically requires a clear distinction between a focal variable $W$ acting as a pre-exposure confounder or a post-exposure mediator.

• The Ambiguity: In many observational studies (e.g., cross-sectional data), the temporal ordering of $W$ relative to the exposure $A$ is unclear. $W$ might be a baseline characteristic affecting both $A$ and $Y$, or it might be an intermediate variable on the causal pathway from $A$ to $Y$.
• The Consequence: Standard estimands differ based on this structural assumption:
  • If $W$ is a confounder, the direct effect is the Average Treatment Effect (ATE), averaging the conditional exposure contrast over the observed distribution of $W$ given covariates $X$ ($F_{W|X}$).
  • If $W$ is a mediator, the standard direct effect is the Natural Direct Effect (NDE), which averages the contrast over the distribution of $W$ under the control condition ($F_{W|A=0, X}$).
• The Core Issue: The paper demonstrates that, in general, no single observed-data functional can recover the ATE under the confounder model and the NDE under the mediator model simultaneously. If a practitioner applies an NDE estimator when $W$ is actually a confounder, the resulting estimate converges to a quantity with no clear causal interpretation, leading to substantively misleading conclusions despite numerical stability.

2. Methodology and Theoretical Framework

2.1 Impossibility Result

The authors prove (Theorem 1) that there is no map $T$ from the observed data law to a real number that identifies the ATE under the confounder model and the NDE under the mediator model for all possible data-generating processes. This establishes that structural ambiguity creates estimand ambiguity.

2.2 The No-Additive-Interaction Condition

The paper identifies a specific condition under which the ATE and NDE coincide: No Additive Interaction between the exposure $A$ and the variable $W$ (Assumption 5).

• If the conditional mean effect $\Delta(w, x) = E[Y|A=1, W=w, X=x] - E[Y|A=0, W=w, X=x]$ does not depend on $w$, then the averaging distribution of $W$ becomes irrelevant.
• Under this condition, the discrepancy between the ATE and NDE is zero.
• The authors derive a sensitivity bound (Proposition 3) quantifying how far the NDE deviates from the ATE when this condition is violated. The bound depends on the range of the conditional effect ($\text{rng}\Delta$) and the total variation distance between the distributions $F_{W|X}$ and $F_{W|A=0, X}$.

2.3 The Model-Robust Estimand ($\psi(P)$)

To resolve the ambiguity, the authors propose a new target parameter, $\psi(P)$, defined as the expected conditional exposure contrast averaged over the observed distribution of $W$ given $X$:
$$\psi(P) = E_P \{ \Delta(W, X) \} = E_P \{ Q(1, W, X) - Q(0, W, X) \}$$
Causal Interpretation:

• Under the Confounder Model: $\psi(P)$ equals the ATE (Total Effect), as there is no mediation pathway.
• Under the Mediator Model: $\psi(P)$ equals an Interventional Direct Effect (IDE). It represents the effect of $A$ on $Y$ if $W$ were set to a value drawn from its observed conditional distribution $F_{W|X}$, independent of the potential outcomes.
• Uniqueness: Theorem 4 proves that within the class of "outcome-free" stochastic direct effects (where the intervention rule for $W$ depends only on $(X, A, W)$ and not on the outcome model), $\psi(P)$ is the unique functional that coincides with the ATE under the confounder model.

2.4 Estimation Strategy

The paper develops a cross-fitted one-step estimator ($\hat{\psi}^{cf}$) for $\psi(P)$:

• Efficient Influence Function: The authors derive the efficient influence function (EIF) for $\psi(P)$, which involves the outcome regression $Q(a, w, x)$ and the propensity score $g(w, x) = P(A=1|W, X)$.
• Double Robustness: The estimator is doubly robust. It is consistent if either the outcome regression model or the propensity score model is correctly specified.
• Cross-Fitting: To avoid overfitting bias and allow for flexible machine learning nuisance estimators, the data is split into folds. Nuisance parameters are estimated on training folds and applied to test folds.
• Inference: Asymptotic normality is established, allowing for Wald-type confidence intervals.

3. Key Results

3.1 Simulation Study

The authors conducted simulations across three scenarios:

1. Confounder with Interaction: $W$ is pre-exposure.
2. Mediator without Interaction: $W$ is post-exposure, no $A \times W$ interaction.
3. Mediator with Interaction: $W$ is post-exposure, with $A \times W$ interaction.

Findings:

• The proposed estimator $\hat{\psi}^{cf}$ was unbiased and achieved nominal 95% coverage in all scenarios.
• A standard mediation formula estimator (targeting NDE) performed well in Scenario 2 (where ATE = NDE) but failed in Scenarios 1 and 3.
• In Scenario 1 (Confounder), the mediation estimator converged to a value $\approx 0.18$ lower than the true ATE, demonstrating estimand mismatch rather than estimation error.
• In Scenario 3 (Mediator with interaction), the mediation estimator converged to a value $\approx 0.14$ lower than $\psi(P)$, highlighting the difference between NDE and IDE when interactions exist.

3.2 Real-World Application (NHANES)

The method was applied to study the direct effect of elevated PFAS burden on uric acid, with kidney function (eGFR) as the focal variable.

• Context: eGFR is structurally ambiguous; it could be a pre-exposure confounder (low kidney function raises PFAS and uric acid) or a post-exposure mediator (PFAS damages kidneys, raising uric acid).
• Results:
  • Model-Robust Estimate ($\hat{\psi}$): 0.25 mg/dL (95% CI: 0.05, 0.45).
  • Mediation Model Comparator (NDE): 0.10 mg/dL (95% CI: -0.03, 0.23).
• Implication: The mediation analysis yielded a materially smaller estimate (59% lower). If eGFR is a confounder, the mediation estimate is causally uninterpretable. The model-robust estimate provides a valid summary regardless of the structural role of eGFR.

4. Key Contributions

1. Formalization of Estimand Ambiguity: The paper rigorously proves that structural ambiguity between confounding and mediation leads to a fundamental lack of a common observed-data target for standard direct effects.
2. Proposal of a Canonical Estimand: It introduces $\psi(P)$, a model-robust direct effect that equals the ATE under confounding and an Interventional Direct Effect under mediation.
3. Uniqueness Theorem: It establishes that $\psi(P)$ is the unique outcome-free stochastic direct effect that maintains causal interpretability (specifically, matching the ATE) under the confounder model.
4. Robust Inference: It provides a doubly robust, cross-fitted estimator that handles nuisance parameter estimation flexibly (e.g., via machine learning) while maintaining valid inference.
5. Practical Guidance: The paper demonstrates that relying on standard mediation formulas in the presence of structural ambiguity can lead to substantively different and potentially misleading conclusions.

5. Significance

This work addresses a critical gap in causal inference where researchers often must choose between treating a variable as a confounder or a mediator without definitive temporal evidence.

• Scientific Impact: It prevents the "garbage in, garbage out" scenario where a statistically stable estimator yields a causally meaningless result due to a wrong structural assumption.
• Methodological Advancement: It shifts the focus from "estimator robustness" (double robustness for a fixed target) to "identification robustness" (a target that remains valid across competing structural models).
• Policy Relevance: In fields like environmental epidemiology (e.g., PFAS studies), where mechanisms are complex and data is often cross-sectional, this approach offers a single, interpretable metric for direct effects that does not require the analyst to commit to a potentially incorrect causal diagram.

The paper concludes that while extensions to non-additive scales and longitudinal settings are needed, the proposed framework provides a necessary tool for robust causal analysis in the face of structural uncertainty.