Longitudinal data analysis using growth curve models.

Abstract

Growth curve analysis is useful in studies with repeated measurements on experimental units. They enable one to fit a response curve over the levels of a repeated factor, this response curve then is useful for prediction purposes. The first part of this dissertation considers a generalization of the Potthoff-Roy growth curve model. It combines two previous extensions, the sum of profiles model (Patel (1986), Verbyla (1988), Bandekar (1998)), and the exchangeably distributed errors model (Weissfield and Kshirsagar (1992)). We present a method of estimating growth curve coefficients of such model, using the method of least squares, as in multivariate regression. This also becomes identical to the method of restricted maximum likelihood. We provide explicit expressions for these estimates and their standard errors, unlike Patel (1986) and Verbyla (1988). Some hypothesis testing problems are considered for these coefficients. We reduce the model with exchangeably distributed error to two independent models, by orthogonal transformation and augmentation of the design matrix. One model takes into account group contrasts and the other a kind of average group effect. The second part of this dissertation is concerned with the use of growth curve models for classification purposes. We have developed allocation rules for data that have structured means as well as structured covariance matrices, we provide an estimate for the chance of misclassification. When the means of the different groups have a Potthoff-Roy type structure, certain linear combinations of the feature variables have zero means. These are not useful for classification and should be removed to reduce the chance of misclassification. We derive allocation rules based on Rao's (1973) scores for two different scenarios: (1) When the variance-covariance matrix is known and assumed to be compound symmetric, and no model-based covariates are included in the allocation rule, and (2) when the covariance structure is unknown. For this second case, we have studied the importance of including or excluding model-based covariates in the allocation rule, and their influence on the misclassification rate. An important contribution of this dissertation is to provide the chance of misclassification using these new allocation rules.