非线性回归模型的经验似然诊断

经验似然方法已经被广泛用于线性模型和广义线性模型.本文基于经验似然方法对非线性回归模型进行统计诊断.首先得到模型参数的极大经验似然估计;其次基于经验似然研究了三种不同的影响曲率度量;最后通过一个实际例子,说明了诊断方法的有效性.     

Randomly censored partially linear single-index models

This paper proposes a method for estimation of a class of partially linear single-index models with randomly censored samples. The method provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure-time models for survival analysis. The estimation procedure involves three stages: first, transform the censored data into synthetic data or pseudo-responses unbiasedly; second, obtain quasi-likelihood estimates of the regression coefficients in both linear and single-index components by an iteratively algorithm; finally, estimate the unknown nonparametric regression function using techniques for univariate censored nonparametric regression. The estimators for the regression coefficients are shown to be jointly root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as all the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodology.     

基于经验似然的部分线性模型的统计诊断

经验似然方法己经被广泛应用于许多模型的统计推断.本文基于经验似然对部分线性模型进行统计诊断.首先给出模型的估计方程,进而得到模型参数的极大经验似然估计;其次,基于经验似然研究了三种不同的影响曲率;最后通过随机模拟和实例分析,说明了统计诊断方法的有效性.     

Asymptotic distributions of two “synthetic data” estimators for censored single-index models

The censored single-index model provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored and the link function is unknown. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure time models for survival analysis. This paper proposes two methods for estimation of single-index models with randomly censored samples. We first transform the censored data into synthetic data or pseudo-responses unbiasedly, then obtain estimates of the index coefficients by the rOPG or rMAVE procedures of Xia (2006) [1]. Finally, we estimate the unknown nonparametric link function using techniques for univariate censored nonparametric regression. The estimators for the index coefficients are shown to be root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodologies.     

The use of ROC for defining the validity of the prognostic index in censored data

The validity of a diagnostic marker can be summarised by using statistical measures either for the goodness of the fit like the deviance, measures of the explained variation like R² or the misclassification rate. Other intuitive measures are sensitivity and specificity in the case of binary response. In the absence of censored data the calculation of these measures is widely used. In the presence of censoring the estimation of time-dependent sensitivity and specificity is not well known. In this article we propose a new method for calculating ROC curves with censored data using the observed number of events and calculating the additional number of expected events for censored observations. The new method is illustrated with data for predicting mortality in patients surviving a myocardial infarction.     

Censored multiple regression by the method of average derivatives

This paper proposes a technique [termed censored average derivative estimation (CADE)] for studying estimation of the unknown regression function in nonparametric censored regression models with randomly censored samples. The CADE procedure involves three stages: firstly-transform the censored data into synthetic data or pseudo-responses using the inverse probability censoring weighted (IPCW) technique, secondly estimate the average derivatives of the regression function, and finally approximate the unknown regression function by an estimator of univariate regression using techniques for one-dimensional nonparametric censored regression. The CADE provides an easily implemented methodology for modelling the association between the response and a set of predictor variables when data are randomly censored. It also provides a technique for “dimension reduction” in nonparametric censored regression models. The average derivative estimator is shown to be root-n consistent and asymptotically normal. The estimator of the unknown regression function is a local linear kernel regression estimator and is shown to converge at the optimal one-dimensional nonparametric rate. Monte Carlo experiments show that the proposed estimators work quite well.     

删失数据下的半参数变系数部分线性回归模型

为了分析删失数据,该文考虑变系数部分线性模型,此模型允许协变量对响应变量存在非线性影响.响应变量与协变量之间关系的统计模型通过线性结构来拟合是非常重要而且有益.对于删失数据,常用的统计方法不能直接应用于此模型.该文首先提出一类数据变换用以建立无偏条件期望.然后利用profile最小二乘方法,给出了模型中参数分量和非参数分量的profile最小二乘估计,并建立了这些估计的渐近正态性.最后通过数值例子来说明该文所提出的方法的有效性.     

单纯形分布模型的统计诊断

该文讨论了单纯形分布广义线性模型和广义非线性模型的影响分析问题, 得到了若干有用的诊断统计量;证明了数据删除模型和均值漂移模型的的等价性定理.同时还研究了该模型的变离差检验,得到了Score检验统计量.最后给出了两个实例, 说明该文方法的应用价值.     

Stochastic integrals of empirical-type processes with applications to censored regression

Motivated by the analysis of linear rank estimators and the Buckley-James nonparametric EM estimator in censored regression models, we study herein the asymptotic properties of stochastic integrals of certain two-parameter empirical processes. Applications of these results on empirical processes and their stochastic integrals to the asymptotic analysis of censored regression estimators are also given.     

A Simple Method for Computing the Observed Information Matrix When Using the EM Algorithm with Categorical Data

Abstract

A simple matrix formula is given for the observed information matrix when the EM algorithm is applied to categorical data with missing values. The formula requires only the design matrices, a matrix linking the complete and incomplete data, and a few simple derivatives. It can be easily programmed using a computer language with operators for matrix multiplication, element-by-element multiplication and division, matrix concatenation, and creation of diagonal and block diagonal arrays. The formula is applicable whenever the incomplete data can be expressed as a linear function of the complete data, such as when the observed counts represent the sum of latent classes, a supplemental margin, or the number censored. In addition, the formula applies to a wide variety of models for categorical data, including those with linear, logistic, and log-linear components. Examples include a linear model for genetics, a log-linear model for two variables and nonignorable nonresponse, the product of a log-linear model for two variables and a logit model for nonignorable nonresponse, a latent class model for the results of two diagnostic tests, and a product of linear models under double sampling.     

半参数非线性模型的统计诊断与影响分析

本文系统研究了半参数非线性回归模型的统计诊断与影响分析方法;得到了一系列诊断统计量,两个实际数值例子验证了本文给出的诊断方法的有效性。     

Semiparametric regression analysis for clustered doubly-censored data

This paper considers clustered doubly-censored data that occur when there exist several correlated survival times of interest and only doubly censored data are available for each survival time. In this situation, one approach is to model the marginal distribution of failure times using semiparametric linear transformation models while leaving the dependence structure completely arbitrary. We demonstrate that the approach of Cai et al. (Biometrika 87:867–878, 2000) can be extended to clustered doubly censored data. We propose two estimators by using two different estimated censoring weights. A simulation study is conducted to investigate the proposed estimators.     

On the relationship between the proportional hazards and accelerated failure time models in survival analysis

The slope parameters in the proportional hazards and accelerated failure time models are shown to be proportional if and only if the error variable in the accelerated failure time model has a generalized extreme value distribution. A differential equation relating the two score functions φ(u) and Φ(u) is established. The error distribution is characterized when φ (u) and Φ(u) are linearly related; subject to this restriction, censored data linear rank procedures are studied.     

线性回归诊断的若干问题

本文对于线性回归诊断提出了几种新的模型和方法。我们首次研究了方差加权和均值漂移的混合模型,得到了相应的诊断统计量。本文还引入了罚函数方法,并以此为工具,讨论了若干有偏估计的影响度量,最后,本文提出了基于重心的诊断统计量,对于识别异常点有较好的效果。     

LAD estimation for nonlinear regression models with randomly censored data

The least absolute deviations (LAD) estimation for nonlinear regression models with randomly censored data is studied and the asymptotic properties of LAD estimators such as consistency, boundedness in probability and asymptotic normality are established. Simulation results show that for the problems with censored data, LAD estimation performs much more robustly than the least squares estimation.     

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本文研究了线性模型响应变量被污染且被区间截断下的参数估计问题, 借助于区间数据的无偏转换, 得到了回归系数和污染系数的估计, 并在一定的条件下得到了这些估计的强相合性. 通过若干模拟例子说明, 尽管数据经过污染和区间截断的双重信息损失, 但用本文提出的方法得到的估计, 仍能取得良好的估计效果     

基于生存数据的半参数变换的删失回归估计

生存数据经过未知的单调变换后等于协变量的线性函数加上随机误差, 随机误差的分布函数已知或是带未知参数的已知函数\bd 本文先给出未知单调变换的一个相合估计, 再对删失数据做变换, 在此基础上给出了协变量系数的最小二乘估计, 并讨论它的大样本性质.     

Influence analysis on linear models with random effects

In this paper,a unified diagnostic method for linear models with random effects basedupon the joint likelihood given by Robinson gin 1991) is presented. The case deletion model is e-quivalent to mean shift outlier model,as well as case weights model. From this point of view,several new diagnostic measures,such as Cook disrance,WK diagnostics are derived. Some pre-vious results are improved. Numerical examples illustrate the method is available.     

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.     

Mesures prédictives dans le modèle de Cox–Dirichlet à censures aléatoiresPredictive measures in the Cox–Dirichlet model with random censures

This paper is devoted to the extension to randomly censored observations of the study of predictive measures in the nonparametric bayesian Cox–Dirichlet model, which has been developed for non censored observations by N. Gouget and J.P. Raoult in [3]. We show that in all cases, the predictive measures stay piecewise regular (as defined in [3]). This implies that there exists a partition of the space of observations in subsets, included in linear subspaces, such that in each of them the predictive measure is absolutely continuous w.r.t. the Lebesgue measure. We compute the so defined densities for right censoring. To cite this article: F. Messaci, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 557–560.