Empirical likelihood inference for the accelerated failure time model

Accelerated failure time (AFT) models are useful regression tools for studying the association between a survival time and covariates. Semiparametric inference procedures have been proposed in an extensive literature. Among these, use of an estimating equation which is monotone in the regression parameter and has some excellent properties was proposed by Fygenson and Ritov (1994). However, there is a serious under-coverage problem for small sample sizes. In this paper, we derive the limiting distribution of the empirical log-likelihood ratio for the regression parameter on the basis of the monotone estimating equations. Furthermore, the empirical likelihood (EL) confidence intervals/regions for the regression parameter are obtained. We conduct a simulation study in order to compare the proposed EL method with the normal approximation method. The simulation results suggest that the empirical likelihood based method outperforms the normal approximation based method in terms of coverage probability. Thus, the proposed EL method overcomes the under-coverage problem of the normal approximation method.     

Empirical likelihood inference for median regression models for censored survival data

Recent advances in median regression model have made it possible to use this model for analyzing a variety of censored survival data. For inference on the model parameter vector, there are now semiparametric procedures based on normal approximation that are valid without strong conditions on the error distribution. However, the accuracy of such procedures can be quite low when the censoring proportion is high. In this paper, we propose an alternative semiparametric procedure based on the empirical likelihood. We define the empirical likelihood ratio for the parameter vector and show that its limiting distribution is a weighted sum of chi-square distributions. Numerical results from a simulation study suggest that the empirical likelihood method is more accurate than the normal approximation based method of Ying et al. (J. Amer. Statist. Assoc. 90 (1995) 178).     

Semiparametric inference for transformation models via empirical likelihood

Recent advances in the transformation model have made it possible to use this model for analyzing a variety of censored survival data. For inference on the regression parameters, there are semiparametric procedures based on the normal approximation. However, the accuracy of such procedures can be quite low when the censoring rate is heavy. In this paper, we apply an empirical likelihood ratio method and derive its limiting distribution via U-statistics. We obtain confidence regions for the regression parameters and compare the proposed method with the normal approximation based method in terms of coverage probability. The simulation results demonstrate that the proposed empirical likelihood method overcomes the under-coverage problem substantially and outperforms the normal approximation based method. The proposed method is illustrated with a real data example. Finally, our method can be applied to general U-statistic type estimating equations.     

Empirical likelihood inference for censored median regression with weighted empirical hazard functions

In recent years, median regression models have been shown to be useful for analyzing a variety of censored survival data in clinical trials. For inference on the regression parameter, there have been a variety of semiparametric procedures. However, the accuracy of such procedures in terms of coverage probability can be quite low when the censoring rate is heavy. In this paper, based on weighted empirical hazard functions, we apply an empirical likelihood (EL) ratio method to the median regression model with censoring data and derive the limiting distribution of EL ratio. Confidence region for the regression parameter can then be obtained accordingly. Furthermore, we compared the proposed method with the standard method through extensive simulation studies. The proposed method almost always outperformed the existing method.     

Empirical likelihood analysis of the Buckley-James estimator

The censored linear regression model, also referred to as the accelerated failure time (AFT) model when the logarithm of the survival time is used as the response variable, is widely seen as an alternative to the popular Cox model when the assumption of proportional hazards is questionable. Buckley and James [Linear regression with censored data, Biometrika 66 (1979) 429-436] extended the least squares estimator to the semiparametric censored linear regression model in which the error distribution is completely unspecified. The Buckley-James estimator performs well in many simulation studies and examples. The direct interpretation of the AFT model is also more attractive than the Cox model, as Cox has pointed out, in practical situations. However, the application of the Buckley-James estimation was limited in practice mainly due to its illusive variance. In this paper, we use the empirical likelihood method to derive a new test and confidence interval based on the Buckley-James estimator of the regression coefficient. A standard chi-square distribution is used to calculate the P-value and the confidence interval. The proposed empirical likelihood method does not involve variance estimation. It also shows much better small sample performance than some existing methods in our simulation studies.     

Empirical likelihood for single-index models

The empirical likelihood method is especially useful for constructing confidence intervals or regions of the parameter of interest. This method has been extensively applied to linear regression and generalized linear regression models. In this paper, the empirical likelihood method for single-index regression models is studied. An estimated empirical log-likelihood approach to construct the confidence region of the regression parameter is developed. An adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square. A simulation study indicates that compared with a normal approximation-based approach, the proposed method described herein works better in terms of coverage probabilities and areas (lengths) of confidence regions (intervals).     

Empirical likelihood inference for linear transformation models

Empirical likelihood inference is developed for censored survival data under the linear transformation models, which generalize Cox's [Regression models and life tables (with Discussion), J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] proportional hazards model. We show that the limiting distribution of the empirical likelihood ratio is a weighted sum of standard chi-squared distribution. Empirical likelihood ratio tests for the regression parameters with and without covariate adjustments are also derived. Simulation studies suggest that the empirical likelihood ratio tests are more accurate (under the null hypothesis) and powerful (under the alternative hypothesis) than the normal approximation based tests of Chen et al. [Semiparametric of transformation models with censored data, Biometrika 89 (2002) 659-668] when the model is different from the proportional hazards model and the proportion of censoring is high.     

区间数据情形下线性模型的经验似然推断

§1Introduction Instatisticalapplications,weoftenencounterintervalcensoreddatawhenafailure timeYcannotbeobserved,butcanonlybedeterminedtolieinanintervalobtainedfroma sequenceofexaminationtimes.Forinstance,themaximumdosagewhichpatientscan endureisconcerned.LetYibethemaximumdosagewhichtheithpatientcanendure,Ui,j(j=1,2,...,k)bethedosagewhichthepatienthasbeentested.ItisobviousthatYiis unobservable.SupposetheithpatientisnormalwhenthedosageisUi,j,andhe(orshe)is abnormalwhenthedosageisUi,j+1.Then…     

Inference for Cox’s regression models via adjusted empirical likelihood

The Cox’s regression model is one of the most popular tools used in survival analysis. Recently, Qin and Jing (Commun Stat Simul Comput 30:79–90, 2001) applied empirical likelihood to study it with the assumption that baseline hazard function is known. However, in the Cox’s regression model the baseline hazard function is unspecified. Thus, their method suffers from severe defect. In this paper, we apply a variant of plug-in empirical likelihood by estimating the cumulative baseline hazard function. Adjusted empirical likelihood (AEL) confidence regions for the vector of regression parameters are obtained. Furthermore, we conduct a simulation study to evaluate the performance of the proposed AEL method by comparing it with normal approximation (NA) based method. The simulation studies show that both methods produce comparable coverage probabilities. The proposed AEL method outperforms the NA method based on power analysis.     

Empirical likelihood-based inferences for the area under the ROC curve with covariates

In the receiver operating characteristic (ROC) analysis,the area under the ROC curve (AUC) is a popular summary index of discriminatory accuracy of a diagnostic test.Incorporating covariates into ROC analysis can improve the diagnostic accuracy of the test.Regression model for the AUC is a tool to evaluate the effects of the covariates on the diagnostic accuracy.In this paper,empirical likelihood (EL) method is proposed for the AUC regression model.For the regression parameter vector,it can be shown that the asymptotic distribution of its EL ratio statistic is a weighted sum of independent chi-square distributions.Confidence regions are constructed for the parameter vector based on the newly developed empirical likelihood theorem,as well as for the covariate-specific AUC.Simulation studies were conducted to compare the relative performance of the proposed EL-based methods with the existing method in AUC regression.Finally,the proposed methods are illustrated with a real data set.     

区间数据均值的经验似然估计

本文提出了估计区间数据均值的经验似然方法,通过构造区间数据的无偏转换,导出了渐近服从χ2分布的对数经验似然函数,从而得到了均值的置信区间.通过若干模拟例子说明,用本文提出的方法得到的估计,优于用渐近正态法得到的估计.     

Empirical Likelihood Semiparametric Regression Analysis under Random Censorship

This paper considers large sample inference for the regression parameter in a partly linear model for right censored data. We introduce an estimated empirical likelihood for the regression parameter and show that its limiting distribution is a mixture of central chi-squared distributions. A Monte Carlo method is proposed to approximate the limiting distribution. This enables one to make empirical likelihood-based inference for the regression parameter. We also develop an adjusted empirical likelihood method which only appeals to standard chi-square tables. Finite sample performance of the proposed methods is illustrated in a simulation study.     

Weighted Empirical Likelihood Ratio Confidence Intervals for the Mean with Censored Data

We propose a procedure to construct the empirical likelihood ratio confidence interval for the mean using a resampling method. This approach leads to the definition of a likelihood function for censored data, called weighted empirical likelihood function. With the second order expansion of the log likelihood ratio, a weighted empirical likelihood ratio confidence interval for the mean is proposed and shown by simulation studies to have comparable coverage accuracy to alternative methods, including the nonparametric bootstrap-t. The procedures proposed here apply in a unified way to different types of censored data, such as right censored data, doubly censored data and interval censored data, and computationally more efficient than the bootstrap-t method. An example of a set of doubly censored breast cancer data is presented with the application of our methods.     

Empirical likelihood for linear models under negatively associated errors

In this paper, we discuss the construction of the confidence intervals for the regression vector β in a linear model under negatively associated errors. It is shown that the blockwise empirical likelihood (EL) ratio statistic for β is asymptotically χ²-type distributed. The result is used to obtain an EL based confidence region for β.     

Empirical likelihood for semiparametric regression model with missing response data

A bias-corrected technique for constructing the empirical likelihood ratio is used to study a semiparametric regression model with missing response data. We are interested in inference for the regression coefficients, the baseline function and the response mean. A class of empirical likelihood ratio functions for the parameters of interest is defined so that undersmoothing for estimating the baseline function is avoided. The existing data-driven algorithm is also valid for selecting an optimal bandwidth. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared. Also, a class of estimators for the parameters of interest is constructed, their asymptotic distributions are obtained, and consistent estimators of asymptotic bias and variance are provided. Our results can be used to construct confidence intervals and bands for the parameters of interest. A simulation study is undertaken to compare the empirical likelihood with the normal approximation-based method in terms of coverage accuracies and average lengths of confidence intervals. An example for an AIDS clinical trial data set is used for illustrating our methods.     

Empirical likelihood for the parametric part in partially linear errors-in-function models

Partially linear errors-in-function models were proposed by Liang (2000), but their inferences have not been systematically studied. This article proposes an empirical likelihood method to construct confidence regions of the parametric components. Under mild regularity conditions, the nonparametric version of the Wilk’s theorem is derived. Simulation studies show that the proposed empirical likelihood method provides narrower confidence regions, as well as higher coverage probabilities than those based on the traditional normal approximation method.     

Asymptotic properties of the maximum likelihood estimator for the proportional hazards model with doubly censored data

Doubly censored data, which include left as well as right censored observations, are frequently met in practice. Though estimation of the distribution function with doubly censored data has seen much study, relatively little is known about the inference of regression coefficients in the proportional hazards model for doubly censored data. In particular, theoretical properties of the maximum likelihood estimator of the regression coefficients in the proportional hazards model have not been proved yet. In this paper, we show the consistency and asymptotic normality of the maximum likelihood estimator and prove its semiparametric efficiency. The proposed methods are illustrated with simulation studies and analysis of an application from a medical study.     

右删失数据下非线性回归模型的经验似然推断

考虑随机右删失数据下非线性回归模型,提出了模型中未知参数的调整的经验对数似然比统计量.在一定的条件下,证明了.所提出的的统计量具有渐近χ~2分布,由此结果构造了兴趣参数的置信域.通过模拟研究,对经典的经验似然、调整的经验似然和非线性最小二乘方法在有限样本下进行了比较,并对氯离子浓度试验数据进行了分析.     

Empirical likelihood for the contrast of two hazard functions with right censoring

In this paper, we consider the standard two-sample framework with right censoring. We construct useful confidence intervals for the ratio or difference of two hazard functions using smoothed empirical likelihood (EL) methods. The empirical log-likelihood ratio is derived and its asymptotic distribution is a standard chi-squared distribution. Bootstrap confidence bands are also proposed. Simulation studies show that the proposed EL confidence intervals have outperformed normal approximation methods in terms of coverage probability. It is concluded that the empirical likelihood methods provide better inference results.     

Empirical likelihood for median regression model with designed censoring variables

We propose a new and simple estimating equation for the parameters in median regression models with designed censoring variables, and then apply the empirical log likelihood ratio statistic to construct confidence region for the parameters. The empirical log likelihood ratio statistic is shown to have a standard chi-square distribution, which makes this method easy to implement. At the same time, another empirical log likelihood ratio statistic is proposed based on an existing estimating equation and the limiting distribution of the empirical likelihood ratio statistic is shown to be a sum of weighted chi-square distributions. We compare the performance of the empirical likelihood confidence region based on the new estimating equation, with that based on the existing estimating equation and a normal approximation method by simulation studies.