Penalized quadratic inference functions for single-index models with longitudinal data

In this paper, we focus on single-index models for longitudinal data. We propose a procedure to estimate the single-index component and the unknown link function based on the combination of the penalized splines and quadratic inference functions. It is shown that the proposed estimation method has good asymptotic properties. We also evaluate the finite sample performance of the proposed method via Monte Carlo simulation studies. Furthermore, the proposed method is illustrated in the analysis of a real data set.     

Improved empirical likelihood inference and variable selection for generalized linear models with longitudinal nonignorable dropouts

Annals of the Institute of Statistical Mathematics - In this paper, we propose improved statistical inference and variable selection methods for generalized linear models based on empirical...     

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.     

Empirical likelihood inference in partially linear single-index models for longitudinal data

The empirical likelihood method is especially useful for constructing confidence intervals or regions of parameters of interest. Yet, the technique cannot be directly applied to partially linear single-index models for longitudinal data due to the within-subject correlation. In this paper, a bias-corrected block empirical likelihood (BCBEL) method is suggested to study the models by accounting for the within-subject correlation. BCBEL shares some desired features: unlike any normal approximation based method for confidence region, the estimation of parameters with the iterative algorithm is avoided and a consistent estimator of the asymptotic covariance matrix is not needed. Because of bias correction, the BCBEL ratio is asymptotically chi-squared, and hence it can be directly used to construct confidence regions of the parameters without any extra Monte Carlo approximation that is needed when bias correction is not applied. The proposed method can naturally be applied to deal with pure single-index models and partially linear models for longitudinal data. Some simulation studies are carried out and an example in epidemiology is given for illustration.     

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.     

Robust and accurate inference for generalized linear models

In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022–1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154–1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically χ²-distributed as the three classical tests but with a relative error of order O(n⁻¹). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.     

Generalized empirical likelihood inference in semiparametric regression model for longitudinal data

In this paper, we consider the semiparametric regression model for longitudinal data. Due to the correlation within groups, a generalized empirical log-likelihood ratio statistic for the unknown parameters in the model is suggested by introducing the working covariance matrix. It is proved that the proposed statistic is asymptotically standard chi-squared under some suitable conditions, and hence it can be used to construct the confidence regions of the parameters. A simulation study is conducted to compare the proposed method with the generalized least squares method in terms of coverage accuracy and average lengths of the confidence intervals.     

Corrected empirical likelihood for a class of generalized linear measurement error models

Generalized linear measurement error models, such as Gaussian regression, Poisson regression and logistic regression, are considered. To eliminate the effects of measurement error on parameter estimation, a corrected empirical likelihood method is proposed to make statistical inference for a class of generalized linear measurement error models based on the moment identities of the corrected score function. The asymptotic distribution of the empirical log-likelihood ratio for the regression parameter is proved to be a Chi-squared distribution under some regularity conditions. The corresponding maximum empirical likelihood estimator of the regression parameter π is derived, and the asymptotic normality is shown. Furthermore, we consider the construction of the confidence intervals for one component of the regression parameter by using the partial profile empirical likelihood. Simulation studies are conducted to assess the finite sample performance. A real data set from the ACTG 175 study is used for illustrating the proposed method.     

Two-step generalised empirical likelihood inference for semiparametric models

This paper shows how the generalised empirical likelihood method can be used to obtain valid asymptotic inference for the finite dimensional component of semiparametric models defined by a set of moment conditions. The results of the paper are illustrated using three well-known semiparametric regression models: partially linear single index, linear transformation with random censoring, and quantile regression with random censoring. Monte Carlo simulations suggest that some of the proposed test statistics have competitive finite sample properties. The results of the paper are applied to test for functional misspecification in a hedonic price model of a housing market.     

Robust estimation in generalized semiparametric mixed models for longitudinal data

In this paper, we consider robust generalized estimating equations for the analysis of semiparametric generalized partial linear mixed models (GPLMMs) for longitudinal data. We approximate the non-parametric function in the GPLMM by a regression spline, and make use of bounded scores and leverage-based weights in the estimating equation to achieve robustness against outliers and influential data points, respectively. Under some regularity conditions, the asymptotic properties of the robust estimators are investigated. To avoid the computational problems involving high-dimensional integrals in our estimators, we adopt a robust Monte Carlo Newton-Raphson (RMCNR) algorithm for fitting GPLMMs. Small simulations are carried out to study the behavior of the robust estimates in the presence of outliers, and these estimates are also compared to their corresponding non-robust estimates. The proposed robust method is illustrated in the analysis of two real data sets.     

Empirical likelihood for partial linear models

In this paper the empirical likelihood method due to Owen (1988,Biometrika,75, 237–249) is applied to partial linear random models. A nonparametric version of Wilks' theorem is derived. The theorem is then used to construct confidence regions of the parameter vector in the partial linear models, which has correct asymptotic coverage. A simulation study is conducted to compare the empirical likelihood and normal approximation based method. Research supported by NNSF of China and a grant to the first author for his excellent Ph.D. dissertation work in China. Research supported by Hong Kong RGC CERG No. HKUST6162/97P.     

Likelihood and conditional likelihood inference for generalized additive mixed models for clustered data

Lin and Zhang (J. Roy. Statist. Soc. Ser. B 61 (1999) 381) proposed the generalized additive mixed model (GAMM) as a framework for analysis of correlated data, where normally distributed random effects are used to account for correlation in the data, and proposed to use double penalized quasi-likelihood (DPQL) to estimate the nonparametric functions in the model and marginal likelihood to estimate the smoothing parameters and variance components simultaneously. However, the normal distributional assumption for the random effects may not be realistic in many applications, and it is unclear how violation of this assumption affects ensuing inferences for GAMMs. For a particular class of GAMMs, we propose a conditional estimation procedure built on a conditional likelihood for the response given a sufficient statistic for the random effect, treating the random effect as a nuisance parameter, which thus should be robust to its distribution. In extensive simulation studies, we assess performance of this estimator under a range of conditions and use it as a basis for comparison to DPQL to evaluate the impact of violation of the normality assumption. The procedure is illustrated with application to data from the Multicenter AIDS Cohort Study (MACS).     

Robust likelihood inference for multivariate correlated count data

A parametric robust approach for analyzing correlated count data is introduced. This method enables one to construct an asymptotically valid likelihood for the regression parameter when knowledge about the joint distribution for data is scarce or not available. We use simulations and real data analysis to demonstrate the merit of the proposed robust likelihood method.     

Estimation in generalized linear models for functional data via penalized likelihood

We analyze in a regression setting the link between a scalar response and a functional predictor by means of a Functional Generalized Linear Model. We first give a theoretical framework and then discuss identifiability of the model. The functional coefficient of the model is estimated via penalized likelihood with spline approximation. The L² rate of convergence of this estimator is given under smoothness assumption on the functional coefficient. Heuristic arguments show how these rates may be improved for some particular frameworks.     

双重广义线性模型的经验似然推断

基于截面经验似然方法,将双重广义线性模型的拟似然估计方程作为截面经验似然比函数的约束条件,构造了均值模型和散度模型未知参数的置信区间.最后通过数据模拟,将该方法与正态逼近方法比较,说明了该方法是有效和可行的.     

Corrected empirical likelihood inference for right-censored partially linear single-index model

This article deals with the inference on a right-censored partially linear single-index model (RCPLSIM). The main focus is the local empirical likelihood-based inference on the nonparametric part in RCPLSIM. With a synthetic data approach, an empirical log-likelihood ratio statistic for the nonparametric part is defined and it is shown that its limiting distribution is not a central chi-squared distribution. To increase the accuracy of the confidence interval, we also propose a corrected empirical log-likelihood ratio statistic for the nonparametric function. The resulting statistic is proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence intervals. A real example is also considered.     

Bayesian inference on partially linear mixed-effects joint models for longitudinal data with multiple features

The relationship between viral load and CD4 cell count is one of the interesting questions in AIDS research. Statistical models are powerful tools for clarifying this important problem. Partially linear mixed-effects (PLME) model which accounts for the unknown function of time effect is one of the important models for this purpose. Meanwhile, the mixed-effects modeling approach is suitable for the longitudinal data analysis. However, the complex process of data collection in clinical trials has made it impossible to rely on one particular model to address the issues. Asymmetric distribution, measurement error and left censoring are features commonly arisen in longitudinal studies. It is crucial to take into account these features in the modeling process to achieve reliable estimation and valid conclusion. In this article, we establish a joint model that accounts for all these features in the framework of PLME models. A Bayesian inferential procedure is proposed to estimate parameters in the joint model. A real data example is analyzed to demonstrate the proposed modeling approach for inference and the results are reported by comparing various scenarios-based models.     

Penalized likelihood regression for generalized linear models with non-quadratic penalties

One of the popular method for fitting a regression function is regularization: minimizing an objective function which enforces a roughness penalty in addition to coherence with the data. This is the case when formulating penalized likelihood regression for exponential families. Most of the smoothing methods employ quadratic penalties, leading to linear estimates, and are in general incapable of recovering discontinuities or other important attributes in the regression function. In contrast, non-linear estimates are generally more accurate. In this paper, we focus on non-parametric penalized likelihood regression methods using splines and a variety of non-quadratic penalties, pointing out common basic principles. We present an asymptotic analysis of convergence rates that justifies the approach. We report on a simulation study including comparisons between our method and some existing ones. We illustrate our approach with an application to Poisson non-parametric regression modeling of frequency counts of reported acquired immune deficiency syndrome (AIDS) cases in the UK.