arxiv: 2511.20985 · v4 · submitted 2025-11-26 · 📊 stat.ME

Two-stage Estimation for Causal Inference Involving a Semi-continuous Exposure

Xiaoya Wang , Richard J. Cook , Yeying Zhu , Tugba Akkaya-Hocagil , R. Colin Carter , Sandra W. Jacobson , Joseph L. Jacobson , Louise M. Ryan This is my paper

Pith reviewed 2026-05-17 05:29 UTC · model grok-4.3

classification 📊 stat.ME

keywords causal inferencesemi-continuous exposuretwo-stage estimationpropensity scoremarginal structural modeldose-responseexposure status

0 comments

The pith

A two-stage estimator separates the causal effect of exposure status from the dose-response among the exposed for semi-continuous variables.

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

The paper develops a framework for causal inference when the exposure variable has a point mass at zero together with a continuous positive component. It combines a two-part propensity score with a marginal structural model so that the effect of being exposed at a reference level can be estimated separately from the effect of increasing the exposure level among those who are exposed. A sequential two-stage procedure targets these quantities in turn and the authors derive consistency, asymptotic normality, and the limiting behavior under misspecification of the propensity models. This matters for observational studies in which exposures such as alcohol intake, pollutant levels, or medication doses are zero for many individuals yet vary continuously for others. A reader who accepts the framework gains the ability to ask distinct scientific questions about initiation versus intensity without forcing the exposure into a purely binary or purely continuous mold.

Core claim

For semi-continuous exposures the authors introduce a two-part propensity structure—one binary model for exposure status and one conditional model for the positive exposure level—embedded in a marginal structural model that identifies both the causal effect of exposure status at a reference dose and the causal dose-response function among the exposed. A two-stage estimation procedure sequentially estimates these quantities, permits flexible choice of propensity methods in the second stage, and yields estimators that are consistent and asymptotically normal when the propensity models are correct while converging to well-characterized limits under misspecification.

What carries the argument

Two-part propensity score model (binary status component plus conditional exposure-level component) inside a marginal structural model that disentangles status and dose effects.

If this is right

The estimators remain consistent and asymptotically normal when the two-part propensity models are correctly specified.
Under misspecification the estimators converge to explicit limiting values that can be interpreted.
The procedure allows different propensity methods to be used in the second stage without losing the overall consistency property.
Applied to prenatal alcohol exposure, the method can answer separate questions about whether any exposure occurred and about the effect of higher amounts among exposed pregnancies.

Where Pith is reading between the lines

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

The same two-part structure could be applied to other zero-inflated or count-valued exposures once suitable conditional models are chosen.
Replacing parametric propensity models with machine-learning alternatives in the second stage would be a direct extension that preserves the paper's asymptotic guarantees.
Policy analyses could use the separated parameters to evaluate interventions aimed only at preventing initiation versus those aimed at reducing intensity among users.

Load-bearing premise

The two-part propensity score models are either correctly specified or their misspecification leaves the target causal parameters unbiased within the limits the paper characterizes.

What would settle it

A simulation in which the propensity models are misspecified in a manner outside the paper's robustness analysis and the resulting estimators fail to recover the known true causal effects.

Figures

Figures reproduced from arXiv: 2511.20985 by Joseph L. Jacobson, Louise M. Ryan, R. Colin Carter, Richard J. Cook, Sandra W. Jacobson, Tugba Akkaya-Hocagil, Xiaoya Wang, Yeying Zhu.

read the original abstract

Methods for causal inference are well developed for binary and continuous exposures, but in many settings, the exposure has a substantial mass at zero-such exposures are called semi-continuous. We propose a general causal framework for such semi-continuous exposures, together with a novel two-stage estimation strategy. A two-part propensity structure is introduced for the semi-continuous exposure, with one component for exposure status (exposed vs unexposed) and another for the exposure level among those exposed, and incorporates both into a marginal structural model that disentangles the effects of exposure status and dose. The two-stage procedure sequentially targets the causal dose-response among exposed individuals and the causal effect of exposure status at a reference dose, allowing flexibility in the choice of propensity score methods in the second stage. We establish consistency and asymptotic normality for the resulting estimators, and characterise their limiting values under misspecification of the propensity score models. Simulation studies evaluate finite sample performance and robustness, and an application to a study of prenatal alcohol exposure and child cognition demonstrates how the proposed methods can be used to address a range of scientific questions about both exposure status and exposure intensity.

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 gives a workable two-stage procedure to separate the causal effect of exposure status from the dose-response among the exposed for semi-continuous data inside marginal structural models, with explicit asymptotics including under misspecification.

read the letter

The main point to take away is that this work gives a clear way to handle causal questions for exposures that are zero for many people but positive for others, by using a two-part model for the propensity and a two-stage estimation that first goes after the dose response in the exposed group and then the effect of exposure at a reference level. The new piece is the two-stage targeting strategy inside the marginal structural model. It allows flexibility in how you model the propensity in the second stage. They prove consistency and asymptotic normality for the estimators and work out what the estimates converge to if the propensity models are not correct. That explicit characterization under misspecification is a solid addition because it tells you the consequences rather than leaving it vague. The simulations are useful for seeing how the method behaves in finite samples and under some misspecification scenarios. The application to prenatal alcohol exposure and child cognition is a good illustration of the kind of scientific question this can answer, like whether any exposure has an effect separate from the amount. On the downside, the application does not include any checks on how well the two-part propensity models fit the data or any balance diagnostics after weighting. The stress test is right on this. Even though the paper gives general results for when the models are wrong, without those checks in the example it is hard to judge how much the reported effects might be influenced by model limitations in this particular study. This is a real but not overwhelming gap. The paper is aimed at researchers in biostatistics and epidemiology who need to analyze observational data with semi-continuous exposures. Someone looking for methods to disentangle binary exposure effects from continuous dose effects would get practical value from it. The thinking is clear and the engagement with causal identification and marginal structural models is honest. There are no obvious internal contradictions. I would bring this to a reading group to talk through the two-stage procedure. I would not cite it in my own work unless I had a similar problem. It should go to peer review. My recommendation is to send it for serious refereeing, but request that the authors add model diagnostics and perhaps sensitivity analyses to the application section.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a general causal framework for semi-continuous exposures that incorporates a two-part propensity score structure (one component for exposure status and one for dose intensity among the exposed) into a marginal structural model separating the effects of status and dose. It introduces a novel two-stage estimation procedure, establishes consistency and asymptotic normality of the resulting estimators, and explicitly characterizes their limiting values under misspecification of the propensity models. The work includes simulation studies assessing finite-sample performance and robustness, plus a real-data application to prenatal alcohol exposure and child cognition.

Significance. If the results hold, the paper offers a practically useful extension of causal methods to a common but under-served exposure type, with theoretical guarantees that include robustness characterizations and a flexible two-stage procedure. The explicit limiting-value analysis under misspecification and the provision of simulation studies are particular strengths that support reliable use in applications where propensity models may be imperfect.

major comments (1)

[§6] §6 (Application): No propensity-score model diagnostics, balance checks, or sensitivity analyses are reported for either component of the two-part propensity model in the prenatal alcohol exposure analysis. Because the central claim includes a demonstration that the method addresses scientific questions about exposure status and intensity in this specific dataset, and because the paper derives limiting values under misspecification, the lack of empirical assessment of model adequacy in the observed data is load-bearing for interpreting the reported causal effects.

minor comments (2)

[§3 and §4] Clarify in the main text how the reference dose is chosen in the two-stage procedure and whether results are sensitive to that choice; a brief sensitivity table would help.
[§5] In the simulation section, label the panels or legends to distinguish clearly between the binary-status effect and the dose-response effect among the exposed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments and for recognizing the strengths of our proposed two-stage estimator and theoretical results for semi-continuous exposures. We address the single major comment below and will incorporate the requested additions in the revised manuscript.

read point-by-point responses

Referee: [§6] §6 (Application): No propensity-score model diagnostics, balance checks, or sensitivity analyses are reported for either component of the two-part propensity model in the prenatal alcohol exposure analysis. Because the central claim includes a demonstration that the method addresses scientific questions about exposure status and intensity in this specific dataset, and because the paper derives limiting values under misspecification, the lack of empirical assessment of model adequacy in the observed data is load-bearing for interpreting the reported causal effects.

Authors: We agree that the application would be strengthened by explicit diagnostics. In the revised Section 6 we will add: (i) overlap and positivity diagnostics for both the binary exposure-status model and the continuous dose model among the exposed; (ii) covariate balance tables (standardized mean differences) before and after weighting for each component; (iii) goodness-of-fit summaries (e.g., Hosmer-Lemeshow for the logistic part and residual diagnostics for the linear part); and (iv) sensitivity analyses that vary the specification of each propensity component and report the resulting changes in the estimated status and dose effects. These additions will directly support interpretation of the reported causal contrasts and will be cross-referenced to the limiting-value characterizations already derived in the theoretical sections. revision: yes

Circularity Check

0 steps flagged

No circularity: estimators derived from standard causal identification plus explicit two-part modeling

full rationale

The paper introduces a two-part propensity score for semi-continuous exposures and a two-stage procedure that sequentially targets dose-response among the exposed and the effect of exposure status. Consistency, asymptotic normality, and explicit limiting values under misspecification are derived from standard marginal structural model identification results together with regularity conditions on the estimating equations. These steps are self-contained against external benchmarks (causal identification theory and M-estimation), with no reduction of the target parameters to fitted inputs by construction, no load-bearing self-citations, and no ansatz smuggled via prior work. The application and simulations serve as illustration rather than the source of the claimed properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard causal assumptions plus modeling choices for the semi-continuous exposure; no new free parameters or invented entities are introduced beyond conventional propensity and outcome models.

axioms (2)

domain assumption No unmeasured confounding between exposure and outcome conditional on observed covariates.
Invoked to identify causal effects in the marginal structural model.
domain assumption Positivity of the two-part propensity scores.
Required for the two-stage estimators to be well-defined.

pith-pipeline@v0.9.0 · 5525 in / 1287 out tokens · 51469 ms · 2026-05-17T05:29:05.831639+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.

two-part propensity structure … marginal structural model … two-stage procedure … consistency and asymptotic normality … limiting values under misspecification
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Assumptions 1-4 (SUTVA, consistency, ignorability, positivity) … MSM Y(A,D)=ψ0+ψ11A+ψ12A(D−c)+Q

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

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