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Poisson regression is a popular tool for modeling count data and is appliedin a vast array of applications from the social to the physical sciences andbeyond. Real data, however, are often over- or under-dispersed and, thus, notconducive to Poisson regression. We propose a regression model based on theConway--Maxwell-Poisson (COM-Poisson) distribution to address this problem. TheCOM-Poisson regression generalizes the well-known Poisson and logisticregression models, and is suitable for fitting count data with a wide range ofdispersion levels. With a GLM approach that takes advantage of exponentialfamily properties, we discuss model estimation, inference, diagnostics, andinterpretation, and present a test for determining the need for a COM-Poissonregression over a standard Poisson regression. We compare the COM-Poisson toseveral alternatives and illustrate its advantages and usefulness using threedata sets with varying dispersion.

A flexible regression model for count data