| Linear mixed models are commonly used models in the analysis of correlated data, in which the observed data are grouped according to one or more clustering factors. The selection of covariates, the variance structure and the correlation structure is crucial to the accuracy of both estimation and prediction in linear mixed models. Information criteria such as Akaike's information criterion, Bayesian information criterion, and the risk inflation criterion are mostly applied to select linear mixed models. Most information criteria penalize an increase in the size of a model through a fixed penalization parameter. In this dissertation, we firstly derive the generalized degrees of freedom for linear mixed models. A resampling technique, data perturbation, is employed to estimate the generalized degrees of freedom of linear mixed models. Further, based upon the generalized degrees of freedom of linear mixed models, we develop an adaptive model selection procedure with a data-adaptive model complexity penalty for selecting linear mixed models. The asymptotic optimality of the adaptive model selection procedure in linear mixed models is shown over a class of information criteria. The performance of the adaptive model selection procedure in linear mixed models is studied by numerical simulations. Simulation results show that the adaptive model selection procedure outperforms information criteria such as Akaike's information criterion and Bayesian information criterion in selecting covariates, the variance structure and the correlation structure in linear mixed models. Finally, an application to diabetic retinopathy is examined. |
