We propose analyzing our data with a model that exhibits errors-in-variables (EIV) in auxiliary information and which has an autoregressive covariance structure using a Bayesian methodology. The incorporation of these components into a model is often necessary and realistic in the study of many statistical problems. However, such an analysis usually mandates many simplifying and restrictive assumptions, especially when using a traditional probabilistic approach. Much research has been accumulated in this area. We will show that by taking a Bayesian approach, we can effectively deal with the complexity of these types of models. Such an approach cannot be found in the literature.
In fitting statistical models for the analysis of growth data, many curves and/or models have been proposed. We have collected an exhaustive list of the most important and frequently used growth curves, some of which are used in our model analysis. A motivating example is presented to show the applicability of our general approach. In addition, auxiliary covariates, both qualitative and quantitative, can be added into our model as an extension. These EIV growth curves with auxiliary covariates provide a very general framework for practical application.
We give several illustrative examples demonstrating how a Bayesian approach using MCMC (Metropolis Hastings in Gibbs sampler) techniques and goodness of fit statistics for model selections can be utilized in our analysis. Highest Density Regions (HDR's) are also used to facilitate Bayesian estimation and inference in these examples. Multivariate growth curve models are presented and detailed to study the relationship of the variables in models.
In the final chapter, we present growth curve models with auxiliary variables containing uncertain data distributions (i.e. mixtures of standard components, such as normal distributions) using Dirichlet Process Priors (DPP, which are composed of discrete and continuous components). We show that DPPs are appropriate in determining the number of components and in estimating the parameters simultaneously, and are especially useful and advantageous in the aforementioned multimodal scenario with respect to the goodness of fit of the model.
