Median regression analysis from data with left and right censored observations

Median regression models provide a robust alternative to regression based on the mean. We propose a methodology for fitting a median regression model from data with both left and right censored observations, in which the left censoring variable is always observed. First we set up an adjusted least absolute deviation estimating function using the inverse censoring weighted approach, whose solution specifies the estimator. We derive the consistency and asymptotic normality of the proposed estimator and describe the inference procedure for the regression parameter. Finally, we check the finite sample performance of the proposed procedure through simulation.     

A semi-parametric estimation of a survival function from incomplete and doubly censored data

The purpose of this paper is to present a semi-parametric estimation of a survival function when analyzing incomplete and doubly censored data. Under the assumption that the chance of censoring is not related to the individual's survivorship, we propose a consistent estimation of survival. The derived estimator treats the uncensored observations nonparametrically and uses parametric models for both right and left censored data. Some asymptotic properties and simulation studies are also presented in order to analyze the behavior of the proposed estimator.     

A Class of Semiparametric Transformation Models for Doubly Censored Failure Time Data

Doubly censored failure time data occur in many areas including demographical studies, epidemiology studies, medical studies and tumorigenicity experiments, and correspondingly some inference procedures have been developed in the literature (Biometrika, 91, 2004, 277; Comput. Statist. Data Anal., 57, 2013, 41; J. Comput. Graph. Statist., 13, 2004, 123). In this paper, we discuss regression analysis of such data under a class of flexible semiparametric transformation models, which includes some commonly used models for doubly censored data as special cases. For inference, the non‐parametric maximum likelihood estimation will be developed and in particular, we will present a novel expectation–maximization algorithm with the use of subject‐specific independent Poisson variables. In addition, the asymptotic properties of the proposed estimators are established and an extensive simulation study suggests that the proposed methodology works well for practical situations. The method is applied to an AIDS study.     

Inverse censoring weighted median regression

We implement semiparametric random censorship model aided inference for censored median regression models. This is based on the idea that, when the censoring is specified by a common distribution, a semiparametric survival function estimator acts as an improved weight in the so-called inverse censoring weighted estimating function. We show that the proposed method will always produce estimates of the model parameters that are as good as or better than an existing estimator based on the traditional Kaplan–Meier weights. We also provide an illustration of the method through an analysis of a lung cancer data set.     

Statistical Inferences Based on Extended Weighted Log-Rank Estimating Function for Censored Regression Model

In the regression model with censored data, it is not straightforward to estimate the covariances of the regression estimators, since their asymptotic covariances may involve the unknown error density function and its derivative. In this article, a resampling method for making inferences on the parameter, based on some estimating functions, is discussed for the censored regression model. The inference procedures are associated with a weight function. To find the best weight functions for the proposed procedures, extensive simulations are performed. The validity of the approximation to the distribution of the estimator by a resampling technique is also examined visually. Implementation of the procedures is discussed and illustrated in a real data example.     

Accelerated Recurrence Time Models

Abstract.  For the analysis with recurrent events, we propose a generalization of the accelerated failure time model to allow for evolving covariate effects. These so-called accelerated recurrence time models postulate that the time to expected recurrence frequency, upon transformation, is a linear function of covariates with frequency-dependent coefficients. This modelling strategy shares the same spirit as quantile regression. An estimation and inference procedure is developed by generalizing the celebrated Powell's ( J. Econometrics 25, 1984, 303; J. Econometrics 32, 1986, 143) estimator for censored quantile regression. Consistency and asymptotic normality of the proposed estimator are established. An algorithm is devised to attain good computational efficiency. Simulations demonstrate that this proposal performs well under practical settings. This methodology is illustrated in an application to the well-known bladder cancer study.     

Confidence Intervals on Regression Models with Censored Data

Stute (1993, Consistent estimation under random censorship when covariables are present. Journal of Multivariate Analysis 45, 89–103) proposed a new method to estimate regression models with a censored response variable using least squares and showed the consistency and asymptotic normality for his estimator. This article proposes a new bootstrap-based methodology that improves the performance of the asymptotic interval estimation for the small sample size case. Therefore, we compare the behavior of Stute's asymptotic confidence interval with that of several confidence intervals that are based on resampling bootstrap techniques. In order to build these confidence intervals, we propose a new bootstrap resampling method that has been adapted for the case of censored regression models. We use simulations to study the improvement the performance of the proposed bootstrap-based confidence intervals show when compared to the asymptotic proposal. Simulation results indicate that, for the new proposals, coverage percentages are closer to the nominal values and, in addition, intervals are narrower.     

On cramér-von mises tests of goodness of fit with censored data

The paper considers the goodness of fit tests with right censored data or doubly censored data. The Fredholm Integral Equation (FIE) method proposed by Ren (1993) is implemented in the simulation studies to estimate the null distribution of the Cramér-von Mises test statistics and the asymptotic covariance function of the self-consistent estimator for the lifetime distribution with right censored data or doubly censored data. We show that for fixed alternatives, the bootstrap method does not estimate the null distribution consistently for doubly censored data. For the right censored case, a comparison between the performance of FIE and the η out of η bootstrap shows that FIE gives better estimation for the null distribution. The application of FIE to a set of right censored Channing House data and to a set of doubly censored breast cancer data is presented.     

Semiparametric mixed-effects models for clustered doubly censored data

The Cox proportional frailty model with a random effect has been proposed for the analysis of right-censored data which consist of a large number of small clusters of correlated failure time observations. For right-censored data, Cai et al. [3] proposed a class of semiparametric mixed-effects models which provides useful alternatives to the Cox model. We demonstrate that the approach of Cai et al. [3] can be used to analyze clustered doubly censored data when both left- and right-censoring variables are always observed. The asymptotic properties of the proposed estimator are derived. A simulation study is conducted to investigate the performance of the proposed estimator.     

Estimation of the association for bivariate doubly censored data

We study the problem of estimating the association between two related survival variables when they follow a copula model and the bivariate doubly censored data is available. A two-stage estimation procedure is proposed and the asymptotic properties of the proposed estimator are established. Simulation studies are conducted to investigate the finite sample properties of the proposed estimate.     

Asymptotic normality of kernel estimators based upon incomplete data

In this paper, we are concerned with nonparametric estimation of the density and the failure rate functions of a random variable X which is at risk of being censored. First, we establish the asymptotic normality of a kernel density estimator in a general censoring setup. Then, we apply our result in order to derive the asymptotic normality of both the density and the failure rate estimators in the cases of right, twice and doubly censored data. Finally, the performance and the asymptotic Gaussian behaviour of the studied estimators, based on either doubly or twice censored data, are illustrated through a simulation study.     

Buckley–James Type Estimator for Censored Data with Covariates Missing by Design

Abstract. The Buckley–James estimator (BJE) is a well‐known estimator for linear regression models with censored data. Ritov has generalized the BJE to a semiparametric setting and demonstrated that his class of Buckley–James type estimators is asymptotically equivalent to the class of rank‐based estimators proposed by Tsiatis. In this article, we revisit such relationship in censored data with covariates missing by design. By exploring a similar relationship between our proposed class of Buckley–James type estimating functions to the class of rank‐based estimating functions recently generalized by Nan, Kalbfleisch and Yu, we establish asymptotic properties of our proposed estimators. We also conduct numerical studies to compare asymptotic efficiencies from various estimators.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

Penalized empirical likelihood inference for sparse additive hazards regression with a diverging number of covariates

High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by applications in high-throughput genomic data analysis. In this paper, we propose a class of regularization methods, integrating both the penalized empirical likelihood and pseudoscore approaches, for variable selection and estimation in sparse and high-dimensional additive hazards regression models. When the number of covariates grows with the sample size, we establish asymptotic properties of the resulting estimator and the oracle property of the proposed method. It is shown that the proposed estimator is more efficient than that obtained from the non-concave penalized likelihood approach in the literature. Based on a penalized empirical likelihood ratio statistic, we further develop a nonparametric likelihood approach for testing the linear hypothesis of regression coefficients and constructing confidence regions consequently. Simulation studies are carried out to evaluate the performance of the proposed methodology and also two real data sets are analyzed.     

Median regression model with doubly truncated data

We study the problem of fitting a heteroscedastic median regression model with doubly truncated data. A self-consistency equation is proposed to obtain an estimator. We set up a least absolute deviation estimating function. We establish the consistency and asymptotic normality for the case when covariates are discrete. The finite sample performance of the proposed estimators are investigated through simulation studies. The proposed method is illustrated using the AIDS Blood Transfusion Data.     

Second-order least squares estimation of censored regression models

This paper proposes the second-order least squares estimation, which is an extension of the ordinary least squares method, for censored regression models where the error term has a general parametric distribution (not necessarily normal). The strong consistency and asymptotic normality of the estimator are derived under fairly general regularity conditions. We also propose a computationally simpler estimator which is consistent and asymptotically normal under the same regularity conditions. Finite sample behavior of the proposed estimators under both correctly and misspecified models are investigated through Monte Carlo simulations. The simulation results show that the proposed estimator using optimal weighting matrix performs very similar to the maximum likelihood estimator, and the estimator with the identity weight is more robust against the misspecification.     

Inference for proportional hazard model with propensity score

Since the publication of the seminal paper by Cox (1972), proportional hazard model has become very popular in regression analysis for right censored data. In observational studies, treatment assignment may depend on observed covariates. If these confounding variables are not accounted for properly, the inference based on the Cox proportional hazard model may perform poorly. As shown in Rosenbaum and Rubin (1983), under the strongly ignorable treatment assignment assumption, conditioning on the propensity score yields valid causal effect estimates. Therefore we incorporate the propensity score into the Cox model for causal inference with survival data. We derive the asymptotic property of the maximum partial likelihood estimator when the model is correctly specified. Simulation results show that our method performs quite well for observational data. The approach is applied to a real dataset on the time of readmission of trauma patients. We also derive the asymptotic property of the maximum partial likelihood estimator with a robust variance estimator, when the model is incorrectly specified.     

Imputation methods for doubly censored HIV data

In medical research, it is common to have doubly censored survival data: origin time and event time are both subject to censoring. In this paper, we review simple and probability-based methods that are used to impute interval censored origin time and compare the performance of these methods through extensive simulations in the one-sample problem, two-sample problem and Cox regression model problem. The use of a bootstrap procedure for inference is demonstrated.     

Imputation methods for doubly censored HIV data

In medical research, it is common to have doubly censored survival data: origin time and event time are both subject to censoring. In this paper, we review simple and probability-based methods that are used to impute interval censored origin time and compare the performance of these methods through extensive simulations in the one-sample problem, two-sample problem and Cox regression model problem. The use of a bootstrap procedure for inference is demonstrated.     

Improving the bandwidth-free inference methods by prewhitening

In this paper we consider inference of parameters in time series regression models. In the traditional inference approach, the heteroskedasticity and autocorrelation consistent (HAC) estimation is often involved to consistently estimate the asymptotic covariance matrix of regression parameter estimator. Since the bandwidth parameter in the HAC estimation is difficult to choose in practice, there has been a recent surge of interest in developing bandwidth-free inference methods. However, existing simulation studies show that these new methods suffer from severe size distortion in the presence of strong temporal dependence for a medium sample size. To remedy the problem, we propose to apply the prewhitening to the inconsistent long-run variance estimator in these methods to reduce the size distortion. The asymptotic distribution of the prewhitened Wald statistic is obtained and the general effectiveness of prewhitening is shown through simulations.