pith. sign in

arxiv: 2305.08284 · v3 · submitted 2023-05-15 · 📊 stat.ME

Model-based standardization using multiple imputation

Antonio Remiro-Az\'ocar , Anna Heath , Gianluca Baio This is my paper

Pith reviewed 2026-05-24 09:02 UTC · model grok-4.3

classification 📊 stat.ME
keywords model-based standardizationmultiple imputationmarginal treatment effectgeneralized linear modelscovariate adjustmentBayesian statisticssimulation study
0
0 comments X

The pith

Multiple imputation marginalizes parametric outcome models to recover marginal treatment effects.

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

The paper introduces multiple imputation marginalization (MIM) as a general-purpose alternative to maximum-likelihood estimation plus bootstrap for model-based standardization. MIM creates synthetic datasets by sampling outcomes from the fitted conditional model and then pools the resulting treatment-effect estimates. When the outcome model is correctly specified, the method produces unbiased estimates of the marginal effect in the target covariate distribution together with valid frequentist coverage. The procedure fits naturally inside a Bayesian workflow and propagates uncertainty without separate resampling steps. Finite-sample simulations with binary outcomes and logistic regression confirm performance comparable to the classical standardization approach.

Core claim

Model-based standardization can be recast as a missing-data problem in which the target population covariate values are treated as unobserved; multiple imputation then averages the fitted outcome model over that distribution, delivering a marginal treatment-effect estimate.

What carries the argument

Multiple imputation marginalization (MIM): a two-stage procedure that first generates completed datasets by drawing from the conditional outcome distribution given treatment and covariates, then analyzes each completed dataset to obtain marginal effect estimates that are combined by Rubin's rules.

If this is right

  • MIM yields unbiased estimates of the marginal treatment effect when the outcome model is correct.
  • Frequentist coverage of the MIM interval estimators reaches nominal levels.
  • Point-estimate precision and efficiency are comparable to the standard bootstrap standardization procedure.
  • The method extends immediately to Bayesian outcome models without additional computational machinery.

Where Pith is reading between the lines

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

  • MIM may reduce implementation effort in settings where bootstrap resampling is computationally expensive or analytically intractable.
  • Because imputation is already a standard tool for missing data, the same code base can handle both covariate standardization and actual missing outcomes.
  • Prior distributions on model parameters are propagated automatically, which could simplify sensitivity analyses that vary the strength of prior information.

Load-bearing premise

The parametric outcome model correctly specifies the conditional expectation of the outcome given treatment and covariates.

What would settle it

A simulation in which the logistic outcome model is deliberately misspecified for binary data would produce biased marginal log-odds-ratio estimates under MIM while the bootstrap comparator also fails.

read the original abstract

When studying the association between treatment and a clinical outcome, a parametric multivariable model of the conditional outcome expectation is often used to adjust for covariates. The treatment coefficient of the outcome model targets a conditional treatment effect. Model-based standardization is typically applied to average the model predictions over the target covariate distribution, and generate a covariate-adjusted estimate of the marginal treatment effect. The standard approach to model-based standardization involves maximum-likelihood estimation and use of the non-parametric bootstrap. We introduce a novel, general-purpose, model-based standardization method based on multiple imputation that is easily applicable when the outcome model is a generalized linear model. We term our proposed approach multiple imputation marginalization (MIM). MIM consists of two main stages: the generation of synthetic datasets and their analysis. MIM accommodates a Bayesian statistical framework, which naturally allows for the principled propagation of uncertainty, integrates the analysis into a probabilistic framework, and allows for the incorporation of prior evidence. We conduct a simulation study to benchmark the finite-sample performance of MIM in conjunction with a parametric outcome model. The simulations provide proof-of-principle in scenarios with binary outcomes, continuous-valued covariates, a logistic outcome model and the marginal log odds ratio as the target effect measure. When parametric modeling assumptions hold, MIM yields unbiased estimation in the target covariate distribution, valid coverage rates, and similar precision and efficiency than the standard approach to model-based standardization.

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

0 major / 3 minor

Summary. The paper proposes multiple imputation marginalization (MIM), a two-stage method (synthetic dataset generation followed by analysis) for model-based standardization to obtain marginal treatment effects from generalized linear outcome models. It positions MIM as a general-purpose alternative to maximum-likelihood estimation plus nonparametric bootstrap that naturally supports Bayesian inference and prior incorporation. A simulation study is presented as proof-of-principle for binary outcomes, continuous covariates, logistic regression, and the marginal log-odds ratio; under correct parametric specification the method is reported to yield unbiased point estimates, nominal coverage, and efficiency comparable to the bootstrap standard approach.

Significance. If the finite-sample results hold, MIM supplies a probabilistically coherent route to marginalization that integrates uncertainty propagation without separate bootstrap resampling and is immediately compatible with Bayesian workflows. The explicit conditioning of all claims on correct outcome-model specification, together with the focused simulation design, is a strength; it avoids overgeneralization while still demonstrating practical performance in the stated regime.

minor comments (3)
  1. Abstract, final sentence: the phrasing 'similar precision and efficiency than the standard approach' should be revised to 'similar precision and efficiency to the standard approach' for grammatical accuracy.
  2. Simulation study description: while the abstract states that the design provides proof-of-principle, the manuscript should ensure that the number of Monte Carlo replications, the precise data-generating process for covariates and outcomes, the number of imputations, and the exact numerical results (bias, coverage, interval width) are reported in a dedicated results subsection or table so readers can directly verify the reported performance.
  3. Methods section on MIM implementation: the precise algorithm for generating the synthetic datasets (e.g., how the posterior predictive draws are obtained and how the marginal contrast is computed across imputations) should be stated with an explicit algorithmic outline or pseudocode to facilitate reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The provided summary accurately reflects the proposed MIM method, its two-stage structure, compatibility with Bayesian workflows, and the simulation design focused on correct parametric specification for binary outcomes and the marginal log-odds ratio.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces MIM as a novel multiple-imputation procedure for marginalizing parametric outcome models. Its central claims are explicitly conditioned on correct specification of the outcome model, and the simulation study is presented only as finite-sample proof-of-principle under logistic regression with binary outcomes. No derivation step equates a fitted quantity to a reported prediction by construction, no load-bearing result rests on self-citation, and no ansatz is imported without independent justification. The method and its performance claims remain self-contained against the stated assumptions and external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the parametric outcome model and the properties of multiple imputation for marginalization.

axioms (1)
  • domain assumption The outcome follows a generalized linear model with correctly specified conditional mean.
    The method targets GLMs and assumes parametric modeling assumptions hold for unbiasedness.

pith-pipeline@v0.9.0 · 5771 in / 1234 out tokens · 62467 ms · 2026-05-24T09:02:46.065272+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages · 1 internal anchor

  1. [1]

    Statistics in medicine 41(28), 5558–5569 (2022)

    Remiro-Az´ ocar, A.: Target estimands for population-adjusted indirect comparisons. Statistics in medicine 41(28), 5558–5569 (2022)

    work page 2022

  2. [2]

    target estimands for population-adjusted indirect comparisons

    Russek-Cohen, E.: Discussion of “target estimands for population-adjusted indirect comparisons” by antonio remiro-azocar. Statistics in medicine 41(28), 5573–5576 (2022)

    work page 2022

  3. [3]

    Statistics in medicine 41(28), 5589–5591 (2022)

    Spieker, A.J.: Comments on the debate between marginal and conditional estimands. Statistics in medicine 41(28), 5589–5591 (2022)

    work page 2022

  4. [4]

    Statistics in medicine 41(28), 5586–5588 (2022)

    Senn, S.: Conditions for success and margins of error: estimation in clinical trials. Statistics in medicine 41(28), 5586–5588 (2022)

    work page 2022

  5. [5]

    Statistics in medicine 41(28), 5570–5572 (2022)

    Schiel, A.: Commentary on” target estimands for population-adjusted indirect comparisons”. Statistics in medicine 41(28), 5570–5572 (2022)

    work page 2022

  6. [6]

    Statistics in medicine 41(28), 5577–5585 (2022)

    Van Lancker, K., Vo, T.-T., Akacha, M.: Estimands in heath technology assessment: a causal inference perspective. Statistics in medicine 41(28), 5577–5585 (2022)

    work page 2022

  7. [7]

    International Statistical Review 79(3), 401–426 (2011)

    Greenland, S., Pearl, J.: Adjustments and their consequences—collapsibility analysis using graphical models. International Statistical Review 79(3), 401–426 (2011)

    work page 2011

  8. [8]

    Annual review of public health 22(1), 189–212 (2001)

    Greenland, S., Morgenstern, H.: Confounding in health research. Annual review of public health 22(1), 189–212 (2001)

    work page 2001

  9. [9]

    Epidemiology 21(4), 490–493 (2010) Remiro-Az´ ocaret al

    Kaufman, J.S.: Marginalia: comparing adjusted effect measures. Epidemiology 21(4), 490–493 (2010) Remiro-Az´ ocaret al. Page 18 of 23

    work page 2010

  10. [10]

    Journal of the Royal Statistical Society: Series B (Methodological) 40(3), 328–340 (1978)

    Whittemore, A.S.: Collapsibility of multidimensional contingency tables. Journal of the Royal Statistical Society: Series B (Methodological) 40(3), 328–340 (1978)

    work page 1978

  11. [11]

    Statistical science 14(1), 29–46 (1999)

    Greenland, S., Pearl, J., Robins, J.M.: Confounding and collapsibility in causal inference. Statistical science 14(1), 29–46 (1999)

    work page 1999

  12. [12]

    Emerging themes in epidemiology 16, 1–5 (2019)

    Huitfeldt, A., Stensrud, M.J., Suzuki, E.: On the collapsibility of measures of effect in the counterfactual causal framework. Emerging themes in epidemiology 16, 1–5 (2019)

    work page 2019

  13. [13]

    Trials 23(328) (2022)

    Morris, T.P., Walker, A.S., Williamson, E.J., White, I.R.: Planning a method for covariate adjustment in individually-randomised trials: a practical guide. Trials 23(328) (2022)

    work page 2022

  14. [14]

    Statistics in medicine 32(16), 2837–2849 (2013)

    Austin, P.C.: The performance of different propensity score methods for estimating marginal hazard ratios. Statistics in medicine 32(16), 2837–2849 (2013)

    work page 2013

  15. [15]

    Research synthesis methods 12(6), 750–775 (2021)

    Remiro-Az´ ocar, A., Heath, A., Baio, G.: Methods for population adjustment with limited access to individual patient data: A review and simulation study. Research synthesis methods 12(6), 750–775 (2021)

    work page 2021

  16. [16]

    Research Synthesis Methods 13(6), 716–744 (2022)

    Remiro-Az´ ocar, A., Heath, A., Baio, G.: Parametric g-computation for compatible indirect treatment comparisons with limited individual patient data. Research Synthesis Methods 13(6), 716–744 (2022)

    work page 2022

  17. [17]

    BMC Medical Research Methodology 22(1), 1–16 (2022)

    Remiro-Az´ ocar, A.: Two-stage matching-adjusted indirect comparison. BMC Medical Research Methodology 22(1), 1–16 (2022)

    work page 2022

  18. [18]

    Statistics in medicine 40(19), 4310–4326 (2021)

    Josey, K.P., Berkowitz, S.A., Ghosh, D., Raghavan, S.: Transporting experimental results with entropy balancing. Statistics in medicine 40(19), 4310–4326 (2021)

    work page 2021

  19. [19]

    Journal of the Royal Statistical Society: Series A (Statistics in Society) 183(3), 1189–1210 (2020)

    Phillippo, D.M., Dias, S., Ades, A., Belger, M., Brnabic, A., Schacht, A., Saure, D., Kadziola, Z., Welton, N.J.: Multilevel network meta-regression for population-adjusted treatment comparisons. Journal of the Royal Statistical Society: Series A (Statistics in Society) 183(3), 1189–1210 (2020)

    work page 2020

  20. [20]

    Mathematical modelling 7(9-12), 1393–1512 (1986)

    Robins, J.: 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(9-12), 1393–1512 (1986)

    work page 1986

  21. [21]

    Communications in Statistics-Theory and methods 38(3), 309–321 (2008)

    Zhang, Z.: Estimating a marginal causal odds ratio subject to confounding. Communications in Statistics-Theory and methods 38(3), 309–321 (2008)

    work page 2008

  22. [22]

    Statistics in medicine 28(1), 39–64 (2009)

    Moore, K.L., van der Laan, M.J.: Covariate adjustment in randomized trials with binary outcomes: targeted maximum likelihood estimation. Statistics in medicine 28(1), 39–64 (2009)

    work page 2009

  23. [23]

    Journal of clinical epidemiology 63(1), 2–6 (2010)

    Austin, P.C.: Absolute risk reductions, relative risks, relative risk reductions, and numbers needed to treat can be obtained from a logistic regression model. Journal of clinical epidemiology 63(1), 2–6 (2010)

    work page 2010

  24. [24]

    The international journal of biostatistics 6(1) (2010)

    Rosenblum, M., Van Der Laan, M.J.: Simple, efficient estimators of treatment effects in randomized trials using generalized linear models to leverage baseline variables. The international journal of biostatistics 6(1) (2010)

    work page 2010

  25. [25]

    American journal of epidemiology 173(7), 731–738 (2011)

    Snowden, J.M., Rose, S., Mortimer, K.M.: Implementation of g-computation on a simulated data set: demonstration of a causal inference technique. American journal of epidemiology 173(7), 731–738 (2011)

    work page 2011

  26. [26]

    BMC medical research methodology 17(1), 1–5 (2017)

    Wang, A., Nianogo, R.A., Arah, O.A.: G-computation of average treatment effects on the treated and the untreated. BMC medical research methodology 17(1), 1–5 (2017)

    work page 2017

  27. [27]

    Biometrical Journal 63(3), 528–557 (2021)

    Daniel, R., Zhang, J., Farewell, D.: Making apples from oranges: Comparing noncollapsible effect estimators and their standard errors after adjustment for different covariate sets. Biometrical Journal 63(3), 528–557 (2021)

    work page 2021

  28. [28]

    arXiv preprint arXiv:2301.09661 (2023)

    Campbell, H., Park, J.E., Jansen, J.P., Cope, S.: Standardization allows for efficient unbiased estimation in observational studies and in indirect treatment comparisons: A comprehensive simulation study. arXiv preprint arXiv:2301.09661 (2023)

    work page arXiv 2023

  29. [29]

    Research synthesis methods 10(4), 582–596 (2019)

    Vo, T.-T., Porcher, R., Chaimani, A., Vansteelandt, S.: A novel approach for identifying and addressing case-mix heterogeneity in individual participant data meta-analysis. Research synthesis methods 10(4), 582–596 (2019)

    work page 2019

  30. [30]

    Rubin, D.B.: Multiple Imputation for Nonresponse in Surveys vol. 81. John Wiley & Sons, ??? (2004)

    work page 2004

  31. [31]

    International journal of epidemiology 44(5), 1731–1737 (2015)

    Westreich, D., Edwards, J.K., Cole, S.R., Platt, R.W., Mumford, S.L., Schisterman, E.F.: Imputation approaches for potential outcomes in causal inference. International journal of epidemiology 44(5), 1731–1737 (2015)

    work page 2015

  32. [32]

    PhD thesis, UCL (University College London) (2022)

    Remiro Az´ ocar, A.: Population-adjusted indirect treatment comparisons with limited access to patient-level data. PhD thesis, UCL (University College London) (2022)

    work page 2022

  33. [33]

    arXiv preprint arXiv:2008.05951 (2020)

    Remiro-Az´ ocar, A., Heath, A., Baio, G.: Marginalization of regression-adjusted treatment effects in indirect comparisons with limited patient-level data. arXiv preprint arXiv:2008.05951 (2020)

    work page arXiv 2008

  34. [34]

    Pharmacoepidemiology and drug safety 28(4), 439 (2019)

    Girman, C.J., Ritchey, M.E., Zhou, W., Dreyer, N.A.: Considerations in characterizing real-world data relevance and quality for regulatory purposes: a commentary. Pharmacoepidemiology and drug safety 28(4), 439 (2019)

    work page 2019

  35. [35]

    Weiss, N.S.: Generalizing from the results of randomized studies of treatment: Can non-randomized studies be of help? European journal of epidemiology 34(8), 715–718 (2019)

    work page 2019

  36. [36]

    Journal of Medical Economics 23(12), 1618–1622 (2020)

    Ramsey, S.D., Adamson, B.J., Wang, X., Bargo, D., Baxi, S.S., Ghosh, S., Meropol, N.J.: Using electronic health record data to identify comparator populations for comparative effectiveness research. Journal of Medical Economics 23(12), 1618–1622 (2020)

    work page 2020

  37. [37]

    to whom do the results of this trial apply?

    Rothwell, P.M.: External validity of randomised controlled trials:“to whom do the results of this trial apply?”. The Lancet 365(9453), 82–93 (2005)

    work page 2005

  38. [38]

    International journal of epidemiology 39(1), 94–96 (2010)

    Rothwell, P.M.: Commentary: External validity of results of randomized trials: disentangling a complex concept. International journal of epidemiology 39(1), 94–96 (2010)

    work page 2010

  39. [39]

    the risk of suicidality among pediatric antidepressant users

    Greenhouse, J.B., Kaizar, E.E., Kelleher, K., Seltman, H., Gardner, W.: Generalizing from clinical trial data: a case study. the risk of suicidality among pediatric antidepressant users. Statistics in medicine 27(11), 1801–1813 (2008)

    work page 2008

  40. [40]

    Clinical Pharmacology & Therapeutics 108(4), 817–825 (2020)

    Happich, M., Brnabic, A., Faries, D., Abrams, K., Winfree, K.B., Girvan, A., Jonsson, P., Johnston, J., Belger, M., 1, I.G.W.P.: Reweighting randomized controlled trial evidence to better reflect real life–a case study of the innovative medicines initiative. Clinical Pharmacology & Therapeutics 108(4), 817–825 (2020)

    work page 2020

  41. [41]

    Journal of educational Psychology 66(5), 688 (1974)

    Rubin, D.B.: Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of educational Psychology 66(5), 688 (1974)

    work page 1974

  42. [42]

    Page 19 of 23 adjustment approaches under model misspecification in individually randomized trials

    Tackney, M.S., Morris, T., White, I., Leyrat, C., Diaz-Ordaz, K., Williamson, E.: A comparison of covariate Remiro-Az´ ocaret al. Page 19 of 23 adjustment approaches under model misspecification in individually randomized trials. Trials 24(1), 1–18 (2023)

    work page 2023

  43. [43]

    Multivariate behavioral research 46(3), 399–424 (2011)

    Austin, P.C.: An introduction to propensity score methods for reducing the effects of confounding in observational studies. Multivariate behavioral research 46(3), 399–424 (2011)

    work page 2011

  44. [44]

    Statistical Science, 538–558 (1994)

    Meng, X.-L.: Multiple-imputation inferences with uncongenial sources of input. Statistical Science, 538–558 (1994)

    work page 1994

  45. [45]

    Statistics in medicine 38(8), 1399–1420 (2019)

    Gabrio, A., Mason, A.J., Baio, G.: A full bayesian model to handle structural ones and missingness in economic evaluations from individual-level data. Statistics in medicine 38(8), 1399–1420 (2019)

    work page 2019

  46. [46]

    In: Proceedings of the Survey Research Methods Section of the American Statistical Association, vol

    Rubin, D.B.: Multiple imputations in sample surveys-a phenomenological bayesian approach to nonresponse. In: Proceedings of the Survey Research Methods Section of the American Statistical Association, vol. 1, pp. 20–34 (1978). American Statistical Association Alexandria, VA, USA

    work page 1978

  47. [47]

    In press, Research synthesis methods (2023)

    Vo, T.-T.: A cautionary note on the use of g-computation in population adjustment. In press, Research synthesis methods (2023)

    work page 2023

  48. [48]

    Statistics in medicine 39(14), 1999–2014 (2020)

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

    work page 1999

  49. [49]

    Journal of statistical software 76(1) (2017)

    Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., Riddell, A.: Stan: A probabilistic programming language. Journal of statistical software 76(1) (2017)

    work page 2017

  50. [50]

    Journal of official statistics 19(1), 1 (2003)

    Raghunathan, T.E., Reiter, J.P., Rubin, D.B.: Multiple imputation for statistical disclosure limitation. Journal of official statistics 19(1), 1 (2003)

    work page 2003

  51. [51]

    Journal of official Statistics 9(2), 461–468 (1993)

    Rubin, D.B.: Statistical disclosure limitation. Journal of official Statistics 9(2), 461–468 (1993)

    work page 1993

  52. [52]

    Journal of Official Statistics 18(4), 531 (2002)

    Reiter, J.P.: Satisfying disclosure restrictions with synthetic data sets. Journal of Official Statistics 18(4), 531 (2002)

    work page 2002

  53. [53]

    Journal of the Royal Statistical Society: Series A (Statistics in Society) 168(1), 185–205 (2005)

    Reiter, J.P.: Releasing multiply imputed, synthetic public use microdata: An illustration and empirical study. Journal of the Royal Statistical Society: Series A (Statistics in Society) 168(1), 185–205 (2005)

    work page 2005

  54. [54]

    Journal of Statistical Theory and Practice 5(2), 335–347 (2011)

    Si, Y., Reiter, J.P.: A comparison of posterior simulation and inference by combining rules for multiple imputation. Journal of Statistical Theory and Practice 5(2), 335–347 (2011)

    work page 2011

  55. [55]

    Journal of the American Statistical Association 102(480), 1462–1471 (2007)

    Reiter, J.P., Raghunathan, T.E.: The multiple adaptations of multiple imputation. Journal of the American Statistical Association 102(480), 1462–1471 (2007)

    work page 2007

  56. [56]

    Journal of Privacy and Confidentiality 7(3), 67–97 (2016)

    Raab, G.M., Nowok, B., Dibben, C.: Practical data synthesis for large samples. Journal of Privacy and Confidentiality 7(3), 67–97 (2016)

    work page 2016

  57. [57]

    NICE DSU technical support document 20 (2019)

    Bujkiewicz, S., Achana, F., Papanikos, T., Riley, R., Abrams, K.: Multivariate meta-analysis of summary data for combining treatment effects on correlated outcomes and evaluating surrogate endpoints. NICE DSU technical support document 20 (2019)

    work page 2019

  58. [58]

    Survey Methodology 29(2), 181–188 (2003)

    Reiter, J.P.: Inference for partially synthetic, public use microdata sets. Survey Methodology 29(2), 181–188 (2003)

    work page 2003

  59. [59]

    Statistics in medicine 38(11), 2074–2102 (2019)

    Morris, T.P., White, I.R., Crowther, M.J.: Using simulation studies to evaluate statistical methods. Statistics in medicine 38(11), 2074–2102 (2019)

    work page 2074

  60. [60]

    Team, R.C., et al.: R: A language and environment for statistical computing (2013)

    work page 2013

  61. [61]

    Statistics in medicine 39(30), 4885–4911 (2020)

    Phillippo, D.M., Dias, S., Ades, A., Welton, N.J.: Assessing the performance of population adjustment methods for anchored indirect comparisons: A simulation study. Statistics in medicine 39(30), 4885–4911 (2020)

    work page 2020

  62. [62]

    Statistics in medicine 26(16), 3078–3094 (2007)

    Austin, P.C.: The performance of different propensity score methods for estimating marginal odds ratios. Statistics in medicine 26(16), 3078–3094 (2007)

    work page 2007

  63. [63]

    Communications in Statistics—Simulation and Computation ® 37(6), 1039–1051 (2008)

    Austin, P.C., Stafford, J.: The performance of two data-generation processes for data with specified marginal treatment odds ratios. Communications in Statistics—Simulation and Computation ® 37(6), 1039–1051 (2008)

    work page 2008

  64. [64]

    Transportability of model-based estimands in evidence synthesis

    Remiro-Az´ ocar, A.: Purely prognostic variables may modify marginal treatment effects for non-collapsible effect measures. arXiv preprint arXiv:2210.01757 (2022)

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  65. [65]

    R package version 2(1) (2020)

    Goodrich, B., Gabry, J., Ali, I., Brilleman, S.: rstanarm: Bayesian applied regression modeling via stan. R package version 2(1) (2020)

    work page 2020

  66. [66]

    R package version 2(21.2) (2020)

    Team, S.D.: Rstan: The r interface to stan. R package version 2(21.2) (2020)

    work page 2020

  67. [67]

    BMC medical research methodology 9, 1–5 (2009)

    Nemes, S., Jonasson, J.M., Genell, A., Steineck, G.: Bias in odds ratios by logistic regression modelling and sample size. BMC medical research methodology 9, 1–5 (2009)

    work page 2009

  68. [68]

    American Journal of Epidemiology 192(9), 1536–1544 (2023)

    Naimi, A.I., Mishler, A.E., Kennedy, E.H.: Challenges in obtaining valid causal effect estimates with machine learning algorithms. American Journal of Epidemiology 192(9), 1536–1544 (2023)

    work page 2023

  69. [69]

    Statistical methods in medical research 27(10), 3183–3204 (2018)

    Keil, A.P., Daza, E.J., Engel, S.M., Buckley, J.P., Edwards, J.K.: A bayesian approach to the g-formula. Statistical methods in medical research 27(10), 3183–3204 (2018)

    work page 2018

  70. [70]

    Environmental Health 13(1), 1–10 (2014)

    Keil, A.P., Daniels, J.L., Hertz-Picciotto, I.: Autism spectrum disorder, flea and tick medication, and adjustments for exposure misclassification: the charge (childhood autism risks from genetics and environment) case–control study. Environmental Health 13(1), 1–10 (2014)

    work page 2014

  71. [71]

    Journal of the American Statistical Association 89(428), 1535–1546 (1994)

    Madigan, D., Raftery, A.E.: Model selection and accounting for model uncertainty in graphical models using occam’s window. Journal of the American Statistical Association 89(428), 1535–1546 (1994)

    work page 1994

  72. [72]

    Biometrics 47(3), 871–881 (1991)

    Dixon, D.O., Simon, R.: Bayesian subset analysis. Biometrics 47(3), 871–881 (1991)

    work page 1991

  73. [73]

    Journal of the Royal Statistical Society: Series A (Statistics in Society) 157(3), 357–387 (1994)

    Spiegelhalter, D.J., Freedman, L.S., Parmar, M.K.: Bayesian approaches to randomized trials. Journal of the Royal Statistical Society: Series A (Statistics in Society) 157(3), 357–387 (1994)

    work page 1994

  74. [74]

    Biometrics 53(2), 456–464 (1997)

    Simon, R., Freedman, L.S.: Bayesian design and analysis of two x two factorial clinical trials. Biometrics 53(2), 456–464 (1997)

    work page 1997

  75. [75]

    Prevention Science 16, 475–485 (2015)

    Stuart, E.A., Bradshaw, C.P., Leaf, P.J.: Assessing the generalizability of randomized trial results to target populations. Prevention Science 16, 475–485 (2015)

    work page 2015

  76. [76]

    Evaluation review 41(4), 357–388 (2017)

    Stuart, E.A., Rhodes, A.: Generalizing treatment effect estimates from sample to population: A case study in the difficulties of finding sufficient data. Evaluation review 41(4), 357–388 (2017)

    work page 2017

  77. [77]

    Research Methods in Medicine Remiro-Az´ ocaret al

    Vuong, M.L., Tu, P.H.T., Duong, K.L., Vo, T.-T.: Development of minimum reporting sets of patient characteristics in epidemiological research: a methodological systematic review. Research Methods in Medicine Remiro-Az´ ocaret al. Page 20 of 23 & Health Sciences, 26320843231191777 (2023)

    work page 2023

  78. [78]

    The Annals of Applied Statistics 11(1), 225–247 (2017)

    Nguyen, T.Q., Ebnesajjad, C., Cole, S.R., Stuart, E.A.: Sensitivity analysis for an unobserved moderator in rct-to-target-population generalization of treatment effects. The Annals of Applied Statistics 11(1), 225–247 (2017)

    work page 2017

  79. [79]

    Standard

    Dahabreh, I.J., Hern´ an, M.A.: Extending inferences from a randomized trial to a target population. European journal of epidemiology 34, 719–722 (2019) Remiro-Az´ ocaret al. Page 21 of 23 Figures Figure 1 Multiple imputation marginalization (MIM). A Bayesian directed acyclic graph representing MIM and its two main stages: (1) synthetic data generation; a...

    work page 2019