Poisson–Birnbaum–Saunders regression model for clustered count data

In this paper we study the number of inpatient admissions by individuals to hospital emergency rooms reported by the 2003 Medical Expenditure Panel Survey (MEPS), which the United States Agency for Health Research and Quality conducts. Explanatory variables such as health status, access, use, and costs of health services in the U.S.A. are considered. Our main goal is to properly model the number of inpatient admissions, according to the geographical U.S. regions, as a tool for measuring the volume of diagnostic procedures in the health care system. In the analysis four clusters were determined according to the regions in the U.S., namely, the midwest, northeast, south, and west. The clustered analysis of this count data from the MEPS is a novel contribution to the best of our knowledge. Our analysis demonstrated that a clustered negative binomial (CNB) regression (Poisson model with latent gamma effects) might not be a suitable choice for analyzing the MEPS data. This fact motivates us to introduce a new regression model to handle clustered count data. To account for correlation within clusters, we propose a Poisson regression model where the observations within the same cluster are driven by the same latent random effect that follows a Birnbaum–Saunders distribution with a parameter that controls the strength of dependence among the individuals. This novel multivariate count model is called Clustered Poisson–Birnbaum–Saunders (CPBS) regression. The CPBS model is analytically tractable, and its moment structure can be explicitly obtained. We also derive theoretical/methodological studies to advise when the Birnbaum–Saunders effect should be preferred over the gamma effect (and vice-versa) in terms of probability tail. Estimation is performed through the maximum likelihood method. Here we also developed an expectation-maximization (EM) algorithm for estimation. Simulation results that evaluate the finite-sample performance of our proposed estimators are presented. Studies on the potential impact of model misspecification were conducted, and comparisons between our model and a CNB regression were also addressed. A full statistical analysis of the MEPS data reveals that, compared to the CNB model, the CPBS regression model produces better results in terms of prediction and goodness-of-fit.