A Flexible Regression Model for Count Data

Abstract

Poisson regression is a popular tool for modeling count data and is applied in a vast array of applications from the social to the physical sciences and beyond. Real data, however, are often over-or under-dispersed and, thus, not conducive to Poisson regression. We propose a regression model based on the Conway-Maxwell-Poisson (COM-Poisson) distribution to address this problem. The COM-Poisson regression generalizes the well-known Poisson and logistic regression models, and is suitable for fitting count data with a wide range of dispersion levels. With a GLM approach that takes advantage of exponential family properties, we discuss model estimation, inference, diagnostics, and interpretation, and present a test for determining the need for a COM-Poisson regression over a standard Poisson regression. We compare the COM-Poisson to several alternatives and illustrate its advantages and usefulness using three data sets with varying dispersion.

The statistical analysis of crash-frequency data: A review and assessment of methodological alternatives

“…The gamma model has been proposed by Oh et al (2006) to analyze crash data exhibiting underdispersion (see also Cameron and Trivedi, 1998 Please see Sellers and Shmueli (2010), Guikema and Coffelt (2008) and for further details on the estimation properties and applications of this model.…”

Evaluating the double Poisson generalized linear model

“…Previous studies have investigated the accuracy of those approximation methods. For the COM-Poisson GLM based on its original parameterization, the estimated means can be approximated using the asymptotic expression but is only accurate for 1 v  or 10v   (Sellers and Shmueli, 2010). Although the re-parameterization of the COM-Poisson distribution proposed by Guikama and Coffelt (2008) was based on a more clear centering parameter (i.e., the mode), it is still not directly comparable to the mean in the DP GLM.…”

“…With the derivation of the likelihood function of the COM-Poisson GLM by Sellers and Shmueli (2010), the maximum likelihood estimation (MLE) of the parameters of a COM-Poisson GLM was greatly simplified when compared to the Bayesian estimating method. The MLEbased COM-Poisson GLM is available in R package (COMPoissonReg on CRAN).…”

“…Then, the mean of those samples was as the mean response. For other methods on obtaining fitted values, the reader is referred to Sellers and Shmueli (2010).…”

Determination of the Affecting Factors of the Number of Babies Born Alive in Multiple Pregnancies with Poisson Models

“…The Conway-Maxwell-Poisson model was introduced in 1962 9 ; then only it was evaluated in the context of a GLM 10,11,12 . The corresponding distribution is generalization of the Poisson distribution with two parameters which makes the model flexible enough to describe a wide range of count data.…”