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Case-Base Neural Networks: survival analysis with time-varying, higher-order interactions

Jesse Islam, Maxime Turgeon, Robert Sladek, Sahir Bhatnagar

TL;DR

CBNNs address non-proportional hazards in single-event survival by incorporating time as an input within a case-base sampling framework, enabling data-driven estimation of time-varying interactions and a flexible baseline hazard. After case-base sampling, a standard feed-forward neural network predicts event odds with an offset to correct sampling bias, yielding a full hazard function that can be transformed into survival probabilities. Across simulations and real-world case studies, CBNNs outperform several regression and neural-network baselines in complex scenarios and two of three datasets, while remaining competitive in the third; this demonstrates a simple, flexible, and easily implementable approach to time-varying survival modeling. The work provides practical software, including R and Python implementations, and demonstrates how case-base sampling can extend neural networks to censored survival data without specialized loss functions. Overall, CBNNs offer a user-friendly, scalable framework for learning time-varying effects and complex baselines in single-event survival analyses with censoring.

Abstract

In the context of survival analysis, data-driven neural network-based methods have been developed to model complex covariate effects. While these methods may provide better predictive performance than regression-based approaches, not all can model time-varying interactions and complex baseline hazards. To address this, we propose Case-Base Neural Networks (CBNNs) as a new approach that combines the case-base sampling framework with flexible neural network architectures. Using a novel sampling scheme and data augmentation to naturally account for censoring, we construct a feed-forward neural network that includes time as an input. CBNNs predict the probability of an event occurring at a given moment to estimate the full hazard function. We compare the performance of CBNNs to regression and neural network-based survival methods in a simulation and three case studies using two time-dependent metrics. First, we examine performance on a simulation involving a complex baseline hazard and time-varying interactions to assess all methods, with CBNN outperforming competitors. Then, we apply all methods to three real data applications, with CBNNs outperforming the competing models in two studies and showing similar performance in the third. Our results highlight the benefit of combining case-base sampling with deep learning to provide a simple and flexible framework for data-driven modeling of single event survival outcomes that estimates time-varying effects and a complex baseline hazard by design. An R package is available at https://github.com/Jesse-Islam/cbnn.

Case-Base Neural Networks: survival analysis with time-varying, higher-order interactions

TL;DR

CBNNs address non-proportional hazards in single-event survival by incorporating time as an input within a case-base sampling framework, enabling data-driven estimation of time-varying interactions and a flexible baseline hazard. After case-base sampling, a standard feed-forward neural network predicts event odds with an offset to correct sampling bias, yielding a full hazard function that can be transformed into survival probabilities. Across simulations and real-world case studies, CBNNs outperform several regression and neural-network baselines in complex scenarios and two of three datasets, while remaining competitive in the third; this demonstrates a simple, flexible, and easily implementable approach to time-varying survival modeling. The work provides practical software, including R and Python implementations, and demonstrates how case-base sampling can extend neural networks to censored survival data without specialized loss functions. Overall, CBNNs offer a user-friendly, scalable framework for learning time-varying effects and complex baselines in single-event survival analyses with censoring.

Abstract

In the context of survival analysis, data-driven neural network-based methods have been developed to model complex covariate effects. While these methods may provide better predictive performance than regression-based approaches, not all can model time-varying interactions and complex baseline hazards. To address this, we propose Case-Base Neural Networks (CBNNs) as a new approach that combines the case-base sampling framework with flexible neural network architectures. Using a novel sampling scheme and data augmentation to naturally account for censoring, we construct a feed-forward neural network that includes time as an input. CBNNs predict the probability of an event occurring at a given moment to estimate the full hazard function. We compare the performance of CBNNs to regression and neural network-based survival methods in a simulation and three case studies using two time-dependent metrics. First, we examine performance on a simulation involving a complex baseline hazard and time-varying interactions to assess all methods, with CBNN outperforming competitors. Then, we apply all methods to three real data applications, with CBNNs outperforming the competing models in two studies and showing similar performance in the third. Our results highlight the benefit of combining case-base sampling with deep learning to provide a simple and flexible framework for data-driven modeling of single event survival outcomes that estimates time-varying effects and a complex baseline hazard by design. An R package is available at https://github.com/Jesse-Islam/cbnn.
Paper Structure (25 sections, 14 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 25 sections, 14 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Survival models that ignore censoring
  4. Cox partial log-likelihood
  5. Discrete-time
  6. Fully parametric framework
  7. Case-base neural network, metrics, hyperparameters and software
  8. Case-base sampling
  9. Neural networks to model the hazard function
  10. Performance metrics
  11. Brier Score (BS)
  12. Integrated Brier Score (IBS)
  13. Index of prediction accuracy (IPA)
  14. Inverse probability censoring weights-adjusted time-dependent area under the receiver operating characteristic curve (AUC_IPCW)
  15. Hyperparameter selection
  16. ...and 10 more sections

Figures (2)

  • Figure 1: Methodological steps involved in CBNN. The first step, case-base sampling, is completed before training begins. Then, we pass this sampled data through a feed-forward neural network, add an offset to adjust for the bias inherent in case-base sampling and apply a sigmoid activation function to estimate a probability. Once the neural network model completes its training, we can convert the probability to a hazard for the survival outcome of interest.
  • Figure 2: Performance of each model in the complex simulation (A, E), multiple myeloma (MM) case study (B, F), free light chain (FLC) case study (C, G) and prostate cancer (Prostate) case study (D, H). The first row shows the IPA for each model in each study over follow-up time. Negative values mean the model performs worse than the null model and positive values mean the model performs better. The second row shows the $AUC_{IPCW}$ for each model in each study over follow-up time, where higher is better. Each model-specific metric in each study shows a 95% confidence interval over 100 iterations. Metrics are shown for six models: Case-Base with Logistic Regression (CBLR), Case-Base Neural Network (CBNN), Cox Proportional Hazard (Cox), DeepHit and DeepSurv. The Kaplan-Meier (KM) model serves as a baseline, predicting the average curve for all individuals. CBLR and Cox have near identical performance, resulting in curves that overlap. The Optimal model (a CBLR model with the exact interaction terms and baseline hazard specified) shows the best performance we can expect on the simulated data.