In this paper, we propose a class of high breakdown point estimators for the linear regression model when the response variable contains censored observations. These estimators are robust against high-leverage outliers and they generalize the LMS (least median of squares), S, MM and $τ$-estimators for linear regression. An important contribution of this paper is that we can define consistent estimators using a bounded loss function (or equivalently, a redescending score function). Since the calculation of these estimators can be computationally costly, we propose an efficient algorithm to compute them. We illustrate their use on an example and present simulation studies that show that these estimators also have good finite sample properties.
In this paper, we study the behavior of a kernel estimator of the regression function in the right censored model with $α$-mixing data . The uniform strong consistency over a real compact set of the estimate is established along with a rate of convergence. Some simulations are carried out to illustrate the behavior of the estimate with different examples for finite sample sizes.
It has been recently shown that nonparametric estimators of the additive regression function could be obtained in the presence of right censoring by coupling the marginal integration method with initial kernel-type Inverse Probability of Censoring Weighted estimators of the multivariate regression function \citeDebbarhViallon3. In this paper, we get the exact rate of strong uniform consistency for such estimators. Our uniform limit laws especially lead to the construction of asymptotic simultaneous 100% confidence bands for the true regression function.
Censored data are quite common in statistics and have been studied in depth in the last years. In this paper we consider censored high-dimensional data. High-dimensional models are in some way more complex than their low-dimensional versions, therefore some different techniques are required. For the linear case appropriate estimators based on penalized regression, have been developed in the last years. In particular in sparse contexts the $l_1$-penalised regression (also known as LASSO) performs very well. Only few theoretical work was done in order to analyse censored linear models in a high-…
Let $ (T_i)_i$ be a sequence of independent identically distributed (i.i.d.) random variables (r.v.) of interest distributed as $ T$ and $(X_i)_i$ be a corresponding vector of covariates taking values on $ \mathbb{R}^d$. In censorship models the r.v. $T$ is subject to random censoring by another r.v. $C$. In this paper we built a new kernel estimator based on the so-called synthetic data of the mean squared relative error for the regression function. We establish the uniform almost sure convergence with rate over a compact set and its asymptotic normality. The asymptotic variance is explicitly…
We develop a unified approach for classification and regression support vector machines for data subject to right censoring. We provide finite sample bounds on the generalization error of the algorithm, prove risk consistency for a wide class of probability measures, and study the associated learning rates. We apply the general methodology to estimation of the (truncated) mean, median, quantiles, and for classification problems. We present a simulation study that demonstrates the performance of the proposed approach.
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…
The approximate Bernstein polynomial model, a mixture of beta distributions, is applied to obtain maximum likelihood estimates of the regression coefficients, and the baseline density and survival functions in an accelerated failure time model based on interval censored data including current status data. The rate of convergence of the proposed estimates are given under some conditions for uncensored and interval censored data. Simulation shows that the proposed method is better than its competitors. The proposed method is illustrated by fitting the Breast Cosmetic Data using the accelerated f…
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 the framework of censored data modeling, the classical linear regression model that assumes normally distributed random errors has received increasing attention in recent years, mainly for mathematical and computational convenience. However, practical studies have often criticized this linear regression model due to its sensitivity to departure from the normality and from the partial nonlinearity. This paper proposes to solve these potential issues simultaneously in the context of the partial linear regression model by assuming that the random errors follow a scale-mixture of normal (SMN) f…
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.
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…
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…
CensSpatial is an R package for analyzing spatial censored data through linear models. It offers a set of tools for simulating, estimating, making predictions, and performing local influence diagnostics for outlier detection. The package provides four algorithms for estimation and prediction. One of them is based on the stochastic approximation of the EM (SAEM) algorithm, which allows easy and fast estimation of the parameters of linear spatial models when censoring is present. The package provides worthy measures to perform diagnostic analysis using the Hessian matrix of the completed log-lik…
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…
Censored broken adaptive ridge rank regression via induced smoothing
School of Mathematics, Statistics and Data Science, Sungshin Women’s University（成均馆大学数学、统计与数据科学学院）；Sungshin Women’s University（诚信女子大学）；Department of Biostatistics, Virginia Commonwealth University（弗吉尼亚联邦大学生物统计系）；Virginia Commonwealth University（弗吉尼亚联邦大学）；Center for Data Science, Sungshin Women’s University（诚信女子大学数据科学中心）
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…
A common assumption regarding the standard tobit model is the normality of the error distribution. However, asymmetry and bimodality may be present and alternative tobit models must be used. In this paper, we propose a tobit model based on the class of log-symmetric distributions, which includes as special cases heavy and light tailed distributions and bimodal distributions. We implement a likelihood-based approach for parameter estimation and derive a type of residual. We then discuss the problem of performing testing inference in the proposed class by using the likelihood ratio and gradient …
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 …
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…
We consider efficient estimation of flexible transformation models with interval-censored data. To reduce the dimension of semi-parametric models, the unknown monotone transformation function is approximated via monotone splines. A penalization technique is used to provide more computationally efficient estimation of all parameters. To accomplish model fitting, a computationally efficient nested iterative expectation-maximization (EM) based algorithm is developed for estimation, and an easily implemented variance-covariance approach is proposed for inference on regression parameters. Theoretic…
Despite the recent advances in mass spectrometry (MS), summarizing and analyzing high-throughput mass-spectrometry data remains a challenging task. This is, on the one hand, due to the complexity of the spectral signal which is measured, and on the other, due to the limit of detection (LOD). The LOD is related to the limitation of instruments in measuring markers at a relatively low level. As a consequence, the outcome data set from the quantification step of proteomic analysis often consists of a reduced list of peaks where any peak intensities below the detection limit threshold are reported…
In this article, we propose some new generalizations of M-estimation procedures for single-index regression models in presence of randomly right-censored responses. We derive consistency and asymptotic normality of our estimates. The results are proved in order to be adapted to a wide range of techniques used in a censored regression framework (e.g. synthetic data or weighted least squares). As in the uncensored case, the estimator of the single-index parameter is seen to have the same asymptotic behavior as in a fully parametric scheme. We compare these new estimators with those based on the …
Herein, we propose a Spearman rank correlation based screening procedure for ultrahigh-dimensional data with censored response case. The proposed method is model-free without specifying any regression forms of predictors or response variable and is robust under the unknown monotone transformations of these response variable and predictors. The sure-screening and rank-consistency properties are established under some mild regularity conditions. Simulation studies demonstrate that the new screening method performs well in the presence of a heavy-tailed distribution, strongly dependent predictors…
We propose a censored quantile regression estimator motivated by unbiased estimating equations. Under the usual conditional independence assumption of the survival time and the censoring time given the covariates, we show that the proposed estimator is consistent and asymptotically normal. We develop an efficient computational algorithm which uses existing quantile regression code. As a result, bootstrap-type inference can be efficiently implemented. We illustrate the finite-sample performance of the proposed method by simulation studies and analysis of a survival data set.
When cross-validating standard or extended Cox models, the commonly used criterion is the cross-validated partial loglikelihood using a naive or a van Houwelingen scheme -to make efficient use of the death times of the left out data in relation to the death times of all the data-. Quite astonishingly, we will show, using a strong simulation study involving three different data simulation algorithms, that these two cross-validation methods fail with the extensions, either straightforward or more involved ones, of partial least squares regression to the Cox model. This is quite an interesting re…
We propose a new continuous-discrete mixture regression model which is useful for describing highly censored data. We motivate our investigation based on a case-study in biometry related to measles vaccines in Haiti. In this case-study, the neutralization antibody level is explained by the type of vaccine used, level of the dosage and gender of the patient. This mixture model allows us to account for excess of censored observations and consists of the Birnbaum-Saunders and Bernoulli distributions. These distributions describe the antibody level and the point mass of the censoring observations.…
Under a single-index regression assumption, we introduce a new semiparametric procedure to estimate a conditional density of a censored response. The regression model can be seen as a generalization of Cox regression model and also as a profitable tool to perform dimension reduction under censoring. This technique extends the results of Delecroix et al. (2003). We derive consistency and asymptotic normality of our estimator of the index parameter by proving its asymptotic equivalence with the (uncomputable) maximum likelihood estimator, using martingales results for counting processes and argu…
This paper considers inference in a linear regression model with random right censoring and outliers. The number of outliers can grow with the sample size while their proportion goes to zero. The model is semiparametric and we make only very mild assumptions on the distribution of the error term, contrary to most other existing approaches in the literature. We propose to penalize the estimator proposed by Stute for censored linear regression by the l1-norm. We derive rates of convergence and establish asymptotic normality of the estimator of the regression coefficients. Our estimator has the s…
The stratified proportional intensity model generalizes Cox's proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also…
Censored quantile regression (CQR) has become a valuable tool to study the heterogeneous association between a possibly censored outcome and a set of covariates, yet computation and statistical inference for CQR have remained a challenge for large-scale data with many covariates. In this paper, we focus on a smoothed martingale-based sequential estimating equations approach, to which scalable gradient-based algorithms can be applied. Theoretically, we provide a unified analysis of the smoothed sequential estimator and its penalized counterpart in increasing dimensions. When the covariate dimen…
We consider quantile regression processes from censored data under dependent data structures and derive a uniform Bahadur representation for those processes. We also consider cases where the dimension of the parameter in the quantile regression model is large. It is demonstrated that traditional penalized estimators such as the adaptive lasso yield sub-optimal rates if the coefficients of the quantile regression cross zero. New penalization techniques are introduced which are able to deal with specific problems of censored data and yield estimates with an optimal rate. In contrast to most of t…
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…
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…
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…
In this paper, we are concerned with nonparametric estimation of the multivariate regression function in the presence of right censored data. More precisely, we propose a statistic that is shown to be asymptotically normally distributed under the additive assumption, and that could be used to test for additivity in the censored regression setting.
We develop inference procedures for longitudinal data where some of the measurements are censored by fixed constants. We consider a semi-parametric quantile regression model that makes no distributional assumptions. Our research is motivated by the lack of proper inference procedures for data from biomedical studies where measurements are censored due to a fixed quantification limit. In such studies the focus is often on testing hypotheses about treatment equality. To this end, we propose a rank score test for large sample inference on a subset of the covariates. We demonstrate the importance …
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 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 …
In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $(\tildeθ_n,\tilde{F}_n)$ for the underlying parameter $θ_0$ and distribution $F_0$ is derived, and the strong consistency of $(\tildeθ_n,\tilde{F}_n)$…
Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile cal…
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
When data are collected subject to a detection limit, observations below the detection limit may be considered censored. In addition, the domain of such observations may be restricted; for example, values may be required to be non-negative. We propose a regression method for censored observations that also accounts for domain restriction. The method finds maximum likelihood estimates assuming an underlying truncated normal distribution. We show that our method, tcensReg, outperforms other methods commonly used for data with detection limits such as Tobit regression and single imputation of the…
This note develops a simple two-stage least squares (2SLS) procedure to estimate the causal effect of some endogenous regressors on a randomly right censored outcome in the linear model. The proposal replaces the usual ordinary least squares regressions of the standard 2SLS by weighted least squares regressions. The weights correspond to the inverse probability of censoring. We show consistency and asymptotic normality of the estimator. The estimator exhibits good finite sample performances in simulations.
In many semiparametric models that are parameterized by two types of parameters---a Euclidean parameter of interest and an infinite-dimensional nuisance parameter---the two parameters are bundled together, that is, the nuisance parameter is an unknown function that contains the parameter of interest as part of its argument. For example, in a linear regression model for censored survival data, the unspecified error distribution function involves the regression coefficients. Motivated by developing an efficient estimating method for the regression parameters, we propose a general sieve M-theorem…
