  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.
