Simultaneous variable selection and estimation in semiparametric modeling of longitudinal/clustered data

Abstract

We consider the problem of simultaneous variable selection and estimation in additive, partially linear models for longitudinal/clustered data. We propose an estimation procedure via polynomial splines to estimate the nonparametric components and apply proper penalty functions to achieve sparsity in the linear part. Under reasonable conditions, we obtain the asymptotic normality of the estimators for the linear components and the consistency of the estimators for the nonparametric components. We further demonstrate that, with proper choice of the regularization parameter, the penalized estimators of the non-zero coefficients achieve the asymptotic oracle property. The finite sample behavior of the penalized estimators is evaluated with simulation studies and illustrated by a longitudinal CD4 cell count data set.kind of data is given bywhere β is a d 1 -dimensional regression parameter, and η l , l = 1, . . . , d 2 , are unknown but smooth functions. We assume ε i = (ε i1 , . . . , ε imi ) T ∼ N (0, Σ i ). For identifiability, both the parametric and nonparametric components must be centered, that is, (1) is simplified to be the partially linear model (PLM) in Lin and Carroll [20]. Model (1) retains the merits of additive models, while it is more flexible than purely additive models by allowing a subset of the covariates to be discrete and/or unbounded. When m i s and Σ i s are the same for all individuals, Carroll et al.[3] considered the efficient estimation of β in model (1) using local linear smooth backfitting. In this paper we consider a more general scenario that both m i and Σ i may vary across subjects or experimental units to allow irregular measurements for individuals. Our goal is to simultaneously select significant variables and efficiently estimate the unknown components for model (1). This is challenging due to the issue of "curse of dimensionality" and the additional complexity of the correlation structures (Wang [34]) introduced by repeated measurements.To alleviate the effect of the "curse of dimensionality," more parsimonious models become desirable in practice; see Fan [10], Hall, Müller and Wang [11] and Wang et al. [32]. Variable selection is fundamental to high-dimensional statistical modeling. In the absence of prior knowledge, a large number of variables may be included at the initial stage of modeling in order to reduce possible model bias. This may lead to a complicated model including many insignificant variables, resulting in less predictive powers and difficulty in interpretation. There is an extensive literature on variable selection via various approaches, for example, the classical information criteria such as the Akaike information criterion (AIC) and Bayesian information criterion (BIC) in Yang [40], the least absolute shrinkage and selection operator (LASSO) proposed in Tibshirani [30,31], the non-negative garrote in Yuan and Liu [41], the difference convex algorithm in Wu and Liu [36], the combination of L 0 and L 1 penalties in Liu and Wu [22], and the nonparametric independe...

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

“…Hence, in our numerical studies, we use κ = 1 to simplify calculations. Following Ma et al (2013), we use BIC criterion to choose λ by minimizing the following function…”

“…The average of bias, SD, SE and P 0.95 for estimating β 1 and β 2 and AD E l for estimating g l (·), l = 1, 2 are reported in Tables 1, 2 and 3 under different settings. Meanwhile, we compare the performance of our method (proposed) with other existing methods, for example, the standard quantile regression method with working independence (qr-indep) and the GEE method (Ma et al 2013) with a specific correlation structure such as independence (gee-indep), exchangeable (gee-exch) and AR(1) (gee-ar1).…”

“…Our goal is to compare the proposed approach (RSGEE) with existing variable selection method, such as the SGEE approach which is similar to Ma et al (2013). Note that Ma et al (2013) only considered variable selection for the linear part with the SCAD penalty.…”

“…The semiparametric model (1) containing both linear components and nonparametric functions makes variable selection quite challenging. Various penalty functions have been used for performing variable selection in model (1) Lian 2012;Ma et al 2013). The most common penalty functions include the Lasso (Tibshirani 1996), the SCAD (Fan and Li 2001), the adaptive Lasso (Zou 2006), and so on.…”

A Semiparametric Regression Model for Longitudinal Data with Non‐stationary Errors

“…A recent line of research for variable selection has also undergone rapid development, see e.g., Fan & Li (2001), Wang et al (2009, Ma et al (2013). Among these studies, a working correlation model for the variance is often assumed.…”