A BFGS-Based Simulation Study of Covariance Effects in a Trivariate Poisson Regression Model

Trivariate count responses are common in real-life applications where several events occur concurrently with dependence among them. The trivariate Poisson regression (TPR) has been proposed in the literature for modelling such correlated count responses in the presence of a constant covariance. In this paper, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is utilized for the maximum likelihood estimation of the TPR model parameters. The behaviour of the estimators in finite-sample conditions was assessed at varying degrees of shared constant covariance between the count outcomes. We performed simulation studies with the TPR model having no covariates, one covariate, two covariates, and three covariates, with the constant covariance varied from 0.0 to 1.2 and sample sizes ranging from 30, 250, and 500. Our results revealed no consistent monotonic association between the empirical bias, empirical standard error, and the size of the constant covariance, but there is higher variability in some cases where sample sizes are limited. The increase in sample sizes leads to better stability of parameter estimates, as expected from the maximum likelihood estimation procedure. The AIC and BIC values provide information about the likelihood-complexity trade-off in the various simulation settings, but their values themselves do not prove that the BFGS algorithm is better or that the addition of covariates will always improve the model fit. This study demonstrates the ability of the BFGS algorithm to deal with high-dimensional Poisson models, giving data analysts and researchers dealing with multivariate count data in applied contexts practical recommendations.