The selection of an adequate and parsimonious model among suitable candidates is an essential aspect of the model-building process. Model selection approaches have been widely studied for the univariate linear model and other models arising from cross-sectional data. As researchers increasingly rely on linear mixed models (LMMs) to characterize longitudinal data, there is a need for improved techniques for selecting among this class of models which requires specification of both fixed and random effects via a mean model and variance-covariance structure. The model selection process for LMMs is further complicated when fixed and/or random effects are nonnested between models. Presently, information criteria such as AIC and BIC dominate model selection criteria used to compare nested and nonnested LMMs. This dissertation explores the development of a hypothesis test to compare nonnested LMMs based on extensions of the work begun by Sir David Cox. Particularly, we address the complex issue of estimating the variance of the Cox test statistics through the use of parametric bootstrapping. Various information criteria have been modified for this purpose, but recent investigations have all led to inconclusive results as to which criterion is the best to select among LMMs. We also consider the use of the Extended Information Criterion (EIC) as an improvement on the more commonly used AIC. Application to observed data demonstrates the viability of both the Cox test and the EIC to select among nonnested LMMs.