Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

A Unified Framework for the Transportability of Population-Level Causal Measures

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that a broad class of population causal effect measures—absolute and relative, collapsible and not—can be identified and consistently estimated in a target population under covariate shift, and provides estimators with…

desk verdict Useful unification and a correct identification formula for non-collapsible measures, but the Section 4.2 influence function is missing a chain-rule factor that breaks the efficiency and double-robustness claims. read the letter →

arxiv 2505.13104 v1 pith:FWPZNQGG submitted 2025-05-19 stat.ME

Ahmed Boughdiri , Clément Berenfeld , Julie Josse (PREMEDICAL) , Erwan Scornet This is my paper · ORCID
classification stat.ME
keywords transportabilitygeneralizabilityfirst-momentcausalmeasurescovariateshifteffectmeasureexchangeabilitynon-collapsibledoublyrobustestimationsemiparametricefficiency
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that one unified machinery can transport almost any commonly reported causal effect measure—not just the risk difference but also risk ratios, odds ratios, number needed to treat, and similar quantities—from a randomized trial to a different target population. Under covariate shift, it identifies the target estimand $\Phi(E_T[Y(1)], E_T[Y(0)])$ whenever either exchangeability in mean holds, or the weaker exchangeability in effect measure holds with access to target control outcomes. It then builds weighting, regression, one-step, and estimating-equation estimators, proving asymptotic normality for the classical ones and double robustness for estimating-equation estimators across the whole class. The practical payoff is that clinical and policy audiences who need several effect measures on different scales could use a single framework instead of per-measure ad hoc methods. A notable corollary is that non-collapsible measures such as the odds ratio become transportable under the weaker assumption, a result the paper says had not been derived before.

What carries the argument

The load-bearing object is the first-moment causal measure, a functional $\Phi(\psi_1,\psi_0)$ of the two marginal potential-outcome means, paired with its effect function $\Gamma(\cdot,\psi_0)$, which inverts $\Phi$ at a fixed baseline. The effect function does the heavy lifting: under exchangeability in effect measure, the paper averages $\Gamma(\tau_{S,\Phi}(X),\mu_T^{(0)}(X))$ over the target covariate distribution, converting a conditional effect on one scale into a conditional potential-outcome mean that can be averaged despite non-collapsibility. The density ratio $r(X)=P_T(X)/P_S(X)$ carries the covariate shift in the reweighting estimators, and the efficient influence function (EIF), computed by the chain rule through $\Phi$ and $\Gamma$, generates both the one-step and estimating-equation estimators and their double-robustness property.

What would settle it

Simulate a binary outcome with source-specific baseline outcome levels but a constant conditional odds ratio across source and target, so Assumption 3 fails while Assumption 5 holds, and compare the proposed Gamma-formula and estimating-equation estimators for the target odds ratio against the true value using correct nuisance functions; if they are biased even with correct $r$ and $\mu_T^{(0)}$, the identification formula fails, whereas unbiasedness while standard reweighting fails would support the paper's claim.

Watch

Extended reading notes

Core claim

The central claim is that for any first-moment population causal measure $\tau_{T,\Phi} = \Phi(E_T[Y(1)], E_T[Y(0)])$, the target-population value is identifiable from a randomized trial plus target covariates under covariate shift. Under exchangeability in mean, identification runs through three equivalent formulas: $E_T[Y(a)] = E_T[\mu_S^{(a)}(X)] = E_S[r(X)Y(a)] = E_S[r(X)\mu_S^{(a)}(X)]$, where $r(X) = P_T(X)/P_S(X)$ is the density ratio between target and source covariate distributions. Under the weaker exchangeability in effect measure, the paper identifies $\tau_{T,\Phi}$ through the effect function $\Gamma$ as $\Phi(E_T[\Gamma(\tau_{S,\Phi}(X),\mu_T^{(0)}(X))], E_T[Y(0)])$, which lets even non-collapsible measures such as the odds ratio be transported once target control outcomes are observed. For both settings the paper provides weighting and regression estimators with closed-form asymptotic variances, plus semiparametric one-step and estimating-equation estimators, and proves the estimating-equation estimators are doubly robust for every measure in the class.

Load-bearing premise

The load-bearing premise is that target-population control outcomes (or their conditional mean $\mu_T^{(0)}(x)$) are available for the exchangeability-in-effect-measure branch, because Equations (12)-(13) and all Section 4 estimators require $E_T[Y(0)]$ and target control information; if the target dataset contains only covariates, that branch cannot be computed.

Editorial extensions

If this is right

  • All causal measures in the first-moment class—including risk difference, risk ratio, odds ratio, number needed to treat, excess risk ratio, survival ratio, and log-odds ratio—can be estimated in a target population under covariate shift from a single estimation recipe.
  • Under exchangeability in effect measure, non-collapsible measures such as the odds ratio become transportable, provided target control outcomes or their conditional mean are available.
  • Estimating-equation estimators are doubly robust for every first-moment measure, so consistency holds if either the outcome regression or the density-ratio model is correctly specified.
  • For linear outcome models, the asymptotic variances satisfy $V_{tG} \le V_{wG} \le V_{wHT}$, meaning the transported G-formula is the most efficient of the classical estimators in that setting.
  • Closed-form asymptotic variances for the weighted Horvitz-Thompson and G-formula estimators enable standard confidence intervals for all transported measures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If target control outcomes are unavailable, as in the paper's Section 2 data setup where the target dataset contains only covariates, the exchangeability-in-effect-measure branch requires data that the framework does not otherwise assume; the weaker assumption is weaker statistically but not cheaper to satisfy.
  • Because one-step and estimating-equation estimators coincide only for linear functionals and diverge for nonlinear ones such as the odds ratio, estimator choice should depend on the reported measure, and one could construct a test of whether the one-step correction is negligible by comparing bootstrap distributions.
  • The paper's stated extension to multiple RCTs suggests a path toward a causal meta-analysis that transports both absolute and relative measures from several trials; formal pooled estimation with multiple source populations is left implicit.
  • The variance-ordering result for linear outcome models may extend to nonparametric outcome regression under oracle rates, which would give measure-specific guidance on when to prefer transported G-formula over weighting estimators.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a unified framework for transporting first-moment population causal measures, i.e., estimands of the form τ_P = Φ(E_P[Y(1)], E_P[Y(0)]) for a broad class of effect measures Φ (RD, RR, OR, NNT, and others), from an RCT source population to a target population under covariate shift. Identification is studied under two assumptions: exchangeability in mean (Assumption 3) and exchangeability in effect measure (Assumption 5). For the former, the paper derives weighted Horvitz-Thompson, weighted/transported G-formula, and one-step/estimating-equation semiparametric estimators, with asymptotic variance expressions and a double-robustness result for the estimating-equation estimator (Proposition 5). For the latter, it gives an identification formula (Eq. (12)) based on target control outcomes, proposes Γ-formula estimators, and states an influence function (Proposition 6) used to build one-step and estimating-equation estimators. The theoretical results are supplemented by simulations and a real-data application on the CRASH-3 trial and Traumabase registry.

Significance. The paper addresses a real and practically important gap: most generalization methods focus on the risk difference, while clinical reporting routinely uses absolute and relative measures. If the Section 4 semiparametric results were correct, the paper would provide a genuinely useful unification, including the transport of non-collapsible measures such as the odds ratio under exchangeability in effect measure. The strengths of the manuscript include the clean identification algebra in Sections 3 and 4.1, the explicit double-robustness proof for the estimating-equation estimator under exchangeability in mean (Proposition 5 with proof in Appendix B.5), the closed-form asymptotic variance computations under logistic/linear nuisance models, and the extensive simulation study. These contributions are substantive. However, the influence function stated in Proposition 6 is incorrect when baseline risks differ between source and target, and this invalidates the main semiparametric claims of Section 4. The identification formula itself appears sound, so the result is correctable, but the Section 4 estimator development and its efficiency/double-robustness claims need substantial rework.

major comments (3)
  1. [Section 4.2, Proposition 6] Proposition 6 is not the influence function of ψ_T1 under Assumption 5. From Eq. (12), ψ_T1 = E_T[Γ(τΦ(X), μ_T0(X))] with τΦ(X) = Φ(μ_S1(X), μ_S0(X)). Differentiating through Γ and τΦ gives a source-treated coefficient of ∂1Γ(τ, μ_T0)∂1Φ(μ_S1, μ_S0) = ∂1Γ(τ, μ_T0)/∂1Γ(τ, μ_S0) and a source-control coefficient of ∂1Γ(τ, μ_T0)∂0Φ(μ_S1, μ_S0) = -∂1Γ(τ, μ_T0)∂0Γ(τ, μ_S0)/∂1Γ(τ, μ_S0). The printed φ1 instead uses coefficient 1 on the treated residual and -∂0Γ(τΦ, μ_T0) on the control residual. These coincide with the correct coefficients only when μ_T0 = μ_S0, i.e., exactly in the regime where Assumption 5 is not weaker than Assumption 3. The proof in Appendix C.2 actually derives the factor ∂1Γ(τ, μ_T0) multiplying IF(τΦ(x)), so the discrepancy is in the displayed formula of Proposition 6.
  2. [Section 4.2 / Appendix C.3] As a consequence of the error in Proposition 6, the one-step and estimating-equation estimators of Section 4, as well as the explicit RD/RR/OR expressions in Appendix C.3, are not first-order correct or efficient under Assumption 5 when baseline risks differ. The estimating-equation estimator may still be consistent when all nuisance models are correctly specified, because the printed moment has mean zero at the truth, but the claimed double robustness and the variance/efficiency statements in Section 4.2 are unsupported; no Proposition-5-type double-robustness proof is supplied for Section 4. The simulation setting of Experiment 2 has μ_T0 ≠ μ_S0 by construction, so the unbiasedness reported there is not evidence for the printed estimators' efficiency or double robustness. The authors should either correct Proposition 6 and all downstream formulas, or restrict the semiparametric claims to the case μ_T0 = μ_S0 and clearly label the general case as an open problem.
  3. [Section 2 vs. Section 4.1] There is an internal inconsistency in the data framework. Section 2 states that the target dataset contains only covariates (X_i)_{i∈[m]}, while Eq. (12), Eq. (13), Definition 4, and all Section 4 estimators require E_T[Y(0)] and μ_T0(X). The prose says Assumption 5 'requires access to control outcomes in the target population,' but this requirement is not incorporated into the formal sampling model. The claimed weakness of Assumption 5 relative to Assumption 3 is therefore misleading: it replaces an outcome-transportability assumption with an additional data requirement. The formal setup should be amended to specify how target control outcomes arise (e.g., an additional untreated sample from the target), or Section 4 should be explicitly presented as a different data regime rather than as a weaker assumption within the same framework.
minor comments (4)
  1. [Throughout appendices] Several assumption references are broken, e.g., 'Assumption 1 to 1' in Propositions 12-14 and 'Assumption 1 to 3.1' in Proposition 9; these should be corrected to the intended numbered assumptions.
  2. [Figure 1 caption] The caption 'Source values are 0.45 / 3.2 / 7.5' does not state which of the three numbers corresponds to the risk difference, risk ratio, and odds ratio; please spell this out.
  3. [Section 2.1, Assumption 1] 'SUTV A' appears to be a typo for 'SUTVA'; please fix throughout.
  4. [Section 4.2, first paragraph] The sentence 'which is is related to φ1' contains a duplicated 'is'; please correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transport formulas and estimators follow from the stated assumptions and the definition of the effect function, not from the target estimand.

full rationale

The paper's central derivation (Section 4.1, Eq. (12)) is a direct consequence of Assumption 5 and the inverse effect function Gamma defined in Definition 1. Given tau_S_Phi(x) = tau_T_Phi(x) and mu_T1(x) = Gamma(tau_S_Phi(x), mu_T0(x)), the target mean E_T[Y(1)] is expressed in terms of the source conditional effect and the target baseline; the target estimand is then Phi of the resulting means. This is a derivation, not a restatement of the conclusion: the only inputs are the explicitly stated exchangeability-in-effect-measure assumption and access to target control outcomes. The Section 3 estimators are standard weighting, G-formula, and EIF constructions with proofs supplied in the appendices; Proposition 5 proves double robustness rather than importing it by citation. Self-citations (Colnet et al. 2023, Boughdiri et al. 2024) are contextual and not load-bearing: collapsible cases are acknowledged as prior work, and the RR one-step robustness remark is auxiliary to the main derivation. No fitted parameter is renamed as a prediction; the nuisance estimators (density ratio, outcome regressions) are estimated from source or target covariates, and the target effect is not used in their construction. Two non-circular flags are noted for the record: (i) the Assumption 5 branch requires target control outcomes, which the Section 2 target dataset (covariates only) does not formally provide, a data-availability gap rather than a circular reduction; and (ii) the Appendix C.2 proof of Proposition 6 appears to omit the chain-rule factor partial_1 Gamma(tau_Phi, mu_T0) multiplying the source-treated influence term, which would affect the correctness or efficiency claims but is a mathematical error, not a circular reduction.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

All central estimators rely on standard causal assumptions (internal validity, overlap) plus a transportability condition. Section 4 additionally requires outcome information in the target population, which is a domain assumption that changes the data setting. Technical regularity conditions, such as sub-Gaussian covariates, a correct logistic selection model, and a correct linear outcome model for the variance ordering, are explicit in the paper but are load-bearing for the asymptotic claims.

assumptions (9)
  • domain assumption Assumption 1: ignorability, SUTVA, positivity and randomized assignment within the trial
    Standard causal identification assumptions; invoked throughout Section 3.
  • domain assumption Assumption 2: overlap P(S=1|X)>0 for x in supp(P_T)
    Needed for density ratio identifiability; Section 2.2.
  • domain assumption Assumption 3: conditional exchangeability in mean, E_S[Y(a)|X]=E_T[Y(a)|X]
    Needed for Section 3 identification formulas (3)-(5).
  • domain assumption Assumption 5: exchangeability in effect measure, τ_S_Φ(x)=τ_T_Φ(x)
    Needed for Section 4 identification Eq. (12).
  • domain assumption Access to control potential outcomes in the target population for Section 4
    Eq. (12) uses μ_T(0)(x) and E_T[Y(0)], requiring outcome data for controls in the target; this is not available in the Section 2 setup.
  • domain assumption Logistic model for P(S=1|X) is correctly specified
    Proposition 1 and variance formulas in Appendix B.3 assume the selection model is logistic; if misspecified, the weighted Horvitz-Thompson estimator does not consistently target the density ratio.
  • domain assumption Linear outcome model (Linear model 1) for G-formula variance ordering
    Proposition 2 and Lemma 2 derive the variance ordering under homoskedastic linear outcome regressions.
  • standard math Regularity conditions: sub-Gaussian X, positive-definite E[XX^T], square-integrability
    Used for M-estimation asymptotic normality in Appendix B.3; technical but conventional.
  • standard math X discrete for influence function computation
    Proofs of Proposition 3 and Proposition 6 compute influence functions assuming categorical X; the continuous case is asserted without a full proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Unified Framework for the Transportability of Population-Level Causal Measures." pith.science (2026). https://pith.science/paper/FWPZNQGG

@misc{pith2026250513104,
  author       = {Pith},
  title        = {Pith review of: A Unified Framework for the Transportability of Population-Level Causal Measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FWPZNQGG}},
  note         = {Machine review of arXiv:2505.13104}
}
read the original abstract

Generalization methods offer a powerful solution to one of the key drawbacks of randomized controlled trials (RCTs): their limited representativeness. By enabling the transport of treatment effect estimates to target populations subject to distributional shifts, these methods are increasingly recognized as the future of meta-analysis, the current gold standard in evidence-based medicine. Yet most existing approaches focus on the risk difference, overlooking the diverse range of causal measures routinely reported in clinical research. Reporting multiple effect measures-both absolute (e.g., risk difference, number needed to treat) and relative (e.g., risk ratio, odds ratio)-is essential to ensure clinical relevance, policy utility, and interpretability across contexts. To address this gap, we propose a unified framework for transporting a broad class of first-moment population causal effect measures under covariate shift. We provide identification results under two conditional exchangeability assumptions, derive both classical and semiparametric estimators, and evaluate their performance through theoretical analysis, simulations, and real-world applications. Our analysis shows the specificity of different causal measures and thus the interest of studying them all: for instance, two common approaches (one-step, estimating equation) lead to similar estimators for the risk difference but to two distinct estimators for the odds ratio.

Figures

Figures reproduced from arXiv: 2505.13104 by the authors.

Figure 1
Figure 1. Comparison of estimators across different causal measures under a non-linear outcome [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Comparison of estimators across different causal measures under a linear outcome model [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Comparison of estimators across different causal measures on the combined CRASH-3 and [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Comparison of estimators across different causal measures under a linear outcome model [PITH_FULL_IMAGE:figures/full_fig_p044_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Uncovering Bias Mechanisms in Observational Studies

    stat.ME 2025-06 conditional novelty 7.0 of 10

    Covariances between the size of causal bias and conditional variances of treatment, selection, and outcome form a fingerprint that distinguishes transportability, confounding, and selection bias mechanisms.

Reference graph

Works this paper leans on

44 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    Josse, and E

    Boughdiri, A., J. Josse, and E. Scornet (2024). Quantifying treatment effects: Estimating risk ratios in causal inference

    work page 2024

  2. [2]

    Campbell, H. and A. Remiro-Azócar (2025). Doubly robust augmented weighting estimators for the analysis of externally controlled single-arm trials and unanchored indirect treatment comparisons

    work page 2025

  3. [3]

    Chetverikov, M

    Chernozhukov, V., D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, W. Newey, and J. Robins (2017). Double/debiased machine learning for treatment and causal parameters

    work page 2017

  4. [4]

    Cole, P. and B. MacMAHON (1971). Attributable risk percent in case-control studies. British journal of preventive & social medicine\/ 25\/ (4), 242

    work page 1971

  5. [5]

    Cole, S. R. and E. A. Stuart (2010). Generalizing evidence from randomized clinical trials to target populations: the actg 320 trial. American journal of epidemiology\/ 172\/ (1), 107--115

    work page 2010

  6. [6]

    Josse, G

    Colnet, B., J. Josse, G. Varoquaux, and E. Scornet (2022, January). Causal effect on a target population: A sensitivity analysis to handle missing covariates. Journal of Causal Inference\/ 10\/ (1), 372–414

    work page 2022

  7. [7]

    Josse, G

    Colnet, B., J. Josse, G. Varoquaux, and E. Scornet (2023). Risk ratio, odds ratio, risk difference... which causal measure is easier to generalize? arXiv preprint arXiv:2303.16008\/

    arXiv 2023

  8. [8]

    Josse, G

    Colnet, B., J. Josse, G. Varoquaux, and E. Scornet (2024). Reweighting the rct for generalization: finite sample error and variable selection

    work page 2024

Show all 44 references
  1. [9]

    Mayer, G

    Colnet, B., I. Mayer, G. Chen, A. Dieng, R. Li, G. Varoquaux, J.-P. Vert, J. Josse, and S. Yang (2023). Causal inference methods for combining randomized trials and observational studies: a review

    2023

  2. [10]

    Mayer, G

    Colnet, B., I. Mayer, G. Chen, A. Dieng, R. Li, G. Varoquaux, J.-P. Vert, J. Josse, and S. Yang (2024). Causal inference methods for combining randomized trials and observational studies: a review. Statistical science\/ 39\/ (1), 165--191

    2024

  3. [11]

    Dahabreh, I. J., S. E. Robertson, L. C. Petito, M. A. Hern \'a n, and J. A. Steingrimsson (2023). Efficient and robust methods for causally interpretable meta-analysis: Transporting inferences from multiple randomized trials to a target population. Biometrics\/ 79\/ (2), 1057--1072

    2023

  4. [12]

    Dahabreh, I. J., S. E. Robertson, J. A. Steingrimsson, E. A. Stuart, and M. A. Hernan (2020). Extending inferences from a randomized trial to a new target population. Statistics in medicine\/ 39\/ (14), 1999--2014

    2020

  5. [13]

    Dahabreh, I. J., J. M. Robins, S. J. Haneuse, and M. A. Hern \'a n (2019). Generalizing causal inferences from randomized trials: counterfactual and graphical identification. arXiv preprint arXiv:1906.10792\/

    2019 arXiv

  6. [14]

    Degtiar, I. and S. Rose (2023). A review of generalizability and transportability. Annual Review of Statistics and Its Application\/ 10\/ (1), 501--524

    2023

  7. [15]

    Demirel, I., A. Alaa, A. Philippakis, and D. Sontag (2024). Prediction-powered generalization of causal inferences. arXiv preprint arXiv:2406.02873\/

    2024 arXiv

  8. [16]

    Fay, M. P. and F. Li (2024, Oct). Causal interpretation of the hazard ratio in randomized clinical trials. Clinical Trials\/ 21\/ (5), 623--635. Epub 2024 Apr 28

    2024

  9. [17]

    Pricing & reimbursement of drugs and hta policies in france

    French Health Authority (2024). Pricing & reimbursement of drugs and hta policies in france

    2024

  10. [18]

    (1987, 05)

    Greenland, S. (1987, 05). Interpretation and choice of effect measures in epidemiologic analysies . American Journal of Epidemiology\/ 125\/ (5), 761--768

    1987

  11. [19]

    Greenland, S., J. M. Robbins, and J. Pearl (1999, 01). Confounding and collapsibility in causal inference. Statistical Science\/ 14 , 29--46

    1999

  12. [20]

    Liu, and E

    Hong, H., L. Liu, and E. A. Stuart (2025, Feb). Estimating target population treatment effects in meta-analysis with individual participant-level data. Statistical Methods in Medical Research\/ 34\/ (2), 355--368. Epub 2025 Jan 19

    2025

  13. [21]

    Horvitz, D. G. and D. J. Thompson (1952). A generalization of sampling without replacement from a finite universe. Journal of the American statistical Association\/ 47\/ (260), 663--685

    1952

  14. [22]

    Huitfeldt, A., M. P. Fox, E. J. Murray, A. Hróbjartsson, and R. M. Daniel (2022). Shall we count the living or the dead?

    2022

  15. [23]

    Stensrud, and E

    Huitfeldt, A., M. Stensrud, and E. Suzuki (2019, 01). On the collapsibility of measures of effect in the counterfactual causal framework. Emerging Themes in Epidemiology\/ 16

    2019

  16. [24]

    Suzuki, and M

    Kanamori, T., T. Suzuki, and M. Sugiyama (2010). Theoretical analysis of density ratio estimation. IEICE transactions on fundamentals of electronics, communications and computer sciences\/ 93\/ (4), 787--798

    2010

  17. [25]

    Kennedy, E. H. (2022). Semiparametric doubly robust targeted double machine learning: a review. arXiv preprint arXiv:2203.06469\/

    2022 arXiv

  18. [26]

    Kennedy, E. H. (2024). Semiparametric doubly robust targeted double machine learning: a review. Handbook of Statistical Methods for Precision Medicine\/ , 207--236

    2024

  19. [27]

    Kern, H. L., E. A. Stuart, J. Hill, and D. P. Green (2016). Assessing methods for generalizing experimental impact estimates to target populations. Journal of research on educational effectiveness\/ 9\/ (1), 103--127

    2016

  20. [28]

    King, N. B., S. Harper, and M. E. Young (2012). Use of relative and absolute effect measures in reporting health inequalities: structured review. Bmj\/ 345

    2012

  21. [29]

    Laupacis, A., D. L. Sackett, and R. S. Roberts (1988). An assessment of clinically useful measures of the consequences of treatment. New England Journal of Medicine\/ 318\/ (26), 1728--1733. PMID: 3374545

    1988

  22. [30]

    Liu, J., K. Zhu, S. Yang, and X. Wang (2025). Robust estimation and inference in hybrid controlled trials for binary outcomes: A case study on non-small cell lung cancer

    2025

  23. [31]

    Hopewell, K

    Moher, D., S. Hopewell, K. F. Schulz, V. Montori, P. C. G tzsche, P. J. Devereaux, D. Elbourne, M. Egger, and D. G. Altman (2010). Consort 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials. BMJ\/ 340

    2010

  24. [32]

    Pearl, J. (2009). Causality\/ (2 ed.). Cambridge University Press

    2009

  25. [33]

    Pearl, J. and E. Bareinboim (2011). Transportability of causal and statistical relations: A formal approach. In Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence , AAAI'11, pp.\ 247–254. AAAI Press

    2011

  26. [34]

    Richardson, T. S., J. M. Robins, and L. Wang (2017). On modeling and estimation for the relative risk and risk difference. Journal of the American Statistical Association\/ 112\/ (519), 1121--1130

    2017

  27. [35]

    Robins, J. (1986). A new approach to causal inference in mortality studies with a sustained exposure period---application to control of the healthy worker survivor effect. Mathematical Modelling\/ 7\/ (9), 1393--1512

    1986

  28. [36]

    Rothwell, P. M. (2005). External validity of randomised controlled trials: ``to whom do the results of this trial apply?''. The Lancet\/ 365 , 82--93

    2005

  29. [37]

    Rott, K. W., G. Bronfort, H. Chu, J. D. Huling, B. Leininger, M. H. Murad, Z. Wang, and J. S. Hodges (2024). Causally interpretable meta-analysis: Clearly defined causal effects and two case studies. Research Synthesis Methods\/ 15\/ (1), 61--72

    2024

  30. [38]

    Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational psychology\/ 66\/ (5), 688--701

    1974

  31. [39]

    Schulz, K. F., D. G. Altman, D. Moher, and C. Group* (2010). Consort 2010 statement: updated guidelines for reporting parallel group randomized trials. Annals of internal medicine\/ 152\/ (11), 726--732

    2010

  32. [40]

    Shirvaikar, V. and C. Holmes (2023). Targeting relative risk heterogeneity with causal forests

    2023

  33. [41]

    Splawa-Neyman, J., D. M. Dabrowska, and T. P. Speed (1990). On the Application of Probability Theory to Agricultural Experiments. Essay on Principles. Section 9 . Statistical Science\/ 5\/ (4), 465 -- 472

    1990

  34. [42]

    Stefanski, L. A. and D. D. Boos (2002). The calculus of m-estimation. The American Statistician\/ 56\/ (1), 29--38

    2002

  35. [43]

    Stuart, E. A., S. R. Cole, C. P. Bradshaw, and P. J. Leaf (2011). The use of propensity scores to assess the generalizability of results from randomized trials. Journal of the Royal Statistical Society Series A: Statistics in Society\/ 174\/ (2), 369--386

    2011

  36. [44]

    Pellegrini, F

    Yadlowsky, S., F. Pellegrini, F. Lionetto, S. Braune, and L. Tian (2021). Estimation and validation of ratio-based conditional average treatment effects using observational data. Journal of the American Statistical Association\/ 116\/ (533), 335--352

    2021

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.