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.
Applications of machine learning in healthcare often require working with time-to-event prediction tasks including prognostication of an adverse event, re-hospitalization or death. Such outcomes are typically subject to censoring due to loss of follow up. Standard machine learning methods cannot be applied in a straightforward manner to datasets with censored outcomes. In this paper, we present auton-survival, an open-source repository of tools to streamline working with censored time-to-event or survival data. auton-survival includes tools for survival regression, adjustment in the presence o…
Montreal Institute for Learning Algorithms（蒙特利尔学习算法研究所）；Université de Montréal（蒙特利尔大学）；Centre Hospitalier de l’Université de Montréal（蒙特利尔大学中心医院）；École Polytechnique de Montréal（蒙特利尔综合理工学院）
Survival analysis is a type of semi-supervised ranking task where the target output (the survival time) is often right-censored. Utilizing this information is a challenge because it is not obvious how to correctly incorporate these censored examples into a model. We study how three categories of loss functions, namely partial likelihood methods, rank methods, and our classification method based on a Wasserstein metric (WM) and the non-parametric Kaplan Meier estimate of the probability density to impute the labels of censored examples, can take advantage of this information. The proposed metho…
Tree methods (recursive partitioning) are a popular class of nonparametric methods for analyzing data. One extension of the basic tree methodology is the survival tree, which applies recursive partitioning to censored survival data. There are several existing survival tree methods in the literature, which are mainly designed for right-censored data. We propose two new survival trees for left-truncated and right-censored (LTRC) data, which can be seen as a generalization of the traditional survival tree for right-censored data. Further, we show that such trees can be used to analyze survival da…
This paper is devoted to robust estimation based on dual divergences estimators for parametric models in the framework of right censored data. We give limit laws of the proposed estimators and examine their asymptotic properties through a simulation study.
Texas A&M University（德克萨斯农工大学）；Department of Industrial and Systems Engineering（工业与系统工程系）；Heart, Artery, and Vein Center of Fresno（弗雷斯诺心脏、动脉和静脉中心）
Accuracies of survival models for life expectancy prediction as well as critical-care applications are significantly compromised due to the sparsity of samples and extreme imbalance between the survival (usually, the majority) and mortality class sizes. While a recent random survival forest (RSF) model overcomes the limitations of the proportional hazard assumption, an imbalance in the data results in an underestimation (overestimation) of the hazard of the mortality (survival) classes. A balanced random survival forests (BRSF) model, based on training the RSF model with data generated from a …
Doubly Robust Conformalized Survival Analysis with Right-Censored Data
Department of Data Sciences and Operations（数据科学与运营系）；Department of Computer Science, University of Southern California（计算机科学系，南加州大学）；Merck & Co., Inc.（默克公司）
We develop flexible multi-parameter regression survival models for interval censored survival data arising in longitudinal prospective studies and longitudinal randomised controlled clinical trials. A multi-parameter Weibull regression survival model, which is wholly parametric, and has non-proportional hazards, is the main focus of the paper. We describe the basic model, develop the interval-censored likelihood and extend the model to include gamma frailty and a dispersion model. We evaluate the models by means of a simulation study and a detailed re-analysis of data from the Signal Tandmobie…
In medical and biological research, longitudinal data and survival data types are commonly seen. Traditional statistical models mostly consider to deal with either of the data types, such as linear mixed models for longitudinal data, and the Cox models for survival data, while they do not adjust the association between these two different data types. It is desirable to have a joint modeling approach which accomadates both data types and the dependency between them. In this paper, we extend traditional single-index models to a new joint modeling approach, by replacing the single-index component…
Learning Survival Models with Right-Censored Reporting Delays
The Graduate University for Advanced Studies（高级研究大学）；Tokio Marine Holdings, Inc.（东京海上日赤保险株式会社）；The Institute of Statistical Mathematics（统计数学研究所）；RIKEN（理化学研究所）
We propose a novel approach for estimating mean survival time in the presence of censored data, in which we divide the population under study into survival-ordered fractions defined by a set of proportions, and compute the mean survival time for each fraction separately. Our approach provides a detailed picture of the distribution of the time variable while preserving the appealing interpretation of the mean. Our measure proves to be of great use in applications, particularly those where we are able to detect differences in mean survival across groups for certain fractions of the population th…
In this paper we investigate the estimation of the unknown parameters of a competing risk model based on a Weibull distributed decreasing failure rate and an exponentially distributed constant failure rate, under right censored data.likelihood estimators.
A model for competing (resp. complementary) risks survival data where the failure time can be left (resp. right) censored is proposed. Product-limit estimators for the survival functions of the individual risks are derived. We deduce the strong convergence of our estimators on the whole real half-line without any additional assumptions and their asymptotic normality under conditions concerning only the observed distribution. When the observations are generated according to the double censoring model introduced by Turnbull, the product-limit estimators represent upper and lower bounds for Turnb…
Self-Consistent Equation-guided Neural Networks for Censored Time-to-Event Data
Harvard Pilgrim Health Care Institute（哈佛朝圣者医疗保健研究所）；Harvard Medical School（哈佛医学院）；Harvard School of Public Health（哈佛公共卫生学院）；North Carolina State University（北卡罗来纳州立大学）
Methods for the evaluation of the predictive accuracy of biomarkers with respect to survival outcomes subject to right censoring have been discussed extensively in the literature. In cancer and other diseases, survival outcomes are commonly subject to interval censoring by design or due to the follow up schema. In this paper, we present an estimator for the area under the time-dependent receiver operating characteristic ROC curve for interval censored data based on a nonparametric sieve maximum likelihood approach. We establish the asymptotic properties of the proposed estimator, and illustrat…
This paper addresses the problem of estimating, in the presence of random censoring as well as competing risks, the extreme value index of the (sub)-distribution function associated to one particular cause, in the heavy-tail case. Asymptotic normality of the proposed estimator (which has the form of an Aalen-Johansen integral, and is the first estimator proposed in this context) is established. A small simulation study exhibits its performances for finite samples. Estimation of extreme quantiles of the cumulative incidence function is also addressed.
We introduce a novel procedure to perform Bayesian non-parametric inference with right-censored data, the \emph{beta-Stacy bootstrap}. This approximates the posterior law of summaries of the survival distribution (e.g. the mean survival time). More precisely, our procedure approximates the joint posterior law of functionals of the beta-Stacy process, a non-parametric process prior that generalizes the Dirichlet process and that is widely used in survival analysis. The beta-Stacy bootstrap generalizes and unifies other common Bayesian bootstraps for complete or censored data based on non-parame…
Density estimation is a classical problem in statistics and has received considerable attention when both the data has been fully observed and in the case of partially observed (censored) samples. In survival analysis or clinical trials, a typical problem encountered in the data collection stage is that the samples may be censored from the right. The variable of interest could be observed partially due to the presence of a set of events that occur at random and potentially censor the data. Consequently, developing a methodology that enables robust estimation of the lifetimes in such setting is…
Interval-censored data analysis is important in biomedical statistics for any type of time-to-event response where the time of response is not known exactly, but rather only known to occur between two assessment times. Many clinical trials and longitudinal studies generate interval-censored data; one common example occurs in medical studies that entail periodic follow-up. In this paper we propose a survival forest method for interval-censored data based on the conditional inference framework. We describe how this framework can be adapted to the situation of interval-censored data. We show that…
Survival analysis is the branch of statistics that studies the relation between the characteristics of living entities and their respective survival times, taking into account the partial information held by censored cases. A good analysis can, for example, determine whether one medical treatment for a group of patients is better than another. With the rise of machine learning, survival analysis can be modeled as learning a function that maps studied patients to their survival times. To succeed with that, there are three crucial issues to be tackled. First, some patient data is censored: we do…
There is a surge in medical follow-up studies that include longitudinal covariates in the modeling of survival data. So far, the focus has been largely on right-censored survival data. We consider survival data that are subject to both left truncation and right censoring. Left truncation is well known to produce biased sample. The sampling bias issue has been resolved in the literature for the case which involves baseline or time-varying covariates that are observable. The problem remains open, however, for the important case where longitudinal covariates are present in survival models. A join…
We propose a general index model for survival data, which generalizes many commonly used semiparametric survival models and belongs to the framework of dimension reduction. Using a combination of geometric approach in semiparametrics and martingale treatment in survival data analysis, we devise estimation procedures that are feasible and do not require covariate-independent censoring as assumed in many dimension reduction methods for censored survival data. We establish the root-$n$ consistency and asymptotic normality of the proposed estimators and derive the most efficient estimator in this …
Deep Learning-Based Survival Analysis with Copula-Based Activation Functions for Multivariate Response Prediction
Statistics Discipline, Division of Science and Mathematics, University of Minnesota-Morris（明尼苏达州立大学莫里斯分校统计学系）；EGADE Business School, Tecnológico de Monterrey（墨西哥蒙特雷技术学院EGADE商学院）；Department of Statistics and Data Science, Pukyong National University（浦项国立大学统计学与数据科学系）；Department of Management Information Systems, Dong-A University（东阿大学管理信息系）
University of California, Los Angeles（加州大学洛杉矶分校）；University of Oxford（牛津大学）；Alan Turing Institute（艾伦·图灵研究所）
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 p…
We propose the interval censored recursive forests (ICRF) which is an iterative tree ensemble method for interval censored survival data. This nonparametric regression estimator makes the best use of censored information by iteratively updating the survival estimate, and can be viewed as a self-consistent estimator with convergence monitored using out-of-bag samples. Splitting rules optimized for interval censored data are developed and kernel-smoothing is applied. The ICRF displays the highest prediction accuracy among competing nonparametric methods in most of the simulations and in an appli…
Balanced representation learning methods have been applied successfully to counterfactual inference from observational data. However, approaches that account for survival outcomes are relatively limited. Survival data are frequently encountered across diverse medical applications, i.e., drug development, risk profiling, and clinical trials, and such data are also relevant in fields like manufacturing (e.g., for equipment monitoring). When the outcome of interest is a time-to-event, special precautions for handling censored events need to be taken, as ignoring censored outcomes may lead to bias…
Survival analysis concerns the study of timeline data where the event of interest may remain unobserved (i.e., censored). Studies commonly record more than one type of event, but conventional survival techniques focus on a single event type. We set out to integrate both multiple independently censored time-to-event variables as well as missing observations. An energy-based approach is taken with a bi-partite structure between latent and visible states, known as harmoniums (or restricted Boltzmann machines). The present harmonium is shown, both theoretically and experimentally, to capture non-l…
An optimal dynamic treatment regime (DTR) consists of a sequence of decision rules in maximizing long-term benefits, which is applicable for chronic diseases such as HIV infection or cancer. In this paper, we develop a novel angle-based approach to search the optimal DTR under a multicategory treatment framework for survival data. The proposed method targets maximization the conditional survival function of patients following a DTR. In contrast to most existing approaches which are designed to maximize the expected survival time under a binary treatment framework, the proposed method solves th…
Carnegie Mellon University（卡内基梅隆大学）；University of Pittsburgh Medical Center（匹兹堡大学医学中心）
Estimation of treatment efficacy of real-world clinical interventions involves working with continuous outcomes such as time-to-death, re-hospitalization, or a composite event that may be subject to censoring. Counterfactual reasoning in such scenarios requires decoupling the effects of confounding physiological characteristics that affect baseline survival rates from the effects of the interventions being assessed. In this paper, we present a latent variable approach to model heterogeneous treatment effects by proposing that an individual can belong to one of latent clusters with distinct res…
ICODEN: Ordinary Differential Equation Neural Networks for Interval-Censored Data
Department of Biostatistics and Health Data Science, University of Pittsburgh, PA, USA（匹兹堡大学生物统计与健康数据科学系）；Center for Applied Statistics and School of Statistics, Renmin University of China, Beijing, China（中国人民大学应用统计中心与统计学院）；Department of Applied Statistics, Konkuk University, Seoul, Republic of Korea（韩国康国民大学应用统计系）
In this paper, we consider survival analysis with right-censored data which is a common situation in predictive maintenance and health field. We propose a model based on the estimation of two-parameter Weibull distribution conditionally to the features. To achieve this result, we describe a neural network architecture and the associated loss functions that takes into account the right-censored data. We extend the approach to a finite mixture of two-parameter Weibull distributions. We first validate that our model is able to precisely estimate the right parameters of the conditional Weibull dis…
Survival analysis of right censored data arises often in many areas of research including medical research. Effect of covariates (and their interactions) on survival distribution can be studied through existing methods which requires to pre-specify the functional form of the covariates including their interactions. Survival trees offer relatively flexible approach when the form of covariates' effects is unknown. Most of the currently available survival tree construction techniques are not based on a formal test of significance; however, recently proposed ctree algorithm (Hothorn et al., 2006) …
McCombs School of Business（麦克库姆斯商学院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
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 not only censoring, but also missing event times or event types. For inference, we develop a Gibbs sampler under data augmentation for moderately sized data, along with a …
Survival analysis has been developed and applied in the number of areas including manufacturing, finance, economics and healthcare. In healthcare domain, usually clinical data are high-dimensional, sparse and complex and sometimes there exists few amount of time-to-event (labeled) instances. Therefore building an accurate survival model from electronic health records is challenging. With this motivation, we address this issue and provide a new survival analysis framework using deep learning and active learning with a novel sampling strategy. First, our approach provides better representation w…
Spatial survival analysis has received a great deal of attention over the last 20 years due to the important role that geographical information can play in predicting survival. This paper provides an introduction to a set of programs for implementing some Bayesian spatial survival models in R using the package spBayesSurv. The function survregbayes includes the three most commonly-used semiparametric models: proportional hazards, proportional odds, and accelerated failure time. All manner of censored survival times are simultaneously accommodated including uncensored, interval censored, curren…
A setting of a trivairate survival function using semi-competing risks concept is proposed. The Stanford Heart Transplant data is reanalyzed using a trivariate Weibull distribution model with the proposed survival function.
Nonparametric and semiparametric methods are commonly used in survival analysis to mitigate the bias due to model misspecification. However, such methods often cannot estimate upper-tail survival quantiles when a sizable proportion of the data are censored, in which case parametric likelihood-based estimators present a viable alternative. In this article, we extend a popular family of parametric survival models which make the Accelerated Failure Time (AFT) assumption to account for heteroscedasticity in the survival times. The conditional variances can depend on arbitrary covariates, thus addi…
This paper focuses on quantifying and estimating the predictive accuracy of prognostic models for time-to-event outcomes with competing events. We consider the time-dependent discrimination and calibration metrics, including the receiver operating characteristics curve and the Brier score, in the context of competing risks. To address censoring, we propose a unified nonparametric estimation framework for both discrimination and calibration measures, by weighting the censored subjects with the conditional probability of the event of interest given the observed data. We demonstrate through simul…
In this paper, we propose a flexible model for survival analysis using neural networks along with scalable optimization algorithms. One key technical challenge for directly applying maximum likelihood estimation (MLE) to censored data is that evaluating the objective function and its gradients with respect to model parameters requires the calculation of integrals. To address this challenge, we recognize that the MLE for censored data can be viewed as a differential-equation constrained optimization problem, a novel perspective. Following this connection, we model the distribution of event time…
Datasets containing large samples of time-to-event data arising from several small heterogeneous groups are commonly encountered in statistics. This presents problems as they cannot be pooled directly due to their heterogeneity or analyzed individually because of their small sample size. Bayesian nonparametric modelling approaches can be used to model such datasets given their ability to flexibly share information across groups. In this paper, we will compare three popular Bayesian nonparametric methods for modelling the survival functions of heterogeneous groups. Specifically, we will first c…
Estimating individualized treatment rules is a central task for personalized medicine. [zhao2012estimating] and [zhang2012robust] proposed outcome weighted learning to estimate individualized treatment rules directly through maximizing the expected outcome without modeling the response directly. In this paper, we extend the outcome weighted learning to right censored survival data without requiring either an inverse probability of censoring weighting or a semiparametric modeling of the censoring and failure times as done in [zhao2015doubly]. To accomplish this, we take advantage of the tree ba…
PyDTS: A Python Package for Discrete-Time Survival Analysis with Competing Risks and Optional Penalization
Faculty of Data and Decision Sciences, Technion - Israel Institute of Technology（数据与决策科学学院，技术离子-以色列理工学院）；Department of Statistics and Operations Research, Tel Aviv University（统计与运筹学系，特拉维夫大学）
