Variable Selection Procedures for Generalized Linear Mixed Models in Longitudinal Data Analysis

Abstract

Model selection is important for longitudinal data analysis. But up to date little work has been done on variable selection for generalized linear mixed models (GLMM). In this paper we propose and study a class of variable selection methods. Full likelihood (FL) approach is proposed for simultaneous model selection and parameter estimation. Due to the intensive computation involved in FL approach, Penalized Quasi-Likelihood (PQL) procedure is developed so that model selection in GLMMs can proceed in the framework of linear mixed models. Since the PQL approach will produce biased parameter estimates for sparse binary longitudinal data, Two-stage Penalized Quasi-Likelihood approach (TPQL) is proposed to bias correct PQL in terms of estimation: use PQL to do model selection at the first stage and existing software to do parameter estimation at the second stage. Marginal approach for some special types of data is also developed. A robust estimator of standard error for the fitted parameters is derived based on a sandwich formula. A bias correction is proposed to improve the estimation accuracy of PQL for binary data.

The sampling performance of four proposed procedures is evaluated through extensive simulations and their application to real data analysis. In terms of model selection, all of them perform closely. As for parameter estimation, FL, AML and TPQL yield similar results. Compared with FL, the other procedures greatly reduce computational load.

The proposed procedures can be extended to longitudinal data analysis involving missing data, and the shrinkage penalty based approach allows them to work even when the number of observations n is less than the number of parameters d.