Modeling variability in generalized linear models

Abstract

This work proposes joint modeling of parameters in the biparametric exponential family, including heteroscedastic linear regression (non linear regression) models; with joint modeling of the mean and precision (the variance) parameters; beta regression models, longitudinal date analysis (including modeling of the covariance matrix) and hierarchical models. This work presents results of the classic approach to fit regression models for both mean and precision parameters in biparametric exponential family of distributions, which includes Bayesian methods for fitting the proposed models. And also extensions of the Bayesian methods to fit nonlinear regression models. Finally, proposes to use a Bayesian approach for modeling the covariance matrix in normal regression models when the observations are not independent. This document includes the following chapters: Chapter 1 is a introduction. Chapter 2 presents a summary of generalized linear models and the classical and Bayesian approaches to the parameters estimation, presenting the Fisher score method and a Bayesian approach using the Metropolis-Hastings algorithm. In Chapter 3, the heteroscedastic normal linear regression models are considered, including summaries of the classic method and Bayesian method proposed to fit these models. Chapter 4 is an extension of Chapter 3, which studies the regression models in the biparametric exponential family of distribution for mean and precision parameters. The following examples are included. 1. Gamma regression models with regression structures in the mean and precision (variance). 2. Beta regression models with regression structures in both mean and dispersion parameter. Several simulation studies were performed to illustrate these models and the proposed Bayesian methods. Chapter 5 discusses normal nonlinear heteroskedastic regression models. Chapter 6 include a Bayesian proposal to fit longitudinal regression models, where regression structures are assumed for the mean and the variance-covariance matrix of observations with Normal distribution (longitudinal data) Chapter 7 presents an extension of the methodology proposed in the previous chapters for adjusting hierarchical models.