Statistical methods for clustered competing risk data when the event types are only available in a training dataset

Molin Wang; Yujie Wu

REVIEW 4 major objections 4 minor 20 references

Statistical methods for clustered competing risk data when the event types are only available in a training dataset

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Cause-specific hazard ratios can be estimated from clustered competing-risks data with missing event types by substituting classifier probabilities for the unobserved event-type indicators in the penalized partial likelihood.

desk verdict Useful extension of cause-specific frailty models to missing event types, but the weights ignore event time; under non-proportional baselines the estimators can be biased, and the simulations don't cover that. read the letter →

arxiv 2505.02217 v1 pith:YFENKQLW submitted 2025-05-04 stat.ME

Yujie Wu , Molin Wang This is my paper · ORCID

classification stat.ME MSC 62N0162N0262H3062P10

keywords competingriskscause-specifichazardsfrailtymodelspenalizedpartiallikelihoodmissingeventtypesweightedestimatingequationsimputationhearingloss

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

Epidemiological studies often know when someone dies or develops a condition but not which of several competing causes was responsible. This paper develops methods for clustered survival data—for example, two ears within a person—where the event type is missing in the main study but available in a small training set together with predictive covariates. The authors propose two approaches: a weighted penalized partial likelihood that substitutes the estimated probability of each event type for the unknown indicator, and an imputation approach that assigns the most probable type. Simulations show both estimators usually have bias below 5% and confidence intervals with coverage close to the nominal 95% when the classifier is accurate. An application to hearing-loss data finds a significant association between tinnitus and metabolic+sensory hearing loss, but not the other two phenotypes.

What carries the argument

The load-bearing object is the penalized partial likelihood (PPLL) for the cause-specific proportional hazards frailty model, where each cluster (a participant) carries correlated random effects across the $K$ event types. In the complete-data score equation, the factor $I(\delta_{ij}=k)$ multiplies the 'observed-minus-expected' covariate contribution of each failed unit. The paper's central move is to condition on the observed censoring indicator $\Delta_{ij}=1$ and the predictive covariates $W_{ij}$, replacing $I(\delta_{ij}=k)$ with $P(\delta_{ij}=k\mid W_{ij},\Delta_{ij}=1)$ in the weighted approach, or with $I(\hat\delta_{ij}=k)$ in the imputed approach. This substitution turns the PPLL into a weighted partial likelihood that can be maximized with standard penalized survival software.

What would settle it

Simulate a main study and a training dataset drawn from different class-conditional distributions (for example, shift the mean audiometric thresholds for a given hearing-loss phenotype by several decibels between the two samples), fit the weighted estimator, and check whether the percent bias of the cause-specific log-hazard ratios exceeds the roughly 5% level reported in the paper; if so, the transportability assumption is violated in a way the current simulations do not capture.

Watch

Extended reading notes

Core claim

The paper's central claim is that the score equations of the penalized partial likelihood remain valid when each unknown event-type indicator $I(\delta_{ij}=k)$ is replaced by the conditional probability $P(\delta_{ij}=k\mid W_{ij},\Delta_{ij}=1)$ estimated from a training set. The resulting weighted estimating equations yield cause-specific hazard ratio estimates whose percent biases in simulations are generally under 5% and whose 95% confidence intervals have coverage close to nominal. A hard-imputation alternative, which assigns $\hat\delta_{ij}=\arg\max_k \hat P(\delta_{ij}=k)$, performs comparably and is sometimes less biased for large effects, while the weighted version is usually more efficient. The paper further claims that analytical variance estimators—sandwich and inverse Hessian—provide usable inference, and that the weighted model can be fitted with existing software through data duplication.

Load-bearing premise

The classifier that supplies the event-type probabilities must be transportable: the relationship between the covariates $W_{ij}$ and the event types in the training dataset must match the relationship in the main study, or the weights and imputed types are systematically wrong and the hazard ratios are biased.

Editorial extensions

If this is right

Careful analysts can now estimate cause-specific hazard ratios in cohort studies where event types are only ascertained in a validation sub-study, without discarding the main cohort.
The derived variance estimators make the method usable for inference, not just point estimation, in the settings the simulations cover.
Because the weighted model can be fit through data duplication in existing software packages, the approach is implementable with tools already used in practice.
In the hearing-loss example, the method yields a stable, statistically significant elevated hazard ratio for tinnitus with metabolic+sensory hearing loss across five classification models.

Reading between the lines

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

The variance formulas ignore that the event-type probabilities are themselves estimated; with small training samples, standard errors may be understated, and a bootstrap or delta-method correction would propagate that uncertainty.
The simulations train and apply the classifier under the same data-generating mechanism, so the transportability assumption is not stress-tested; a distribution-shift simulation would quantify how miscalibration translates into bias.
The weighted estimator is a form of inverse-probability weighting, suggesting a doubly robust extension that remains consistent if either the survival model or the classification model is misspecified.
The paper notes its event-time measurement-error assumption may fail in self-reported hearing loss data; correcting for both outcome and event-time error is a natural next step.

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.

Desk Editor's Note

read the letter

Colleague,

This paper is worth a look if you work on competing risks with missing causes. It extends cause-specific proportional hazards frailty models to the case where event types are missing in the main study but available in a training set, using weighted or imputed estimating equations. That combination is new in the cited literature, and the implementation via coxme is a practical plus.

What it does well: clear setup, standard penalized partial likelihood machinery, simulations show low bias and good coverage when baselines are equal and the classifier is accurate. They give analytical variance formulas. The real application to hearing loss is interesting, and the stacking ensemble is a thoughtful touch.

The biggest soft spot is in Equation (6). The weight is P(delta=k | W, Delta=1), marginal over event time. For the score to be unbiased at the true beta, you need the conditional expectation of I(delta=k) given the full data, including T. That conditional expectation depends on the cause-specific baseline hazards at t. If the lambda_0k(t) are not proportional across causes, the time-marginal weight is misspecified and the estimating equation has nonzero mean. The simulation sets lambda_0k(t)=1 for all k, so the bias mechanism never appears. That is central, not minor.

Second, the variance estimator deliberately ignores uncertainty in the estimated event probabilities. With training samples of 50 or 100 in simulations and 338 in the application, confidence intervals could be anti-conservative. The paper actually states it ignores this, so the authors are candid, but the consequence is understated.

Third, the real-data analysis transports a classifier from MUSC to the AAA cohort. The paper notes the training set is external but does not discuss possible differences in phenotype definitions or population. The simulation bakes in transportability by generating training and main data from the same mechanism.

The authors do flag the measurement error assumption and the ignored probability estimation, so they are engaged. They just miss the time-dependence issue, which is the more serious one.

Bottom line: the paper addresses a real problem and the methods are coherent, but the weighted estimator needs a correction or at least a stated condition of proportional cause-specific baseline hazards, and the real-data conclusions should be framed as conditional on that plus transportability. I would send it to a serious referee, but I would ask the referee to focus on the time-dependence of the weights and to require simulations with non-proportional baselines.

Referee Report

4 major / 4 minor

Summary. The manuscript develops weighted and imputed estimating equations for cause-specific hazard ratios in clustered competing risks data when event types are missing in the main study but are available in a training dataset. In the weighted approach, the unobserved indicators I(δij=k) in the penalized partial likelihood score are replaced by classifier-based probabilities P(δij=k | W, Δ=1); in the imputed approach, event types are replaced by the argmax of those predicted probabilities. The authors derive a sandwich-type variance estimator and also recommend the inverse-Hessian estimator from the coxme package, evaluate finite-sample performance in simulations with constant and equal cause-specific baseline hazards, and apply the methods to estimate associations between tinnitus and metabolic, sensory, and metabolic+sensory hearing loss in the CHEARS AAA cohort using a classifier trained on the MUSC database.

Significance. If the proposed estimators are valid, the paper fills a practical gap: clustered competing risks data with event types available only in an auxiliary training set are common in epidemiologic studies, and the proposed use of existing software (coxme) would make the methods accessible. The real-data application and the explicit discussion of the event-time measurement error limitation are welcome. However, the central estimator is not shown to be an unbiased estimating equation under the stated model unless strong extra conditions hold; the simulation design uses equal constant baseline hazards and generates training and main data from the same mechanism, so the reported bias and coverage results do not establish general validity. The significance of the contribution is therefore conditional on a substantial reworking of the weighting scheme and the simulation evidence.

major comments (4)

[Section 2.2, Eq. (6); Section 2.3; Section 3; Tables 2–3] The weighted estimating equation in Eq. (6) replaces I(δij=k) with P(δij=k | Wij, Δij=1), but the quantity needed for an unbiased conditional score is P(δij=k | Tij, Xij, vi, Wij, Δij=1). Under the assumed cause-specific frailty model, this conditional probability is proportional to λ0k(Tij) exp(Xij^T βk + vik) times the classifier's contribution from Wij, divided by the sum over causes. It therefore depends on the event time through the baseline hazard ratio λ0k(Tij)/λ0l(Tij) and on the covariates and frailty through exp{Xij^T(βk-βl) + vik - vil}. Equation (6) uses a time-marginal probability that conditions only on W and Δ, so the weighted score generally has nonzero expectation at the true β. The imputed estimator in Section 2.3 inherits the same problem because the hard assignments also ignore event time, covariates, and frailty. The simulation in Section 3 sets λ0k(t)=1 for all k and generates training and main data from the same mechanism, so this bias mechanism is absent from every simulation setting. The real-data analysis in Section 4 uses hearing-loss phenotypes whose age-at-onset hazards are likely to differ, so the cause-specific hazard ratios in Tables 2–3 may be biased. The authors should either replace the weights with estimates of the conditional probability given T, X, and v, or provide a formal asymptotic justification with clearly stated conditions under which Eq. (6) is an unbiased estimating equation.
[Section 4, MUSC training data; Section 2.2] There is a contradiction between the method statement and the application. Section 2.2 explicitly says the classification model is fitted among participants that have experienced the events (Δij=1), but the real-data analysis builds the classifier on all 338 MUSC ears, 11% of which are labelled 'older-normal (i.e. without hearing loss)'. Those older-normal ears are censored observations (Δ=0), not observed event types, so including them as a fourth class means the classifier does not estimate P(δ=k | W, Δ=1); the predicted probabilities for the three hearing-loss types are not the conditional event-type probabilities required by Eq. (6). This label contamination can bias the hazard ratio estimates in Tables 2 and 3. The analysis should either restrict the training sample to ears with hearing loss and treat older-normal ears as censored, or explicitly model the conditional probability of event type given that an event occurred.
[Section 2.2, Eq. (7); Table 1] The variance estimator in Eq. (7) treats the estimated probabilities P(δ|W) as known, and the paper cites Lok (2021) to argue that ignoring this estimation is conservative. In the present setting, the probabilities are estimated from a training sample that is separate from the main study, and ignoring that estimation step generally understates the variability of the estimating equation rather than overstating it. The coverage probabilities in Table 1 are close to nominal in most cells, but several cells fall to 0.92–0.93 (e.g., (1000,50), (log(1.5), log(1.75)), γ=2.5, weighted β2; (1000,100), (log(1.5), log(1.75)), γ=2.5, imputed β2). The authors should either derive a variance estimator that accounts for the estimated classifier or empirically evaluate the magnitude of the undercoverage by reporting the coverage when training samples are redrawn across replicates, which is already the case in the simulation but is not reflected in the inference.
[Section 3; Section 2.2] The simulation design bakes in transportability of the classifier: the training and main data are generated from the same data-generating mechanism, and the classifier (multinomial logistic regression on W) is correctly specified for the conditional distribution of W given δ. The real-data application instead requires transporting a classifier from the MUSC database to the CHEARS AAA cohort, where the relationship between audiometric thresholds and hearing-loss phenotypes, or the phenotype definitions themselves, may differ. A simulation that trains the classifier under one mechanism and applies it to main-study data generated under a different mechanism (e.g., different W|δ distributions or different cause-specific baseline hazards) is needed to assess the method's robustness. Without such a sensitivity analysis, the real-data results in Tables 2 and 3 cannot be interpreted as reliable estimates of the cause-specific hazard ratios.

minor comments (4)

[Section 3] The text says the regression parameters are chosen among {(log(1.25), log(1.5)), (log(1.5), log(1.75)), (log(1.5), log(1.75))}, but the second and third pairs are identical; presumably one of them should be (log(1.5), log(1.5)) as used in Table 1.
[Section 4, Tables 2–3] The confidence intervals for the sensory hearing loss associations are extremely wide (for example, 0.55–4.89 in the QDA-weighted row), so the statement that the associations 'are not substantially different' is not well supported; reporting p-values for the difference or a joint test would be more informative.
[Appendix] The information matrix formulas use tij in the denominators while the main text uses Tij; the notation should be made consistent.
[Figure 1 and Section 4] The figure shows four groups including older-normal, but the main-study analysis is for three hearing-loss phenotypes; the authors should clarify how predicted older-normal probabilities are handled for participants with hearing loss, since the method in Section 2.2 conditions on Δ=1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the event-type weights come from an independent training-set classifier and are not functions of the main-study regression parameters.

full rationale

The paper's central proposal replaces the unobserved event-type indicator I(delta_ij=k) in the full-data score with the classifier-based probability P(delta_ij=k | W_ij, Delta_ij=1), estimated from a separate training dataset. These weights are fixed inputs when the main-study parameters beta and v are estimated, so no fitted target parameter is reused as a prediction. The simulation trains the classifier on data generated from the same mechanism as the main study, but this exercises transportability and classification accuracy rather than making the estimator circular. The analytical variance formula deliberately ignores the fact that the probabilities are estimated, which is an acknowledged inferential simplification, not a circular reduction. The self-citations, such as Wang et al. (2020) for the penalized partial likelihood and Wu and Wang (2023) for a discussion remark, support standard machinery or future extensions and are not load-bearing for the identification of the effect estimates. The main statistical concern, that the conditional expectation of the score should also condition on the observed event time through the risk-set terms unless cause-specific baseline hazards are proportional across causes, is a potential bias or model-misspecification issue, not a circularity. No equation in the paper reduces by construction to its own inputs, and no fitted parameter is renamed as a prediction, so the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The method does not introduce new physical or statistical entities or ad hoc fitted constants. It relies on standard survival-model assumptions, on the missing-data and transportability assumption for the classifier, and on measurement-error-free event times. The classifier probabilities are estimated from a separate training set, not fitted to the target parameters.

assumptions (4)

domain assumption Cause-specific proportional hazards with log-normal frailty: λijk(t)=λ0k(t) exp(Xij^T βk + vik), with vi multivariate normal with mean 0 and covariance D(θ).
Invoked in Section 2.1 as the model for the true event process. The validity of the proposed estimators assumes this model or a good approximation for the main study.
standard math Independent censoring: Cij is independent of the event time and event type given covariates and random effects.
Needed for the partial-likelihood construction in Section 2.1; standard but not stated explicitly.
domain assumption Transportability of the classifier: P(δ=k|W,Δ=1) estimated in the training dataset equals the corresponding probability in the main study population.
Used in Section 2.2 to replace I(δij=k) with classifier probabilities and in Section 4 where the MUSC classifier is applied to CHEARS AAA. This is the weakest assumption and is not validated.
domain assumption Event times are measured without error.
Acknowledged as a limitation in Section 5: 'One limitation of our methods is the assumption of no measurement error in the event time.' Violation can bias hazard-ratio estimates.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Statistical methods for clustered competing risk data when the event types are only available in a training dataset." pith.science (2026). https://pith.science/paper/YFENKQLW

@misc{pith2026250502217,
  author       = {Pith},
  title        = {Pith review of: Statistical methods for clustered competing risk data when the event types are only available in a training dataset},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFENKQLW}},
  note         = {Machine review of arXiv:2505.02217}
}

read the original abstract

We develop methods to analyze clustered competing risks data when the event types are only available in a training dataset and are missing in the main study. We propose to estimate the exposure effects through the cause-specific proportional hazards frailty model where random effects are introduced into the model to account for the within-cluster correlation. We propose a weighted penalized partial likelihood method where the weights represent the probabilities of the occurrence of events, and the weights can be obtained by fitting a classification model for the event types on the training dataset. Alternatively, we propose an imputation approach where the missing event types are imputed based on the predictions from the classification model. We derive the analytical variances, and evaluate the finite sample properties of our methods in an extensive simulation study. As an illustrative example, we apply our methods to estimate the associations between tinnitus and metabolic, sensory and metabolic+sensory hearing loss in the Conservation of Hearing Study Audiology Assessment Arm.

Figures

Figures reproduced from arXiv: 2505.02217 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 19 canonical work pages

[1]

R. V. Craiu and T. Duchesne. Inference based on the em algorithm for the competing risks model with masked causes of failure. Biometrika, 91 0 (3): 0 543--558, 2004

work page 2004
[2]

S. G. Curhan, M. Wang, R. D. Eavey, M. J. Stampfer, and G. C. Curhan. Adherence to healthful dietary patterns is associated with lower risk of hearing loss in women. The Journal of nutrition, 148 0 (6): 0 944--951, 2018

work page 2018
[3]

S. G. Curhan, C. Halpin, M. Wang, R. D. Eavey, and G. C. Curhan. Tinnitus and 3-year change in audiometric hearing thresholds. Ear and hearing, 42 0 (4): 0 886, 2021

work page 2021
[4]

S. H. Dharmarajan, D. E. Schaubel, and R. Saran. Evaluating center performance in the competing risks setting: Application to outcomes of wait-listed end-stage renal disease patients. Biometrics, 74 0 (1): 0 289--299, 2018

work page 2018
[5]

J. R. Dubno, M. A. Eckert, F.-S. Lee, L. J. Matthews, and R. A. Schmiedt. Classifying human audiometric phenotypes of age-related hearing loss from animal models. Journal of the Association for Research in Otolaryngology, 14: 0 687--701, 2013

work page 2013
[6]

Goetghebeur and L

E. Goetghebeur and L. Ryan. Analysis of competing risks survival data when some failure types are missing. Biometrika, 82 0 (4): 0 821--833, 1995

work page 1995
[7]

R. J. Gray. Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. Journal of the American Statistical Association, 87 0 (420): 0 942--951, 1992

work page 1992
[8]

Gupta, R

S. Gupta, R. D. Eavey, M. Wang, S. G. Curhan, and G. C. Curhan. Type 2 diabetes and the risk of incident hearing loss. Diabetologia, 62: 0 281--285, 2019

work page 2019

Show all 20 references

[9]

Hougaard

P. Hougaard. Frailty models for survival data. Lifetime data analysis, 1: 0 255--273, 1995

1995
[10]

S. Hyun, J. Lee, and Y. Sun. Proportional hazards model for competing risks data with missing cause of failure. Journal of statistical planning and inference, 142 0 (7): 0 1767--1779, 2012

2012
[11]

B. M. Lin, M. Wang, K. M. Stankovic, R. Eavey, M. J. McKenna, G. C. Curhan, and S. G. Curhan. Cigarette smoking, smoking cessation, and risk of hearing loss in women. The American journal of medicine, 133 0 (10): 0 1180--1186, 2020

2020
[12]

J. J. Lok. How estimating nuisance parameters can reduce the variance (with consistent variance estimation). arXiv preprint arXiv:2109.02690, 2021

2021 arXiv
[13]

Lu and A

K. Lu and A. A. Tsiatis. Multiple imputation methods for estimating regression coefficients in the competing risks model with missing cause of failure. Biometrics, 57 0 (4): 0 1191--1197, 2001

2001
[14]

Patil and G

P. Patil and G. Parmigiani. Training replicable predictors in multiple studies. Proceedings of the National Academy of Sciences, 115 0 (11): 0 2578--2583, 2018

2018
[15]

R. L. Prentice, J. D. Kalbfleisch, A. V. Peterson Jr, N. Flournoy, V. T. Farewell, and N. E. Breslow. The analysis of failure times in the presence of competing risks. Biometrics, pages 541--554, 1978

1978
[16]

Ripatti and J

S. Ripatti and J. Palmgren. Estimation of multivariate frailty models using penalized partial likelihood. Biometrics, 56 0 (4): 0 1016--1022, 2000

2000
[17]

Sagi and L

O. Sagi and L. Rokach. Ensemble learning: A survey. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 8 0 (4): 0 e1249, 2018

2018
[18]

T. M. Therneau, P. M. Grambsch, and V. S. Pankratz. Penalized survival models and frailty. Journal of computational and graphical statistics, 12 0 (1): 0 156--175, 2003

2003
[19]

L. Wang, K. He, and D. E. Schaubel. Penalized survival models for the analysis of alternating recurrent event data. Biometrics, 76 0 (2): 0 448--459, 2020

2020
[20]

Wu and M

Y. Wu and M. Wang. Outcome measurement error correction for survival analyses with multiple failure types: application to hearing loss studies. arXiv preprint arXiv:2306.10568, 2023

2023 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

R. V. Craiu and T. Duchesne. Inference based on the em algorithm for the competing risks model with masked causes of failure. Biometrika, 91 0 (3): 0 543--558, 2004

work page 2004

[2] [2]

S. G. Curhan, M. Wang, R. D. Eavey, M. J. Stampfer, and G. C. Curhan. Adherence to healthful dietary patterns is associated with lower risk of hearing loss in women. The Journal of nutrition, 148 0 (6): 0 944--951, 2018

work page 2018

[3] [3]

S. G. Curhan, C. Halpin, M. Wang, R. D. Eavey, and G. C. Curhan. Tinnitus and 3-year change in audiometric hearing thresholds. Ear and hearing, 42 0 (4): 0 886, 2021

work page 2021

[4] [4]

S. H. Dharmarajan, D. E. Schaubel, and R. Saran. Evaluating center performance in the competing risks setting: Application to outcomes of wait-listed end-stage renal disease patients. Biometrics, 74 0 (1): 0 289--299, 2018

work page 2018

[5] [5]

J. R. Dubno, M. A. Eckert, F.-S. Lee, L. J. Matthews, and R. A. Schmiedt. Classifying human audiometric phenotypes of age-related hearing loss from animal models. Journal of the Association for Research in Otolaryngology, 14: 0 687--701, 2013

work page 2013

[6] [6]

Goetghebeur and L

E. Goetghebeur and L. Ryan. Analysis of competing risks survival data when some failure types are missing. Biometrika, 82 0 (4): 0 821--833, 1995

work page 1995

[7] [7]

R. J. Gray. Flexible methods for analyzing survival data using splines, with applications to breast cancer prognosis. Journal of the American Statistical Association, 87 0 (420): 0 942--951, 1992

work page 1992

[8] [8]

Gupta, R

S. Gupta, R. D. Eavey, M. Wang, S. G. Curhan, and G. C. Curhan. Type 2 diabetes and the risk of incident hearing loss. Diabetologia, 62: 0 281--285, 2019

work page 2019