Model Detection for Additive Models with Longitudinal Data

Jian Wu,Liugen Xue

doi:10.4236/ojs.2014.410082

Abstract

In this paper, we consider the problem of variable selection and model detection in additive models with longitudinal data. Our approach is based on spline approximation for the components aided by two Smoothly Clipped Absolute Deviation (SCAD) penalty terms. It can perform model selection (finding both zero and linear components) and estimation simultaneously. With appropriate selection of the tuning parameters, we show that the proposed procedure is consistent in both variable selection and linear components selection. Besides, being theoretically justified, the proposed method is easy to understand and straightforward to implement. Extensive simulation studies as well as a real dataset are used to illustrate the performances.

Highlights

Longitudinal data arise frequently in biological and economic applications
We propose a penalized method for variable selection and model detection in model (1.1) and show that the proposed method can correctly select the nonzero components with probability approaching one as the sample size goes to infinity
We make several novel contributions: 1) We develop a new strategies for model selection and variable selection in additive model with longitudinal data; 2) We develop theoretical properties for our procedure

Summary

Introduction

Longitudinal data arise frequently in biological and economic applications. The challenge in analyzing longitudinal data is that the likelihood function is difficult to specify or formulate for non-normal responses with large cluster size. Statistical inference of additive models with longitudinal data has been considered by some authors. By extending the generalized estimating equations approach, [2] studied the estimation of additive model with longitudinal data. [3] focuses on a nonparametric additive time-varying regression model for longitudinal data. We make several novel contributions: 1) We develop a new strategies for model selection and variable selection in additive model with longitudinal data; 2) We develop theoretical properties for our procedure. We show that the procedure can select the true model with probability approaching one, and show that newly proposed method estimates the non-zero function components in the model with the same optimal mean square convergence rate as the standard spline estimators.

Additive Models with Two Fold Penalized Splines

Asymptotic Properties

Simulation Study

Real Data Analysis

Findings

Concluding Remark