Development Study of GLMM-GEE-Tree Regression Modelling for Beta Distribution Response Data

Abstract

Economic inequality remains one of the most persistent challenges faced by Indonesia as a developing country. Previous studies have predominantly employed conventional models such as Ordinary Least Squares (OLS) or Panel Least Squares. However, these models are often inappropriate, as they fail to account for the bounded nature of inequality indices such as the Gini ratio, which ranges between 0 and 1. Beta regression offers a more appropriate alternative. In the context of panel data, Generalized Linear Mixed Models (GLMM) and Generalized Estimating Equations (GEE) are commonly used to handle correlated data; however, their integration with nonlinear models for longitudinal Beta-distributed responses remains limited. This study proposes a novel GLMM-GEE-Tree modeling approach for Beta-distributed response data. The proposed model combines GLMM (to capture individual random effects), GEE (to handle temporal correlation and provide robust marginal estimates), and Regression Trees (to address nonlinear relationships and complex interactions). The aim is to simultaneously tackle the challenges of proportional responses, panel structure, random effects, correlation, and nonlinearity. Empirical validation uses Gini ratio data from 34 Indonesian provinces spanning 2018 to 2024. The findings reveal that in this empirical data, the GLMM-GEE-Tree model outperforms alternative models, achieving an R² of 0.472 and a QIC of 13.435 and yielding the lowest AIC and BIC values