[Paper Review] Deep Survival Machines: Fully Parametric Survival Regression and Representation Learning for Censored Data with Competing Risks

Chirag Nagpal, Xinyu Rachel Li+1 more|arXiv (Cornell University)|Mar 2, 2020

Statistical Methods and InferenceMathematics21 citations

TL;DR

Deep Survival Machines (DSM) proposes a fully parametric survival regression model that uses deep neural networks to learn nonlinear representations of covariates and models the survival function as a mixture of parametric distributions, enabling time-varying hazards without proportional hazards assumptions. It outperforms state-of-the-art methods like DeepHit and DeepSurv in predictive performance and knowledge transfer across competing risks, with faster training and lower parameter complexity.

ABSTRACT

We describe a new approach to estimating relative risks in time-to-event prediction problems with censored data in a fully parametric manner. Our approach does not require making strong assumptions of constant proportional hazard of the underlying survival distribution, as required by the Cox-proportional hazard model. By jointly learning deep nonlinear representations of the input covariates, we demonstrate the benefits of our approach when used to estimate survival risks through extensive experimentation on multiple real world datasets with different levels of censoring. We further demonstrate advantages of our model in the competing risks scenario. To the best of our knowledge, this is the first work involving fully parametric estimation of survival times with competing risks in the presence of censoring.

Motivation & Objective

• To address the limitations of proportional hazards assumptions in traditional survival models like Cox Proportional Hazards.
• To develop a fully parametric survival model that can handle censored data and competing risks without relying on semi-parametric assumptions.
• To enable representation learning via deep neural networks that transfer knowledge across multiple competing risk events.
• To reduce model complexity and improve inference efficiency compared to deep learning-based survival models.

Proposed method

• The model estimates the conditional survival function as a mixture of individual parametric survival distributions (e.g., Weibull, Gumbel) using a hierarchical graphical model.
• Neural networks are used to parameterize the mixture weights and distribution parameters as functions of input covariates, enabling nonlinear representation learning.
• A differentiable loss function is designed to handle both censored and uncensored observations, optimizing survival function estimation directly.
• The model jointly learns deep representations of covariates and survival distributions, allowing knowledge transfer across competing risks.
• The approach avoids proportional hazards assumptions by modeling time-varying hazard ratios through flexible parametric mixtures.
• Model training uses backpropagation with gradient-based optimization, and inference is efficient due to the parametric form of the survival distributions.

Experimental results

Research questions

• RQ1 Can a fully parametric survival model outperform semi-parametric and non-parametric methods in time-to-event prediction with censored data?
• RQ2 Does the ability to model time-varying hazards without proportional hazards assumptions improve predictive performance?
• RQ3 Can deep representations learned by the model transfer knowledge across competing risks in multi-event survival settings?
• RQ4 How does the model’s computational efficiency and parameter complexity compare to deep learning-based survival models like DeepHit and DeepSurv?

Key findings

• DSM achieved a C-index of 0.7724 ± 0.0025 on the SYNTHETIC dataset for knowledge transfer from Event 1 to Event 2, outperforming DeepHit (0.7688 ± 0.0040), DeepSurv (0.6988 ± 0.0038), and other representation baselines.
• The model demonstrated superior performance on real-world datasets including METABRIC and SUPPORT, with consistent improvements in concordance index over classical and deep learning baselines.
• DSM required fewer parameters than DeepHit and DeepSurv, with a 30–50% reduction in model size depending on architecture, indicating lower memory and computational cost.
• Training time for DSM was significantly faster than DeepHit and DeepSurv, and it scaled better with increasing data size, outperforming random survival forests on larger datasets.
• The model successfully handled competing risks by learning shared representations that generalized across events, even when training data for each event was disjoint.
• In the absence of proportional hazards, DSM maintained strong predictive performance, demonstrating robustness to violations of the Cox model’s core assumption.