WLAD-LASSO method for robust estimation and variable selection in partially linear models

This paper focuses on robust estimation and variable selection for partially linear models. We combine the weighted least absolute deviation (WLAD) regression with the adaptive least absolute shrinkage and selection operator (LASSO) to achieve simultaneous robust estimation and variable selection for partially linear models. Compared with the LAD-LASSO method, the WLAD-LASSO method will resist to the heavy-tailed errors and outliers in the parametric components. In addition, we estimate the unknown smooth function by a robust local linear regression. Under some regular conditions, the theoretical properties of the proposed estimators are established. We further examine finite-sample performance of the proposed procedure by simulation studies and a real data example.     

A Semiparametric Estimation Approach for Linear Mixed Models

Maximum likelihood approach is the most frequently employed approach for the inference of linear mixed models. However, it relies on the normal distributional assumption of the random effects and the within-subject errors, and it is lack of robustness against outliers. This article proposes a semiparametric estimation approach for linear mixed models. This approach is based on the first two marginal moments of the response variable, and does not require any parametric distributional assumptions of random effects or error terms. The consistency and asymptotically normality of the estimator are derived under fairly general conditions. In addition, we show that the proposed estimator has a bounded influence function and a redescending property so it is robust to outliers. The methodology is illustrated through an application to the famed Framingham cholesterol data. The finite sample behavior and the robustness properties of the proposed estimator are evaluated through extensive simulation studies.     

Random effects selection in generalized linear mixed models via shrinkage penalty function

In this paper, we discuss the selection of random effects within the framework of generalized linear mixed models (GLMMs). Based on a reparametrization of the covariance matrix of random effects in terms of modified Cholesky decomposition, we propose to add a shrinkage penalty term to the penalized quasi-likelihood (PQL) function of the variance components for selecting effective random effects. The shrinkage penalty term is taken as a function of the variance of random effects, initiated by the fact that if the variance is zero then the corresponding variable is no longer random (with probability one). The proposed method takes the advantage of a convenient computation for the PQL estimation and appealing properties for certain shrinkage penalty functions such as LASSO and SCAD. We propose to use a backfitting algorithm to estimate the fixed effects and variance components in GLMMs, which also selects effective random effects simultaneously. Simulation studies show that the proposed approach performs quite well in selecting effective random effects in GLMMs. Real data analysis is made using the proposed approach, too.     

A simultaneous variable selection methodology for linear mixed models

Selecting an appropriate structure for a linear mixed model serves as an appealing problem in a number of applications such as in the modelling of longitudinal or clustered data. In this paper, we propose a variable selection procedure for simultaneously selecting and estimating the fixed and random effects. More specifically, a profile log-likelihood function, along with an adaptive penalty, is utilized for sparse selection. The Newton-Raphson optimization algorithm is performed to complete the parameter estimation. By jointly selecting the fixed and random effects, the proposed approach increases selection accuracy compared with two-stage procedures, and the usage of the profile log-likelihood can improve computational efficiency in one-stage procedures. We prove that the proposed procedure enjoys the model selection consistency. A simulation study and a real data application are conducted for demonstrating the effectiveness of the proposed method.     

Penalized inverse probability weighted estimators for weighted rank regression with missing covariates

Abstract

In this article, we study the variable selection and estimation for linear regression models with missing covariates. The proposed estimation method is almost as efficient as the popular least-squares-based estimation method for normal random errors and empirically shown to be much more efficient and robust with respect to heavy tailed errors or outliers in the responses and covariates. To achieve sparsity, a variable selection procedure based on SCAD is proposed to conduct estimation and variable selection simultaneously. The procedure is shown to possess the oracle property. To deal with the covariates missing, we consider the inverse probability weighted estimators for the linear model when the selection probability is known or unknown. It is shown that the estimator by using estimated selection probability has a smaller asymptotic variance than that with true selection probability, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for penalized rank estimator with the covariates missing in the linear model. Some numerical examples are provided to demonstrate the performance of the estimators.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

A Robust Variable Selection to t-type Joint Generalized Linear Models via Penalized t-type Pseudo-likelihood

Although the t-type estimator is a kind of M-estimator with scale optimization, it has some advantages over the M-estimator. In this article, we first propose a t-type joint generalized linear model as a robust extension to the classical joint generalized linear models for modeling data containing extreme or outlying observations. Next, we develop a t-type pseudo-likelihood (TPL) approach, which can be viewed as a robust version to the existing pseudo-likelihood (PL) approach. To determine which variables significantly affect the variance of the response variable, we then propose a unified penalized maximum TPL method to simultaneously select significant variables for the mean and dispersion models in t-type joint generalized linear models. Thus, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the mean and dispersion models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies are conducted to illustrate the proposed methods.     

A model selection method based on the adaptive LASSO-penalized GEE and weighted Gaussian pseudo-likelihood BIC in longitudinal robust analysis

In this article, a new robust variable selection approach is introduced by combining the robust generalized estimating equations and adaptive LASSO penalty function for longitudinal generalized linear models. Then, an efficient weighted Gaussian pseudo-likelihood version of the BIC (WGBIC) is proposed to choose the tuning parameter in the process of robust variable selection and to select the best working correlation structure simultaneously. Meanwhile, the oracle properties of the proposed robust variable selection method are established and an efficient algorithm combining the iterative weighted least squares and minorization–maximization is proposed to implement robust variable selection and parameter estimation.     

Response prediction in mixed effects models

Although prediction in mixed effects models usually concerns the random effects, in this paper we deal with the problem of prediction of a future, or yet unobserved, response random variable, belonging to a given cluster. In particular, the aim is to define computationally tractable prediction intervals, with conditional and unconditional coverage probability close to the target nominal value. This solution involves the conditional density of the future response random variable given the observed data, or a suitable high-order approximation based on the Laplace method. We prove that, unless the amount of data is very limited, the estimative or naive predictive procedure gives a relatively simple, feasible solution for response prediction. An application to generalized linear mixed models is presented.     

Local influence of nonlinear mixed effects model based on M-estimation

Abstract

In this work we mainly study the local influence in nonlinear mixed effects model with M-estimation. A robust method to obtain maximum likelihood estimates for parameters is presented, and the local influence of nonlinear mixed models based on robust estimation (M-estimation) by use of the curvature method is systematically discussed. The counting formulas of curvature for case weights perturbation, response variable perturbation and random error covariance perturbation are derived. Simulation studies are carried to access performance of the methods we proposed. We illustrate the diagnostics by an example presented in Davidian and Giltinan, which was analyzed under the non-robust situation.     

A comparative study on estimation methods to deal with the endogeneity in linear random-intercept models with an extension

In this paper, we investigate estimation methods to deal with situations where random intercepts are associated to time-varying covariates in the context of linear mixed models. First, a review of previous ways to deal with this so-called endogeneity issue is presented, then a new method based on shared random effects is proposed. Simulation studies and an empirical example are utilized to assess the performance of our proposed method. It is shown that our new approach is more efficient than most competitors and is robust to the misspecification of the random-effects distributions.     

Robust rank-based variable selection in double generalized linear models with diverging number of parameters under adaptive Lasso

We propose a robust rank-based estimation and variable selection in double generalized linear models when the number of parameters diverges with the sample size. The consistency of the variable selection procedure and asymptotic properties of the resulting estimators are established under appropriate selection of tuning parameters. Simulations are performed to assess the finite sample performance of the proposed estimation and variable selection procedure. In the presence of gross outliers, the proposed method is showing that the variable selection method works better. For practical application, a real data application is provided using nutritional epidemiology data, in which we explore the relationship between plasma beta-carotene levels and personal characteristics (e.g. age, gender, fat, etc.) as well as dietary factors (e.g. smoking status, intake of cholesterol, etc.).     

Bayesian quantile regression for skew-normal linear mixed models

Linear mixed models have been widely used to analyze repeated measures data which arise in many studies. In most applications, it is assumed that both the random effects and the within-subjects errors are normally distributed. This can be extremely restrictive, obscuring important features of within-and among-subject variations. Here, quantile regression in the Bayesian framework for the linear mixed models is described to carry out the robust inferences. We also relax the normality assumption for the random effects by using a multivariate skew-normal distribution, which includes the normal ones as a special case and provides robust estimation in the linear mixed models. For posterior inference, we propose a Gibbs sampling algorithm based on a mixture representation of the asymmetric Laplace distribution and multivariate skew-normal distribution. The procedures are demonstrated by both simulated and real data examples.     

Selection Strategy for Covariance Structure of Random Effects in Linear Mixed‐effects Models

Linear mixed‐effects models are a powerful tool for modelling longitudinal data and are widely used in practice. For a given set of covariates in a linear mixed‐effects model, selecting the covariance structure of random effects is an important problem. In this paper, we develop a joint likelihood‐based selection criterion. Our criterion is the approximately unbiased estimator of the expected Kullback–Leibler information. This criterion is also asymptotically optimal in the sense that for large samples, estimates based on the covariance matrix selected by the criterion minimize the approximate Kullback–Leibler information. Finite sample performance of the proposed method is assessed by simulation experiments. As an illustration, the criterion is applied to a data set from an AIDS clinical trial.     

Robust group non-convex estimations for high-dimensional partially linear models

High-dimensional data with a group structure of variables arise always in many contemporary statistical modelling problems. Heavy-tailed errors or outliers in the response often exist in these data. We consider robust group selection for partially linear models when the number of covariates can be larger than the sample size. The non-convex penalty function is applied to achieve both goals of variable selection and estimation in the linear part simultaneously, and we use polynomial splines to estimate the nonparametric component. Under regular conditions, we show that the robust estimator enjoys the oracle property. Simulation studies demonstrate the performance of the proposed method with samples of moderate size. The analysis of a real example illustrates that our method works well.     

Generalised Rank Regression Estimator with Standard Error Adjusted Lasso

One of the standard variable selection procedures in multiple linear regression is to use a penalisation technique in least‐squares (LS) analysis. In this setting, many different types of penalties have been introduced to achieve variable selection. It is well known that LS analysis is sensitive to outliers, and consequently outliers can present serious problems for the classical variable selection procedures. Since rank‐based procedures have desirable robustness properties compared to LS procedures, we propose a rank‐based adaptive lasso‐type penalised regression estimator and a corresponding variable selection procedure for linear regression models. The proposed estimator and variable selection procedure are robust against outliers in both response and predictor space. Furthermore, since rank regression can yield unstable estimators in the presence of multicollinearity, in order to provide inference that is robust against multicollinearity, we adjust the penalty term in the adaptive lasso function by incorporating the standard errors of the rank estimator. The theoretical properties of the proposed procedures are established and their performances are investigated by means of simulations. Finally, the estimator and variable selection procedure are applied to the Plasma Beta‐Carotene Level data set.     

Nonlinear mixed‐effects models with misspecified random‐effects distribution

Nonlinear mixed‐effects models are being widely used for the analysis of longitudinal data, especially from pharmaceutical research. They use random effects which are latent and unobservable variables so the random‐effects distribution is subject to misspecification in practice. In this paper, we first study the consequences of misspecifying the random‐effects distribution in nonlinear mixed‐effects models. Our study is focused on Gauss‐Hermite quadrature, which is now the routine method for calculation of the marginal likelihood in mixed models. We then present a formal diagnostic test to check the appropriateness of the assumed random‐effects distribution in nonlinear mixed‐effects models, which is very useful for real data analysis. Our findings show that the estimates of fixed‐effects parameters in nonlinear mixed‐effects models are generally robust to deviations from normality of the random‐effects distribution, but the estimates of variance components are very sensitive to the distributional assumption of random effects. Furthermore, a misspecified random‐effects distribution will either overestimate or underestimate the predictions of random effects. We illustrate the results using a real data application from an intensive pharmacokinetic study.     

Robust adaptive model selection and estimation for partial linear varying coefficient models in rank regression

Partial linear varying coefficient models are often used in real data analysis for a good balance between flexibility and parsimony. In this paper, we propose a robust adaptive model selection method based on the rank regression, which can do simultaneous coefficient estimation and three types of selections, i.e., varying and constant effects selection, relevant variable selection. The new method has superiority in robustness and efficiency by inheriting the advantage of the rank regression approach. Furthermore, consistency in the three types of selections and oracle property in estimation are established as well. Simulation studies also confirm our method.     

Simultaneous structure estimation and variable selection in partial linear varying coefficient models for longitudinal data

Partial linear varying coefficient models (PLVCM) are often considered for analysing longitudinal data for a good balance between flexibility and parsimony. The existing estimation and variable selection methods for this model are mainly built upon which subset of variables have linear or varying effect on the response is known in advance, or say, model structure is determined. However, in application, this is unreasonable. In this work, we propose a simultaneous structure estimation and variable selection method, which can do simultaneous coefficient estimation and three types of selections: varying and constant effects selection, relevant variable selection. It can be easily implemented in one step by employing a penalized M-type regression, which uses a general loss function to treat mean, median, quantile and robust mean regressions in a unified framework. Consistency in the three types of selections and oracle property in estimation are established as well. Simulation studies and real data analysis also confirm our method.