text: Survival analysis in the presence of multiple possible adverse events, i.e., competing risks, is a pervasive problem in many industries (healthcare, finance, etc.). Since only one event is typically observed, the incidence of an event of interest is often obscured by other related competing events. This nonidentifiability, or inability to estimate true cause-specific survival curves from empirical data, further complicates competing risk survival analysis. We introduce Siamese Survival Prognosis Network (SSPN), a novel deep learning architecture for estimating personalized risk scores in the presence of competing risks. SSPN circumvents the nonidentifiability problem by avoiding the estimation of cause-specific survival curves and instead determines pairwise concordant time-dependent risks, where longer event times are assigned lower risks. Furthermore, SSPN is able to directly optimize an approximation to the C-discrimination index, rather than relying on well-known metrics which are unable to capture the unique requirements of survival analysis with competing risks.
  @cite_26: An accurate model of patient-specific kidney graft survival distributions can help to improve shared-decision making in the treatment and care of patients. In this paper, we propose a deep learning method that directly models the survival function instead of estimating the hazard function to predict survival times for graft patients based on the principle of multi-task learning. By learning to jointly predict the time of the event, and its rank in the cox partial log likelihood framework, our deep learning approach outperforms, in terms of survival time prediction quality and concordance index, other common methods for survival analysis, including the Cox Proportional Hazards model and a network trained on the cox partial log-likelihood.
  @cite_4: CONTINUOUS FAILURE TIMES AND THEIR CAUSES Basic Probability Functions Some Small Data Sets Hazard Functions Regression Models PARAMETRIC LIKELIHOOD INFERENCE The Likelihood for Competing Risks Model Checking Inference Some Examples Masked Systems LATENT FAILURE TIMES: PROBABILITY DISTRIBUTIONS Basic Probability Functions Some Examples Marginal vs. Sub-Distributions Independent Risks A Risk-Removal Model LIKELIHOOD FUNCTIONS FOR UNIVARIATE SURVIVAL DATA Discrete and Continuous Failure Times Discrete Failure Times: Estimation Continuous Failure Times: Random Samples Continuous Failure Times: Explanatory Variables Discrete Failure Times Again Time-Dependent Covariates DISCRETE FAILURE TIMES IN COMPETING RISKS Basic Probability Functions Latent Failure Times Some Examples Based on Bernoulli Trials Likelihood Functions HAZARD-BASED METHODS FOR CONTINUOUS FAILURE TIMES Latent Failure Times vs. Hazard Modelling Some Examples of Hazard Modelling Nonparametric Methods for Random Samples Proportional Hazards and Partial Likelihood LATENT FAILURE TIMES: IDENTIFIABILITY CRISES The Cox-Tsiatis Impasse More General Identifiability Results Specified Marginals Discrete Failure Times Regression Case Censoring of Survival Data Parametric Identifiability MARTINGALE COUNTING PROCESSESES IN SURVIVAL DATA Introduction Back to Basics: Probability Spaces and Conditional Expectation Filtrations Martingales Counting Processes Product Integrals Survival Data Non-parametric Estimation Non-parametric Testing Regression Models Epilogue APPENDIX 1: Numerical Maximisation of Likelihood Functions APPENDIX 2: Bayesian Computation Bibliography Index
  @cite_22: We introduce a new approach to competing risks using random forests. Our method is fully non-parametric and can be used for selecting event-specific variables and for estimating the cumulative incidence function. We show that the method is highly effective for both prediction and variable selection in high-dimensional problems and in settings such as HIV AIDS that involve many competing risks.
  @cite_7: 
  @cite_8: 
  @cite_21: Abstract With explanatory covariates, the standard analysis for competing risks data involves modeling the cause-specific hazard functions via a proportional hazards assumption. Unfortunately, the cause-specific hazard function does not have a direct interpretation in terms of survival probabilities for the particular failure type. In recent years many clinicians have begun using the cumulative incidence function, the marginal failure probabilities for a particular cause, which is intuitively appealing and more easily explained to the nonstatistician. The cumulative incidence is especially relevant in cost-effectiveness analyses in which the survival probabilities are needed to determine treatment utility. Previously, authors have considered methods for combining estimates of the cause-specific hazard functions under the proportional hazards formulation. However, these methods do not allow the analyst to directly assess the effect of a covariate on the marginal probability function. In this article we pro...
  @cite_1: Previous research has shown that neural networks can model survival data in situations in which some patients' death times are unknown, e.g. right-censored. However, neural networks have rarely been shown to outperform their linear counterparts such as the Cox proportional hazards model. In this paper, we run simulated experiments and use real survival data to build upon the risk-regression architecture proposed by Faraggi and Simon. We demonstrate that our model, DeepSurv, not only works as well as other survival models but actually outperforms in predictive ability on survival data with linear and nonlinear risk functions. We then show that the neural network can also serve as a recommender system by including a categorical variable representing a treatment group. This can be used to provide personalized treatment recommendations based on an individual's calculated risk. We provide an open source Python module that implements these methods in order to advance research on deep learning and survival analysis.
  @cite_15: 
summary: Previous work on classical survival analysis has demonstrated the advantages of deep learning over statistical methods @cite_26 @cite_1 @cite_8 . Cox proportional hazards model @cite_15 is the baseline statistical model for survival analysis, but is limited since the dependent risk function is the product of a linear covariate function and a time dependent function, which is insufficient for modeling complex non-linear medical data. @cite_1 replaced the linear covariate function with a feed-forward neural network as input for the Cox PH model and demonstrated improved performance. The current literature addresses competing risks based on statistical methods (the Fine Gray model @cite_21 ), classical machine learning (Random Survival Forest @cite_7 @cite_22 ), multi-task learning @cite_4 ) etc., with limited success. These existing competing risk models are challenged by computational scalability issues for datasets with many patients and multiple covariates. To address this challenge, we propose a deep learning architecture for survival analysis with competing risks to optimize the time-dependent discrimination index. This is not trivial and will be elaborated in the next section.