Abstract

In biomedical research, one major objective is to identify risk factors and study their risk impacts, as this identification can help clinicians to both properly make a decision and increase efficiency of treatments and resource allocation. A two-step penalized-based procedure is proposed to select linear regression coefficients for linear components and to identify significant nonparametric varying-coefficient functions for semiparametric varying-coefficient partially linear Cox models. It is shown that the penalized-based resulting estimators of the linear regression coefficients are asymptotically normal and have oracle properties, and the resulting estimators of the varying-coefficient functions have optimal convergence rates. A simulation study and an empirical example are presented for illustration.

1. Introduction

To balance model flexibility and specificity, a variety of semiparametric models have been proposed in survival analysis. For example, Huang [1] studied partially linear Cox models and Cai et al. [2] and Fan et al. [3] studied the Cox proportional hazard model with varying coefficients. Tian et al. [4] proposed an estimation procedure for the Cox model with time-varying coefficients. Perhaps the most appropriate model is the semiparametric varying-coefficient model; its merit includes easy interpretation, flexible structure, potential interaction between covariates, and dimension reduction of nonparametric components models. Recently, a lot of effort has been made in this direction. Cai et al. [2] considered the Cox proportional hazard model with a semiparametric varying-coefficient structure. Yin et al. [5] proposed the semiparametric additive hazard model with varying coefficients.

In biomedical research and clinical trials, one major objective is to identify risk factors and study their risk impacts, as this identification can help clinicians to properly make a decision and increase efficiency of treatment and resource allocation. This is essentially a variable selection procedure in spirit. When the outcomes are censored, selection of significant risk factors becomes more complicated and challenging than with the case where all outcomes are complete. However, important progress has been made in recent literature. For instance, Fan and Li [6] extended the nonconcave penalized likelihood approach proposed by Fan and Li [7] to the Cox proportional hazards model and the Cox proportional hazards frailty model. Johnson et al. [8, 9] proposed procedures for selecting variables in semiparametric linear regression models for censored data. However, these papers did not consider variable selection with functional coefficients; while Wang et al. [10] extended the application of penalized likelihood to the varying-coefficient setting and studied the asymptotic distributions of the estimators, they only focused on the SCAD penalty for completely observed data. Du et al. [11] proposed a penalized variable selection procedure for Cox models with a semiparametric relative risk.

In this paper, we study variable selection for both regression parameters and varying coefficients in the semiparametric varying-coefficient additive hazards model: where is a vector of covariates with a linear effect on the logarithm of the hazard function , is the main exposure variable of interest whose effect on the logarithm of the hazard might be nonlinear, and is a vector of covariates that may interact with the exposure covariate . is the baseline hazard ratio function and is an unspecified smooth function. is a -vector of constant parameters. Let to assure that the model is identifiable. This model covers generally used basic semiparametric model settings. For instance, if and , model (1) reduces to Lin and Ying’s additive hazard model [12] and Aalen’s additive hazard model when the baseline hazard function also equals to zero [13, 14]. Furthermore, when the exposure covariate is time , then model (1) reduces to the partly parametric additive hazard model if [15, 16] and reduces to the time-varying coefficient additive hazard model [17].

To select significant elements of and coefficient functions , we require an estimation procedure for the regression parameters and . However, this poses a challenge because it is impossible to simultaneously get the root- consistent penalized estimators for the nonzero components of by penalizing and . To achieve this goal, we propose the following two-step estimation procedure for selecting the relevant variables. In Step 1, we estimate by using the profile penalized least square function after locally approximating the nonparametric functions and . In Step 2, given the penalized estimator of in Step 1, we develop a penalized least square function for by using the basis expansion. We will demonstrate that the proposed estimation procedures have oracle properties, and all penalized estimators of nonzero components can achieve their optimal convergence rate.

This paper is organized as follows. Section 2 introduces the SCAD-based variable selection procedure for the parametric components and establishes the asymptotic normality for the resulting penalized estimators. Section 3 discusses a variable selection procedure for the coefficient functions by approximating these functions by using the B-spline approach. The simulation studies and an application of the proposed methods in a real data example are included in Sections 4 and 5, respectively. The technical lemmas and proofs of the main results are given in the appendix.

2. Penalized Least Squares Based Variable Selection for Parametric Components

Let denote the potential failure time, let denote the potential censoring time, and let denote the observed time for the th individual, . Let be an indicator which equals 1 if is a failure time and 0 otherwise. Let represent the failure, censoring, and covariate information up to time for the th individual. The observed data structure is . Assume that and are conditionally independent given covariates and that the observation period is , where is a constant denoting the time for the end of the study.

Let denote the counting process corresponding to and let . Let the filtration be the history up to time ; that is, , . Write . Then is a martingale with respect to . For ease of presentation, we drop the dependence of covariates on time. The methods and proofs in this paper are applicable to external time-dependent covariates [18].

Here we use the techniques of local linear fitting [19]. Suppose that and are smooth enough to allow Taylor’s expansion as follows: for each which is the support of and for in a neighborhood of , Then we can approximate the hazard ratio function given in (1) by where , , and .

Let be a diagonal matrix with the first diagonal elements 1 and the rest . If is given, then by using the counting-process notation, similarly to [5], we can obtain an estimator of at each by solving the estimating equations whose objective function is given as follows: where with being a kernel function and being a bandwidth, and . To avoid the technicality of tail problems, only data up to a finite time point are frequently used.

Denote the solution of by , which can be expressed as follows: Here and below, for any vector and for .

As a consequence, we can estimate and presuming that is known:

2.1. Variable Selection

Notice that if is given, then from (7) we can estimate by Given the observed data and based on (7) and (8), , , and the cumulative function can be approximately expressed as functions of . These expressions inspire us to estimate by minimizing the following objective function subject to : Accordingly, our penalized objective function is defined as follows: Thus, minimizing with respect to results in a penalized least squares estimator of .

Let and . Now we establish the oracle properties for the resulting estimators. Let be the true value of , the first subvector of length , containing all nonzero elements of and .

We introduce an additional notation as follows. Let . For , let ,   ,  . Denote where is the density of . Denote   and   , where For , , let , and ,   for any vector .

Theorem 1. If Assumptions (A.i)–(A.vi) are satisfied and as , then there exist local minimizers of such that .

Remark 2. Theorem 1 indicates that by choosing a proper , which leads to , there exists a root- consistent penalized least squares estimator of .

Theorem 3. Assume that Under Assumptions (A.i)–(A.vi), if and as , then the local root-n consistent local minimizer in Theorem 1 with probability tending to 1 satisfies(1); and(2)asymptotic normality where and are the first submatrix of n and , respectively, and

3. Variable Selection for the Varying Coefficients

When some variables of are not relevant in the additive hazard regression model, the corresponding functional coefficients are zero. In this section, by using basis function expansion techniques, we try to estimate those zero functional coefficients as identically zero via nonconcave penalty functions.

For a simplified expression, we denote as and . Hence, we can rewrite the additive hazard regression model (1) as where and . Assume is a given root- consistent estimator of . From the arguments in Section 2, such an is available.

3.1. Penalized Function

We approximate each element of by the basis expansion; that is, for . This approximation indicates that the hazard function given in (16) can be approximated as follows: Let and . Write and . Then, by using similar arguments as in Section 2, we estimate the basic hazard function by and estimate by minimizing the following least square function with respect to : Suppose that ,  , as mentioned before. We are trying to correctly identify these zero functional coefficients via nonconcave penalty functions, that is, via minimizing the following penalized least square function with respect to : where and with .

Let be the minimizer of penalized least square function (21) and define .

Remark 4. Various basis systems, including Fourier basis, polynomial basis, and B-spline basis, can be used in the basis expansion. We focus on the B-spline basis and examine asymptotic properties of .

3.2. Asymptotic Properties

Define and let be the minimizer of in (20). Hence, we can obtain by resolving is the derivative of with respect to , the usual score function of without the penalty.

For any square integrable functions and on , let denote the norm of on , and define as the distance between and . Let denote all functions that have the form ,   for B-spline base. and . Denote .

Let and .

Theorem 5. Suppose that (13) and Assumptions (A.i) and (A.vii)–(A.iv) hold. If and as , then one has(a),  , with probability approaching ; and(b),  .

Remark 6. Let be a space of splines with a degree no less than 1 and with equally spaced interior knots, where ,  . Note that [20, Theorem 6.21]. Hence, if , Theorem 5 implies that ,  .

Remark 7. Suppose that ,  . Because , which implies that for the hard thresholding and SCAD penalty function, the convergent rate of penalized least square estimator is . This is the optimal rate for nonparametric regression [21] if . However, for the penalty, ; hence is required to lead to the optimal rate. On the other hand, the oracle property in part (b) of Theorem 5 requires , which contradicts . As a consequence, for the penalty, the oracle property does not hold.

Next, we demonstrate the asymptotic normality of . First, we analyze , that is, the penalized estimator of . Let , , and denote the selected columns of , , and , respectively, corresponding to . Similarly, let denote selected diagonal blocks of ,  . Define and . Then by part (a) of Theorem 5 and (21), is the local solution of Recall that is root- consistent. It follows from (23) that