Variable selection for generalized linear mixed models by L 1-penalized estimation

Generalized linear mixed models are a widely used tool for modeling longitudinal data. However, their use is typically restricted to few covariates, because the presence of many predictors yields unstable estimates. The presented approach to the fitting of generalized linear mixed models includes an L ₁-penalty term that enforces variable selection and shrinkage simultaneously. A gradient ascent algorithm is proposed that allows to maximize the penalized log-likelihood yielding models with reduced complexity. In contrast to common procedures it can be used in high-dimensional settings where a large number of potentially influential explanatory variables is available. The method is investigated in simulation studies and illustrated by use of real data sets.     

Consistent fractional Bayes factor for nested normal linear models

In the Bayesian approach to parametric model comparison, the use of improper priors is problematic due to the indeterminacy of the resulting Bayes factor (BF). The need for developing automatic and robust methods for model comparison has led to the introduction of alternative BFs. Intrinsic Bayes factors (Berger and Pericchi, 1996a) and fractional Bayes factors (FBF) (O'Hagan, 1995) are two alternative strategies for default model selection. We show in this paper that the FBF can be inconsistent. To overcome this problem, we propose a generalization of the FBF approach that leads to the usual FBF or to some variants of it in some special cases. As an important problem, we consider and discuss this generalization for model selection in nested linear models.     

The predictive Lasso

We propose a shrinkage procedure for simultaneous variable selection and estimation in generalized linear models (GLMs) with an explicit predictive motivation. The procedure estimates the coefficients by minimizing the Kullback-Leibler divergence of a set of predictive distributions to the corresponding predictive distributions for the full model, subject to an l ₁ constraint on the coefficient vector. This results in selection of a parsimonious model with similar predictive performance to the full model. Thanks to its similar form to the original Lasso problem for GLMs, our procedure can benefit from available l ₁-regularization path algorithms. Simulation studies and real data examples confirm the efficiency of our method in terms of predictive performance on future observations.     

On variable selection in generalized linear and related regression models

This paper is concerned with selection of explanatory variables in generalized linear models (GLM). The class of GLM's is quite large and contains e.g. the ordinary linear regression, the binary logistic regression, the probit model and Poisson regression with linear or log-linear parameter structure. We show that, through an approximation of the log likelihood and a certain data transformation, the variable selection problem in a GLM can be converted into variable selection in an ordinary (unweighted) linear regression model. As a consequence no specific computer software for variable selection in GLM's is needed. Instead, some suitable variable selection program for linear regression can be used. We also present a simulation study which shows that the log likelihood approximation is very good in many practical situations. Finally, we mention briefly possible extensions to regression models outside the class of GLM's.     

Clustering gene expression time course data using mixtures of multivariate t-distributions

Clustering gene expression time course data is an important problem in bioinformatics because understanding which genes behave similarly can lead to the discovery of important biological information. Statistically, the problem of clustering time course data is a special case of the more general problem of clustering longitudinal data. In this paper, a very general and flexible model-based technique is used to cluster longitudinal data. Mixtures of multivariate t-distributions are utilized, with a linear model for the mean and a modified Cholesky-decomposed covariance structure. Constraints are placed upon the covariance structure, leading to a novel family of mixture models, including parsimonious models. In addition to model-based clustering, these models are also used for model-based classification, i.e., semi-supervised clustering. Parameters, including the component degrees of freedom, are estimated using an expectation-maximization algorithm and two different approaches to model selection are considered. The models are applied to simulated data to illustrate their efficacy; this includes a comparison with their Gaussian analogues—the use of these Gaussian analogues with a linear model for the mean is novel in itself. Our family of multivariate t mixture models is then applied to two real gene expression time course data sets and the results are discussed. We conclude with a summary, suggestions for future work, and a discussion about constraining the degrees of freedom parameter.     

Generalized additive mixed models

Following the extension from linear mixed models to additive mixed models, extension from generalized linear mixed models to generalized additive mixed models is made, Algorithms are developed to compute the MLE's of the nonlinear effects and the covariance structures based on the penalized marginal likelihood. Convergence of the algorithms and selection of the smooth param¬eters are discussed.     

Optimal designs for discriminating between some extensions of the Michaelis–Menten model

In this paper some results on the computation of optimal designs for discriminating between nonlinear models are provided. In particular, some typical deviations of the Michaelis–Menten model are considered. A common deviation of this pharmacokinetic model consists on adding a linear term. If two linear models differ in one parameter the T-optimal design for discriminating between them is c-optimal for estimating the added linear term. This is not the case for nonlinear models.     

Inferences from biased samples with a memory effect

Biased sampling occurs often in observational studies. With one biased sample, the problem of nonparametrically estimating both a target density function and a selection bias function is unidentifiable. This paper studies the nonparametric estimation problem when there are two biased samples that have some overlapping observations (i.e. recaptures) from a finite population. Since an intelligent subject sampled previously may experience a memory effect if sampled again, two general 2-stage models that incorporate both a selection bias and a possible memory effect are proposed. Nonparametric estimators of the target density, selection bias, and memory functions, as well as the population size are developed. Asymptotic properties of these estimators are studied and confidence bands for the selection function and memory function are provided. Our procedures are compared with those ignoring the memory effect or the selection bias in finite sample situations. A nonparametric model selection procedure is also given for choosing a model from the two 2-stage models and a mixture of these two models. Our procedures work well with or without a memory effect, and with or without a selection bias. The paper concludes with an application to a real survey data set.     

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.     

Model selection consistency of U-statistics with convex loss and weighted lasso penalty

In the paper we consider minimisation of U-statistics with the weighted Lasso penalty and investigate their asymptotic properties in model selection and estimation. We prove that the use of appropriate weights in the penalty leads to the procedure that behaves like the oracle that knows the true model in advance, i.e. it is model selection consistent and estimates nonzero parameters with the standard rate. For the unweighted Lasso penalty, we obtain sufficient and necessary conditions for model selection consistency of estimators. The obtained results strongly based on the convexity of the loss function that is the main assumption of the paper. Our theorems can be applied to the ranking problem as well as generalised regression models. Thus, using U-statistics we can study more complex models (better describing real problems) than usually investigated linear or generalised linear models.     

Stochastic correlation coefficient ensembles for variable selection

In this paper, we propose a novel Max-Relevance and Min-Common-Redundancy criterion for variable selection in linear models. Considering that the ensemble approach for variable selection has been proven to be quite effective in linear regression models, we construct a variable selection ensemble (VSE) by combining the presented stochastic correlation coefficient algorithm with a stochastic stepwise algorithm. We conduct extensive experimental comparison of our algorithm and other methods using two simulation studies and four real-life data sets. The results confirm that the proposed VSE leads to promising improvement on variable selection and regression accuracy.     

Nonlinear GCV and quasi-GCV for shrinkage models

The generalized cross-validation (GCV) method has been a popular technique for the selection of tuning parameters for smoothing and penalty, and has been a standard tool to select tuning parameters for shrinkage models in recent works. Its computational ease and robustness compared to the cross-validation method makes it competitive for model selection as well. It is well known that the GCV method performs well for linear estimators, which are linear functions of the response variable, such as ridge estimator. However, it may not perform well for nonlinear estimators since the GCV emphasizes linear characteristics by taking the trace of the projection matrix. This paper aims to explore the GCV for nonlinear estimators and to further extend the results to correlated data in longitudinal studies. We expect that the nonlinear GCV and quasi-GCV developed in this paper will provide similar tools for the selection of tuning parameters in linear penalty models and penalized GEE models.     

Improved Liu estimator in a linear regression model

In this paper, we mainly aim to introduce the notion of improved Liu estimator (ILE) in the linear regression model y=Xβ+e. The selection of the biasing parameters is investigated under the PRESS criterion and the optimal selection is successfully derived. We make a simulation study to show the performance of ILE compared to the ordinary least squares estimator and the Liu estimator. Finally, the main results are applied to the Hald data.     

Likelihood-based approaches for multivariate linear models under inequality constraints for incomplete data

In this paper, we consider a multivariate linear model with complete/incomplete data, where the regression coefficients are subject to a set of linear inequality restrictions. We first develop an expectation/conditional maximization (ECM) algorithm for calculating restricted maximum likelihood estimates of parameters of interest. We then establish the corresponding convergence properties for the proposed ECM algorithm. Applications to growth curve models and linear mixed models are presented. Confidence interval construction via the double-bootstrap method is provided. Some simulation studies are performed and a real example is used to illustrate the proposed methods.     

Design optimality in multi-factor generalized linear models in the presence of an unrestricted quantitative factor

In this paper we show that product type designs are optimal in partially heteroscedastic multi-factor linear models. This result is applied to obtain locally D-optimal designs in multi-factor generalized linear models by means of a canonical transformation. As a consequence we can construct optimal designs for direct logistic response as well as for Bradley–Terry type paired comparison experiments.     

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.     

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.     

Estimation and variable selection for generalised partially linear single-index models

In this paper, we study the problem of estimation and variable selection for generalised partially linear single-index models based on quasi-likelihood, extending existing studies on variable selection for partially linear single-index models to binary and count responses. To take into account the unit norm constraint of the index parameter, we use the ‘delete-one-component’ approach. The asymptotic normality of the estimates is demonstrated. Furthermore, the smoothly clipped absolute deviation penalty is added for variable selection of parameters both in the nonparametric part and the parametric part, and the oracle property of the variable selection procedure is shown. Finally, some simulation studies are carried out to illustrate the finite sample performance.     

An alternate version of the conceptual predictive statistic based on a symmetrized discrepancy measure

The conceptual predictive statistic, C_p, is a widely used criterion for model selection in linear regression. C_p serves as an estimator of a discrepancy, a measure that reflects the disparity between the generating model and a fitted candidate model. This discrepancy, based on scaled squared error loss, is asymmetric: an alternate measure is obtained by reversing the roles of the two models in the definition of the measure. We propose a variant of the C_p statistic based on estimating a symmetrized version of the discrepancy targeted by C_p. We claim that the resulting criterion provides better protection against overfitting than C_p, since the symmetric discrepancy is more sensitive towards detecting overspecification than its asymmetric counterpart. We illustrate our claim by presenting simulation results. Finally, we demonstrate the practical utility of the new criterion by discussing a modeling application based on data collected in a cardiac rehabilitation program at University of Iowa Hospitals and Clinics.     

P-splines regression smoothing and difference type of penalty

P-splines regression provides a flexible smoothing tool. In this paper we consider difference type penalties in a context of nonparametric generalized linear models, and investigate the impact of the order of the differencing operator. Minimizing Akaike’s information criterion we search for a possible best data-driven value of the differencing order. Theoretical derivations are established for the normal model and provide insights into a possible ‘optimal’ choice of the differencing order and its interrelation with other parameters. Applications of the selection procedure to non-normal models, such as Poisson models, are given. Simulation studies investigate the performance of the selection procedure and we illustrate its use on real data examples.