Learning association from multiple intermediate events for dynamic prediction of survival: an application to cardiovascular disease prognosis

Abstract: Cardiovascular diseases are major causes of mortality globally. They often co-occur and are interrelated, leading to partial-order relationships among their onset times. However, these onset times are subject to informative censoring due to the occurrence of death, posing significant challenges for survival prediction. In this article, we propose a novel copula-based framework that learns dependence among multiple correlated marginal components through a pseudo-likelihood for estimation. We adopt nonparametric marginals, alleviating the reliance on marginal distribution assumptions typically required in conventional copula models, and estimate the association between the onsets of intermediate cardiovascular diseases and death by solving a concordance estimating equation. Under this framework, a renewable risk assessment method is developed for dynamic survival prediction, leveraging information on disease onset times and the maximum follow-up duration. Our proposed method yields estimators with well-established properties, and its flexibility and predictive effectiveness are demonstrated through extensive simulation studies. We apply the method to data from a heart disease study, demonstrating the benefits of incorporating the associations among various cardiovascular diseases and their synergistic effects on mortality for dynamic prediction of overall survival.

• The paper introduces a multistage copula-based framework that models the complex dependencies between intermediate cardiovascular events and death.
• It employs a two-stage estimation process using nonparametric marginals and maximum pseudo-likelihood to address informative censoring and partial event ordering.
• The proposed algorithm enhances dynamic survival prediction by providing calibrated, individualized risk estimates and outperforming standard landmarking models.

Copula-Based Association Learning for Dynamic Survival Prediction: Application to Cardiovascular Disease Prognosis

Problem Motivation and Clinical Context

Cardiovascular diseases encompass multiple interconnected conditions (e.g., hypertension, angina pectoris, coronary heart disease, stroke, myocardial infarction, and fatal cerebrovascular events) with complex temporal and stochastic relationships to mortality. Their onset times are subject to informative censoring via death, resulting in partially ordered event times and observable regions constrained by semicompeting risks structure. Conventional survival models inadequately capture the intricate dependence among these intermediate events and their association with death. Improvements in dynamic survival prediction demand flexible modeling that integrates individualized disease histories and onset times, beyond the limited use of occurrence indicators or recurrent-event aggregation.

Copula-Based Modeling Framework

This paper introduces a multistage copula-based methodology for learning association structures and facilitating dynamic survival prediction in the presence of multiple dependent intermediate events. The architecture leverages Archimedean copulas due to their versatility with multivariate right-censored data and their capacity to encode asymmetric/nonexchangeable dependence via generator functions.

• Bivariate Layer: Each intermediate event time $T_k$ is paired with terminal event (death) time $D$, and their joint survival is modeled using an Archimedean copula parameterized by $\theta_k$. The bivariate survival function $S_{k,D}(t_k, t)$ is strictly defined for $t_k \leq t$.
• Multivariate Layer: The conditional joint survival of all intermediate events, given death at time $y$, is encoded by a global association parameter $\alpha$. The nested structure ensures compatibility with the observable region defined by $T_k \leq D$ for all $k$.
• Partial ordering and informative censoring: The joint model is rigorously restricted to the upper wedge of the event space, acknowledging that intermediate events cannot be observed post-mortem.

  Figure 1: A flowchart of the copula-based learning framework for association analysis and dynamic prediction; event colors are aligned across training/test data and survival functions.

The unique parameterization allows capturing disease-specific risk contributions and conditional dependencies, overcoming limitations in existing semicompeting risks, multi-state, and landmarking models.

Estimation and Association Learning

Estimation proceeds via a two-stage procedure:

1. Marginals and Bivariate Associations: Marginal survival functions ($S_k$, $D$0) are estimated nonparametrically; $D$1 is estimated by concordance techniques exploiting order statistics and pseudo-likelihoods tailored to the copula structure. Informative censoring is handled with pseudo self-consistency equations for marginals.
2. Multivariate Association: The global parameter $D$2 is estimated by maximum pseudo-likelihood, integrating information on onset histories and observable event orderings. Computational efficiency is achieved by leveraging Laplace transforms for higher-order derivatives in the likelihood contributions, allowing for consistent estimation as sample size and event count grow.

Strong theoretical guarantees are established, including consistency and asymptotic normality for $D$3 and plug-in joint survival function estimators, even with complex censoring patterns and partial ordering.

Figure 2: Association structure of event times and the flow of conditional dependence in estimation.

Dynamic Survival Prediction Algorithm

The proposed approach dynamically predicts overall survival given an individual's observed history of intermediate event onsets and maximal follow-up. Unlike standard landmarking (which incorporates only indicator variables or fixed covariates), this methodology computes conditional survival probabilities and restricted mean residual life for arbitrary combinations of event times using the learned joint copula structure.

• Prediction quantities: Conditional survival probabilities, restricted mean residual life, and conditional quantiles of survival time are provided, anchored at individualized landmark times.
• Interval prediction and calibration: Individual prediction intervals are constructed from conditional quantiles, enabling reliable coverage and assessment of uncertainty.

Empirical accuracy is assessed via mean squared prediction error (MSPE), quantile prediction error (QPE), integrated Brier score (IBS), and time-dependent AUC. The methodology supports robust performance evaluation for both in-sample and out-of-sample predictions.

Figure 3: Pairwise Kendall correlations illustrating complex nonexchangeable dependence among simulated intermediate and terminal events.

Numerical Results and Cardiovascular Data Application

Extensive simulation experiments verify the robustness and accuracy of the estimators and the dynamic prediction algorithm under variable association structures, levels of censoring, and copula mis-specification. Numerical results include:

• Consistent estimation: Parameters ($D$4, $D$5) are recovered with low bias and proper coverage across different $D$6 and censoring rates.
• Prediction superiority: Dynamic prediction (DP) outperforms competing methods (landmarking, single-event models, and aggregate-recurrent analysis) in both MSPE and IBS, especially when the number of intermediate events is large or their associations are heterogeneous.

The methodology is applied to the Framingham Heart Study cohort. Empirical estimates show:

• Highly differentiated associations of specific cardiovascular events with death, with MIFC, STRK, and CVD showing strongest effects (e.g., Clayton $D$7).
• Joint-model-based dynamic survival predictions are demonstrably more accurate and provide better-calibrated individualized survival probabilities.
• Reliability and calibration are maintained across training and repeated random splits, with predicted survival probabilities closely matching observed event times.

  Figure 4: Comparative predictive performance for various algorithms, with DP methods achieving lower MSPE/QPE and higher AUC, demonstrating robustness across random splits.

Practical and Theoretical Implications

This research advances the modeling and prediction landscape for survival analysis subject to informative censoring and semicompeting risks. The copula-based framework:

• Enables disease-specific risk quantification, supporting precision medicine and adaptive risk stratification.
• Captures complex conditional and unconditional dependencies among intermediate events, offering insights beyond aggregated recurrent-event models or longitudinal landmarking approaches.
• Unlocks path toward scalable, computationally tractable individualized survival prediction for multi-pathogenesis clinical cohorts.

In theoretical terms, the parameterization and estimation framework generalize beyond semi-competing risks, allowing for systematic handling of partial ordering, dynamic prediction, and informative censoring in multivariate survival contexts. Methodological extensions are feasible for recurrent events, incorporation of covariates, and alternative copula structures.

Future Directions

Potential advancements include:

• Extension to non-Archimedean copula families and time-varying global association structures.
• Integration of genomic or biomarker panels for further risk stratification [meyre2025biomarker].
• Algorithmic improvements for high-dimensional survival data, parallel computing, and efficient C++ implementations.
• Theoretical exploration of joint model estimation with pseudo-observations and handling mixtures of event types.

Conclusion

The copula-based multistage conditional dependence framework developed in this work substantially improves individualized dynamic prediction for survival under multiple intermediate events and informative censoring. It resolves major limitations of existing approaches, accommodates the complex stochastic nature of cardiovascular disease progression, and provides a rigorous foundation for personalized risk estimation in diverse clinical contexts. The methodological rigor and empirical robustness suggest broad applicability and further extension for advanced AI-driven prognosis systems.