Toward Joint Prediction of a Longitudinal Marker and a Terminal Event: A bivariate discrete-time framework

Daniel Kramer; Harrison Reeder; Rajarshi Mukherjee; Rui Duan; Sebastien Haneuse; Stephanie Armbruster

REVIEW 3 major objections 6 minor 300 references

Toward Joint Prediction of a Longitudinal Marker and a Terminal Event: A bivariate discrete-time framework

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper claims that discretizing follow-up time turns joint prediction of a terminal event and a longitudinal marker into a single Bayesian model whose marker trajectory is conditional on being alive—and demonstrates it on quality of lif

desk verdict A well-executed framework paper with a real clinical motivation, but the partial-information missingness strategy has an unaddressed gap that directly affects the SCD-HeFT death-hazard terms. read the letter →

arxiv 2607.09931 v2 pith:O2QPS2Y5 submitted 2026-07-10 stat.ME

Stephanie Armbruster , Daniel Kramer , Rui Duan , Rajarshi Mukherjee , Sebastien Haneuse , Harrison Reeder This is my paper

classification stat.ME MSC 62N0162F1562P10

keywords Discrete-timepartitionTerminaleventoutcomeLongitudinalmarkertrajectoryRandomeffectAutoregressionSharedfrailtyBayesianestimationHamiltonianMonteCarlo

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that partitioning follow-up time into discrete intervals makes it possible to model death and a repeatedly measured health marker as one bivariate process, and to issue patient-specific joint predictions of both. The marker trajectory is defined conditional on being alive at each interval—a 'partly conditional' mean—so predictions do not imagine an immortal cohort in which death never happens. The framework lets the death hazard depend on recent marker values, with either constant or time-varying strength, plus a shared patient frailty, and estimates all of this in a Bayesian way. Applied to heart-failure patients with implantable defibrillators, it produces distinct predicted journeys: good quality of life with low mortality, stable poor quality of life with rising mortality, rapid deterioration, and high baseline burden. If this works, a clinician could weigh mortality against quality of life in a single conversation instead of relying on separate univariate predictions.

What carries the argument

The key object is the bivariate discrete-time process (N_{k,i}, Y_{k,i}) built on a partition of continuous study time. At each interval k, the joint density factorizes as a Bernoulli terminal-event indicator with hazard pi_{k,i} = P(N_{k,i}=1 | N_{k-1,i}=0, history) times a marker distribution f_Y(y_{k,i} | N_{k,i}=0, history), whose mean mu_{k,i} is the partly conditional trajectory. This sequential conditioning on survival is what makes the marker trajectory interpretable as the expected value among patients still alive, and it lets the terminal hazard depend on lagged marker values through global and local dependence terms, on a shared frailty, and on autoregressive and random-effect str

What would settle it

In the SCD-HeFT data, fit a model for whether a patient missed a scheduled visit as a function of baseline covariates, the last observed marker value, and subsequent death status; if missingness predicts death beyond the observed marker history, non-informative missingness is false and the framework's estimates are suspect. Alternatively, a sensitivity analysis that systematically imputes missed markers as worse than the last observed value should materially change the joint predictions if the assumption is load-bearing.

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Extended reading notes

Core claim

The central claim is that the bivariate discrete-time framework provides joint prediction of time to terminal event and longitudinal marker trajectory by conditioning each interval's marker distribution on survival through that interval, thereby avoiding the implicit 'immortal cohort' extrapolation of standard joint models. The paper shows in simulations that correctly specified models recover regression parameters with near-nominal coverage and that marginal Bayesian model-selection criteria can pick a parsimonious model from a candidate set. In the SCD-HeFT application, the base model with global dependence and shared frailty is preferred, and patient-specific posterior predictions span a

Load-bearing premise

The whole analysis hinges on the assumption that a patient who skips a follow-up visit is, on average, no sicker than one who attends once baseline information is accounted for—if missed visits actually reflect declining health, the predicted trajectories and the link between quality of life and mortality will be biased.

Editorial extensions

If this is right

Joint posterior draws respect death: each predicted marker trajectory stops at the predicted time of death, so summaries such as 'expected quality of life at 24 months' are averages over patients alive at 24 months.
For ICD patients, the framework separates those who will maintain good quality of life with low mortality from those who will deteriorate or die, information that can directly inform ICD continuation, deactivation, or palliative care discussions.
Marginal BIC, WAIC, and ELPD-based model selection favor parsimonious specifications in simulations, giving analysts a principled way to choose among global dependence, local dependence, autoregression, and shared frailty options.
The partial-information strategy for intermittent missingness avoids the artificial autocorrelation introduced by last-observation-carried-forward and linear interpolation in the autoregressive parameter.
Correctly specified and flexibly parameterized models show small bias and near-nominal credible-interval coverage in the simulation study, while misspecification of the terminal event submodel or dependence structure biases both submodels.

Reading between the lines

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

My inference: the same survival-conditional factorization should extend naturally to competing risks or multiple longitudinal markers, because the argument does not depend on the number of markers or event types; the paper mentions such extensions as future work.
My inference: the choice of discrete-time partition is not just a technical nuisance but a substantive decision about clinical timescale, so sensitivity analyses over interval widths could matter as much as model selection.
My inference: because the longitudinal predictions are partly conditional, a reader must interpret them as 'if you are alive then,' not as a unconditional forecast; a decision aid built on these predictions should state that interpretation explicitly.
My inference: the paper's own observation that no joint predictive-accuracy metric exists points to a concrete testable next step—define a proper scoring rule for pairs of survival time and marker trajectory and compare the bivariate model against univariate benchmarks.

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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

The core idea here is genuinely useful: discretize time so that the longitudinal trajectory is modeled only among survivors, giving 'partly conditional' predictions that don't extrapolate past death. The framework is flexible — autoregression, global and local dependence, shared frailty — and the Bayesian machinery is handled cleanly, with posterior predictive sampling that produces the kinds of joint patient trajectories a clinician might actually use. The simulation study shows the estimator behaves well when the model is correct, and the SCD-HeFT application is a nice demonstration, even if the clinical vignettes are somewhat illustrative rather than validated. The main problem is in the missing-data mechanics, and it is more specific than the reader's concern about untested MAR assumptions. Appendix I.1 writes the partial-information likelihood as a product over intervals with observed lagged markers, times the final death contribution π_{k_i^r,i}^{Δ_i}. That death term depends on Y_{k_i^r-1,i}. If the marker at the visit before death is missing, the term is not computable from the observed data. The paper never says how these cases are handled: last-value imputation? integration over the missing marker? dropping such deaths? Given that 62% of patients have at least one missing MLHFQ item and that missed visits plausibly precede death, this is not a corner case. It can bias the dependence parameter ϑ and the joint predictions in Figures 5–6. This is a load-bearing gap, not a stylistic nit. Other soft spots: no comparison with existing joint longitudinal-survival or semi-competing risk models, which the authors explicitly defer but which limits any claim of added predictive or interpretational value. The non-informative intermittent missingness assumption is stated but untested, and no sensitivity analysis is provided. The package and code are mentioned but no URL or hash is given, which is annoying for reproducibility. That said, the paper is clearly written, the modeling options are thoughtfully laid out, and the idea is worth pursuing. The issue is fixable — spell out the imputation or integration for the death interval, and run some sensitivity checks. Who is this for? Biostatisticians working on joint modeling or semi-competing risks, and applied researchers in cardiac outcomes who want to move beyond univariate prediction. It deserves a serious referee, but the referee should push hard on the missingness gap before publication. I'd send it to review, with the expectation of major revision.

Referee Report

3 major / 6 minor

Summary. The paper proposes a Bayesian bivariate discrete-time framework for jointly modeling a longitudinal marker and a terminal event, with the explicit goal of patient-specific joint prediction. The key design feature is discretizing time so that the longitudinal mean is defined conditionally on being alive, i.e., a partly-conditional trajectory, thereby avoiding implicit extrapolation of the marker beyond death. The framework allows flexible dependence structures: global time-invariant and local time-varying dependence of the hazard on lagged marker values, autoregressive marker dynamics, and a shared frailty. Estimation is carried out in Stan, with model selection via marginal BIC, WAIC, and PSIS leave-one-out criteria. A simulation study shows good parameter recovery and model-selection behavior for correctly specified or slightly overparameterized models. The SCD-HeFT application produces joint posterior predictions of MLHFQ quality-of-life trajectories and mortality for hypothetical ICD patients, and the authors discuss how these predictions could inform ICD continuation/deactivation decisions.

Significance. If the identified gaps are resolved, the framework would be a useful addition to the joint-modeling and dynamic-prediction literature, particularly for clinical settings where death truncates the longitudinal trajectory and both outcomes are of intrinsic interest. The partly-conditional interpretation of the longitudinal mean is clinically natural, and the unified Bayesian computation in Stan with an accompanying R package is a practical strength. The paper is also honest about its scope, presenting the work as a framework rather than a fully validated prediction tool. However, the central predictive claims currently rest on an incompletely specified likelihood when lagged markers are missing, and on a strong untested missingness assumption in the application. These issues are load-bearing for the SCD-HeFT results and for the joint predictions in Figures 5–6.

major comments (3)

[§3.5, Eq. (1); Appendix I.1] The partial-information likelihood is not computable when the lagged marker is missing in the death interval. Eq. (1) and the Appendix I.1 product include the death contribution π_{k_r,i}^{Δ_i} unconditionally, even if M_{k_r-1,i}=0. Since π_{k,i} contains global/local dependence on Y_{k-1,i}, the death term then depends on an unobserved value. The paper never states whether such deaths are omitted, whether a last-observation value is substituted, or whether integration over missing Y is performed. With 87 deaths in the partial-information analytic sample, this is not a corner case; it directly affects the estimated dependence ϑ and the joint predictions used in Figures 5–6. The authors must specify the evaluable likelihood contribution and, ideally, conduct a sensitivity analysis.
[§8.2, Appendix I.1] The analysis assumes that intermittent missingness of the MLHFQ score is non-informative: independent of time to death and the marker value conditional on baseline covariates. The paper reports that 62.3% of patients missed at least one MLHFQ question. If sicker patients are more likely to miss visits, the estimated partly-conditional trajectories, the dependence parameter ϑ, and the joint predictions for ICD decision-making may be biased. The assumption is stated but not tested, and no sensitivity analysis is provided for informative missingness. A pattern-mixture or shared-parameter extension would be one way to assess robustness; at minimum, a discussion of the direction and plausible magnitude of bias is needed.
[§7, Tables A13–A14] Although the stated primary goal is prediction, the simulation study evaluates only parameter recovery and model selection among candidate models from the same bivariate discrete-time family. There is no assessment of predictive calibration or discrimination (e.g., Brier score, AUC, interval scores for the joint outcome) and no comparison with existing dynamic-prediction alternatives such as landmarking, joint longitudinal–survival models, or separate univariate predictions. The claim that the framework 'provides' joint predictions that are clinically useful would be materially strengthened by evidence that the posterior predictive distributions are calibrated and at least competitive with simpler approaches. This is a substantive gap, not merely a presentation issue.

minor comments (6)

[§2.2] Typographical errors: 'whcih' and 'trajectoryies' in Section 2.3. 'SCD-Heft' also appears in Section 6.1; use consistent SCD-HeFT.
[Figure A3] The flow diagram reports N=784 for Case 3 (linear imputation), exceeding the complete-case base N=612. This appears to be an error or a different sample construction; please explain or correct.
[§4.1] The prior for σ_α is described as 'weakly informative,' but the actual value is not reported in the main text or simulation summary. The simulations use σ_α=20 in Table A5; the application likely uses a different value. Report the values used for the SCD-HeFT analysis.
[Table 2] The notation RCS(k=3) is not defined in the table footnote. Clarify that k denotes the number of knots and state the knot locations or that they are placed at quantiles.
[Figures 5–6] The figures show 50 predicted trajectories with 'a random subset in bold,' but the legend does not identify which subset. A simple statement in the caption or legend would improve readability.
[§3.4.3] The shared frailty linear predictor αγ_i is added on the link scale in the hazard but on the mean scale in the longitudinal submodel. This asymmetry is not discussed. A short explanation of why this is the intended parameterization would help readers.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the partly-conditional property is built into the model definition and the Nevo et al. citation is not load-bearing.

full rationale

The paper's central claim—that discretizing time yields partly-conditional longitudinal predictions without extrapolating past death—is a direct consequence of the model construction, not a derived empirical result. In Section 3.2, Y_{k,i} is defined only when N_{k,i}=0, and in Section 3.3, the estimand µ_{k,i} is explicitly defined as E(Y_{k,i}|N_{k,i}=0,H_{k,i}); the 'partly-conditional' interpretation is therefore the target of estimation, not a prediction smuggled in from external inputs. The self-cited Nevo et al. (2022) is used for the global/local dependence formulation and for observation-attribution options, but the paper's estimation strategy, simulation study, and SCD-HeFT application are implemented and evaluated independently here, so the self-citation is not load-bearing. The only substantive concern is in Appendix I.1, where the partial-information likelihood retains π^{Δ_i}_{k_i^r,i} even if the lagged marker Y_{k_i^r-1,i} is missing, making that term not directly computable from observed data; however, this is an estimation gap or missing-data assumption issue, not circularity, because nothing in the framework's derivation reduces its predictions to its fitted inputs. Overall, the derivation chain is self-contained and no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The framework is a modeling construct, so the ledger lists the distributional and missingness assumptions required for its validity plus user-chosen design parameters (partition, attribution, priors) that affect the estimands. No new physical entities are introduced; the shared frailty is a latent statistical variable without independent evidence.

free parameters (3)

Discrete-time partition (6-month intervals for SCD-HeFT) = 6 months (0, 6, ..., 72 months)
Chosen to match the SCD-HeFT visit schedule; determines the timescale of hazards and lag distances. The paper gives guidance but does not fit the partition to the data.
Attribution rule for continuous-time observations = Ceiling (carry time-to-death forward)
Chosen for the SCD-HeFT analysis to retain all observations; affects interval assignment of marker values and death times, and can alter hazard and dependence estimates.
Prior standard deviation for scale multiplier α (σ_α) = Not explicitly reported; prior α ∼ N(0, σ_α²)
A weakly informative prior on the shared-frailty weight; its scale is a user-chosen hyperparameter that influences shrinkage of α and is part of the Bayesian model specification.

assumptions (5)

domain assumption Non-informative right censoring: (N_{k,i}, Y_{k,i}) ⊥ C_i | X_i
Used to factor the observed-data likelihood (Appendix D); standard in survival analysis but not directly testable from data.
domain assumption Non-informative intermittent missingness: (N_{k,i}, Y_{k,i}) ⊥ M_{k,i} | X_i
Assumed for the SCD-HeFT application (Section 8.2, Appendix I.1); required for validity of the partial-information likelihood when visits are missed.
domain assumption Sequential factorization of the joint density: p(N_k,Y_k|N_{k-1}=0,H_k) = π_k if N_k=1 and (1-π_k)f_Y(Y_k|N_k=0,H_k) if N_k=0
Defines the joint model (Section 3.3); implies no contemporaneous residual association between Y_k and N_k beyond the hazard's dependence on history.
domain assumption The longitudinal marker follows a parametric distribution whose mean is the partly-conditional mean (e.g., Gaussian with identity link in the implementation)
Required for the likelihood; the paper illustrates several distributions (Appendix B), but the simulation and application use Gaussian/identity link.
ad hoc to paper Identifiability of latent factors (b_i, γ_i) via subtraction of Z_i^T b_i and γ_i from lagged marker terms in dependence structures
The authors introduce this parameterization to ensure identifiability and to preserve a marginal interpretation of latent factors (Section 3.4.4). It is a modeling constraint specific to this framework.

invented entities (1)

Shared frailty γ_i
purpose: Captures unobserved patient-specific residual dependence between the longitudinal marker trajectory and the terminal event hazard beyond observed covariates and direct marker dependence.
A latent variable with no directly falsifiable measurement; its existence is inferred only through model fit. It is a standard construct in joint modeling, not a newly discovered physical entity, but it is a postulated statistical entity without independent external evidence.

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Cite this review

Pith. "Pith review of Toward Joint Prediction of a Longitudinal Marker and a Terminal Event: A bivariate discrete-time framework." pith.science (2026). https://pith.science/paper/O2QPS2Y5

@misc{pith2026260709931,
  author       = {Pith},
  title        = {Pith review of: Toward Joint Prediction of a Longitudinal Marker and a Terminal Event: A bivariate discrete-time framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O2QPS2Y5}},
  note         = {Machine review of arXiv:2607.09931}
}

read the original abstract

Sudden cardiac death (SCD) is a leading cause of death in the U.S. Patients at elevated risk of SCD are primarily treated with an implantable cardioverter-defibrillator (ICD), which may prevent death from cardiovascular causes but may cause severe side effects, such as reduced quality of life from shock-induced pain. Decisions about ICD treatment therefore involve complex personal trade-offs across multiple health events, including mortality and quality of life. While prediction tools could help weigh these trade-offs, they commonly focus on univariate outcomes; at best, they treat other clinical endpoints as inputs, so trade-offs cannot be directly informed. To address this, we propose a novel general Bayesian framework that jointly models a terminal event and a longitudinal marker as a bivariate process over discrete time, for settings where prediction is the primary goal. Discretization of study time lets the framework capture the dynamic interplay between outcomes while avoiding implicit extrapolation beyond truncation by a terminal event. The framework flexibly accommodates phenomena arising in applied contexts, including global time-invariant and local time-dependent dependence structures between the terminal event and the longitudinal marker, and latent association via a shared frailty term. Estimation proceeds via the Bayesian paradigm, yielding patient-specific joint posterior predictions for the time to terminal event and the future marker trajectory. We introduce the framework with a focus on its modeling flexibility, provide guidance on discretization and on Bayesian model construction and selection, and discuss insights from the joint posterior predictions. Finally, we demonstrate the framework's clinical relevance and practicality for SCD and ICD therapy using data from the Sudden Cardiac Death in Heart Failure Trial (SCD-HeFT), an important ICD-related benchmark trial.

Figures

Figures reproduced from arXiv: 2607.09931 by the authors.

**Figure 2.** Comparison of mean MLHFQ score trajectories for different patient cohorts and of MLHFQ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 2.** Bivariate discrete-time process for terminal event indicator [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Figures from the paper (8 more)

**Figure 3.** Figure 3: Bivariate discrete-time process for terminal event indicator [PITH_FULL_IMAGE:figures/full_fig_p007_3.png]

**Figure 4.** Figure 4: Graphical representation of the bivariate discrete-time framework; [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 4.** Figure 4: Illustration of the different approaches (Nearest Neighbor (NN), Ceiling, and Floor) to [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Illustration of the different approaches (Nearest Neighbor (NN), Ceiling, and Floor) to [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 5.** Figure 5: Patient-specific conditional joint posterior predictions for the base model under partial [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

**Figure 6.** Figure 6: Patient-specific characterization of conditional joint posterior predictive distribution for [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: Patient-specific conditional joint posterior predictions for the base model under partial [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: Conditional joint prediction for patient 2 under partial information strategy for intermittent [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

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Biometrics , volume=

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

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2013 , publisher=

Counting processes and Survival Analysis , author=. 2013 , publisher=

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International journal of forecasting , volume=

Another look at measures of forecast accuracy , author=. International journal of forecasting , volume=. 2006 , publisher=

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Regression models for expected length of stay , author=. Statistics in Medicine , volume=. 2016 , publisher=

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