The Gamma-Poisson model, i.e., a Poisson distribution where the parameter lambda is Gamma distributed, has been suggested as a statistical method for determining whether or not micro-organisms are randomly distributed in a food matrix. In this study, we analyse the Gamma-Poisson model to explore some of the properties of the Gamma-Poisson model left unexplored by the previous study. The conclusion of our analysis is that the Gamma-Poisson model distinguishes poorly between variation at the Poisson level and the Gamma level. Estimated parameter values from simulated data-sets showed large variation around the true values, even for moderate sample sizes (n = 100). Furthermore, at these sample sizes the likelihood ratio is not a good test statistic for discriminating between the Gamma-Poisson distribution and the Poisson distribution. Hence, to determine if data are randomly distributed, i.e., Poisson distributed, the Gamma-Poisson distribution is not a good choice. However, the ratio between variation at the Poisson level and the Gamma level does provide a measure of the amount of overdispersion. (c) 2005 Elsevier Ltd. All rights reserved.
Toft, N., Innocent, G. T., Mellor, D. J., & Reid, S. W. J. (2006). The Gamma-Poisson model as a statistical method to determine if micro-organisms are randomly distributed in a food matrix. Food Microbiology, 23(1), 90-94. https://doi.org/10.1016/j.fm.2005.01.014