Novos desenvolvimentos para dados de contagem

Abstract

This work investigates new developments in count data analysis, focusing on two methodologies: Item Response Theory (IRT) models and Generalized Linear Mixed Models (GLMM). The research focuses on the development and application of classical and Bayesian methods to improve the analysis of existing count models and to propose new models. The chapters of this thesis comprise manuscripts developed throughout the doctoral program. First, a study is presented with the Rasch Poisson count model, already existing in the literature, aiming at a better understanding and introducing a new approach through the method of nested and integrated Laplace approximations. Techniques for residual analysis are shown through graphical visualization, using randomized quantile residuals, and the methodology is applied in the field of Psychology. We explore alternative models for count responses, which overcome some of the limitations of the Rasch-Poisson model. We detail its formulation and estimation methods, under both Classical and Bayesian approaches. Additionally, we demonstrate the potential of residual analysis using graphs applied to data from an attention test. Next, when considering mixed models, we introduce a new proposal for count responses, based on the one-parameter Bell distribution, explicitly detailing its formulation and estimation under Classical and Bayesian approaches. We evaluate the parameter recovery of the proposed estimation methodology through a simulation study and also show the potential of its use in an application to epileptic seizure data. Finally, we propose a new mixed regression model based on the Bell-Touchard distribution parameterized by the mean. Simulation studies are presented, and the methodology is applied in a neurophysiological experiment. The various studies and applications throughout the text show that the proposals yield good results and have potential use by researchers in various fields, with the codes used for parameter estimation made available.