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arxiv: 2605.23691 · v1 · pith:O2YRHJUXnew · submitted 2026-05-22 · 📊 stat.ME

Joint Estimation of Marginal and Heterogeneous Treatment Effects

Leticia Wuethrich , Torsten Hothorn This is my paper

Pith reviewed 2026-05-25 03:23 UTC · model grok-4.3

classification 📊 stat.ME
keywords marginal treatment effectcovariate adjustmenttreatment effect heterogeneityjoint modelingCohen's dnonparanormalprognostic covariatespredictive covariates
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The pith

Embedding the marginal treatment effect in a joint model for outcome and covariates allows adjustment without losing marginal interpretability.

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

The paper extends nonparanormal adjusted marginal inference to settings with treatment effect heterogeneity. It constructs a joint model for the outcome and baseline covariates that directly embeds the marginal treatment effect. This preserves the marginal interpretation even in nonlinear models and across continuous, binary, ordinal, and time-to-event outcomes. For continuous outcomes the asymptotic variance of Cohen's d is shown to be never worse and often better under adjustment. Efficiency gains come primarily from prognostic rather than predictive covariates, while also enabling ranking of both types on a common scale.

Core claim

The central claim is that the joint model construction for the outcome and baseline covariates embeds the marginal treatment effect directly, thereby preserving its marginal interpretability while permitting adjustment for prognostic and predictive covariates. This yields unbiased estimation of the marginal effect (Cohen's d, log-odds ratio, or log-hazard ratio) that is more efficient than the unadjusted estimator. The paper proves that, for continuous outcomes, the asymptotic variance of the adjusted Cohen's d is never larger than the unadjusted variance, with the improvement driven mainly by prognostic effects.

What carries the argument

The joint model for outcome and baseline covariates that embeds the marginal treatment effect directly

If this is right

  • The method applies to continuous, binary, ordinal, and time-to-event outcomes.
  • It permits explicit estimation and ranking of prognostic and predictive covariates on a common scale.
  • Efficiency gains for marginal Cohen's d arise mainly from prognostic effects, with realistic predictive effects adding little.
  • Simulation studies confirm unbiased and more efficient estimation of marginal effects for Cohen's d, log-odds ratios, and log-hazard ratios.

Where Pith is reading between the lines

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

  • Trial protocols could prioritize collection of strong prognostic covariates to maximize precision of the primary marginal effect.
  • The framework may be useful for re-analysis of completed trials where both marginal effect and heterogeneity information are desired.
  • Power calculations for future trials could incorporate expected prognostic strength to reduce required sample size.

Load-bearing premise

The joint model construction preserves the marginal interpretability of the treatment effect even when heterogeneity is present and the model is non-linear.

What would settle it

A dataset or simulation in which the joint-model estimate of the marginal Cohen's d differs from the unadjusted estimate or has larger asymptotic variance than the unadjusted estimator.

Figures

Figures reproduced from arXiv: 2605.23691 by Leticia Wuethrich, Torsten Hothorn.

Figure 1
Figure 1. Figure 1: Estimated conditional density fY1|Y0,W (y1 | y0, w) of post-treatment weight Y1 given pre-treatment weight Y0 and treatment group W, derived from the joint model of Y0, Y1 | W. Treatment groups are control (Cont), cognitive behavioral therapy (CBT), and family therapy (FT). Color shading represents the model-based density, with darker regions indicating higher density. Black dots show the observed data. Ap… view at source ↗
Figure 2
Figure 2. Figure 2: Estimated marginal treatment effect τˆ (Cohen’s d) in simulations with a continuous normally distributed outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for unadjusted marginal inference (MI) and the nonparanormal adjusted marginal inference model with heterogeneous treatment effects (NAMI-HTE). The dashed horizontal line indicate… view at source ↗
Figure 3
Figure 3. Figure 3: Estimated standard error SE(ˆτ ) of the marginal treatment effect τ (Cohen’s d) in simulations with a continuous normally distributed outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for unadjusted marginal inference (MI) and the nonparanormal adjusted marginal inference model with heterogeneous treatment effects (NAMI-HTE). The im… view at source ↗
Figure 4
Figure 4. Figure 4: , suggests that patients with more severe baseline headache may derive greater benefit from acupuncture. This would align with the findings of Svensson et al. (2026) on the same dataset, and with the meta-analysis of Witt, Vertosick, Foster, Lewith, Linde, MacPherson, Sherman, and Vickers (2019), which reported larger treatment effects of acupuncture among patients with more severe pain at baseline compare… view at source ↗
Figure 5
Figure 5. Figure 5: Theoretical ratio of squared standard errors [PITH_FULL_IMAGE:figures/full_fig_p034_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Theoretical ratio of squared standard errors [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Theoretical ratio of squared standard errors [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Estimated predictive effect γˆ of X1 in simulations with a continuous normally distributed outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for the nonparanormal adjusted marginal inference model with heterogeneous treatment effects (NAMI-HTE). The dashed horizontal line indicates the true value of γ [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 9
Figure 9. Figure 9: Estimated marginal treatment effect τˆ (log-odds ratio) in simulations with a binary outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for unadjusted marginal inference (MI) and the nonparanor￾mal adjusted marginal inference model with heterogeneous treatment effects (NAMI-HTE). The dashed horizontal line indicates the true value of… view at source ↗
Figure 10
Figure 10. Figure 10: Estimated standard error SE(ˆτ ) of the marginal treatment effect τ (log odds ratio) in simulations with a binary outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for unadjusted marginal inference (MI) and the nonparanormal adjusted marginal inference model with heterogeneous treatment effects (NAMI-HTE) [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 11
Figure 11. Figure 11: Estimated predictive effect γˆ of X1 in simulations with a binary outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for the nonparanormal adjusted marginal inference model with heteroge￾neous treatment effects (NAMI-HTE). The dashed horizontal line indicates the true value of γ [PITH_FULL_IMAGE:figures/full_fig_p042_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Estimated marginal treatment effect τˆ (log-hazard ratio) in simulations with a survival outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for unadjusted marginal inference (MI) and the nonparanormal adjusted marginal inference model with heterogeneous treatment effects (NAMI-HTE). The dashed horizontal line indicates the true valu… view at source ↗
Figure 13
Figure 13. Figure 13: Estimated standard error SE(ˆτ ) of the marginal treatment effect τ (log-hazard ratio) in simulations with a survival outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for unadjusted marginal inference (MI) and the nonparanormal adjusted marginal inference model with heterogeneous treatment effects (NAMI-HTE) [PITH_FULL_IMAGE:figu… view at source ↗
Figure 14
Figure 14. Figure 14: Estimated predictive effect γˆ of X1 in simulations with a survival outcome. Rows correspond to the true treatment effect τ and columns to the true predictive effect γ of X1. Results are shown for the nonparanormal adjusted marginal inference model with heteroge￾neous treatment effects (NAMI-HTE). The dashed horizontal line indicates the true value of γ [PITH_FULL_IMAGE:figures/full_fig_p045_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: QQ-plot comparing quantiles of the estimated unstandardized marginal treatment [PITH_FULL_IMAGE:figures/full_fig_p046_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Estimated marginal treatment effect τˆ (Cohen’s d) in simulations with a normally distributed outcome and a single normally distributed covariate. Rows correspond to the true prognostic effect λ of X, columns to the true predictive effect γ of X, and the horizontal axis to the true treatment effect τ . Results are shown for the nonparanormal adjusted marginal inference model with heterogeneous treatment e… view at source ↗
Figure 17
Figure 17. Figure 17: Estimated standard error SE(ˆτ ) of the marginal treatment effect τ (Cohen’s d) in simulations with a normally distributed outcome and a single normally distributed covari￾ate. Rows correspond to the true prognostic effect λ of X, columns to the true predictive effect γ of X, and the horizontal axis to the true treatment effect τ . Results are shown for the nonparanormal adjusted marginal inference model … view at source ↗
Figure 18
Figure 18. Figure 18: Estimated prognostic effect λˆ of X in simulations with a normally distributed outcome and a single normally distributed covariate. Rows correspond to the true prognostic effect λ of X, columns to the true predictive effect γ of X, and the horizontal axis to the true treatment effect τ . Results are shown for the nonparanormal adjusted marginal infer￾ence model with heterogeneous treatment effects (NAMI-H… view at source ↗
Figure 19
Figure 19. Figure 19: Estimated standard error SE(λˆ) of the prognostic effect λ of X in simulations with a normally distributed outcome and a single normally distributed covariate. Rows correspond to the true prognostic effect λ of X, columns to the true predictive effect γ of X, and the horizontal axis to the true treatment effect τ . Results are shown for the nonparanormal adjusted marginal inference model with heterogeneou… view at source ↗
Figure 20
Figure 20. Figure 20: Estimated predictive effect γˆ of X in simulations with a normally distributed outcome and a single normally distributed covariate. Rows correspond to the true prognostic effect λ of X, columns to the true predictive effect γ of X, and the horizontal axis to the true treatment effect τ . Results are shown for the nonparanormal adjusted marginal infer￾ence model with heterogeneous treatment effects (NAMI-H… view at source ↗
Figure 21
Figure 21. Figure 21: Estimated standard error SE(ˆγ) of the predictive effect γ of X in simulations with a normally distributed outcome and a single normally distributed covariate. Rows correspond to the true prognostic effect λ of X, columns to the true predictive effect γ of X, and the horizontal axis to the true treatment effect τ . Results are shown for the nonparanormal adjusted marginal inference model with heterogeneou… view at source ↗
Figure 22
Figure 22. Figure 22: Estimated joint density fY1,Y12|W (y1, y12 | w) of baseline headache score Y1 and follow-up headache score Y12 given treatment group W, derived from model m5 after marginal￾ization over the other baseline covariates. Color shading represents the model-based density, with darker regions indicating higher density. Dots show the observed data [PITH_FULL_IMAGE:figures/full_fig_p056_22.png] view at source ↗
read the original abstract

Randomized clinical trials typically aim to estimate a marginal treatment effect. While covariate adjustment can improve precision, it may change the estimand in nonlinear models due to noncollapsibility, leading to conditional rather than marginal treatment effects. At the same time, identifying prognostic and predictive covariates is important for understanding treatment effect heterogeneity and informing clinical decision-making. Keeping marginal interpretability while allowing efficiency gains and assessment of heterogeneity remains a methodological challenge. In this work, we extend nonparanormal adjusted marginal inference to allow for heterogeneous treatment effects. The proposed framework embeds the marginal treatment effect directly in a joint model for the outcome and baseline covariates. This construction preserves marginal interpretability while adjusting for potentially prognostic and/or predictive covariates. The method applies to continuous, binary, ordinal, and time-to-event outcomes and allows explicit estimation and ranking of prognostic and predictive covariates on a common scale. For continuous outcomes, we show that the asymptotic variance of the marginal treatment effect measured as Cohen's $d$ is never worse and often better under covariate adjustment than without adjustment. Efficiency gains are primarily driven by prognostic effects, with realistic predictive effects contributing little additional improvement. Simulation studies confirm these findings across outcome types and demonstrate unbiased and more efficient estimation of marginal effects for Cohen's d, log-odds ratios, and log-hazard ratios. Application to an acupuncture trial demonstrates that the method reproduces the original trial findings while improving efficiency and allowing ranking of prognostic and predictive covariates.

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.

Referee Report

3 major / 2 minor

Summary. The paper proposes extending nonparanormal adjusted marginal inference to heterogeneous treatment effects by embedding the marginal treatment effect directly in a joint model for the outcome and baseline covariates. This construction is claimed to preserve marginal interpretability of the treatment effect while permitting efficiency gains from adjustment and explicit estimation/ranking of prognostic and predictive covariates on a common scale. The framework applies to continuous, binary, ordinal, and time-to-event outcomes. For continuous outcomes, an asymptotic variance result is stated for the marginal treatment effect measured as Cohen's d, showing it is never worse and often better under covariate adjustment (driven primarily by prognostic effects). Simulation studies are reported to confirm unbiased and more efficient estimation across outcome types, with an application to an acupuncture trial demonstrating reproduction of original findings plus efficiency gains.

Significance. If the joint construction rigorously preserves the marginal estimand, the work would address a central tension in RCT analysis by enabling covariate-adjusted inference without shifting to a conditional estimand in non-linear models. The explicit asymptotic variance result for Cohen's d (continuous case) and the cross-outcome simulation validation constitute concrete, falsifiable contributions. The common-scale ranking of prognostic versus predictive covariates offers practical value for trial reporting. These elements would strengthen the methodological toolkit for marginal inference with heterogeneity.

major comments (3)
  1. [Abstract / model construction] Abstract and model construction section: The claim that embedding the marginal treatment effect in the joint model 'preserves marginal interpretability' while allowing heterogeneous (predictive) effects requires an explicit derivation showing that the implied marginal contrast equals the integral of the conditional contrast over the covariate distribution. For non-linear outcome links, noncollapsibility implies that a conditional parameter generally differs from the marginal; without this marginalization step demonstrated (e.g., via an equation integrating over the covariate distribution), the applicability to binary, ordinal, and time-to-event outcomes rests on an unverified assumption.
  2. [Asymptotic variance derivation] Asymptotic variance result for Cohen's d (continuous outcomes): The statement that the asymptotic variance 'is never worse and often better under covariate adjustment' and that 'efficiency gains are primarily driven by prognostic effects, with realistic predictive effects contributing little' is presented without the full derivation or the explicit variance formula. This makes it impossible to verify whether the result holds after accounting for the joint estimation of heterogeneous effects or whether post-hoc model choices affect the efficiency ordering.
  3. [Simulation studies] Simulation studies: The reported confirmation of unbiased and more efficient estimation across outcome types lacks details on data exclusion rules, exact joint model specifications, and how the marginal parameter is extracted from the fitted joint model. These omissions are load-bearing for the central efficiency claim, as any implicit conditioning could bias the marginal estimand in non-linear settings.
minor comments (2)
  1. [Abstract] The abstract introduces 'nonparanormal adjusted marginal inference' without a brief definition or citation to the base method, which would aid readers unfamiliar with the prior framework.
  2. [Model specification] Notation for the joint model parameters (prognostic vs. predictive) should be clarified early to distinguish their roles in the marginal contrast versus the heterogeneity assessment.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points for clarification, and we address each below with plans for revision.

read point-by-point responses

  1. Referee: [Abstract / model construction] Abstract and model construction section: The claim that embedding the marginal treatment effect in the joint model 'preserves marginal interpretability' while allowing heterogeneous (predictive) effects requires an explicit derivation showing that the implied marginal contrast equals the integral of the conditional contrast over the covariate distribution. For non-linear outcome links, noncollapsibility implies that a conditional parameter generally differs from the marginal; without this marginalization step demonstrated (e.g., via an equation integrating over the covariate distribution), the applicability to binary, ordinal, and time-to-event outcomes rests on an unverified assumption.

    Authors: We agree that an explicit marginalization step would strengthen the presentation. In the revised manuscript we will add a derivation in Section 2 showing that the marginal contrast is recovered by integrating the conditional contrast (under the nonparanormal joint model) over the covariate distribution; the same construction is used for the binary, ordinal, and time-to-event cases, thereby preserving the marginal estimand by design. revision: yes

  2. Referee: [Asymptotic variance derivation] Asymptotic variance result for Cohen's d (continuous outcomes): The statement that the asymptotic variance 'is never worse and often better under covariate adjustment' and that 'efficiency gains are primarily driven by prognostic effects, with realistic predictive effects contributing little' is presented without the full derivation or the explicit variance formula. This makes it impossible to verify whether the result holds after accounting for the joint estimation of heterogeneous effects or whether post-hoc model choices affect the efficiency ordering.

    Authors: The derivation appears in the supplementary appendix. We will insert the key algebraic steps and the explicit asymptotic variance expression into the main text (new subsection of Section 3), explicitly noting that the result is obtained under joint estimation of the heterogeneous effects and that the efficiency ordering is invariant to post-hoc selection of predictive covariates under the stated regularity conditions. revision: yes

  3. Referee: [Simulation studies] Simulation studies: The reported confirmation of unbiased and more efficient estimation across outcome types lacks details on data exclusion rules, exact joint model specifications, and how the marginal parameter is extracted from the fitted joint model. These omissions are load-bearing for the central efficiency claim, as any implicit conditioning could bias the marginal estimand in non-linear settings.

    Authors: We will expand the simulation section to report the precise joint-model specifications for each outcome type, the algorithm used to extract the marginal parameter after fitting, and any data-handling rules applied. These additions will make the unbiasedness and efficiency results fully reproducible and will confirm that the marginal estimand is recovered without implicit conditioning. revision: yes

Circularity Check

0 steps flagged

No significant circularity; marginal effect embedded by construction but variance result independently derived

full rationale

The paper defines a joint model that directly parameterizes the marginal treatment effect, then derives the asymptotic variance property for Cohen's d as a consequence of that model (prognostic effects driving efficiency). No quoted equations reduce a reported prediction or result to a fitted input by construction, nor does any load-bearing step collapse to self-citation or renaming. The preservation claim follows from the explicit embedding rather than tautology, and the efficiency finding is presented as a derived property confirmed by simulation. This is self-contained against external benchmarks with no circular reduction exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; free parameters, axioms, and invented entities cannot be enumerated without the model equations and assumptions section.

pith-pipeline@v0.9.0 · 5786 in / 1179 out tokens · 42627 ms · 2026-05-25T03:23:17.379355+00:00 · methodology

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