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arxiv: 2607.00222 · v1 · pith:QLQEO6INnew · submitted 2026-06-30 · 📊 stat.ME · math.ST· stat.TH

Causal Inference for All: Marginal Estimands for Outcomes Truncated by Death

Ruixuan Zhao , Mats Stensrud , Linbo Wang This is my paper

Pith reviewed 2026-07-02 17:25 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords causal inferencetruncated by deathseparable effectsmarginal estimandslongitudinal datasurvival outcomestreatment effectsprostate cancer
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The pith

Single-world marginal separable effects deliver causally interpretable summaries for the entire population when outcomes are truncated by death.

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

Studies with outcomes observed only among survivors face a choice between targeting a latent subgroup or using summaries that resist causal interpretation. This paper defines a new family of marginal estimands that apply to everyone in the study population while remaining causally meaningful. The key objects are single-world marginal separable effects, which extend earlier conditional versions by averaging over the full population with the help of longitudinal measurements. Identification and estimation procedures are derived under standard assumptions, and the methods are illustrated by reanalyzing a prostate cancer trial where the choice of estimand changes the apparent treatment conclusion.

Core claim

The paper claims that single-world marginal separable effects generalize conditional separable effects to full-population summaries. These estimands concern the entire population, remain causally interpretable, and are identified from observed longitudinal data under the usual consistency, positivity, and no-unmeasured-confounding assumptions without further restrictions on the outcome process.

What carries the argument

single-world marginal separable effects, which separate the effect of treatment on the non-mortality outcome from its effect on survival while producing population-level summaries.

If this is right

  • The estimands apply to the entire study population rather than only survivors.
  • They preserve a direct causal reading on the non-mortality outcome.
  • Identification and estimation use only the longitudinal data routinely collected in such studies.
  • Different estimands can produce different conclusions about treatment in the same trial data.

Where Pith is reading between the lines

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

  • The same construction could be tried in other settings where an outcome is undefined after an intercurrent event.
  • Head-to-head comparisons with survivor-average causal effects on real datasets would clarify when the marginal versions change substantive conclusions.
  • The framework may extend naturally to time-varying treatments or multiple truncated outcomes.

Load-bearing premise

Standard causal assumptions together with the observed longitudinal data structure suffice to identify the marginal separable effects.

What would settle it

A simulation study with a known data-generating process in which the true full-population effects are calculated directly; the new estimators should recover those values when the identifying assumptions hold.

Figures

Figures reproduced from arXiv: 2607.00222 by Linbo Wang, Mats Stensrud, Ruixuan Zhao.

Figure 1
Figure 1. Figure 1: Illustration of the “while guaranteed-survival” and “while extended-survival” estimands, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Longitudinal data structure with outcomes truncated by death. Survival status [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two causal diagrams illustrating the separable effects framework: panel (a) shows the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A toy example with two individuals to illustrate our proposed estimands. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the area under the potential outcome curves for individual [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the areas under the potential outcome curves for two estimands. The blue [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Estimated survival curve under two chemotherapies. [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The left panel shows the estimated SACE(t) at three time points together with the corre [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
read the original abstract

In longitudinal studies, outcomes of interest are often truncated by death, meaning that they are only observed or well-defined conditional on intercurrent events such as survival. Existing strategies face a trade-off: causally interpretable estimands, such as survivor average causal effects, target a latent subgroup, whereas while-alive and composite summaries apply to the full population but are difficult to interpret as causal effects on the non-mortality outcome. We address these challenges by introducing methodology for a new set of estimands that (i) concern the entire population, (ii) remain causally interpretable, and (iii) leverage the longitudinal data commonly available in studies with outcomes truncated by death. The set of estimands includes single-world marginal separable effects that generalize conditional separable effects to full-population summaries. We develop identification and estimation results for these estimands and apply the methodology in a reanalysis of a prostate cancer trial, highlighting how different estimands can yield different treatment conclusions.

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 / 2 minor

Summary. The manuscript introduces a new class of estimands called single-world marginal separable effects for causal inference when outcomes are truncated by death. These targets apply to the full population (rather than a latent survivor subgroup), remain causally interpretable, and are identified from longitudinal data under standard assumptions (consistency, positivity, no unmeasured confounding). The paper develops identification and estimation results that generalize prior conditional separable effects, and illustrates the methods via reanalysis of a prostate cancer trial where different estimands produce differing treatment conclusions.

Significance. If the identification and estimation results are correct, the work supplies a practically useful addition to the truncated-outcomes literature by delivering full-population causal effects without requiring stronger restrictions on the outcome process or additional measurements. The explicit generalization from conditional to marginal versions, together with the real-data demonstration, could influence how analysts summarize treatment effects on non-mortality outcomes in clinical trials.

minor comments (2)
  1. [Abstract] The abstract states that the marginal effects 'leverage the longitudinal data commonly available,' but the main text should include an explicit statement (perhaps in §3 or §4) confirming that no additional post-baseline covariates beyond those already measured are required for identification.
  2. Notation for the separable effects (e.g., the distinction between single-world and multi-world versions) should be introduced with a small illustrative numerical example early in the paper to aid readability before the formal identification theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation for minor revision. No major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces single-world marginal separable effects as new full-population estimands that generalize conditional separable effects, with identification and estimation derived from longitudinal data under standard causal assumptions (consistency, positivity, no unmeasured confounding). No equations, definitions, or steps in the abstract or described methodology reduce the target estimands to quantities fitted from the same data, self-definitions, or load-bearing self-citations by construction. The derivation remains self-contained and independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard causal identification assumptions that are not derived in the paper; no free parameters or new invented entities are described in the abstract.

axioms (1)
  • domain assumption Standard causal assumptions (consistency, positivity, conditional exchangeability/no unmeasured confounding) hold for identification of the marginal separable effects from longitudinal observed data.
    Required for any identification result in causal inference with truncation by death; invoked implicitly when claiming the estimands are identifiable.

pith-pipeline@v0.9.1-grok · 5700 in / 1350 out tokens · 35464 ms · 2026-07-02T17:25:03.198142+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

209 extracted references · 14 canonical work pages · 5 internal anchors

  1. [2]

    V., Roychoudhury, S., Schmidli, H., Shentu, Y., and Wolbers, M

    Bornkamp, B., Rufibach, K., Lin, J., Liu, Y., Mehrotra, D. V., Roychoudhury, S., Schmidli, H., Shentu, Y., and Wolbers, M. (2021). Principal stratum strategy: potential role in drug development. Pharmaceutical Statistics , 20(4):737--751

    2021

  2. [3]

    M., Fayers, P., and Brown, J

    Cocks, K., King, M., Velikova, G., de Castro Jr, G., St-James, M. M., Fayers, P., and Brown, J. (2012). Evidence-based guidelines for interpreting change scores for the European Organisation for the Research and Treatment of Cancer Quality of Life Questionnaire Core 30 . European Journal of Cancer , 48(11):1713--1721

    2012

  3. [4]

    Comment, L., Mealli, F., Haneuse, S., and Zigler, C. M. (2025). Survivor average causal effects for continuous time: a principal stratification approach to causal inference with semicompeting risks. Biometrical Journal , 67(2):e70041

    2025

  4. [5]

    Ding, P., Geng, Z., Yan, W., and Zhou, X.-H. (2011). Identifiability and estimation of causal effects by principal stratification with outcomes truncated by death. Journal of the American Statistical Association , 106(496):1578--1591

    2011

  5. [6]

    B., Bosch, R

    Gilbert, P. B., Bosch, R. J., and Hudgens, M. G. (2003). Sensitivity analysis for the assessment of causal vaccine effects on viral load in HIV vaccine trials. Biometrics , 59(3):531--541

    2003

  6. [7]

    Grossi, G., Mariani, M., Mattei, A., and Mealli, F. (2025). Bayesian principal stratification with longitudinal data and truncation by death. Econometrics and Statistics , pages 1--12. In press

    2025

  7. [8]

    Hern \'a n, M. A. and Scharfstein, D. (2018). Cautions as regulators move to end exclusive reliance on intention to treat. Annals of Internal Medicine , 168(7):515--516

    2018

  8. [9]

    ICH E9(R1): Addendum on Estimands and Sensitivity Analysis in Clinical Trials to the Guideline on Statistical Principles for Clinical Trials

    ICH (2019). ICH E9(R1): Addendum on Estimands and Sensitivity Analysis in Clinical Trials to the Guideline on Statistical Principles for Clinical Trials . Technical report, ICH. Available at https://database.ich.org/sites/default/files/E9-R1_Step4_Guideline_2019_1203.pdf

    2019

  9. [10]

    G., Ryalen, P

    Janvin, M., Young, J. G., Ryalen, P. C., and Stensrud, M. J. (2024). Causal inference with recurrent and competing events. Lifetime Data Analysis , 30(1):59--118

    2024

  10. [11]

    Lachin, J. M. (2020). Worst-rank score methods—a nonparametric approach to informatively missing data. JAMA , 324(16):1670--1671

    2020

  11. [12]

    J., D'Agostino, R., Cohen, M

    Little, R. J., D'Agostino, R., Cohen, M. L., Dickersin, K., Emerson, S. S., Farrar, J. T., Frangakis, C., Hogan, J. W., Molenberghs, G., Murphy, S. A., et al. (2012). The prevention and treatment of missing data in clinical trials. New England Journal of Medicine , 367(14):1355--1360

    2012

  12. [13]

    J., and Tchetgen Tchetgen, E

    Park, C., Stensrud, M. J., and Tchetgen Tchetgen, E. J. (2026). Proximal causal inference for conditional separable effects. Journal of the Royal Statistical Society Series B: Statistical Methodology , page qkag051. Advance article

    2026

  13. [14]

    P., Tangen, C

    Petrylak, D. P., Tangen, C. M., Hussain, M. H., Lara Jr, P. N., Jones, J. A., Taplin, M. E., Burch, P. A., Berry, D., Moinpour, C., Kohli, M., et al. (2004). Docetaxel and estramustine compared with mitoxantrone and prednisone for advanced refractory prostate cancer. New England Journal of Medicine , 351(15):1513--1520

    2004

  14. [15]

    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(9-12):1393--1512

    1986

  15. [16]

    Robins, J. M. and Richardson, T. S. (2010). Alternative graphical causal models and the identification of direct effects. In Shrout, P. E., Keyes, K. M., and Ornstein, K., editors, Causality and Psychopathology: Finding the Determinants of Disorders and their Cures , pages 103--158. Oxford University Press, Oxford, UK

    2010

  16. [17]

    Rubin, D. B. (2006). Causal inference through potential outcomes and principal stratification: application to studies with ``censoring'' due to death. Statistical Science , 21(3):299--309

    2006

  17. [18]

    Stensrud, M. J. and Dukes, O. (2022). Translating questions to estimands in randomized clinical trials with intercurrent events. Statistics in Medicine , 41(16):3211--3228

    2022

  18. [19]

    J., Robins, J

    Stensrud, M. J., Robins, J. M., Sarvet, A., Tchetgen Tchetgen, E. J., and Young, J. G. (2023). Conditional separable effects. Journal of the American Statistical Association , 118(544):2671--2683

    2023

  19. [20]

    J., Young, J

    Stensrud, M. J., Young, J. G., Didelez, V., Robins, J. M., and Hern \'a n, M. A. (2022). Separable effects for causal inference in the presence of competing events. Journal of the American Statistical Association , 117(537):175--183

    2022

  20. [21]

    F., Osoba, D., Stockler, M

    Tannock, I. F., Osoba, D., Stockler, M. R., Ernst, D. S., Neville, A. J., Moore, M. J., Armitage, G. R., Wilson, J. J., Venner, P. M., Coppin, C., et al. (1996). Chemotherapy with mitoxantrone plus prednisone or prednisone alone for symptomatic hormone-resistant prostate cancer: a Canadian randomized trial with palliative end points. Journal of Clinical O...

    1996

  21. [22]

    Troyanskaya, O., Cantor, M., Sherlock, G., Brown, P., Hastie, T., Tibshirani, R., Botstein, D., and Altman, R. B. (2001). Missing value estimation methods for DNA microarrays. Bioinformatics , 17(6):520--525

    2001

  22. [23]

    S., and Zhou, X.-H

    Wang, L., Richardson, T. S., and Zhou, X.-H. (2017a). Causal analysis of ordinal treatments and binary outcomes under truncation by death. Journal of the Royal Statistical Society Series B: Statistical Methodology , 79(3):719--735

  23. [24]

    Wang, L., Zhou, X.-H., and Richardson, T. S. (2017b). Identification and estimation of causal effects with outcomes truncated by death. Biometrika , 104(3):597--612

  24. [25]

    Wei, J., Muetze, T., Jahn-Eimermacher, A., and Roger, J. (2023). Properties of two while-alive estimands for recurrent events and their potential estimators. Statistics in Biopharmaceutical Research , 15(2):257--267

    2023

  25. [26]

    and Ding, P

    Yang, F. and Ding, P. (2018). Using survival information in truncation by death problems without the monotonicity assumption. Biometrics , 74(4):1232--1239

    2018

  26. [27]

    J., Klaassen, C

    Bickel, P. J., Klaassen, C. A., Bickel, P. J., Ritov, Y., Klaassen, J., Wellner, J. A., and Ritov, Y. (1993). Efficient and adaptive estimation for semiparametric models , volume 4. Springer

    1993

  27. [28]

    E., and Beresford, S

    Diehr, P., Patrick, D., Hedrick, S., Rothman, M., Grembowski, D., Raghunathan, T. E., and Beresford, S. (1995). Including deaths when measuring health status over time. Medical Care , 33(4 Suppl):AS164--AS172

    1995

  28. [29]

    Frangakis, C. E. and Rubin, D. B. (2002). Principal stratification in causal inference. Biometrics , 58(1):21--29

    2002

  29. [30]

    F., Johnson, L

    Kurland, B. F., Johnson, L. L., Egleston, B. L., and Diehr, P. H. (2009). Longitudinal data with follow-up truncated by death: match the analysis method to research aims. Statistical Science , 24(2):211--222

    2009

  30. [31]

    J., Bosch, R

    Lok, J. J., Bosch, R. J., Benson, C. A., Collier, A. C., Robbins, G. K., Shafer, R. W., Hughes, M. D., et al. (2010). Long-term increase in CD4+ T -cell counts during combination antiretroviral therapy for HIV-1 infection. AIDS , 24(12):1867--1876

    2010

  31. [33]

    Rosenbaum, P. R. (2006). Comment: the place of death in the quality of life. Statistical Science , 21(3):313--316

    2006

  32. [34]

    Tsiatis, A. A. (2006). Semiparametric theory and missing data . Springer

    2006

  33. [35]

    J., and Lok, J

    Xiang, Q., Bosch, R. J., and Lok, J. J. (2023). The survival-incorporated median vs the median in the survivors or in the always-survivors: What are we measuring? and why? Statistics in Medicine , 42(29):5479--5490

    2023

  34. [36]

    G., Stensrud, M

    Young, J. G., Stensrud, M. J., Tchetgen Tchetgen, E. J., and Hern \'a n, M. A. (2020). A causal framework for classical statistical estimands in failure-time settings with competing events. Statistics in Medicine , 39(8):1199--1236

    2020

  35. [37]

    Center for the Statistics and the Social Sciences, University of Washington Series

    Single world intervention graphs (SWIGs): A unification of the counterfactual and graphical approaches to causality , author=. Center for the Statistics and the Social Sciences, University of Washington Series. Working Paper , volume=. 2013 , publisher=

    2013

  36. [38]

    The International Journal of Biostatistics , volume=

    Principal stratification—uses and limitations , author=. The International Journal of Biostatistics , volume=

  37. [39]

    Epidemiology , volume=

    Identified versus interesting causal effects in fertility trials and other settings with competing or truncation events , author=. Epidemiology , volume=. 2021 , publisher=

    2021

  38. [40]

    Annals of Epidemiology , volume=

    Does water kill? A call for less casual causal inferences , author=. Annals of Epidemiology , volume=. 2016 , publisher=

    2016

  39. [41]

    Statistics in Medicine , volume=

    A causal framework for classical statistical estimands in failure-time settings with competing events , author=. Statistics in Medicine , volume=. 2020 , publisher=

    2020

  40. [42]

    Long-term increase in

    Lok, Judith J and Bosch, Ronald J and Benson, Constance A and Collier, Ann C and Robbins, Gregory K and Shafer, Robert W and Hughes, Michael D and others , journal=. Long-term increase in. 2010 , publisher=

    2010

  41. [43]

    Statistics in Medicine , volume=

    The survival-incorporated median vs the median in the survivors or in the always-survivors: What are we measuring? and Why? , author=. Statistics in Medicine , volume=. 2023 , publisher=

    2023

  42. [44]

    Miguel A. Hern. Causal Inference: What If , year =

  43. [45]

    Statistics in Medicine , volume=

    Translating questions to estimands in randomized clinical trials with intercurrent events , author=. Statistics in Medicine , volume=. 2022 , publisher=

    2022

  44. [46]

    Statistical Science , volume=

    Comment: the place of death in the quality of life , author=. Statistical Science , volume=. 2006 , publisher=

    2006

  45. [47]

    Statistics in Biopharmaceutical Research , volume=

    Estimands for recurrent event endpoints in the presence of a terminal event , author=. Statistics in Biopharmaceutical Research , volume=. 2023 , publisher=

    2023

  46. [48]

    Statistics in Biopharmaceutical Research , volume=

    A general framework for treatment effect estimators considering patient adherence , author=. Statistics in Biopharmaceutical Research , volume=. 2020 , publisher=

    2020

  47. [49]

    Statistics in Biopharmaceutical Research , volume=

    Hypothetical estimands in clinical trials: a unification of causal inference and missing data methods , author=. Statistics in Biopharmaceutical Research , volume=. 2023 , publisher=

    2023

  48. [50]

    JAMA , volume=

    Worst-rank score methods—A nonparametric approach to informatively missing data , author=. JAMA , volume=. 2020 , publisher=

    2020

  49. [51]

    JAMA , volume=

    Effect of doxycycline on aneurysm growth among patients with small infrarenal abdominal aortic aneurysms: a randomized clinical trial , author=. JAMA , volume=. 2020 , publisher=

    2020

  50. [52]

    New England Journal of Medicine , volume=

    Phase 3 trial of sotatercept for treatment of pulmonary arterial hypertension , author=. New England Journal of Medicine , volume=. 2023 , publisher=

    2023

  51. [53]

    Statistics in Biopharmaceutical Research , volume=

    Estimands and their estimators for clinical trials impacted by the COVID-19 pandemic: a report from the NISS Ingram Olkin Forum Series on unplanned clinical trial disruptions , author=. Statistics in Biopharmaceutical Research , volume=. 2023 , publisher=

    2023

  52. [54]

    Journal of Educational and Behavioral Statistics , volume=

    Estimation of causal effects via principal stratification when some outcomes are truncated by ``death'' , author=. Journal of Educational and Behavioral Statistics , volume=. 2003 , publisher=

    2003

  53. [55]

    Sensitivity analysis for the assessment of causal vaccine effects on viral load in

    Gilbert, Peter B and Bosch, Ronald J and Hudgens, Michael G , journal=. Sensitivity analysis for the assessment of causal vaccine effects on viral load in. 2003 , publisher=

    2003

  54. [56]

    Sensitivity analyses comparing outcomes only existing in a subset selected post-randomization, conditional on covariates, with application to

    Shepherd, Bryan E and Gilbert, Peter B and Jemiai, Yannis and Rotnitzky, Andrea , journal=. Sensitivity analyses comparing outcomes only existing in a subset selected post-randomization, conditional on covariates, with application to. 2006 , publisher=

    2006

  55. [57]

    Journal of the Royal Statistical Society: Series B , year =

    Lu, Shan and Jiang, Zhichao and Ding, Peng , title =. Journal of the Royal Statistical Society: Series B , year =

  56. [58]

    Journal of the Royal Statistical Society Series A: Statistics in Society , volume=

    Matching methods for truncation by death problems , author=. Journal of the Royal Statistical Society Series A: Statistics in Society , volume=. 2023 , publisher=

    2023

  57. [59]

    Pharmaceutical Statistics , volume=

    Principal stratum strategy: potential role in drug development , author=. Pharmaceutical Statistics , volume=. 2021 , publisher=

    2021

  58. [60]

    Annals of Internal Medicine , volume=

    Cautions as regulators move to end exclusive reliance on intention to treat , author=. Annals of Internal Medicine , volume=. 2018 , publisher=

    2018

  59. [61]

    Clinical Trials , volume=

    A constructive critique of the draft ICH E9 addendum , author=. Clinical Trials , volume=. 2019 , publisher=

    2019

  60. [62]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Causal analysis of ordinal treatments and binary outcomes under truncation by death , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2017 , publisher=

    2017

  61. [63]

    2019 , publisher=

    Statistical analysis with missing data , author=. 2019 , publisher=

    2019

  62. [64]

    Biometrika , volume=

    Inference and missing data , author=. Biometrika , volume=. 1976 , publisher=

    1976

  63. [65]

    Technical Report , year =

    Demystifying the separable effects , author=. Technical Report , year =

  64. [66]

    2024 , publisher=

    The estimands framework: a primer on the ICH E9 (R1) addendum , author=. 2024 , publisher=

    2024

  65. [67]

    Lifetime Data Analysis , volume=

    Causal inference with recurrent and competing events , author=. Lifetime Data Analysis , volume=. 2024 , publisher=

    2024

  66. [68]

    Statistical Science , volume=

    Causal inference through potential outcomes and principal stratification: application to studies with ``censoring'' due to death , author=. Statistical Science , volume=. 2006 , publisher=

    2006

  67. [69]

    The International Journal of Biostatistics , volume=

    Principal stratification—a goal or a tool? , author=. The International Journal of Biostatistics , volume=

  68. [70]

    Estimating treatment effects with competing intercurrent events in randomized controlled trials

    Estimating treatment effects with competing intercurrent events in randomized controlled trials , author=. arXiv preprint arXiv:2503.03049 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  69. [71]

    New England Journal of Medicine , volume=

    Docetaxel plus prednisone or mitoxantrone plus prednisone for advanced prostate cancer , author=. New England Journal of Medicine , volume=. 2004 , publisher=

    2004

  70. [72]

    Current Treatment Options in Oncology , volume=

    Therapeutic strategies for bone metastases and their clinical sequelae in prostate cancer , author=. Current Treatment Options in Oncology , volume=. 2012 , publisher=

    2012

  71. [73]

    Urology , volume=

    Future directions in the treatment of androgen-independent prostate cancer , author=. Urology , volume=. 2005 , publisher=

    2005

  72. [74]

    Journal of Statistical Software , volume=

    Imputation with the R Package VIM , author=. Journal of Statistical Software , volume=

  73. [75]

    arXiv preprint arXiv:2509.10067 , year=

    Weakening assumptions in the evaluation of treatment effects in longitudinal randomized trials with truncation by death or other intercurrent events , author=. arXiv preprint arXiv:2509.10067 , year=

    work page arXiv

  74. [76]

    Statistics in Biopharmaceutical Research , volume=

    Properties of two while-alive estimands for recurrent events and their potential estimators , author=. Statistics in Biopharmaceutical Research , volume=. 2023 , publisher=

    2023

  75. [77]

    Biometrics , volume=

    Using survival information in truncation by death problems without the monotonicity assumption , author=. Biometrics , volume=. 2018 , publisher=

    2018

  76. [78]

    2006 , publisher=

    Semiparametric theory and missing data , author=. 2006 , publisher=

    2006

  77. [79]

    1993 , publisher=

    Efficient and adaptive estimation for semiparametric models , author=. 1993 , publisher=

    1993

  78. [80]

    Econometrica: Journal of the econometric society , pages=

    Large sample properties of generalized method of moments estimators , author=. Econometrica: Journal of the econometric society , pages=. 1982 , publisher=

    1982

  79. [81]

    Biometrics , pages=

    Estimating medical costs from incomplete follow-up data , author=. Biometrics , pages=. 1997 , publisher=

    1997

  80. [82]

    New England Journal of Medicine , volume=

    Docetaxel and estramustine compared with mitoxantrone and prednisone for advanced refractory prostate cancer , author=. New England Journal of Medicine , volume=. 2004 , publisher=

    2004

Showing first 80 references.