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.
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.
@cite_26 @cite_4 @cite_22 @cite_7 @cite_8 @cite_21 @cite_1 @cite_15
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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 patients 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 patients covariates and their risk of failure. We then show how DeepSurv models the relationship between a patients 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 its 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, u 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. 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 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. We propose Lomax delegate racing (LDR) to explicitly model the mechanism of survival under competing risks and to interpret how the covariates accelerate or decelerate the time to event. LDR explains non-monotonic covariate effects by racing a potentially infinite number of sub-risks, and consequently relaxes the ubiquitous proportional-hazards assumption which may be too restrictive. Moreover, LDR is naturally able to model censoring and missing event times or types. For inference, we develop a Gibbs sampler under data augmentation for moderately sized data, along with a stochastic gradient descent maximum a posteriori inference algorithm for big data applications. Illustrative experiments are provided on both synthetic and real datasets, and comparison with various benchmark algorithms for survival analysis with competing risks demonstrates distinguished performance of LDR. 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... The electronic health record (EHR) provides an unprecedented opportunity to build actionable tools to support physicians at the point of care. In this paper, we investigate survival analysis in the context of EHR data. We introduce deep survival analysis, a hierarchical generative approach to survival analysis. It departs from previous approaches in two primary ways: (1) all observations, including covariates, are modeled jointly conditioned on a rich latent structure; and (2) the observations are aligned by their failure time, rather than by an arbitrary time zero as in traditional survival analysis. Further, it (3) scalably handles heterogeneous (continuous and discrete) data types that occur in the EHR. We validate deep survival analysis model by stratifying patients according to risk of developing coronary heart disease (CHD). Specifically, we study a dataset of 313,000 patients corresponding to 5.5 million months of observations. When compared to the clinically validated Framingham CHD risk score, deep survival analysis is significantly superior in stratifying patients according to their risk. A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results.