The SiZer method is extended to nonparametric hazard estimation and also to censored density and hazard estimation. The new method allows quick, visual statistical inference about the important issue of statistically significant increases and decreases in the smooth curve estimate. This extension has required the opening of a new avenue of research on the interface between statistical inference and scale space.
In reliability and life testing studies, the topic of estimating hazard rate has received great attention in recent years since an estimate of hazard rate is a quite useful tool for making decisions. Some works have included nonparametric approaches while some have considered parametric structural models for complete as well as censored data sets; see Meeker et al. (1992), Antoniadis and Grégoire (1999), Rai and Singh (2003), Bezandry et al. (2005), Brunel and Comte (2008), and Mahapatra et al. (2012). Depending on the shapes of the hazard rate, efficiencies differ markedly across proposed est…
The proportional hazards assumption in the commonly used Cox model for censored failure time data is often violated in scientific studies. Yang and Prentice (2005) proposed a novel semiparametric two-sample model that includes the proportional hazards model and the proportional odds model as sub-models, and accommodates crossing survival curves. The model leaves the baseline hazard unspecified and the two model parameters can be interpreted as the short-term and long-term hazard ratios. Inference procedures were developed based on a pseudo score approach. Although extension to accommodate cova…
We express the joint Rényi entropy of progressively censored order statistics in terms of an incomplete integral of the hazard function, and provide a simple estimate of the joint Rényi entropy of progressively Type-II censored data. Then we establish a goodness of fit test statistic based on the Rényi Kullback-Leibler information with the progressively Type-II censored data, and compare its performance with the leading test statistic. A Monte Carlo simulation study shows that the proposed test statistic shows better powers than the leading test statistic against the alternatives with monotone…
Multistate models can be used to describe transitions over time across states. In the presence of interval-censored times for transitions, the likelihood is constructed using transition probabilities. Models are specified using proportional hazards model for the transitions. Time-dependency is usually defined by parametric models, which can be too restrictive. Nonparametric hazards specification with splines allow for flexible modelling of time-dependency without making strong model assumptions. Penalised maximum likelihood is used to estimate the models. Selecting the optimal amount of smooth…
Many insurance premium principles are defined and various estimation procedures introduced in the literature. In this paper, we focus on the estimation of the excess-of-loss reinsurance premium when the risks are randomly right-censored. The asymptotic normality of the proposed estimator is established under suitable conditions and its performance evaluated through sets of simulated data.
This paper considers the nonparametric maximum likelihood estimator (MLE) for the joint distribution function of an interval censored survival time and a continuous mark variable. We provide a new explicit formula for the MLE in this problem. We use this formula and the mark specific cumulative hazard function of Huang and Louis (1998) to obtain the almost sure limit of the MLE. This result leads to necessary and sufficient conditions for consistency of the MLE which imply that the MLE is inconsistent in general. We show that the inconsistency can be repaired by discretizing the marks. Our the…
Cox's proportional hazards model is one of the most popular statistical models to evaluate associations of exposure with a censored failure time outcome. When confounding factors are not fully observed, the exposure hazard ratio estimated using a Cox model is subject to unmeasured confounding bias. To address this, we propose a novel approach for the identification and estimation of the causal hazard ratio in the presence of unmeasured confounding factors. Our approach is based on a binary instrumental variable, and an additional no-interaction assumption in a first stage regression of the tre…
Interval-censored competing risks data arise when each study subject may experience an event or failure from one of several causes and the failure time is not observed exactly but rather known to lie in an interval between two successive examinations. We formulate the effects of possibly time-varying covariates on the cumulative incidence or sub-distribution function (i.e., the marginal probability of failure from a particular cause) of competing risks through a broad class of semiparametric regression models that captures both proportional and non-proportional hazards structures for the sub-d…
In this paper the regression discontinuity design is adapted to the survival analysis setting with right-censored data, studied in an intensity based counting process framework. In particular, a local polynomial regression version of the Aalen additive hazards estimator is introduced as an estimator of the difference between two covariate dependent cumulative hazard rate functions. Large-sample theory for this estimator is developed, including confidence intervals that take into account the uncertainty associated with bias correction. As is standard in the causality literature, the models and …
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 deep generative approach to nonparametric estimation of conditional survival and hazard functions with right-censored data. The key idea of the proposed method is to first learn a conditional generator for the joint conditional distribution of the observed time and censoring indicator given the covariates, and then construct the Kaplan-Meier and Nelson-Aalen estimators based on this conditional generator for the conditional hazard and survival functions. Our method combines ideas from the recently developed deep generative learning and classical nonparametric estimation in surviva…
Interpretable Deep Regression Models with Interval-Censored Failure Time Data
School of Mathematics, Jilin University（吉林大学数学学院）；School of Economics and Statistics, Guangzhou University（广州大学经济与统计学院）；Department of Statistics, Chinese University of Hong Kong（香港中文大学统计系）；School of Data Science, Fudan University（复旦大学数据科学学院）
This work presents novel and powerful tests for comparing non-proportional hazard functions, based on sample-space partitions. Right censoring introduces two major difficulties which make the existing sample-space partition tests for uncensored data non-applicable: (i) the actual event times of censored observations are unknown; and (ii) the standard permutation procedure is invalid in case the censoring distributions of the groups are unequal. We overcome these two obstacles, introduce invariant tests, and prove their consistency. Extensive simulations reveal that under non-proportional alter…
When the observations are not quantified and are known to be less than a threshold value, the concept of left censoring needs to be included in the analysis of such datasets. In many real multi component lifetime systems left censored data is very common. The usual assumption that components which are part of a system, work independently seems not appropriate in a number of applications. For instance it is more realistic to acknowledge that the working status of a component affects the remaining components. When you have left-censored data, it is more meaningful to use the reversed hazard rate…
Case-cohort design, an outcome-dependent sampling design for censored survival data, is increasingly used in biomedical research. The development of asymptotic theory for a case-cohort design in the current literature primarily relies on counting process stochastic integrals. Such an approach, however, is rather limited and lacks theoretical justification for outcome-dependent weighted methods due to non-predictability. Instead of stochastic integrals, we derive asymptotic properties for case-cohort studies based on a general Z-estimation theory for semiparametric models with bundled parameter…
We consider parameter estimation in a regression model corresponding to an iid sequence of censored observations of a finite state modulated renewal process. The model assumes a similar form as in Cox regression except that the baseline intensities are functions of the backwards recurrence time of the process and a time dependent covariate. As a result of this it falls outside the class of multiplicative intensity models. We use kernel estimation to construct estimates of the regression coefficints and the baseline cumulative hazards. We give conditions for consistency and asymptotic normality…
In this paper a new distribution is proposed. This new model provides more flexibility to modeling data with upside-down bathtub hazard rate function. A significant account of mathematical properties of the new distribution is presented. The maximum likelihood estimators for the parameters in the presence of complete and censored data are presented. Two corrective approaches are considered to derive modified estimators that are bias-free to second order. A numerical simulation is carried out to examine the efficiency of the bias correction. Finally, an application using a real data set is pres…
Continuous-time multi-state survival models can be used to describe health-related processes over time. In the presence of interval-censored times for transitions between the living states, the likelihood is constructed using transition probabilities. Models can be specified using parametric or semi-parametric shapes for the hazards. Semi-parametric hazards can be fitted using $P$-splines and penalised maximum likelihood estimation. This paper presents a method to estimate flexible multi-state models which allows for parametric and semi-parametric hazard specifications. The estimation is based…
The proportional hazards (PH) model is arguably one of the most popular models used to analyze time to event data arising from clinical trials and longitudinal studies, among many others. In many such studies, the event time of interest is not directly observed but is known relative to periodic examination times; i.e., practitioners observe either current status or interval-censored data. The analysis of data of this structure is often fraught with many difficulties. Further exacerbating this issue, in some such studies the observed data also consists of instantaneous failures; i.e., the event…
A new method for the analysis of time to ankylosis complication on a dataset of replanted teeth is proposed. In this context of left-censored, interval-censored and right-censored data, a Cox model with piecewise constant baseline hazard is introduced. Estimation is carried out with the EM algorithm by treating the true event times as unobserved variables. This estimation procedure is shown to produce a block diagonal Hessian matrix of the baseline parameters. Taking advantage of this interesting feature of the estimation method a L0 penalised likelihood method is implemented in order to autom…
This paper develops a general theory on rates of convergence of penalized spline estimators for function estimation when the likelihood functional is concave in candidate functions, where the likelihood is interpreted in a broad sense that includes conditional likelihood, quasi-likelihood, and pseudo-likelihood. The theory allows all feasible combinations of the spline degree, the penalty order, and the smoothness of the unknown functions. According to this theory, the asymptotic behaviors of the penalized spline estimators depends on interplay between the spline knot number and the penalty pa…
In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions…
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…
We propose inference procedures for general nonparametric factorial survival designs with possibly right-censored data. Similar to additive Aalen models, null hypotheses are formulated in terms of cumulative hazards. Thereby, deviations are measured in terms of quadratic forms in Nelson-Aalen-type integrals. Different to existing approaches this allows to work without restrictive model assumptions as proportional hazards. In particular, crossing survival or hazard curves can be detected without a significant loss of power. For a distribution-free application of the method, a permutation strate…
Statistical methods are developed for analysis of clinical and virus genetics data from phase 3 randomized, placebo-controlled trials of vaccines against novel coronavirus COVID-19. Vaccine efficacy (VE) of a vaccine to prevent COVID-19 caused by one of finitely many genetic strains of SARS-CoV-2 may vary by strain. The problem of assessing differential VE by viral genetics can be formulated under a competing risks model where the endpoint is virologically confirmed COVID-19 and the cause-of-failure is the infecting SARS-CoV-2 genotype. Strain-specific VE is defined as one minus the cause-spec…
In the study of life tables the random variable of interest is usually assumed discrete since mortality rates are studied for integer ages. In dynamic life tables a time domain is included to account for the evolution effect of the hazard rates in time. In this article we follow a survival analysis approach and use a nonparametric description of the hazard rates. We construct a discrete time stochastic processes that reflects dependence across age as well as in time. This process is used as a bayesian nonparametric prior distribution for the hazard rates for the study of evolutionary life tabl…
High-dimensional sparse modeling with censored survival data is of great practical importance, and several methods have been proposed for variable selection based on different models. However, the impact of biased sample caused by left-truncation and covariates measurement error to variable selection is not fully explored. In this paper, we mainly focus on the additive hazards model and analyze the high-dimensional survival data subject to left-truncation and measurement error in covariates. We develop the three-stage procedure to correct the error effect, select variables, and estimate the pa…
This paper considers the problem of semi-parametric proportional hazards model fitting for interval, left and right censored survival times. We adopt a more versatile penalized likelihood method to estimate the baseline hazard and the regression coefficients simultaneously, where the penalty is introduced in order to regularize the baseline hazard estimate. We present asymptotic properties of our estimate, allowing for the possibility that it may lie on the boundary of the parameter space. We also provide a computational method based on marginal likelihood, which allows the regularization para…
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…
Estimation of reliability and hazard rate is one of the most important problems raised in many applications especially in engineering studies as well as human lifetime. In this regard, different methods of estimation have been used. Each method exploits various tools and suffers from problems such as complexity of computations, low precision, and so forth. This study is employed the E-Bayesian method, for estimating the parameter and survival functions of the Weibull Generalized Exponential distribution. The estimators are obtained under squared error and LINEX loss functions under progressive…
Most work in neural networks focuses on estimating the conditional mean of a continuous response variable given a set of covariates.In this article, we consider estimating the conditional distribution function using neural networks for both censored and uncensored data. The algorithm is built upon the data structure particularly constructed for the Cox regression with time-dependent covariates. Without imposing any model assumption, we consider a loss function that is based on the full likelihood where the conditional hazard function is the only unknown nonparametric parameter, for which uncon…
Studies involving both randomized experiments as well as observational data typically involve time-to-event outcomes such as time-to-failure, death or onset of an adverse condition. Such outcomes are typically subject to censoring due to loss of follow-up and established statistical practice involves comparing treatment efficacy in terms of hazard ratios between the treated and control groups. In this paper we propose a statistical approach to recovering sparse phenogroups (or subtypes) that demonstrate differential treatment effects as compared to the study population. Our approach involves m…
Factorial survival designs with right-censored observations are commonly inferred by Cox regression and explained by means of hazard ratios. However, in case of non-proportional hazards, their interpretation can become cumbersome; especially for clinicians. We therefore offer an alternative: median survival times are used to estimate treatment and interaction effects and null hypotheses are formulated in contrasts of their population versions. Permutation-based tests and confidence regions are proposed and shown to be asymptotically valid. Their type-1 error control and power behavior are inve…
Instrumental variables (IV) are a useful tool for estimating causal effects in the presence of unmeasured confounding. IV methods are well developed for uncensored outcomes, particularly for structural linear equation models, where simple two-stage estimation schemes are available. The extension of these methods to survival settings is challenging, partly because of the nonlinearity of the popular survival regression models and partly because of the complications associated with right censoring or other survival features. We develop a simple causal hazard ratio estimator in a proportional haza…
Individuals often respond differently to identical treatments, and characterizing such variability in treatment response is an important aim in the practice of personalized medicine. In this article, we describe a non-parametric accelerated failure time model that can be used to analyze heterogeneous treatment effects (HTE) when patient outcomes are time-to-event. By utilizing Bayesian additive regression trees and a mean-constrained Dirichlet process mixture model, our approach offers a flexible model for the regression function while placing few restrictions on the baseline hazard. Our non-p…
We consider a Kendall's tau measure between a binary group indicator and the continuous variable under investigation to develop a thorough two-sample comparison procedure. The measure serves as a useful alternative to the hazard ratio whose applicability depends on the proportional hazards assumption. For right censored data, we propose a weighted log-rank statistic with weights adapted to the censoring distributions and develop theoretical properties of the derived estimators. In absence of censoring, the proposed estimator reduces to the WMW statistic. The proposed methodology is applied to …
