Generalized mixed spatiotemporal modeling with a continuous response and random effect via factor analysis

This work focuses on Generalized Linear Mixed Models that incorporate spatiotemporal random effects structured via Factor Model (FM) with nonlinear interaction between latent factors. A central aspect is to model continuous responses from Normal, Gamma, and Beta distributions. Discrete cases (Bernoulli and Poisson) have been previously explored in the literature. Spatial dependence is established through Conditional Autoregressive (CAR) modeling for the columns of the loadings matrix. Temporal dependence is defined through an Autoregressive AR(1) process for the rows of the factor scores matrix. By incorporating the nonlinear interaction, we can capture more detailed associations between regions and factors. Regions are grouped based on the impact of the main factors or their interaction. It is important to address identification issues arising in the FM, and this study discusses strategies to handle this obstacle. To evaluate the performance of the models, a comprehensive simulation study, including a Monte Carlo scheme, is conducted. Lastly, a real application is presented using the Beta model to analyze a nationwide high school exam called ENEM, administered between 2015 and 2021 to students in Brazil. ENEM scores are accepted by many Brazilian universities for admission purposes. The real analysis aims to estimate and interpret the behavior of the factors and identify groups of municipalities that share similar associations with them.