Modelling over-dispersed count data using the poisson and negative binomial distributions

Abstract

The negative binomial model is generally used as an alternative to the Poisson regression model when real overdispersion is significant. The Poisson regression model has obvious appeal and much simpler to implement than the negative binomial model which simultaneously can be interpreted as a generalization of the basic Poisson regression model. Consequently, Poisson applications are more popular amongst researchers, unlike the Negative Binomial model which might provide similar predictions in certain circumstances. However, the assumptions made by the Poisson model are more restrictive than those made by the Negative Binomial (NB) model, and consequently motivated researchers to propose the NB model. In fact the mean of the Poisson model is allowed to vary in an NB model by specifying it to be a random variable. There exist various parameterization forms of the negative binomial model with differing mean variance relationships, yet the most popular forms are those with linear and quadratic relationships. The specification of the Poisson mean as a random variable and having a gamma distribution leads to the NB model. NB models with a linear variance can be parameterized into NB 1 models; whereas those with a quadratic variance can be parameterized as NB2 or NB-C models. The NB models with a quadratic variance are considered as a generalized linear model given that the shape parameter is known. The aim of this research study is to compare two models for count data, assuming either a Poisson or a Negative Binomial distribution, to predict the number of ECTS resits for a university student in the Faculty of Science given a number of course-related predictors and students' attainment in A level subjects. Each student in the 2002-2006 Faculty of Science cohort had to choose two subjects from five available science subjects. This data set is categorized by year of study and subject chosen by the student. It was noted that students with a low attainment score tended to have more ECTS resits than their counterparts with a high attainment score.