Transformation and selection of covariates using generalized estimating equations

The selection of a suitable model from a large class of possible submodels is an important problem in applied statistics. In the regression setting, the choices often involve the two questions: which covariates should be included in the model and if they are included, what form they should take? This dissertation attempts to deal with these two practical issues when the responses are cluster-correlated and marginally distributed according to the generalized linear models (GLMs) of McCulloch and Nelder (1989). Specifically, the situation addressed herein is the one where the effect of the covariates on the marginal responses is of primary interest and the cluster-correlation is a nuisance characteristic of the data. In other words, the situation in which the generalized estimating equations (GEEs) of Liang and Zeger (1986) are usually applied. While GEEs have become commonly used in the past 15 years or so, few model selection techniques have been extended to this setting to date. The first part of this thesis proposes an iterative technique for the estimation of the parameters of covariate transformations when the form of the transformations is known. The fractional polynomial transformation of Royston and Altman (1994) is emphasized, though it is applicable to more general situations as well. A hypothesis test is proposed for testing between two nested covariate transformation models. The question of covariate subset selection is addressed in the second part of this thesis. An extension of the Mallows' C_p procedure (Mallows, 1973) is proposed for the GEE setting. Its properties are shown to be similar to that of the usual Mallows' C_p procedure in the classical linear regression setting. A generalization and a small-sample adjustment to this extended Mallows' C_p procedure are presented. Also considered is selection of the working correlation to be used in the GEES for regression parameter estimation. The thesis concludes with an analysis of two real datasets applying the covariate transformation and selection methods.