Modelos de resposta discreta com funções de ligação da família gumbel

Abstract

The present study focuses on the introduction and development of asymmetrical statistical models to address imbalanced data in binomial regressions and within Item Response Theory (IRT). Initially, we delve into the complementary log-log link function, introduced by Fisher in 1922, as an asymmetrical alternative to the logit and probit link functions. We propose flexible variations of this function to model binomial regression, incorporating additional parameters that account for imbalances in the binomial outcomes. For model inference, we develop a Bayesian approach employing Monte Carlo Markov chain methods. Furthermore, we investigate the relationship between asymmetrical Item Characteristic Curves (ICCs) within IRT for imbalanced binary response data. We propose new IRT models with asymmetrical ICCs as their primary feature, including the cloglog IRT model as a special case. We emphasize the significance of these models in educational data analysis and compare their efficacy against other models proposed in the IRT literature. Additionally, we introduce two new item response theory models based on the Generalized Extreme Value (GEV) distribution. We discuss Bayesian estimation methods for these models and demonstrate their applicability through simulation studies and analysis of real-world data from mathematical tests in public schools in Peru. These models show promise in handling imbalances and asymmetries in binary data, providing a robust and adaptable statistical approach across various domains, including healthcare, education, and test assessment.