Model-Robust Direct Effect Under Confounder-Mediator Ambiguity

AmirEmad Ghassami

arxiv: 2604.08978 · v1 · submitted 2026-04-10 · 📊 stat.ME

Model-Robust Direct Effect Under Confounder-Mediator Ambiguity

AmirEmad Ghassami This is my paper

Pith reviewed 2026-05-10 18:18 UTC · model grok-4.3

classification 📊 stat.ME

keywords direct effectsconfounder-mediator ambiguitymodel-robust estimationnatural direct effectaverage treatment effectdoubly robust estimatorsensitivity analysisobservational data

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The pith

A single observed-data functional recovers the average treatment effect when a focal variable is a pre-exposure confounder and an interventional direct effect when it is a post-exposure mediator.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Direct effect analyses require deciding if a focal variable acts as a confounder before exposure or a mediator after, but observational data often leaves this ambiguous due to coarse timing or evolving processes. The paper shows that in general no single observed-data quantity can recover both the average treatment effect in the confounder case and the natural direct effect in the mediator case. It identifies a no-additive-interaction condition under which the two quantities coincide, supplies sensitivity bounds for violations, and introduces an alternative model-robust estimand. This estimand stays causally interpretable in both structural roles and, within the class of outcome-free stochastic direct effects, is the unique such functional with that property. The authors derive its efficient influence function and a doubly robust estimator, then illustrate the approach on NHANES data linking PFAS burden, kidney function, and uric acid.

Core claim

We show that no single observed-data estimand generally recovers both the average treatment effect when the focal variable is a confounder and the natural direct effect when it is a mediator. Under a no-additive-interaction condition the two quantities coincide; we develop sensitivity bounds for departures from the condition and propose an alternative model-robust estimand. This estimand equals the ATE when the variable is pre-exposure and an interventional direct effect when it is post-exposure. Within a natural class of outcome-free stochastic direct effects, it is the unique observed-data functional that remains causally interpretable under both structural roles of the focal variable. We

What carries the argument

The model-robust estimand, defined as the unique observed-data functional within the class of outcome-free stochastic direct effects that equals the ATE under a confounder role and an interventional direct effect under a mediator role.

If this is right

Under the no-additive-interaction condition the new estimand recovers both the ATE and the NDE, so practitioners no longer need to guess the variable's structural role.
The estimator remains consistent if either the outcome model or the treatment and mediator models are correctly specified, giving double robustness at the estimation stage.
Sensitivity bounds quantify how far the reported quantity can deviate when additive interaction is present.
In applications with coarse timing measurements the method supplies a single number that retains a causal reading regardless of whether the focal variable is treated as pre- or post-exposure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same construction could be applied to other variables whose temporal status is uncertain in longitudinal or registry data, such as biomarkers or behaviors whose onset is imprecisely dated.
Reporting the model-robust quantity together with its sensitivity bounds would give readers a transparent interval rather than a point estimate that silently depends on an untestable role assumption.
In settings with multiple candidate focal variables the approach might be iterated, producing a set of robust direct-effect estimates whose joint pattern could itself be informative.

Load-bearing premise

The no-additive-interaction condition between the focal variable and the treatment in their joint effect on the outcome, which makes the average treatment effect and natural direct effect coincide.

What would settle it

In a simulation or real dataset where the focal variable's role is known with certainty, compute both the pure ATE estimator and the pure interventional-direct-effect estimator; if the proposed model-robust quantity fails to match the known-role estimator in either case, the uniqueness and dual-interpretability claim is refuted.

Figures

Figures reproduced from arXiv: 2604.08978 by AmirEmad Ghassami.

**Figure 2.** Figure 2: Monte Carlo distributions of the cross-fitted estimator [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: Empirical bias relative to ψ(P) for the cross-fitted estimator and the mediation formula comparator, together with empirical 95% confidence interval coverage for ψb cf. includes PFOA, PFOS, PFHxS, and PFNA, NASEM recommends forming the sum from those committee analytes [National Academies of Sciences, Engineering, and Medicine, 2022, Agency for Toxic Substances and Disease Registry, 2024]. We set W equal t… view at source ↗

read the original abstract

Direct effect analyses usually require deciding whether a focal variable is a pre-exposure confounder or a post-exposure mediator. In observational studies, that distinction may be unclear because timing is measured coarsely or the variable reflects an evolving process. Considering the average treatment effect (ATE) and the natural direct effect (NDE) as a common notion of the direct effect when the focal variable is a confounder and a mediator, respectively, we show that, in general, no single observed-data estimand recovers both the ATE when the focal variable is a confounder and the NDE when it is a mediator. Consequently, if a practitioner applies an NDE estimator when the variable is actually pre-exposure, the resulting estimate may have no clear causal interpretation. We identify a no-additive-interaction condition under which these quantities coincide, develop sensitivity bounds for departures from that condition, and propose an alternative model-robust estimand. This estimand equals the ATE when the variable is pre-exposure and an interventional direct effect when it is post-exposure. Moreover, within a natural class of outcome-free stochastic direct effects, it is the unique observed-data functional that remains causally interpretable under both structural roles of the focal variable. We derive an efficient influence function and a doubly robust estimator, yielding robustness at two levels: the estimand is model-robust across the two causal scenarios, and the estimator is doubly robust with respect to nuisance estimation. In simulations and in an NHANES application on elevated PFAS burden, kidney function, and uric acid, mediation-based analyses yielded materially different reported estimates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper supplies a new observed-data estimand for direct effects that stays interpretable whether the focal variable is a confounder or mediator, unique inside its stated class, plus sensitivity bounds and a doubly robust estimator.

read the letter

The main thing to know is that the authors construct an estimand which equals the ATE under a confounder reading and an interventional direct effect under a mediator reading. Within the class of outcome-free stochastic direct effects, it is the unique functional that keeps a causal interpretation in both cases. They also identify the no-additive-interaction condition that makes standard ATE and NDE coincide, supply sensitivity bounds for when it fails, and derive an efficient influence function with a doubly robust estimator. The NHANES example on PFAS, kidney function, and uric acid shows that ordinary mediation estimators can produce materially different numbers, which illustrates the practical stakes. On reading the full paper the derivations line up without circularity or post-hoc restrictions that undermine the claim. The uniqueness result is stated only inside the defined class, which is a limitation but not a hidden flaw. The no-additive-interaction condition is treated as an assumption rather than a default, and the sensitivity analysis is presented as the way to handle departures. That keeps the argument proportionate. The simulations and application are used to show divergence rather than to claim superiority in every setting, which is fair. This is for causal inference people who work with observational data where timing is coarse or roles are ambiguous. A reader who needs a default estimand with built-in robustness across the two interpretations will find concrete tools here. The formal grounding and the absence of load-bearing gaps mean it deserves a serious referee rather than a desk reject.

Referee Report

2 major / 3 minor

Summary. The paper addresses ambiguity in whether a focal variable acts as a pre-exposure confounder or post-exposure mediator in direct-effect estimation. It shows that no single observed-data functional generally recovers both the ATE (under confounder interpretation) and the NDE (under mediator interpretation). Under a no-additive-interaction condition the two coincide; the authors supply sensitivity bounds for departures and propose an alternative estimand that equals the ATE when the variable is pre-exposure and an interventional direct effect when post-exposure. Within the class of outcome-free stochastic direct effects this estimand is claimed to be the unique observed-data functional that remains causally interpretable under both structural roles. An efficient influence function and doubly-robust estimator are derived; the approach is illustrated in simulations and an NHANES analysis of PFAS burden, kidney function, and uric acid.

Significance. If the derivations hold, the work supplies a practical, model-robust solution to a frequent observational-data problem where timing or process ambiguity prevents clean confounder/mediator classification. The double robustness—both at the level of the target estimand across causal scenarios and at the level of nuisance estimation—is a clear strength. The uniqueness result inside the stated class, together with the explicit no-additive-interaction condition and sensitivity analysis, adds rigor. The empirical example demonstrates that standard mediation estimators can produce materially different numerical conclusions. These features could influence routine practice in epidemiology and social science whenever variable roles are uncertain.

major comments (2)

[§3.2] §3.2, definition of the outcome-free stochastic direct effects class: the uniqueness claim is load-bearing for the central recommendation, yet the precise characterization of the class (which functionals are included or excluded) is not fully spelled out before the uniqueness theorem; without an explicit enumeration or axiomatic description it is difficult to verify that no other functional satisfies the same dual-interpretability property.
[§4.1, Eq. (12)] §4.1, Eq. (12) (sensitivity bounds): the bounds are derived under the no-additive-interaction condition, but the paper does not show that they remain valid when the focal variable is continuous or when the interaction term is allowed to depend on covariates; this could affect the practical utility of the bounds in the NHANES application where uric acid is continuous.

minor comments (3)

[Abstract] Abstract: the phrase 'model-robust at two levels' is used but the two levels (estimand robustness across scenarios and estimator double robustness) are not distinguished in the abstract itself; a single clarifying sentence would help readers.
[§5] §5 (simulations): coverage probabilities and interval widths for the proposed doubly-robust estimator are not reported; adding these would strengthen the finite-sample evidence for the efficiency claim.
[Notation] Notation: the symbol for the interventional direct effect is introduced without an explicit contrast to the natural direct effect in the main text; a short comparison table would reduce reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments, which have helped us improve the clarity of the manuscript. We respond to each major comment below and indicate the revisions that will be made.

read point-by-point responses

Referee: [§3.2] §3.2, definition of the outcome-free stochastic direct effects class: the uniqueness claim is load-bearing for the central recommendation, yet the precise characterization of the class (which functionals are included or excluded) is not fully spelled out before the uniqueness theorem; without an explicit enumeration or axiomatic description it is difficult to verify that no other functional satisfies the same dual-interpretability property.

Authors: We appreciate the referee drawing attention to this issue. The class is introduced in the text of §3.2 immediately before the uniqueness result, but we agree that a more formal and self-contained characterization would strengthen verifiability. In the revision we will insert, prior to the theorem, an explicit axiomatic definition of the outcome-free stochastic direct effects class (functionals expressible solely in terms of the conditional law of the outcome given the focal variable and covariates, under stochastic interventions that do not involve the outcome regression) together with a short enumeration of included and excluded members. This addition will make the scope of the uniqueness claim transparent without changing any derivations or results. revision: yes
Referee: [§4.1, Eq. (12)] §4.1, Eq. (12) (sensitivity bounds): the bounds are derived under the no-additive-interaction condition, but the paper does not show that they remain valid when the focal variable is continuous or when the interaction term is allowed to depend on covariates; this could affect the practical utility of the bounds in the NHANES application where uric acid is continuous.

Authors: The referee is correct that the derivation of the sensitivity bounds in Equation (12) is presented under the no-additive-interaction assumption with a binary focal variable and no covariate dependence in the interaction term. The NHANES illustration treats uric acid as continuous and applies the bounds after a simple discretization for expository purposes. We will revise §4.1 to state these scope conditions explicitly, add a brief discussion of the resulting limitations for continuous exposures, and note that practitioners may adapt the bounds via discretization or local approximations when the focal variable is continuous. A full general derivation for arbitrary continuous variables and covariate-dependent interactions lies beyond the present scope but can be flagged as a natural direction for follow-up work. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central derivation identifies a no-additive-interaction condition under which ATE and NDE coincide, then constructs an alternative observed-data functional that equals the ATE under a confounder interpretation and an interventional direct effect under a mediator interpretation. Uniqueness is asserted only inside an explicitly delimited class of outcome-free stochastic direct effects, with the functional derived from matching the two causal interpretations rather than by re-labeling fitted parameters or importing a self-citation theorem. The efficient influence function and doubly-robust estimator follow standard semiparametric constructions without reducing the target estimand to its own inputs. No load-bearing self-citation, ansatz smuggling, or self-definitional loop is present in the argument structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard causal identification assumptions plus the newly introduced no-additive-interaction condition and the definition of the outcome-free stochastic direct effect class.

axioms (2)

domain assumption Standard causal assumptions (consistency, positivity, no unmeasured confounding) for identification of ATE and NDE
Required for any causal interpretation of the estimands in observational data.
ad hoc to paper No additive interaction between treatment and the focal variable on the outcome scale
The condition identified in the paper under which ATE and NDE coincide.

pith-pipeline@v0.9.0 · 5582 in / 1352 out tokens · 39150 ms · 2026-05-10T18:18:18.728994+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4 (Uniqueness of ψ(P)). Let P↦ν_P(·|x) be outcome-free... then ν_P(·|X)=F_{W|X}(·|X) P-almost surely. Consequently Ψ_ν(P)=E_P{Δ(W,X)}=ψ(P).
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3 (The model-robust direct effect). (i) under C ... ψ(P)=θ_ATE; (ii) under M ... ψ(P)=θ_IDE.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

National health and nutrition examination survey 2015–2016: Standard biochemistry profile (biopro i) data documentation, codebook, and fre- quencies, 2024a

Centers for Disease Control and Prevention. National health and nutrition examination survey 2015–2016: Standard biochemistry profile (biopro i) data documentation, codebook, and fre- quencies, 2024a. URLhttps://wwwn.cdc.gov/Nchs/Data/Nhanes/Public/2015/DataFiles/ BIOPRO_I.htm. Centers for Disease Control and Prevention. National health and nutrition exam...

work page 2015
[2]

Double/debiased machine learning for treatment and structural parameters

ISSN 1368-4221. URLhttps: //doi.org/10.1111/ectj.12097. Radhika Dhingra, Andrea Winquist, Lyndsey A. Darrow, Mitchel Klein, and Kyle Steenland. A study of reverse causation: Examining the associations of perfluorooctanoic acid serum levels with two outcomes.Environmental Health Perspectives, 125(3):416–421,

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doi: 10.1093/ehjopen/oeaf070

ISSN 2752-4191. doi: 10.1093/ehjopen/oeaf070. URLhttps://doi.org/10.1093/ehjopen/oeaf070. Isabel R. Fulcher, Ilya Shpitser, Stella Marealle, and Eric J. Tchetgen Tchetgen. Robust inference on population indirect causal effects: the generalized front door criterion.Journal of the Royal Statistical Society: Series B (Statistical Methodology), 82(1):199–214,

work page doi:10.1093/ehjopen/oeaf070
[4]

Perfluoroalkyl substances (pfass) exposure and kidney damage: Causal interpreta- tion using the us 2003–2018 national health and nutrition examination survey (nhanes) datasets

Jinyoung Moon. Perfluoroalkyl substances (pfass) exposure and kidney damage: Causal interpreta- tion using the us 2003–2018 national health and nutrition examination survey (nhanes) datasets. Environmental Pollution, 288:117707,

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Annals of statistics , author =

doi: 10.1214/12-AOS990. Anastasios A. Tsiatis.Semiparametric Theory and Missing Data. Springer Series in Statistics. Springer, New York, NY,

work page doi:10.1214/12-aos990
[6]

Therefore the test that rejects whenp max < αhas size at mostαunder the union null

holds, the same argument usingp M shows P(pmax < α)≤α. Therefore the test that rejects whenp max < αhas size at mostαunder the union null. D Survey-Weighted Implementation and PSU Bootstrap for the NHANES Application The theory in Sections 2–4 is developed for independent and identically distributed observations from an observed-data lawP. NHANES, however...

work page 2013

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National health and nutrition examination survey 2015–2016: Standard biochemistry profile (biopro i) data documentation, codebook, and fre- quencies, 2024a

Centers for Disease Control and Prevention. National health and nutrition examination survey 2015–2016: Standard biochemistry profile (biopro i) data documentation, codebook, and fre- quencies, 2024a. URLhttps://wwwn.cdc.gov/Nchs/Data/Nhanes/Public/2015/DataFiles/ BIOPRO_I.htm. Centers for Disease Control and Prevention. National health and nutrition exam...

work page 2015

[2] [2]

Double/debiased machine learning for treatment and structural parameters

ISSN 1368-4221. URLhttps: //doi.org/10.1111/ectj.12097. Radhika Dhingra, Andrea Winquist, Lyndsey A. Darrow, Mitchel Klein, and Kyle Steenland. A study of reverse causation: Examining the associations of perfluorooctanoic acid serum levels with two outcomes.Environmental Health Perspectives, 125(3):416–421,

work page doi:10.1111/ectj.12097

[3] [3]

doi: 10.1093/ehjopen/oeaf070

ISSN 2752-4191. doi: 10.1093/ehjopen/oeaf070. URLhttps://doi.org/10.1093/ehjopen/oeaf070. Isabel R. Fulcher, Ilya Shpitser, Stella Marealle, and Eric J. Tchetgen Tchetgen. Robust inference on population indirect causal effects: the generalized front door criterion.Journal of the Royal Statistical Society: Series B (Statistical Methodology), 82(1):199–214,

work page doi:10.1093/ehjopen/oeaf070

[4] [4]

Perfluoroalkyl substances (pfass) exposure and kidney damage: Causal interpreta- tion using the us 2003–2018 national health and nutrition examination survey (nhanes) datasets

Jinyoung Moon. Perfluoroalkyl substances (pfass) exposure and kidney damage: Causal interpreta- tion using the us 2003–2018 national health and nutrition examination survey (nhanes) datasets. Environmental Pollution, 288:117707,

work page 2003

[5] [5]

Annals of statistics , author =

doi: 10.1214/12-AOS990. Anastasios A. Tsiatis.Semiparametric Theory and Missing Data. Springer Series in Statistics. Springer, New York, NY,

work page doi:10.1214/12-aos990

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Therefore the test that rejects whenp max < αhas size at mostαunder the union null

holds, the same argument usingp M shows P(pmax < α)≤α. Therefore the test that rejects whenp max < αhas size at mostαunder the union null. D Survey-Weighted Implementation and PSU Bootstrap for the NHANES Application The theory in Sections 2–4 is developed for independent and identically distributed observations from an observed-data lawP. NHANES, however...

work page 2013