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Discretizing a continuous mediator in causal functionals induces a first-order coarsening bias, and evaluating the outcome regression at within-bin conditional means removes that leading term.

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2026-08-02 20:48 UTC pith:U4OMOMTQ

load-bearing objection Useful, simple bias correction for discretized causal functionals; the influence-function part has a real gap and needs repair. the 3 major comments →

arxiv 2602.22083 v2 pith:U4OMOMTQ submitted 2026-02-25 stat.ME cs.LGstat.ML

Coarsening Bias from Variable Discretization in Causal Functionals

Xiaxian Ou , Razieh Nabi This is my paper
classification stat.ME cs.LGstat.ML MSC 62D2062G0562G20
keywords coarsening biasmediator discretizationcausal functionalswithin-bin conditional meanssecond-order approximationinfluence functionfront-door functionalmediation analysis
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies what happens when a continuous mediator is binned before computing causal functionals such as the mediation functional E(Y(a1, M(a0))) or the front-door functional E(Y(a0)). It shows that the naive discretized version of the integral is a different population parameter: its bias relative to the true functional is first order in the bin width, O(w_max), even when identification and nuisance estimation are perfect. The proposed fix replaces, in each bin, the within-bin outcome mean with the outcome regression evaluated at the conditional mean of the mediator under the treatment-referent level, m_k(a0,c). A Taylor expansion removes the leading term, leaving a second-order error O(w_max^2) — O(1/K^2) for equal-width bins — under twice-differentiability of the outcome regression. A kernel-smoothed variant remains second-order in bin width and bandwidth while restoring pathwise differentiability, so one-step estimators and confidence intervals become available.

Core claim

The paper establishes that the coarsening error of the naive discretized functional, Δ_h(Q)(c) = Σ_k {μ_k(a1,c) − μ_{k,a1}(a0,c)} g_k(a0,c), is first order in the bin width, and that replacing the within-bin outcome mean μ_k(a1,c) by μ(m_k(a0,c), a1,c) — the outcome regression evaluated at the treatment-a0 within-bin conditional mediator mean — removes the leading term. The remaining error is bounded by half the sup curvature of μ times the within-bin variance, giving O(w_max^2); with equal-width bins this is O(1/K^2). The paper further shows that a kernel-smoothed version of the corrected functional has combined error O(w_max^2 + b^2) and is pathwise differentiable, so one-step estimators c

What carries the argument

The key object is the within-bin conditional mean m_k(a,c) = E(M | A=a, C=c, bin k), used as the expansion point in a Taylor series of the outcome regression μ(·, a1, c). The exact coarsening-error identity, Δ_h(Q)(c) = Σ_k {μ_k(a1,c) − μ_{k,a1}(a0,c)} g_k(a0,c), isolates the bias; the debiased functional replaces μ_k(a1,c) with μ(m_k(a0,c), a1,c), so the difference becomes a centered second-order remainder bounded by the second derivative of μ and the within-bin variance, which is at most w_k^2/4. A kernel-smoothed local average around m_k(a0,c) restores pathwise differentiability, enabling influence-function-based estimation.

Load-bearing premise

The central claim collapses if the outcome regression is not twice continuously differentiable in the mediator with uniformly bounded second derivative within each bin (Lemma 4.1), and the statistical claims further assume O_p(n^{-1/2}) nuisance estimation error for fixed bins (stated before Eq. 9).

What would settle it

Generate data with a known mediator–outcome regression that is continuous but not differentiable at a bin boundary (e.g., μ(m) = |m| or μ(m) = max(0,m) inside a bin), fit the debiased coarsened functional for K=2,4,8,..., and measure its error against the exact integral. If the error decays at first order, or if it does not decay at all, the claim that the within-bin-mean correction eliminates the leading term fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • For equal-width bins, the same precision requires roughly the square root of the number of bins: approximation error drops from 1/K to 1/K^2, so K=10 gives about the error that naive binning needs K=100 to reach.
  • One-step estimators built on the influence function correct statistical estimation bias but not discretization bias; the target functional itself must be corrected, which is what the debiased functional does.
  • With the corrected functional, the smoothed one-step estimator is asymptotically equivalent to the original undiscretized functional provided nuisance estimators converge and the bin width and bandwidth go to zero fast enough.
  • The construction carries over to multiple mediators with a bound in terms of each mediator's maximum bin width, so the correction is not limited to univariate binning.
  • Simulations with correctly specified nuisance models show the debiased plug-in estimator already achieves near-nominal coverage and small MSE, so the main benefit is available without implementing influence functions.

Where Pith is reading between the lines

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

  • The same within-bin-mean correction should apply to any causal functional that integrates a smooth regression against a reference conditional distribution — for example, other path-specific effects or g-computation formulas that currently discretize continuous covariates — as long as the reference-level bin means are estimable.
  • The covariance view of the bias (Remark 3.2) suggests a practical diagnostic: estimate the within-bin covariance between the outcome surface and the treatment-induced density ratio; a large nonzero covariance predicts that the naive coarsened estimate will be materially biased and the debiased version is needed.
  • A testable extension: in an applied dataset, compute both naive and debiased coarsened estimates at several bin counts and compare how the difference shrinks; the paper's simulations show the predicted 1/K versus 1/K^2 decay, which practitioners can reproduce to decide whether binning is safe.
  • Flagged from the text: the rate example for K_n stated after the smoothed estimator appears to have the inequality direction reversed — the condition 1/K_n^2 = o(n^{-1/2}) requires K_n to grow faster than n^{1/4}, not slower.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the population-level approximation error induced by discretizing a continuous mediator in causal functionals of the form θ(Q)(c)=∫ µ(m,a₁,c) f_{M|A,C}(m|a₀,c) dm, as used in mediation and front-door estimands. It defines the naive coarsened functional θ_h, shows under one-time differentiability that its coarsening error is O(w_max,K) (Lemma 3.1), and proposes a debiased coarsened functional θ~_h that evaluates the outcome regression at the within-bin conditional mean under A=a₀, achieving O(w²_max,K) under two-time differentiability (Lemma 4.1). A smoothed variant θ~_{h,b} is introduced to restore pathwise differentiability, and an influence-function-based one-step estimator is derived in Theorem 5.2. Simulations and a stroke-data application compare plug-in and one-step estimators for the naive and debiased functionals.

Significance. If the main approximation results are correct, the proposed debiased functional is a simple and practically valuable correction: binning a continuous mediator can be made nearly bias-free by a within-bin-mean evaluation of the outcome regression, reducing coarsening error from first order to second order in bin width. The population-level decomposition (coarsening error versus estimation error) is clearly articulated, and the simulation design in Section 6 usefully isolates the population coarsening error by using a large Monte Carlo sample. The paper does not provide machine-checked proofs or reproducible code for all experiments, but the Taylor-expansion arguments behind Lemmas 4.1 and 5.1 are transparent. However, the influence-function derivation in Theorem 5.2 contains a load-bearing error, and the statistical-estimation contribution is therefore not reliable as written.

major comments (3)
  1. [§5.2, Theorem 5.2 and Appendix B.6, Part (I)] The EIF for μ_{b,k}(a₁,c) is incorrect because the denominator D_k(c)=E[K_b(M−m_k(a₀,c))|A=a₁,C=c] is itself a functional of P_{M|A=a₁,C}. In Part (I) the derivation treats ω_{b,k} as fixed and then adds a chain-rule correction only for m_k(a₀,c), but that correction does not account for the pathwise variation of the normalizing denominator. Along a submodel with score s for the conditional law under A=a₁, the correct gradient is E[(Y−μ_{b,k})ω_{b,k}·s], so the EIF term should be (Y−μ_{b,k})ω_{b,k}, not Yω_{b,k}−μ_{b,k}. The difference μ_{b,k}(ω_{b,k}−1) has conditional mean zero but is not orthogonal to the tangent space; for example, if Y is independent of M, tilting the law of M can make this term contribute a nonzero pathwise derivative while the true functional is unchanged. Consequently, Theorem 5.2's displayed EIF is not the efficient influence function, and the asymptotic-lineari
  2. [Appendix B.2, Eq. (37)] The inequality |µ_k(a₁,c)−µ_{k,a₁}(a₀,c)| ≤ L(c)|m_k(a₁,c)−m_k(a₀,c)| is not valid in general. Two distributions can have the same conditional mean inside a bin while giving different expectations of a function with bounded derivative; e.g., with a tent-shaped µ on [0,1] (slope ±L), P₀ putting mass 1/2 at 0 and 1/2 at 1, and P₁ a point mass at 1/2, both means are 1/2 but the expectations differ by L/2. This is a step in the proof of Lemma 3.1, although the resulting O(w_max,K) bound is recoverable by replacing the inequality with a bound such as |µ_k(a₁,c)−µ_{k,a₁}(a₀,c)| ≤ 2L(c)w_k(c). The proof should be corrected.
  3. [§5.2 and §6] The paper derives a one-step estimator for the smoothed functional θ~_{h,b} (Theorem 5.2) but never simulates this estimator. The one-step estimators used in Section 6, in particular ψ~⁺_{h2} obtained by replacing θ(Q̂) with θ~_h(Q̂) in Eq. (20), target the non-smoothed debiased functional θ~_h, which Section 5 explicitly states is not pathwise differentiable in the nonparametric model. Thus the theoretical guarantees of Section 5 do not cover the debiased one-step estimator whose finite-sample performance is reported. Either the simulations should use the smoothed estimator whose theory is developed, or the claims about one-step estimation for the non-smoothed functional should be clearly labeled as heuristic without asymptotic justification.
minor comments (5)
  1. [§5.2, Theorem 5.2 statement] The phrase 'when m_k(a₀,k) is fixed' appears to contain a typo; it should read 'm_k(a₀,c)'.
  2. [Lemma 4.1, final sentence] The final sentence states '∆_h(Q)(c)=O(1/K²)'; this should be '˜∆_h(Q)(c)=O(1/K²)' for the debiased functional.
  3. [§6, Simulation #1 and #2] The theoretical scaling O(1/K) and O(1/K²) is derived under equal-width bins, but the simulations use equal-frequency bins. Equal-frequency bins need not have equal widths, especially under skewed mediator distributions; the text should clarify why the equal-width theory is expected to apply or provide a separate argument.
  4. [§6, Eqs. (20)–(21)] The one-step formulas for ψ⁺ and ψ⁺_h are stated without derivation. Since these estimators play a key role in the simulation comparisons, a reference to the standard derivation or a brief appendix entry would improve readability.
  5. [§7, real data application] The outcome mRS is ordinal but treated as continuous. A sentence acknowledging this simplification and its potential impact on the front-door estimand would be appropriate.

Circularity Check

0 steps flagged

No significant circularity: the approximation-error claims are derived from explicit definitions and Taylor expansions, not fitted to the target.

full rationale

The paper's central claims (Lemmas 3.1, 4.1, and 5.1) are derived from explicit definitions of the coarsened and debiased functionals via Taylor expansions; the debiased functional is constructed around the within-bin mean, and the proof shows the first-order term vanishes because E[M - m_k(a0,c) | A=a0, C=c, M-tilde=k] = 0 by definition of m_k. This is a mathematical identity, not a circular empirical prediction. Simulations compute the theoretical coarsening error from the known DGP rather than fitting the target functional, so there is no fitted-input-called-prediction pattern. Self-citations (e.g., [8], [29]) are used only as background for standard identification results, which are also re-derived in Appendix B.1; they are not load-bearing for the novel approximation claims. The possible concern about the EIF derivation in Theorem 5.2 concerning pathwise differentiation of the kernel normalizing denominator, if valid, would be a correctness or regularity gap, not a circular reduction of the paper's results to their inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The population-level bias results rest on smoothness, causal identification assumptions, and a variance bound; no free parameters are fitted to data, but K, b, and the kernel are user choices. The EIF derivation additionally assumes regularity/nuisance-rate conditions that are not demonstrated.

free parameters (3)
  • Number of bins K = user-chosen
    Discretization resolution; error rates O(1/K) vs O(1/K^2) are stated in terms of K, but K is not determined by the theory.
  • Smoothing bandwidth b = user-chosen
    Bandwidth for kernel localized mean in Section 5; bias O(b^2) requires b→0 with n, no data-driven selection given.
  • Smoothing kernel K = user-chosen (symmetric, second-order)
    Assumed symmetric with ∫uK=0, ∫u^2K<∞; influences the constant in O(b^2).
axioms (7)
  • domain assumption μ(m,a1,c) is twice continuously differentiable in m on each bin with uniformly bounded second derivative
    Used in Lemma 4.1 and 5.1; if false, second-order error claim fails (only first order).
  • domain assumption f_{M|A,C}(m|a1,c) is continuous and bounded away from zero near mk(a0,c)
    Required for kernel expansion in Lemma 5.1 to control denominator; fails near support boundaries without boundary correction.
  • domain assumption Causal identification assumptions: consistency, positivity, conditional ignorability / front-door no-direct-effect (Appendix B.1)
    Established assumptions from prior literature; target estimands equal causal effects only under these.
  • ad hoc to paper For fixed h, plug-in estimation error is Op(n^{-1/2}) in L2(P_C)
    Assumed before Eq (9) and (14); requires sufficiently fast nuisance convergence, not guaranteed by flexible ML in general.
  • ad hoc to paper Regularity conditions for asymptotic linearity of one-step: n^{-1/4} nuisance rates and cross-fitting
    Stated as 'sufficient regularity conditions' after Theorem 5.2; not verified in simulations.
  • standard math Kernel symmetry: ∫uK(u)du=0, ∫u^2K(u)du<∞
    Standard second-order kernel assumption in Lemma 5.1.
  • standard math Within-bin variance bound Var(M|bin) ≤ w^2/4
    Used in Lemma 4.1 bound; holds because conditional law supported on interval of width w.

pith-pipeline@v1.3.0-alltime-deepseek · 21818 in / 18422 out tokens · 177743 ms · 2026-08-02T20:48:01.934664+00:00 · methodology

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Cite this review

Pith. "Pith review of Coarsening Bias from Variable Discretization in Causal Functionals." pith.science (2026). https://pith.science/paper/U4OMOMTQ

@misc{pith2026260222083,
  author       = {Pith},
  title        = {Pith review of: Coarsening Bias from Variable Discretization in Causal Functionals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U4OMOMTQ}},
  note         = {Machine review of arXiv:2602.22083}
}
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read the original abstract

Causal identification functionals often require integration over conditional densities of continuous variables, such as those arising in nonparametric identification theory of total and mediated causal effects in DAGs with hidden variables. Estimating these densities and evaluating the resulting integrals can be statistically and computationally demanding. A common workaround is to discretize the continuous variable and replace integrals with finite sums. Although convenient, discretization alters the population-level functional and can induce non-negligible approximation bias, even when identification is correct. Under smoothness conditions, we show that the resulting coarsening error is first order in the bin width and arises at the level of the target functional, distinct from statistical estimation error. We propose a simple debiased coarsened functional that evaluates the outcome regression at within-bin conditional means, eliminating the leading coarsening error term and yielding a second-order approximation error. We derive plug-in and one-step estimators for this debiased coarsened functional. Simulations demonstrate substantial bias reduction and near-nominal confidence interval coverage, even under coarse binning. Our results provide a simple framework for controlling the impact of variable discretization on both parameter approximation and statistical estimation.

Figures

Figures reproduced from arXiv: 2602.22083 by Razieh Nabi, Xiaxian Ou.

Figure 1
Figure 1. Figure 1: Within-bin differences in conditional mediator means, [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Within-bin covariance between µ(M, a1, C) and rk(M | C) at C = 0, as formalized in Remark 3.2. Finer discretization reduces the magnitude of this covariance. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bias of the coarsened plug-in estimator ψh(Qb) and the debiased plug-in estimator ψe h(Qb) as functions of sample size for several discretization levels K. For fixed K, the bias of ψh(Qb) persists as n increases, reflecting population coarsening error, whereas ψe h(Qb) exhibits substantially reduced bias across sample sizes [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The performance of the coarsened plug-in estimator [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of plug-in and one-step estimators under nuisance-model misspecification [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of plug-in estimators γh(Qb), γeh(Qb) and γs(Qb) in B_PROUD study for estimating γ(Q) = E[Y (a0)] thrombolytic therapy. We adjust for two baseline continuous covariates: systolic blood pressure and stroke severity. Missing data are handled using multiple imputation. This dataset was previously analyzed by [27] employing a front-door approach to estimate the causal effect of MSU dispatch on 3-mon… view at source ↗
Figure 7
Figure 7. Figure 7: Estimated within-bin differences in conditional mediator means, [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

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Reference graph

Works this paper leans on

30 extracted references · 4 linked inside Pith

  1. [1]

    Direct and indirect effects

    Judea Pearl. Direct and indirect effects. InProceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (UAI-01), pages 411–420. Morgan Kaufmann, San Francisco, 2001

    2001

  2. [2]

    Mediation analysis with multiple mediators

    Tyler VanderWeele and Stijn Vansteelandt. Mediation analysis with multiple mediators. Epidemiologic methods, 2(1):95–115, 2014

    2014

  3. [3]

    Identifiability of path-specific effects

    Chen Avin, Ilya Shpitser, and Judea Pearl. Identifiability of path-specific effects. In Proceedings of the IJCAI, pages 357–363, 12 2005

    2005

  4. [4]

    Counterfactual graphical models for longitudinal mediation analysis with unobserved confounding.Cognitive science, 37(6):1011–1035, 2013

    Ilya Shpitser. Counterfactual graphical models for longitudinal mediation analysis with unobserved confounding.Cognitive science, 37(6):1011–1035, 2013

    2013

  5. [5]

    A general identification condition for causal effects

    Jin Tian and Judea Pearl. A general identification condition for causal effects. InEighteenth National Conference on Artificial Intelligence, pages 567–573, 2002

    2002

  6. [6]

    Nested markov properties for acyclic directed mixed graphs.The Annals of Statistics, 51(1):334–361, 2023

    Thomas S Richardson, Robin J Evans, James M Robins, and Ilya Shpitser. Nested markov properties for acyclic directed mixed graphs.The Annals of Statistics, 51(1):334–361, 2023

    2023

  7. [7]

    Semiparametric inference for causal effects in graphical models with hidden variables.Journal of Machine Learning Research, 23:1–76, 2022

    Rohit Bhattacharya, Razieh Nabi, and Ilya Shpitser. Semiparametric inference for causal effects in graphical models with hidden variables.Journal of Machine Learning Research, 23:1–76, 2022

    2022

  8. [8]

    Flexible nonparametric inference for causal effects under the front-door model.arXiv preprint arXiv:2312.10234, 2023

    Anna Guo, David Benkeser, and Razieh Nabi. Flexible nonparametric inference for causal effects under the front-door model.arXiv preprint arXiv:2312.10234, 2023

    Pith/arXiv arXiv 2023

  9. [9]

    napkin graph

    Anna Guo, David Benkeser, and Razieh Nabi. Causal inference with the" napkin graph". arXiv preprint arXiv:2512.19861, 2025

    Pith/arXiv arXiv 2025

  10. [10]

    Discovery and mediation analysis of cross-phenotype associations between asthma and body mass index in 12q13

    Yasmmyn D Salinas, Zuoheng Wang, and Andrew T DeWan. Discovery and mediation analysis of cross-phenotype associations between asthma and body mass index in 12q13. 2. American Journal of Epidemiology, 190(1):85–94, 2021

    2021

  11. [11]

    Jiasheng Huang, Yehua Li, Angelique G Brellenthin, Duck-chul Lee, Xuemei Sui, and Steven N Blair. Causal mediation analysis between resistance exercise and reduced risk of 22 cardiovascular disease based on the aerobics center longitudinal study.Journal of Applied Statistics, 49(14):3750–3767, 2022

    2022

  12. [12]

    Revisiting the obesity–anaemia paradox: Inflammation and iron homeostasis in the bmi–haemoglobin relationship.Endocrinology, Diabetes & Metabolism, 8(5):e70110, 2025

    Ali Hemade and Pascale Salameh. Revisiting the obesity–anaemia paradox: Inflammation and iron homeostasis in the bmi–haemoglobin relationship.Endocrinology, Diabetes & Metabolism, 8(5):e70110, 2025

    2025

  13. [13]

    A multiple mediator model: Power analysis based on monte carlo simulation.American Journal of Applied Psychology, 3(3):72–79, 2014

    Ze-wei Ma and Wei-nan Zeng. A multiple mediator model: Power analysis based on monte carlo simulation.American Journal of Applied Psychology, 3(3):72–79, 2014

    2014

  14. [14]

    Mediation: R package for causal mediation analysis.Journal of statistical software, 59:1–38, 2014

    Dustin Tingley, Teppei Yamamoto, Kentaro Hirose, Luke Keele, and Kosuke Imai. Mediation: R package for causal mediation analysis.Journal of statistical software, 59:1–38, 2014

    2014

  15. [15]

    The role of body mass index at diagnosis of colorectal cancer on black–white disparities in survival: a density regression mediation approach.Biostatistics, 23(2):449–466, 2022

    Katrina L Devick, Linda Valeri, Jarvis Chen, Alejandro Jara, Marie-Abèle Bind, and Brent A Coull. The role of body mass index at diagnosis of colorectal cancer on black–white disparities in survival: a density regression mediation approach.Biostatistics, 23(2):449–466, 2022

    2022

  16. [16]

    Simulation-based estimators of analytically intractable causal effects

    Antonio R Linero. Simulation-based estimators of analytically intractable causal effects. Biometrics, 78(3):1001–1017, 2022

    2022

  17. [17]

    Effect measure modification by covariates in mediation: extending regression-based causal mediation analysis.Epidemiology, 34(5):661–672, 2023

    Yi Li, Maya B Mathur, Daniel H Solomon, Paul M Ridker, Robert J Glynn, and Kazuki Yoshida. Effect measure modification by covariates in mediation: extending regression-based causal mediation analysis.Epidemiology, 34(5):661–672, 2023

    2023

  18. [18]

    A bayesian approach to the g-formula via iterative conditional regression.Statistics in Medicine, 44(13-14):e70123, 2025

    Ruyi Liu, Liangyuan Hu, Francis Perry Wilson, Joshua L Warren, and Fan Li. A bayesian approach to the g-formula via iterative conditional regression.Statistics in Medicine, 44(13-14):e70123, 2025

    2025

  19. [19]

    Targeted maximum likelihood based estimation for longitudinal mediation analysis.Journal of Causal Inference, 13(1):20230013, 2025

    Zeyi Wang, Lars van der Laan, Maya Petersen, Thomas Gerds, Kajsa Kvist, and Mark van der Laan. Targeted maximum likelihood based estimation for longitudinal mediation analysis.Journal of Causal Inference, 13(1):20230013, 2025

    2025

  20. [20]

    Ratio of mediator probability weighting for estimating natural direct and indirect effects.arXiv preprint arXiv:2506.03284, 2025

    Guanglei Hong. Ratio of mediator probability weighting for estimating natural direct and indirect effects.arXiv preprint arXiv:2506.03284, 2025

    Pith/arXiv arXiv 2025

  21. [21]

    Xiang Zhou. Semiparametric estimation for causal mediation analysis with multiple causally 23 ordered mediators.Journal of the Royal Statistical Society Series B: Statistical Methodology, 84(3):794–821, 2022

    2022

  22. [22]

    Two-stage nuisance function estimation for causal mediation analysis.arXiv preprint arXiv:2404.00735, 2024

    Chang Liu and AmirEmad Ghassami. Two-stage nuisance function estimation for causal mediation analysis.arXiv preprint arXiv:2404.00735, 2024

    arXiv 2024

  23. [23]

    Cambridge university press, 2000

    Aad W Van der Vaart.Asymptotic statistics, volume 3. Cambridge university press, 2000

    2000

  24. [24]

    Targeted maximum likelihood learning.The International Journal of Biostatistics, 2(1), 2006

    Mark J van der Laan and Daniel Rubin. Targeted maximum likelihood learning.The International Journal of Biostatistics, 2(1), 2006

    2006

  25. [25]

    Mu-Chi Chung, Hui-Tsung Hsu, Yan-Chiao Mao, Chin-Ching Wu, Chih-Te Ho, Chiu-Shong Liu, and Chi-Jung Chung. Association and mediation analyses among multiple metals exposure, plasma folate, and community-based impaired estimated glomerular filtration rate in central taiwan.Environmental Health, 21(1):44, 2022

    2022

  26. [26]

    Discretizing continuous variables in nutrition and obesity research: a practice that needs to be cut short.Nutrition & Diabetes, 13(1):20, 2023

    Osvaldo F Morera, Mosi I Dane’el, Brandt A Smith, Alisha H Redelfs, Sarah L Ruiz, Kristopher J Preacher, and Leah D Whigham. Discretizing continuous variables in nutrition and obesity research: a practice that needs to be cut short.Nutrition & Diabetes, 13(1):20, 2023

    2023

  27. [27]

    The effect of mobile stroke unit care on functional outcomes: an application of the front-door formula

    Marco Piccininni, Tobias Kurth, Heinrich J Audebert, and Jessica L Rohmann. The effect of mobile stroke unit care on functional outcomes: an application of the front-door formula. Epidemiology, 34(5):712–720, 2023

    2023

  28. [28]

    Causal diagrams for empirical research.Biometrika, 82(4):669–688, 1995

    Judea Pearl. Causal diagrams for empirical research.Biometrika, 82(4):669–688, 1995

    1995

  29. [29]

    Average causal effect estimation in dags with hidden variables: Beyond back-door and front-door criteria.arXiv preprint arXiv:2409.03962, 2024

    Anna Guo and Razieh Nabi. Average causal effect estimation in dags with hidden variables: Beyond back-door and front-door criteria.arXiv preprint arXiv:2409.03962, 2024

    Pith/arXiv arXiv 2024

  30. [30]

    Berlin prehospital or usual delivery of acute stroke care–study protocol

    Martin Ebinger, Peter Harmel, Christian H Nolte, Ulrike Grittner, Bob Siegerink, and Heinrich J Audebert. Berlin prehospital or usual delivery of acute stroke care–study protocol. International Journal of Stroke, 12(6):653–658, 2017. 24 Coarsening Bias from Variable Discretization in Causal Functionals (Supplementary Material) This Supplementary Material ...

    2017