[Paper Review] DeepHazard: neural network for time-varying risks

Denise Rava, Jelena Bradić|arXiv (Cornell University)|Jul 26, 2020

Statistical Methods and Inference40 references4 citations

TL;DR

DeepHazard proposes a deep neural network for survival analysis that models time-varying hazards without assuming proportional hazards, using a flexible, additive-in-time hazard formulation. It outperforms state-of-the-art methods on METABRIC, GBSG, and ACTG datasets, achieving superior predictive performance via a C-index improvement through end-to-end training with time-dependent covariates and dynamic risk scoring.

ABSTRACT

Prognostic models in survival analysis are aimed at understanding the relationship between patients' covariates and the distribution of survival time. Traditionally, semi-parametric models, such as the Cox model, have been assumed. These often rely on strong proportionality assumptions of the hazard that might be violated in practice. Moreover, they do not often include covariate information updated over time. We propose a new flexible method for survival prediction: DeepHazard, a neural network for time-varying risks. Our approach is tailored for a wide range of continuous hazards forms, with the only restriction of being additive in time. A flexible implementation, allowing different optimization methods, along with any norm penalty, is developed. Numerical examples illustrate that our approach outperforms existing state-of-the-art methodology in terms of predictive capability evaluated through the C-index metric. The same is revealed on the popular real datasets as METABRIC, GBSG, and ACTG.

Motivation & Objective

• To address the limitations of traditional survival models like the Cox model, which rely on strong proportional hazards assumptions that are often violated in real-world data.
• To develop a flexible deep learning framework that models non-proportional, time-varying hazards while preserving the continuous nature of survival time.
• To incorporate time-dependent covariates by structuring the model as a sequence of time-frame-specific neural networks that share information across intervals.
• To unify feature extraction and survival prediction into a single end-to-end trainable architecture, avoiding separate optimization steps.
• To improve predictive performance in survival analysis by replacing restrictive parametric or discrete-time assumptions with a continuous, additive hazard formulation.

Proposed method

• The model uses a sequence of time-frame-specific neural networks, each trained on individuals still at risk at that time interval, enabling dynamic hazard estimation over time.
• Each time-frame network takes as input the current covariates and the output (risk score) from the previous time-frame network, capturing temporal dependencies across intervals.
• The hazard function is modeled as an additive combination of a baseline hazard and a time-varying component learned by the neural network, allowing flexible, nonparametric hazard forms.
• The loss function is derived from the additive hazards model, minimizing the difference between observed and predicted cumulative hazards via a survival-specific objective function.
• The method supports various optimization techniques and regularization via norm penalties, enabling robust training and generalization.
• The model estimates the conditional survival function using the cumulative hazard, which can then be used to compute predictive metrics like the C-index and explained variation.

Experimental results

Research questions

• RQ1 Can a deep neural network effectively model time-varying hazards without assuming proportional hazards?
• RQ2 How does the proposed method perform compared to existing state-of-the-art survival models in terms of predictive accuracy?
• RQ3 To what extent can the model incorporate time-dependent covariates and dynamic risk updates in a continuous survival framework?
• RQ4 Does the end-to-end training of feature extraction and survival prediction improve predictive performance compared to decoupled approaches?
• RQ5 Can the model maintain robustness and flexibility across diverse real-world datasets with non-proportional hazards?

Key findings

• DeepHazard significantly outperforms existing state-of-the-art methods in predictive performance, as measured by the C-index, on the METABRIC, GBSG, and ACTG real-world survival datasets.
• The model achieves superior predictive capability by modeling non-proportional hazards through a flexible, additive-in-time hazard formulation without relying on proportional hazards assumptions.
• The integration of time-dependent covariates via sequential time-frame networks with cross-time information flow enhances dynamic risk prediction accuracy.
• The end-to-end training of the neural network, combining feature learning and survival modeling, leads to better generalization and predictive performance than separate optimization of features and survival models.
• The model's ability to estimate the conditional survival function enables downstream analysis such as expected survival time and explained variation, supporting clinical and biological interpretation.