A multivariate correlated poisson generalized inverse gaussian regression model for dependent count data: Estimation and testing procedures

Regression modeling for multivariate count data often struggles with assumption of overdispersion and correlation among response variables. To address these issues, this study proposes a new model called Multivariate Correlated Poisson Generalized Inverse Gaussian Regression (MCPGIGR), which integrates random effects through common shock variables and allows for flexible mean structures via a log-link function. This research develops a Maximum Likelihood Estimation (MLE) and Maximum Likelihood Ratio Tests (MLRT) to evaluate both simultaneous and partial significance of predictors. We conduct simulation studies to assess the consistency and performance of the proposed estimators. Furthermore, in an application to maternal and neonatal mortality across 38 districts/cities in East Java (Indonesia), MCPGIGR substantially improves model fit relative to a Multivariate Poisson Regression (MPR) baseline (AICc decreases from 2378.63 to 1924.60 for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mrow><mml:mn>1</mml:mn> <mml:mspace></mml:mspace> <mml:mo>/</mml:mo> <mml:mspace></mml:mspace> <mml:mn>2</mml:mn></mml:mrow> </mml:mrow> </mml:math> ). The proposed framework provides a practical and flexible tool for analyzing correlated, overdispersed multivariate counts in public health and related domains. The highlights of this research are: • The MCPGIGR model introduces a correlated multivariate count regression framework with exposure adjustment. • It provides robust parameter estimation and hypothesis testing via MLE and MLRT. • MCPGIGR demonstrates improved model fit and practical interpretability in public health applications.