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DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network

Jared Katzman, Uri Shaham, Jonathan Bates, Alexander Cloninger, Tingting Jiang, Yuval Kluger

arXiv:1606.00931v3stat.MLcs.NE

TL;DR

Survival models need to capture nonlinear relationships between patient features, risk, and treatment effectiveness, but prior neural-network approaches had not shown reliable advantages or treatment-recommendation capability. DeepSurv addresses this with a nonlinear Cox proportional hazards network, matching or exceeding other survival methods and producing personalized recommendations that increase population survival time.

  • Problem

    Linear Cox models can be too simplistic for nonlinear risk and treatment interactions, while prior neural networks had not demonstrated advantages over standard survival methods or effective treatment recommendation.

  • Method

    DeepSurv is a nonlinear Cox proportional hazards deep neural network that predicts individual risk and uses treatment-specific risk estimates to recommend treatments.

  • Results

    DeepSurv performs as well as or better than other survival methods and provides personalized recommendations that increase population survival time, outperforming random survival forests for treatment recommendation.

  • Takeaways & Limitations

    DeepSurv supports using deep neural networks to model complex patient–treatment relationships and generate individualized treatment recommendations in survival analysis.

  • Takeaways & Limitations

    The treatment recommender assumes that each treatment has an independent risk function.

Abstract

from arXiv · show

Medical practitioners use survival models to explore and understand the relationships between patients' covariates (e.g. clinical and genetic features) and the effectiveness of various treatment options. Standard survival models like the linear Cox proportional hazards model require extensive feature engineering or prior medical knowledge to model treatment interaction at an individual level. While nonlinear survival methods, such as neural networks and survival forests, can inherently model these high-level interaction terms, they have yet to be shown as effective treatment recommender systems. We introduce DeepSurv, a Cox proportional hazards deep neural network and state-of-the-art survival method for modeling interactions between a patient's covariates and treatment effectiveness in order to provide personalized treatment recommendations. We perform a number of experiments training DeepSurv on simulated and real survival data. We demonstrate that DeepSurv performs as well as or better than other state-of-the-art survival models and validate that DeepSurv successfully models increasingly complex relationships between a patient's covariates and their risk of failure. We then show how DeepSurv models the relationship between a patient's features and effectiveness of different treatment options to show how DeepSurv can be used to provide individual treatment recommendations. Finally, we train DeepSurv on real clinical studies to demonstrate how it's personalized treatment recommendations would increase the survival time of a set of patients. The predictive and modeling capabilities of DeepSurv will enable medical researchers to use deep neural networks as a tool in their exploration, understanding, and prediction of the effects of a patient's characteristics on their risk of failure.

1 Introduction

DeepSurv is proposed as a Cox proportional hazards deep neural network for modeling nonlinear covariate-risk and treatment interactions. The paper aims to establish its survival-prediction performance and demonstrate personalized treatment recommendations for medical research.

  • Motivation: The Cox proportional hazards model assumes that a patient’s failure risk is a linear combination of covariates, which may be too simplistic for personalized treatment recommendations.This limitation motivates richer survival models for applications involving nonlinear risk functions.
  • Motivation: Risk-predicting neural networks learn complex nonlinear relationships between prognostic features and failure risk without prior feature selection or domain expertise.They can provide personalized treatment recommendations from the computed risk of each treatment option.
  • Problem: Previous neural-network survival studies produced mixed results and failed to demonstrate improvements beyond the linear Cox model.The paper identifies this unresolved performance gap as a motivation for applying modern deep learning to survival analysis.
  • Contributions: The paper proposes DeepSurv, a modern Cox proportional hazards deep neural network designed as the basis for a personalized treatment recommender system.The stated goals are to match or exceed other survival methods in risk prediction and support personalized treatment recommendations.
  • Contributions: DeepSurv is evaluated on survival data with linear and nonlinear risk functions, with treatment-group information included to illustrate learning complex relationships.These contributions address both survival-analysis performance and treatment-effect modeling.

2 Background

Survival analysis represents each patient with baseline data, an event time, and an event indicator, while modeling survival and failure risk through survival and hazard functions. The linear Cox proportional hazards model is useful but becomes impractical for complex nonlinear interactions, motivating nonlinear survival approaches.

  • Survival data: Survival data contains baseline data x, failure event time T, and event indicator E.Observed events have E = 1; unobserved events are right-censored with E = 0.
  • Survival data: Right-censoring records time until last patient contact and can introduce bias when standard regression discards it as missing data.For censored observations, T measures time from baseline-data collection to the last contact or study end.
  • Survival functions: Survival and hazard functions describe, respectively, the probability of surviving beyond time t and the risk of death at time t.The survival function is S(t) = Pr(T > t), while greater hazard indicates greater death risk.
  • Proportional hazards models: A proportional hazards model combines a baseline hazard λ0(t) with a covariate-dependent risk function h(x), yielding λ(t|x) = λ0(t) · eh(x).The CPH estimates h(x) linearly as ˆhβ(x) = βT x and tunes β using the Cox partial likelihood.
  • Model limitations: The CPH model requires high-level interaction terms for nonlinear relationships, becoming prohibitively expensive as features and interactions increase.This limitation motivates more complex nonlinear survival models, including neural-network and random-survival-forest approaches.

3 Methods

DeepSurv is a nonlinear Cox proportional hazards neural network that predicts patient risk and derives personalized treatment recommendations from treatment-specific hazard estimates. Its recommender function compares log hazards without requiring a priori treatment interaction terms.

  • DeepSurv architecture and training: DeepSurv is a multi-layer perceptron with a single output node estimating the patient risk function ĥ_θ(x), trained using negative log partial likelihood.The network weights θ parameterize the estimated risk function.
  • DeepSurv architecture and training: The methodology permits deep architectures and uses regularization, ReLU with batch normalization, SELU, dropout, gradient-based optimization, momentum, gradient clipping, and learning-rate scheduling.Hyper-parameters are tuned through a random hyper-parameter optimization search.
  • Personalized treatment recommendation: Patients are assigned to one of n treatment groups, with each treatment i modeled by an independent risk function h_i(x).The study-level hazard function is generalized from these treatment-specific risks.
  • Personalized treatment recommendation: DeepSurv defines the recommender function rec_ij(x) as the difference between log hazards for treatments i and j under a shared baseline hazard λ_0(t).The function represents the personal risk of prescribing one treatment option over another.
  • Personalized treatment recommendation: A positive rec_ij(x) means treatment i has higher death risk than treatment j, whereas a negative recommendation indicates that treatment i is more effective.Recommendations are obtained by passing a patient through the network under each treatment and comparing the outputs.
  • Personalized treatment recommendation: Unlike CPH, DeepSurv computes treatment recommendations without prespecifying interaction terms, avoiding the extensive experimentation or prior biological knowledge otherwise needed to discover them.The CPH model produces a constant recommender function unless treatment interaction terms are added.

4 Results … 4.2 Simulated survival data

DeepSurv is evaluated on simulated survival data, real survival data, and treatment-recommendation tasks against linear CPH and RSF baselines. In simulated survival experiments, it matches or exceeds competitors in predictive accuracy while more accurately recovering linear and nonlinear risk functions.

  • 4 Results: The evaluation comprises simulated survival, real survival, simulated treatment, and real treatment experiments designed to assess risk learning, prediction, and treatment modeling.The simulated experiments test whether DeepSurv learns the population’s true risk function, while treatment experiments assess multiple treatment-specific risk functions.
  • 4.1 Evaluation: DeepSurv is compared with linear CPH and RSF, although RSF cannot be evaluated for recovering a simulated dataset’s true risk function h(x).CPH provides a linear baseline, while RSF provides a state-of-the-art nonlinear survival comparison for predictive accuracy.
  • Survival data: The C-index measures how well models order patients by death time, with c = 0.5 representing random ranking and c = 1 a perfect ranking.Confidence intervals are obtained by bootstrapping test-set samples with replacement.
  • Treatment recommendations: DeepSurv and RSF recommend treatments individually, whereas CPH would recommend the same treatment to every patient without preselected treatment-interaction terms.The individualized recommenders predict each patient’s risk under each treatment option.
  • Treatment recommendations: Patients are divided into Recommendation and Anti-Recommendation subsets, whose median survival times and log-rank test are used to assess whether recommendations improve survival.The Recommendation subset receives the model’s recommended treatment; the Anti-Recommendation subset does not.
  • 4.2 Simulated survival data: Simulated survival experiments use linear and nonlinear Gaussian risk functions to test whether DeepSurv models true risk rather than overfitting noise.The risk function depends on two of ten covariates, allowing assessment of whether DeepSurv identifies relevant features among noise.
  • 4.2.1 Linear risk experiment: In the linear-risk experiment, DeepSurv performs as well as linear Cox regression and better than RSF in predictive ability.DeepSurv also estimates the true risk function more accurately than CPH, despite similar predictive abilities.
  • 4.2.1 Linear risk experiment: 20.197 versus 0.126: CPH and DeepSurv obtain these MSEs, respectively, when predicting the true risk in the linear-risk experiment.The larger CPH error is especially evident for patients with high positive risk.

4.3 Real survival data experiments · 4.4 Study to Understand Prognoses Preferences Outcomes and Risks of Treatment (SUPPORT) · 4.5 Treatment recommender system experiments

DeepSurv is evaluated on real survival datasets, where it matches or exceeds established survival baselines, and in treatment-recommendation experiments demonstrating personalized risk modeling and improved survival among recommended patients. The experiments span simulated treatment effects and breast cancer clinical-study data.

  • 4.3 Real survival data experiments: DeepSurv is compared with CPH on WHAS, SUPPORT, and METABRIC to assess state-of-the-art predictive ability on real survival data.The studies cover myocardial-infarction survival, seriously ill hospitalized adults, and breast-cancer gene-expression and clinical data.
  • 4.3.1 Worcester Heart Attack Study (WHAS): WHAS contains 1,638 observations and 5 patient features, with 42.12 percent of patients dying and a median death time of 516.0 days.Twenty percent of the dataset is reserved for testing, and the passage reports that DeepSurv outperforms CPH.
  • 4.4 Study to Understand Prognoses Preferences Outcomes and Risks of Treatment (SUPPORT): DeepSurv performs as well as RSF and better than CPH on the larger SUPPORT study, validating its ability to rank patient risks on real survival data.SUPPORT includes 9,105 seriously ill hospitalized adults and 14 features with almost all entries observed.
  • 4.4.1 Molecular Taxonomy of Breast Cancer International Consortium (METABRIC): On METABRIC, DeepSurv performs better than both CPH and RSF, modeling risk effects from gene-expression data and showing potential as a prognostic tool comparable to IHC4+C.The dataset contains 1,980 patients, and preprocessing joins four gene indicators with clinical features in line with IHC4+C.
  • 4.5 Treatment recommender system experiments: The treatment-recommender experiments use simulated treatment data and a real breast-cancer study of hormone-treatment effects to test personalized recommendations.The simulated experiment adds a treatment covariate, while the real-data experiment trains on Rotterdam data and recommends treatments for GBSG patients.
  • 4.5.1 Simulated treatment data: In simulated data, DeepSurv models constant risk under τ = 0 and Gaussian covariate effects under τ = 1, with λmax = 10 and r = 0.5 for the treated group.The treatment groups are assigned uniformly, with τ = 0 representing unaffected patients and τ = 1 receiving Gaussian treatment effects.
  • 4.5.2 Rotterdam & German Breast Cancer Study Group (GBSG): On GBSG breast-cancer data, DeepSurv improves predictive ability relative to CPH and RSF, and its recommendations significantly distinguish survival curves while increasing median survival time.The GBSG testing data contain 686 randomized-trial patients, with 56 percent censored; recommendations are evaluated against anti-recommendations using Kaplan-Meier curves.

5 Conclusion

DeepSurv demonstrates that deep learning can improve survival-analysis performance through model flexibility and provide effective individualized treatment recommendations. Across increasingly complex datasets, it predicts patient risk mostly as well as or better than linear and nonlinear survival methods.

  • 5 Conclusion: Deep learning enables higher survival-analysis performance through model flexibility.The conclusion identifies model flexibility as a basis for improved performance.
  • 5 Conclusion: DeepSurv provides effective treatment recommendations by predicting how treatment options affect an individual’s risk.Recommendations are based on the predicted treatment effects for each patient.
  • 5 Conclusion: DeepSurv predicts patients’ risk mostly as well as or better than other linear and nonlinear survival methods across increasingly complex survival datasets.The conclusion reports validation on datasets of increasing complexity.

Appendix · A Experimental Details

The experiments use established survival-analysis and machine-learning packages within Docker containers for reproducibility. Network configurations are optimized through randomized Sobol sampling and three-fold cross-validation.

  • A Experimental Details: Linear CPH regression, Kaplan–Meier estimation, c-index statistics, and log-rank tests use the Lifelines Python package.
  • A Experimental Details: DeepSurv is implemented in Theano with the Lasagne Python package.
  • A Experimental Details: Random survival forests are fitted using the R package randomForestSRC.
  • A Experimental Details: All experiments run in Docker containers to support reproducibility.The DeepSurv container uses the FloydHub base image.
  • A Experimental Details: The network hyper-parameters include depth, size, learning rate, ℓ2 regularization, dropout, exponential learning-rate decay, and momentum.
  • A Experimental Details: Random hyper-parameter optimization uses the method proposed in and the Python package Optunity.
  • A Experimental Details: The Sobol solver samples each hyper-parameter from a predefined range.
  • A Experimental Details: Configuration performance is evaluated using k-means cross-validation with k = 3.

A.1 Model Hyper-parameters

DeepSurv’s experimental hyper-parameters were selected through random search, with learning rates adjusted by inverse time decay at each epoch.

  • Hyper-parameter tuning: DeepSurv’s hyper-parameters were tuned using a random hyper-parameter search with the Python package Optunity.The resulting settings were summarized for each experiment’s DeepSurv network.
  • Learning-rate schedule: The learning rate was adjusted at each epoch using inverse time decay.The passage gives the decay relation as “decayed LR := LR 1 + epoch · lr decay rate.”

B CPH Recommender Function

The CPH recommender models treatment as a feature weight within a linear risk function, but its recommendation is constant across patients. It therefore recommends the same treatment option to everyone based on the sign of β0 rather than patient-specific features.

  • CPH risk function: The CPH risk function is a linear combination of patient features, with treatment represented as x0 = τ and weighted by β0.Its form is ˆhβ(x) = βT x = β0τ + β1x1 + ... + βnxn.
  • Recommender function: The CPH recommender function is constant and independent of the patient’s features.This follows from the model’s linear treatment weighting in the recommender formulation.
  • Treatment recommendation: The CPH recommends the same treatment option to all patients according to whether β0 is positive or negative, rather than providing personalized recommendations.It determines whether treatment is effective and, if so, recommends it universally.

C Simulated Data Generation

The simulated datasets draw patient covariates uniformly and, when applicable, assign treatment randomly. Event times follow a Cox-based construction and are right-censored at a trial end time chosen to retain 90% observed deaths.

  • Data generation: Baseline covariates x are drawn uniformly from [−1, 1)^d, and treatment status τ_x is sampled from Bernoulli(p = 0.5) when treatment is included.Treatment assignment is independent randomization with probability 0.5.
  • Event-time generation: The initial death time u follows an exponential distribution with mean µ = 5 under a shared baseline hazard function λ_0(t).The Cox proportional hazards model assumes λ_0(t) is shared across patients.
  • Event-time generation: With treatment, each individual death time is generated as T = u e^(τ_xh(x)).The treatment variable and patient-specific function h(x) modify the initial death time.
  • Censoring: Event times are right-censored at trial end time T_0, which is chosen so 90 percent of people have an observed death time.The final observed outcome time is Z = min(T, T_0).
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