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 data with time-varying covariates, essentially building a time-varying covariates survival tree. Implementation of the methods is easy, and simulations and real data analysis results show that the proposed methods work well for LTRC data and survival data with time-varying covariates, respectively.
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…
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…
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…
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…
Feature screening is an important method to reduce the dimension and capture informative variables in ultrahigh-dimensional data analysis. Many methods have been developed for feature screening. These methods, however, are challenged by complex features pertinent to the data collection as well as the nature of the data themselves. Typically, incomplete response caused by right-censoring and covariates measurement error are often accompanying with survival analysis. Even though there are many methods have been proposed for censored data, little work has been available when both incomplete respo…
On a penalised likelihood approach for joint modelling of longitudinal covariates and partly interval-censored data -- an application to the Anti-PD1 brain collaboration trial
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…
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) …
In medical studies, the collected covariates usually contain underlying outliers. For clustered /longitudinal data with censored observations, the traditional Gehan-type estimator is robust to outliers existing in response but sensitive to outliers in the covariate domain, and it also ignores the within-cluster correlations. To take account of within-cluster correlations, varying cluster sizes, and outliers in covariates, we propose weighted Gehan-type estimating functions for parameter estimation in the accelerated failure time model for clustered data. We provide the asymptotic properties of…
To select outcomes for clinical trials testing experimental therapies for Huntington disease, a fatal neurodegenerative disorder, analysts model how potential outcomes change over time. Yet, subjects with Huntington disease are often observed at different levels of disease progression. To account for these differences, analysts include time to clinical diagnosis as a covariate when modeling potential outcomes, but this covariate is often censored. One popular solution is imputation, whereby we impute censored values using predictions from a model of the censored covariate given other data, the…
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…
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）
Individualized treatment rules can lead to better health outcomes when patients have heterogeneous responses to treatment. Very few individualized treatment rule estimation methods are compatible with a multi-treatment observational study with right censored survival outcomes. In this paper we extend policy evaluation methods to the right censored data setting. Existing approaches either make restrictive assumptions about the structure of the data, or use inverse weighting methods that increase the variance of the estimator resulting in decreased performance. We propose a method which uses bal…
In contrast to the popular Cox model which presents a multiplicative covariate effect specification on the time to event hazards, the semiparametric additive risks model (ARM) offers an attractive additive specification, allowing for direct assessment of the changes or the differences in the hazard function for changing value of the covariates. The ARM is a flexible model, allowing the estimation of both time-independent and time-varying covariates. It has a nonparametric component and a regression component identified by a finite-dimensional parameter. This chapter presents an efficient appro…
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…
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.
Analysis of random censored life-time data along with some related stochastic covariables is of great importance in many applied sciences like medical research, population studies and planning etc. The parametric estimation technique commonly used under this set-up is based on the efficient but non-robust likelihood approach. In this paper, we propose a robust parametric estimator for the censored data with stochastic covariates based on the minimum density power divergence approach. The resulting estimator also has competitive efficiency with respect to the maximum likelihood estimator under …
Extended cure survival models enable to separate covariates that affect the probability of an event (or `long-term' survival) from those only affecting the event timing (or `short-term' survival). We propose to generalize the bounded cumulative hazard model to handle additive terms for time-varying (exogenous) covariates jointly impacting long- and short-term survival. The selection of the penalty parameters is a challenge in that framework. A fast algorithm based on Laplace approximations in Bayesian P-spline models is proposed. The methodology is motivated by fertility studies where women's …
The restricted mean survival time is a clinically easy-to-interpret measure that does not require any assumption of proportional hazards. We focus on two ways to directly model the survival time and adjust the covariates. One is to calculate the pseudo-survival time for each subject using leave-one-out, and then perform a model analysis using all pseudo-values to adjust for covariates. The pseudo-survival time is used to reflect information of censored subjects in the model analysis. The other method adjusts for covariates using subjects for whom the time-to-event was observed while adjusting …
We propose a novel personalized concept for the optimal treatment selection for a situation where the response is a multivariate vector, that could contain right-censored variables such as survival time. The proposed method can be applied with any number of treatments and outcome variables, under a broad set of models. Following a working semiparametric Single Index Model that relates covariates and responses, we first define a patient-specific composite score, constructed from individual covariates. We then estimate conditional means of each response, given the patient score, correspond to ea…
Interval censoring arises frequently in clinical, epidemiological, financial, and sociological studies, where the event or failure of interest is known only to occur within an interval induced by periodic monitoring. We formulate the effects of potentially time-dependent covariates on the interval-censored failure time through a broad class of semiparametric transformation models that encompasses proportional hazards and proportional odds models. We consider nonparametric maximum likelihood estimation for this class of models with an arbitrary number of monitoring times for each subject. We de…
Maximum approximate Bernstein likelihood estimates of the baseline density function and the regression coefficients in the proportional hazard regression models based on interval-censored event time data are proposed. This results in not only a smooth estimate of the survival function which enjoys faster convergence rate but also improved estimates of the regression coefficients. Simulation shows that the finite sample performance of the proposed method is better than the existing ones. The proposed method is illustrated by real data applications.
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 …
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…
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…
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 demonstra…
A bivariate cure copula model with zero-inflated gamma frailty for right-censored data
Graduate Institute for Advanced Studies, SOKENDAI（SOKENDAI 高级研究研究生院）；Institute of Statistical Mathematics（统计数学研究所）；School of Informatics and Data Science, Hiroshima University（广岛大学信息科学与数据科学学院）
We introduce a flexible individual frailty model for clustered right-censored data, in which covariate effects can be marginally interpreted as log failure odds ratios. Flexible correlation structures can be imposed by introducing multivariate exponential distributed frailties, constructed from a set of multivariate Gaussian random variables. Finite and infinite dimensional parameters are consistently estimated by maximizing a composite contributing marginal likelihood and a consistent estimate for their asymptotic covariance is proposed. Parameter estimation is implemented through a hybrid ex…
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…
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…
In survival analysis, the hazard function often depends on a set of covariates. Martingale and deviance residual are most widely used for examining the validity of the function form of covariates by checking whether there is a discernible trend in their scatterplot against continuous covariates. However, visual inspection of martingale and deviance residuals is often subjective. In addition, these residuals lack a reference distribution due to censoring. It is therefore challenging to derive numerical statistical tests based on martingale or deviance residuals. In this paper, we extend the ide…
In many randomized clinical trials of therapeutics for COVID-19, the primary outcome is an ordinal categorical variable, and interest focuses on the odds ratio (active agent vs. control) under the assumption of a proportional odds model. Although at the final analysis the outcome will be determined for all subjects, at an interim analysis, the status of some participants may not yet be determined, e.g., because ascertainment of the outcome may not be possible until some pre-specified follow-up time. Accordingly, the outcome from these subjects can be viewed as censored. A valid interim analysi…
In two harmonized efficacy studies to prevent HIV infection through multiple infusions of the monoclonal antibody VRC01, a key objective is to evaluate whether the serum concentration of VRC01, which changes cyclically over time along with the infusion schedule, is associated with the rate of HIV infection. Simulation studies are needed in the development of such survival models. In this paper, we consider simulating event time data with a continuous time-varying covariate whose values vary with time through multiple drug administration cycles, and whose effect on survival changes differently …
