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Penalized Likelihood Optimization for Adaptive Neighborhood Clustering in Time-to-Event Data with Group-Level Heterogeneity

Alessandra Ragni, Lara Cavinato, Francesca Ieva

TL;DR

The study tackles identifying latent patient subgroups with distinct hazard trajectories in hierarchical time-to-event data by embedding patient clustering directly inside a shared-frailty survival model. It introduces a penalized likelihood framework that learns a risk-driven similarity graph via adaptive neighbors and spectral clustering, coupling this with a parametric frailty model to reflect hospital-level dependence. Simulation results demonstrate robust recovery of the true cluster structure and reasonable parameter estimation under appropriate penalization, while the case study on Enhance-Heart data from Lombardy reveals three clinically interpretable subgroups and substantial inter-hospital heterogeneity in outcomes. Overall, the approach provides a flexible, interpretable tool for risk-based patient stratification in multi-center survival analyses, with the accompanying R implementation HazClust publicly available.

Abstract

The identification of patient subgroups with comparable event-risk dynamics plays a key role in supporting informed decision-making in clinical research. In such settings, it is important to account for the inherent dependence that arises when individuals are nested within higher-level units, such as hospitals. Existing survival models account for group-level heterogeneity through frailty terms but do not uncover latent patient subgroups, while most clustering methods ignore hierarchical structure and are not estimated jointly with survival outcomes. In this work, we introduce a new framework that simultaneously performs patient clustering and shared-frailty survival modeling through a penalized likelihood approach. The proposed methodology adaptively learns a patient-to-patient similarity matrix via a modified version of spectral clustering, enabling cluster formation directly from estimated risk profiles while accounting for group membership. A simulation study highlights the proposed model's ability to recover latent clusters and to correctly estimate hazard parameters. We apply our method to a large cohort of heart-failure patients hospitalized with COVID-19 between 2020 and 2021 in the Lombardy region (Italy), identifying clinically meaningful subgroups characterized by distinct risk profiles and highlighting the role of respiratory comorbidities and hospital-level variability in shaping mortality outcomes. This framework provides a flexible and interpretable tool for risk-based patient stratification in hierarchical data settings.

Penalized Likelihood Optimization for Adaptive Neighborhood Clustering in Time-to-Event Data with Group-Level Heterogeneity

TL;DR

The study tackles identifying latent patient subgroups with distinct hazard trajectories in hierarchical time-to-event data by embedding patient clustering directly inside a shared-frailty survival model. It introduces a penalized likelihood framework that learns a risk-driven similarity graph via adaptive neighbors and spectral clustering, coupling this with a parametric frailty model to reflect hospital-level dependence. Simulation results demonstrate robust recovery of the true cluster structure and reasonable parameter estimation under appropriate penalization, while the case study on Enhance-Heart data from Lombardy reveals three clinically interpretable subgroups and substantial inter-hospital heterogeneity in outcomes. Overall, the approach provides a flexible, interpretable tool for risk-based patient stratification in multi-center survival analyses, with the accompanying R implementation HazClust publicly available.

Abstract

The identification of patient subgroups with comparable event-risk dynamics plays a key role in supporting informed decision-making in clinical research. In such settings, it is important to account for the inherent dependence that arises when individuals are nested within higher-level units, such as hospitals. Existing survival models account for group-level heterogeneity through frailty terms but do not uncover latent patient subgroups, while most clustering methods ignore hierarchical structure and are not estimated jointly with survival outcomes. In this work, we introduce a new framework that simultaneously performs patient clustering and shared-frailty survival modeling through a penalized likelihood approach. The proposed methodology adaptively learns a patient-to-patient similarity matrix via a modified version of spectral clustering, enabling cluster formation directly from estimated risk profiles while accounting for group membership. A simulation study highlights the proposed model's ability to recover latent clusters and to correctly estimate hazard parameters. We apply our method to a large cohort of heart-failure patients hospitalized with COVID-19 between 2020 and 2021 in the Lombardy region (Italy), identifying clinically meaningful subgroups characterized by distinct risk profiles and highlighting the role of respiratory comorbidities and hospital-level variability in shaping mortality outcomes. This framework provides a flexible and interpretable tool for risk-based patient stratification in hierarchical data settings.
Paper Structure (17 sections, 25 equations, 2 figures, 5 tables)

This paper contains 17 sections, 25 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Notation
  4. Model
  5. Optimization procedure
  6. Estimation details
  7. Simulation Study
  8. Data generating process
  9. Results
  10. Case Study
  11. Cohort selection
  12. Model setup and results
  13. Discussion
  14. Derivation of the marginal likelihood
  15. Derivation of closed-form solution of (\ref{['eq:step3longform']})
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Boxplots of parameter estimates ($\theta, \rho, \xi, \beta$) across penalization levels $\gamma$ (x-axis) under administrative censoring for $k=20$ and $C=3$. The true parameter values are shown as a piecewise-constant horizontal line.
  • Figure 2: Boxplots of parameter estimates ($\theta, \rho, \xi, \beta$) across penalization levels $\gamma$ (x-axis) under normal censoring for $k=20$ and $C=3$. The true parameter values are shown as a piecewise-constant horizontal line.