Abstract

Semiparametric generalized varying coefficient partially linear models with longitudinal data arise in contemporary biology, medicine, and life science. In this paper, we consider a variable selection procedure based on the combination of the basis function approximations and quadratic inference functions with SCAD penalty. The proposed procedure simultaneously selects significant variables in the parametric components and the nonparametric components. With appropriate selection of the tuning parameters, we establish the consistency, sparsity, and asymptotic normality of the resulting estimators. The finite sample performance of the proposed methods is evaluated through extensive simulation studies and a real data analysis.

1. Introduction

Identifying the significant variables is of great significance in all regression analysis. In practice, a number of variables are available for an initial analysis, but many of them may not be significant and should be excluded from the final model in order to increase the accuracy of prediction. Various procedures and criteria, such as stepwise selection and subset selection with Akaike information criterion (AIC), Mallows Cp, and Bayesian information criterion (BIC), have been developed. Nevertheless, these selection methods suffer from expensive computational costs. Many shrinkage methods have been developed for the purpose of computational efficiency, e.g., the nonnegative garrote [1], the LASSO [2], the bridge regression [3], the SCAD [4], and the one-step sparse estimator [5]. Among those, the SCAD possesses the virtues of continuity, unbiasedness, and sparsity. There are a number of works on the SCAD estimation methods in various regression models, e.g., [6–9]. Zhao and Xue [8] proposed a variable selection method to select significant variables in the parametric components and the nonparametric components simultaneously for the varying coefficient partially linear models (VCPLMs).

On the other hand, longitudinal data occurs frequently in biology, medicine, and life science, in which it is often necessary to make repeated measurements of subjects over time. The responses from different subjects are independent, but the responses from the same subject are very likely to be correlated. This feature is called “within-cluster correlation”. Qu et al. [10] proposed a method of quadratic inference functions (QIFs) to treat the longitudinal data. The QIF can efficiently take the within-cluster correlation into account and is more efficient than the generalized estimating equation (GEE) [11] approach when the working correlation is misspecified. The QIF approach has been applied to many models, including varying coefficient models (VCM) [12, 13], partially linear models (PLM) [14], varying coefficient partially linear models (VCPLMs) [15], and generalized partially linear models (GPLM) [16]. Wang et al. [13] proposed a group SCAD procedure for variable selection of VCM with longitudinal data. More recently, Tian et al. [15] proposed a QIF-based SCAD penalty for the variable selection for VCPLM with longitudinal data.

As introduced in Li and Liang [17], the generalized partially linear varying coefficient model (GPLVCM) possesses the great flexibility of a nonparametric regression model and provides the explanatory power of a generalized linear regression model, which arises naturally due to categorical covariates. Many models are the special case of GPLVCM, e.g., VCM, VCPLM, PLM, and GLM. Li and Liang [17] studied variable selection for GPLVCM, where the parametric components are identified via the SCAD but the nonparametric components are selected via a generalized likelihood ratio test instead of shrinkage. In this paper, we extend the QIF-based group SCAD variable selection procedure to GPLVCM with longitudinal data, and the B-spline methods are adopted to approximate the nonparametric component in the model. With suitable chosen tuning parameters, the proposed variable selection procedure is consistent, and the estimators of regression coefficients have oracle property, i.e., the estimators of the nonparametric components achieve the optimal convergence rate, and the estimators of the parametric components have the same asymptotic distribution as that based on the correct submodel.

The rest of this paper is organized as follows. In Section 2, we propose a variable selection procedure for the GPLVCM with longitudinal data. Asymptotic properties of the resulting estimators and an iteration algorithm are presented in Section 3. In Section 4, we carry out simulation studies to assess the finite sample performance of the method. A real data analysis is given in Section 5 to illustrate the proposed methodology. The details of proofs are provided in the appendix.

2. Methodology

2.1. GPLVCM with Longitudinal Data

In this article, we consider a longitudinal study with subjects and observations over time for the th subject () for a total of observations. Each observation consists of a response variable and the predicator variables , where and is a scalar. We assume that the observations from different subjects are independent, but those within the same subject are dependent. The generalized varying coefficient partially linear model (GPLVM) with longitudinal data takes the form where is the expectation of when , , and are given, is an unknown regression coefficient vector, is a known smooth link function, and is a unknown monotonic smooth function vector. Without loss of generality, we assume .

We approximate by B-spline basis functions with the order of , where and is the number of interior knots, i.e., where is a vector of unknown regression coefficients. Accordingly, is approximated by where and “” is the Kronecker product. We use the B-spline basis functions because they are numerically stable and have bounded support [18]. The spline approach also treats a nonparametric function as a linear function with the basis functions as pseudodesign variables, and thus, any computational algorithm for the generalized linear models can be used for the GPLVCMs.

To incorporate the within-cluster correlation, we apply the QIFs to estimate and , respectively. Denote , we define the extended score as follows: where , is the marginal variance matrix of subject , and are the base matrices to represent the inverse of the working correlation matrix in GEE approach. Following Qu et al. [10], we define the quadratic inference functions to be where . Note that depends on . The QIF estimate is then given by

2.2. Penalized QIF

In real data analysis, the true regression model is always unknown. An overfitted model lowers the efficiency of estimation while an underfitted one leads to a biased estimator. A popular approach to identify the relevant predictors while estimating the nonzero parameters and functions in model (1) simultaneously is to exert some kind of “penalty” on the original objective function. Here, we choose the smoothly clipped absolute deviation (SCAD) penalty because it has several advantages such as unbiasedness, sparsity, and continuity. The SCAD-penalized quadratic inference function (PQIF) is defined as follows: where and is the SCAD penalty function, where the derivative is defined as where ; here, we choose as in [4].

Note that This group-wised penalization ensures that the spline coefficient vector of the same nonparametric component is treated as an entire group in model selection.

Denote to be the penalized estimator obtained by minimizing the penalized objective function of (7). Then, is the estimator of the parameter and the estimator of the nonparametric function is calculated by , where .

3. Asymptotic Properties

3.1. Oracle Property

We next establish the asymptotic properties of the resulting penalized QIF estimators. We first introduce some notations. Let and denote the true values of and . In addition, is the spline coefficient vector from the spline approximation to . Without loss of generality, we assume that and , i.e., only the first component of is nonzero. Similarly, we assume that and , i.e., only the first component of is nonzero. For convenience and simplicity, let denote a positive constant that may have different values at each appearance throughout this paper and denote the modulus of the largest singular value of matrix or vector . Before the proof of our main theorems, we list some regularity conditions used in this paper.

Assumption (A1). The spline regression parameter is identifiable, that is, is the spline coefficient vector from the spline approximation to . In addition, there is a unique satisfying , where is the parameter space.

Assumption (A2). The weight matrix converges almost surely to a constant matrix , where is invertible.

Assumption (A3). The covariate matrices and , , satisfy and .

Assumption (A4). The error satisfies , and there exists a positive constant such that .

Assumption (A5). All marginal variances and .

Assumption (A6). is a bounded sequence of positive integers.

Assumption (A7). is th continuous differentiable on , where .

Assumption (A8). The inner knots satisfy where .

Assumption (A9). The link function is th continuous differentiable and for some .

Assumption (A10). as , where

Theorem 1 indicates that the estimator of nonparametric components achieve the optimal convergence rate.

Theorem 1. Assume that Assumptions (A.1)–(A.10) hold and the number of knots , then Furthermore, under suitable condition, Theorem 1 shows that the penalized QIF estimator has the sparsity property.

Theorem 2. Assume that the conditions in Theorem 1 hold and as , with probability approaching 1, where .
Theorems 1 and 2 indicate that with the tune parameter being suitably chosen, the proposed selection method possesses model selection consistency. Next, we establish the asymptotic property for the estimator of the nonzero parametric components. Let and let and denote their true value, respectively. In addition, let and denote the spline coefficient vector of and , respectively, and let and denote their correspondent covariate. Let , and where , is a block matrix with its block taking the form Theorem 3 states that is asymptotically normally distributed.

Theorem 3. Suppose that Assumptions (A.1)–(A.9) hold and the number of knots , then where and represents the convergence in distribution.

3.2. Selection of Tuning Parameters

Theorems 1–3 imply that the proposed variable selection procedure possessed the oracle property. However, this attractive feature relies on the choice of tuning parameters . The popular criteria to choose include cross-validation, generalized cross-validation, AIC, and BIC. Wang et al. [19] suggested using BIC for the SCAD estimator in linear models and partially linear models and proved its model selection consistency property, i.e., the optimal parameter chosen by BIC can identify the true model with probability tending to one. Tian proved that for partially linear models. Hence, we adopt BIC to choose the optimal . Following [19–21], we simplify the tuning parameters as where and are the unpenalized QIF estimates. Consequently, the original two-dimensional problem becomes a univariate problem about , which can be selected according to the following BIC-type criterion: where is the regression coefficient estimated by minimizing the penalized QIF in (2.8) for a given and is the number of nonzero coefficients of and . Thus, the tuning parameter is obtained by

From Theorem 4 of Tian et al. [15], the BIC tuning parameter selector enables us to select the true model consistently.

3.3. An Algorithm Using Local Quadratic Approximation

Based on Fan and Li’s local quadratic approximating approach [4], we propose an iterative algorithm to minimize the PQIF (7). Similar with Tian et al. [15], we choose the unpenalized QIF estimator as the initial estimator. Let be the value of at the th iteration. If (or ) is close to 0 (or ), i.e., (or ) with some small threshold value , then we set (or ). We consider in our simulations.

Suppose , for , and , for , and , where are the nonzero parametric components and . Similarly, let , where and correspond to zero functions and zero functions, respectively. Let denote a vector which has the same length and same partition with .

For the parametric term, if , the penalty function at is approximated by

Similarly, to the nonparametric component, if , the penalty function at is approximated by where is the first-order derivative of the penalty function . This leads to the local approximation of the PQIF by a quadratic function: where , , with , and

Minimizing the quadratic function (22), we obtain . The Newton-Raphson method then iterates the following process to convergence: