Source-linked AI summary
Deep Survival Machines: Fully Parametric Survival Regression and Representation Learning for Censored Data with Competing Risks
Chirag Nagpal, Xinyu Rachel Li, Artur Dubrawski
TL;DR
The paper addresses censored time-to-event prediction without relying on constant proportional hazards. It introduces Deep Survival Machines, a neural hierarchical graphical model with mixture-based parametric survival distributions, and reports advantages across single-event and competing-risks experiments. The framework is evaluated under right censoring and remains open to extensions beyond the paper’s censoring setting.
Problem
Existing survival methods face dimensionality or strong proportional-hazards assumptions, while fully parametric survival estimation with competing risks and censoring remains an identified gap.
Method
Deep Survival Machines uses a neural-network-parameterized hierarchical graphical model that represents survival functions as weighted mixtures of parametric distributions and handles censored and uncensored data.
Results
DSM outperformed or was competitive with comparison methods across reported single-event and competing-risks experiments, including longer event horizons and both synthetic competing events.
Takeaways & Limitations
Learned representations can support knowledge transfer across competing risks while providing fully parametric time-to-event estimates without strong proportional-hazards assumptions.
Takeaways & Limitations
The paper assumes right censoring and notes that extending the framework to left truncation and interval censoring remains future work.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
Survival regression estimates time-to-event risks despite censoring and competing events, but established non-parametric and semi-parametric approaches face dimensionality and modeling-assumption challenges. Deep Survival Machines addresses these challenges with a fully parametric, neural graphical-model approach designed for time-varying risks and competing risks.
- Motivation: Survival regression estimates the probability that an event occurs beyond a future time given input covariates, while censoring stops observation before some events occur.Multiple event types produce the competing-risks setting.
- Limitations of prior methods: Non-parametric survival methods may suffer from the curse of dimensionality, while semi-parametric methods typically depend on strong modeling assumptions.The proportional-hazards assumption is specifically identified as potentially unrealistic in practical applications.
- Proposed approach: Deep Survival Machines estimates time-to-event outcomes under censoring using a hierarchical graphical model parameterized by neural networks.The model learns distributional representations of the input covariates.
- Proposed approach: DSM represents the conditional survival function as a mixture of parametric survival distributions, avoids strong proportional-hazards assumptions, and enables time-varying risks.Its learned distributed representations are also intended to support knowledge transfer across competing risks.
- Novelty: The paper positions DSM as the first fully parametric survival-regression method for competing risks in the presence of censoring.Prior work is grouped into semi-parametric proportional-hazards methods and non-parametric similarity- or kernel-based models.
III. APPROACH: ‘Deep Survival Machines’
Deep Survival Machines models conditional survival times as mixtures of parametric positive-support distributions whose weights and parameters depend on neural representations of input covariates. The architecture uses an MLP to produce distributional representations and combines K primitive survival distributions for each individual.
- Data assumptions: The model assumes right-censored survival data, observing either an event time or censoring time for each individual.The censoring process is assumed independent of the actual failure time.
- Primitive distributions: DSM models P(T|X=x) as a mixture of K well-defined parametric primitive distributions with positive support and closed-form CDFs.The paper experiments with Weibull and Log-Normal primitives.
- Architecture: The pipeline applies a softmax over mixture size K before describing P(T|X=x) as a mixture of primitive distributions.This is the architecture shown in Figure 1.
- Architecture: Input covariates pass through an MLP, whose representation determines mixture weights and the parameters of K underlying survival distributions.The final individual survival distribution is their weighted average.
C. Parameter Estimation
DSM estimates parameters using loss components for uncensored and censored observations, then combines them into a joint objective. The method also addresses bias from the long tails of positive-support survival distributions.
- Uncensored loss: For uncensored observations, DSM uses a maximum-likelihood objective and applies Jensen’s inequality in deriving the relevant bound.The derivation is presented as the uncensored loss component.
- Censoring loss: For censored observations, the method constructs a lower-bound loss based on the probability that survival exceeds the observed censoring time.The censored-observation objective is introduced separately from the uncensored likelihood term.
- Long-tail bias: DSM mitigates long-tail bias by considering a bounded tail probability P(tmax > T > t) instead of only P(T > t) for censored instances.The upper threshold tmax can be tuned as a hyperparameter, although the paper notes a simplified treatment.
- Combined loss: The combined loss joins the individual loss components for censored and uncensored observations.A scalar α trades off regression loss against the evidence lower bound for uncensored observations.
D. Handling Multiple Competing Risks
DSM adapts its neural representation to competing risks by sharing an MLP while using risk-specific distribution parameters. Experiments include synthetic and SEER data, with censoring and multiple event types represented explicitly.
- Competing-risks model: DSM uses one shared MLP representation for multiple risks, then separate parameters to describe each competing event distribution.Maximum likelihood treats occurrence of a competing event before another as independent censoring.
- Evaluation: The experiments evaluate single-event risk estimation, robustness under increased censoring, and representation transfer across competing events.These evaluations are described as the study’s experimental goals.
- Datasets: Single-event experiments use the SUPPORT and METABRIC medical datasets, while the competing-risks experiments use synthetic and SEER datasets.The dataset descriptions identify distinct single-event and competing-risks evaluation settings.
- Synthetic data: The synthetic dataset contains 30,000 patients, with 50% subjected to random right censoring and two dependent event times generated from covariates.The setup is designed to test whether knowledge of one event improves prediction of the other.
- SEER data: SEER contains 65,481 breast-cancer patients, with breast-cancer death and cardiovascular death treated as competing risks.The reported event counts are 13,564 breast-cancer deaths and 4,245 cardiovascular deaths.
B. Baselines
DSM is compared with semi-parametric, nonparametric, deep proportional-hazards, discrete-time competing-risks, and cause-specific baselines. The competing-risks comparisons include models fitted separately for each event.
- Baseline models: The baseline set includes Cox Proportional Hazards, Random Survival Forests, DeepSurv, DeepHit, and Fine-Gray.These span semi-parametric, nonparametric, deep, discrete-time, and competing-risks approaches.
- Proportional-hazards baselines: Cox Proportional Hazards and DeepSurv assume a constant baseline hazard or constant proportional hazard structure.DeepSurv is described as making the familiar constant-baseline-hazard assumption, as does CPH.
- Competing-risks baselines: DeepHit jointly models competing event times but discretizes the output space to a fixed set of failure times.The paper contrasts this discrete-time design with DSM’s parametric survival distributions.
- Competing-risks baselines: For SYNTHETIC and SEER, DSM is compared with cause-specific CPH and RSF models that fit separate survival regressions for each competing event.The other event is treated as censored in each cause-specific regression.
C. Performance Metrics
DSM is evaluated with time-dependent concordance across event horizons, calibration via censoring-weighted Brier scores, and robustness tests under artificial censoring. Across datasets and horizons, DSM generally outperforms or matches the comparison methods, especially where risks vary over time or data are sparse.
- Evaluation Metrics: Time-dependent concordance evaluates pairwise relative-risk ordering at multiple event horizons, reducing reliance on constant proportional hazards.Ctd is reported at the 25%, 50%, and 75% event-horizon quantiles.
- Evaluation Metrics: Censoring-weighted Brier scores assess DSM calibration against baselines at each event quantile.
- SUPPORT: DSM significantly outperformed RSF at longer SUPPORT event horizons, while RSF remained comparable at the 25% quantile across censoring levels.
- SUPPORT: DSM significantly outperformed DeepSurv at shorter SUPPORT horizons, although DeepSurv was competitive at longer horizons.
- METABRIC: DSM significantly outperformed deep-learning baselines on METABRIC and outperformed RSF on average in 10-fold cross-validation.
- Interpretation: DSM is more robust when data are sparse at longer horizons and can capture flexible survival patterns without the constant proportional-hazard assumption.
F. Competing Risks Scenario
In competing-risks experiments, DSM performs strongly across synthetic and SEER datasets while learning representations transferable between risks. Its advantages are clearest at longer horizons and for the majority SEER risk, though performance is less conclusive for CVD.
- SYNTHETIC: DSM outperformed all other baselines on SYNTHETIC at the 25%, 50%, and 75% event-horizon quantiles while remaining competitive with DeepHit.
- SEER: On SEER, DSM significantly outperformed all other baselines for the majority risk, Breast Cancer.
- SEER: SEER results for CVD were less conclusive because DeepHit was competitive at the 25% quantile, which the authors attribute to class imbalance.
- Representation Learning: DSM learns representations for knowledge transfer across competing risks by training on one event and using the learned embeddings to predict another event with a linear CPH model.
- Representation Learning: On SYNTHETIC, DSM outperformed competing representation-learning baselines in held-out Event 2 relative-risk prediction.
VI. MODEL COMPLEXITY AND SCALABILITY
Deep Survival Machines offers competitive predictive performance with lower computational and parameter complexity than considered deep-learning alternatives. Its runtime scales better with data size, although RSF is faster on METABRIC before scaling poorly on larger SUPPORT data.
- DSM trains faster than other deep-learning approaches and scales better as data size increases.
- DSM requires fewer learnable parameters than the considered deep-learning alternatives.
- DSM’s runtime and space complexity advantages are visible against DeepHit and DeepSurv, with CPH included as a classical baseline.
- RSF trains faster on METABRIC but scales poorly with increasing data, as shown by slower runtime on SUPPORT.
- The paper reports benefits across classical and state-of-the-art survival regression approaches and knowledge transfer across competing risks.
APPENDIX
DSM parameterizes survival distributions from neural representations and trains with objectives designed for censored and uncensored observations. In competing-risks settings, other events are treated as censoring for risk-specific loss computation.
- At test time, DSM represents survival as a weighted mixture of K survival distribution primitives, with weights produced by a neural-network softmax.
- The model’s loss function handles both censored and uncensored observations.
- For each observation, covariates x_i parameterize primitive-distribution parameters through a multilayer perceptron.
- The activation function depends on the chosen primitive: SELU for Weibull and Tanh for Log-Normal distributions.
- With M competing risks, risk-specific objectives are computed by treating other events as censoring before forming the total loss.
B. Results in Tabular Format
The tabular experiments evaluate DSM and baselines across event-time horizons, censoring levels, single-risk datasets, and competing-risks datasets. Results use time-dependent concordance and cross-validation, with baseline settings selected through documented tuning procedures.
- B. Results in Tabular Format: Ctd is evaluated at the 25%, 50%, and 75% event-time quantiles, with means and 90% confidence intervals computed using 5-fold cross-validation.
- B. Results in Tabular Format: SUPPORT and METABRIC provide single-risk results across different censoring levels, including artificially added censoring of 25% or 50% of originally uncensored training observations.
- B. Results in Tabular Format: SYNTHETIC and SEER provide competing-risks results, with cause-specific CPH and RSF included as baselines.
- C. Hyperparameter Tuning for the Baselines: DSM is compared with several competing approaches, and RSF and DeepHit hyperparameters are selected using validation Ctd.
- C. Hyperparameter Tuning for the Baselines: CPH and Fine-Gray use default settings, while DeepSurv uses hyperparameters from its GitHub repository.
- B. Results in Tabular Format: The SUPPORT and METABRIC tables report performance at event-time quantiles and different censoring levels.
D. Data Preprocessing
The preprocessing appendix documents missing-data patterns and imputation for SUPPORT and SEER, alongside dataset-specific tables and computational settings. Missing values are imputed using fixed physiologic values or training-set statistics, depending on feature and dataset.
- The appendix reports missing-value statistics for SUPPORT and SEER and dataset-specific tables for performance and DeepHit hyperparameters.
- SUPPORT has missing values in 21 of 30 features, while SEER has missing data in 5 of 21 patient covariates.
- Seven SUPPORT features use suggested normal physiologic values for imputation.
- Remaining SUPPORT features and the five SEER features are imputed with training-set means for numeric variables and modes for categorical variables.
- Most experiments use a Linux machine with an eight-core Intel i7 CPU and 32 GB RAM, whereas DeepHit uses a TITAN X GPU cluster.