Simultaneous Variable Selection and Estimation in Generalized Semiparametric Mixed Effects Modeling of Longitudinal Data

Abstract

Longitudinal data frequently arises in biological, medical and epidemiological studies, and the main characteristic of it is that repeated measurements from the same subjects are correlated over time. This chapter considers the problem of simultaneous variable selection and estimation in the generalized semiparametric mixed effects model (GSMM) for longitudinal data. The GSMM is a natural extension of the semiparametric mixed effects model where accommodates response variables that follow distributions other than the normal, presents an arbitrary nonparametric smooth function to model the complicated time trend and account for the within subject correlation using the random effects. When a large number of variables are available in the data, it is of critical importance to select the best subset of variables in order to develop an informative yet parsimonious model. The challenge in analyzing longitudinal data when responses are non-normal is the difficulty to specify the full likelihood function. A standard approach to deal with this is to use the generalized estimating equations (GEE). We propose a penalization type of GEE while using regression spline to approximation the nonparametric component. This approach apply the penalty functions such as SCAD to the estimating equation objective function in order to simultaneously estimate parameters and select the important variables. The proposed penalized estimation technique involves the specification of the posterior distribution of the random effects, which cannot be evaluated in closed form. However, it is possible to approximate this posterior distribution by producing random draws from the distribution using a Metropolis algorithm, which does not require the specification of the posterior distribution. Moreover, we discuss how to select the regularization parameters and the model selection procedure for assessing the fits of candidate models is also addressed. For practical implementation, we adopt an appropriate iterative algorithm to select the significant variables and estimate the nonzero coefficient functions. Performance of the proposed penalization technique is analyzed through a simulation study along with the analysis of HIV data.