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.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS306 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Prognostication and related clinical decision making in the majority of patients with melanoma can be improved now using the validated, SEER-based classification. Tumor cell mitotic rate should be incorporated into the next iteration of AJCC staging.
Modulation of adipokine signaling may contribute to the insulin resistant, atherogenic state associated with human inflammatory syndromes. Targeting of individual adipokines or their upstream regulation may prove effective in preventing acute and chronic inflammation-related metabolic complications.
The Poisson distribution is a popular distribution used to describe count information, from which control charts involving count data have been established. Several works recognize the need for a generalized control chart to allow for data over-dispersion; however, analogous arguments can also be made to account for potential underdispersion. The Conway-Maxwell-Poisson (COM-Poisson) distribution is a general count distribution that relaxes the equi-dispersion assumption of the Poisson distribution, and in fact encompasses the special cases of the Poisson, geometric, and Bernoulli distributions. Accordingly, a flexible control chart is developed that encompasses the classical Shewart charts based on the Poisson, Bernoulli (or binomial), and geometric (or negative binomial) distributions.
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.
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