This dissertation investigated the optimal strategy for the model specification search in the latent growth modeling. Although developing an initial model based on the theory from prior research is favored, sometimes researchers may need to specify the starting model in the absence of theory. In this simulation study, the effectiveness of the start models in searching for the true population model was examined. The four possible start models adopted in this study were: the simplest mean and covariance structure model, the simplest mean and the most complex covariance structure model, the most complex mean and the simplest covariance structure model, and the most complex mean and covariance structure model. Six model selection criteria were used to determine the recovery of the true model: Likelihood ratio test (LRT), DeltaCFI, DeltaRMSEA, DeltaSRMR, DeltaAIC, and DeltaBIC. The results showed that specifying the most complex covariance structure (UN) with the most complex mean structure recovered the true mean trajectory most successfully with the average hit rate above 90% using the DeltaCFI, DeltaBIC, DeltaAIC, and DeltaSRMR. In searching for the true covariance structure, LRT, DeltaCFI, DeltaAIC, and DeltaBIC performed successfully regardless of the searching method with different start models.

In a microarray experiment, it is expected that there will be correlations between the expression levels of different genes under study. These correlation structures are of great interest from both biological and statistical points of view. From a biological perspective, the identification of correlation structures can lead to an understanding of genetic pathways involving several genes, while the statistical interest, and the emphasis of this thesis, lies in the development of statistical methods to identify such structures. However, the data arising from microarray studies is typically very high-dimensional, with an order of magnitude more genes being considered than there are samples of each gene. This leads to difficulties in the estimation of the dependence structure of all genes under study. Graphical models and Bayesian networks are often used in these situations, providing flexible frameworks in which dependence structures for high-dimensional data sets can be considered. The current methods for the estimation of dependence structures for high-dimensional data sets typically assume the presence of independent and identically distributed samples of gene expression values. However, often the data available will have a complex mean structure and additional components of variance. Given such data, the application of methods that assume independent and identically distributed samples may result in incorrect biological conclusions being drawn. In this thesis, methods for the estimation of Bayesian networks for gene expression data sets that contain additional complexities are developed and implemented. The focus is on the development of score metrics that take account of these complexities for use in conjunction with score-based methods for the estimation of Bayesian networks, in particular the High-dimensional Bayesian Covariance Selection algorithm. The necessary theory relating to Gaussian graphical models and Bayesian networks is reviewed, as are the methods currently available for the estimation of dependence structures for high-dimensional data sets consisting of independent and identically distributed samples. Score metrics for the estimation of Bayesian networks when data sets are not independent and identically distributed are then developed and explored, and the utility and necessity of these metrics is demonstrated. Finally, the developed metrics are applied to a data set consisting of samples of grape genes taken from several different vineyards. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2010