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Unformatted text preview: Overdispersion and Poisson Regression * Richard Berk John MacDonald Department of Statistics Department of Criminology University of Pennsylvania November 19, 2007 Abstract This article discusses the use of regression models for count data. A claim is often made in criminology applications that the negative binomial distribution is the conditional distribution of choice when for a count response variable there is evidence of overdispersion. Some go on to assert that the overdisperson problem can be “solved” when the negative binomial distribution is used instead of the more conven- tional Poisson distribution. In this paper, we review the assumptions required for both distributions and show that only under very special circumstances are these claims true. 1 Introduction Count data are common in criminological research. When the response vari- able is a count, one option is to employ Poisson regression as a special case of the generalized linear model, whether characterized as a causal model or not. The Poisson formulation has obvious appeal. It is relatively simple to interpret because the right hand side is the familiar linear combination of * Richard Berk’s work on this paper was funded by a grant from the National Science Foundation: SES-0437169, ”Ensemble methods for Data Analysis in the Behavioral, Social and Economic Sciences. 1 predictors and because when exponentiated, the regression coefficients are can interpreted as multipliers. 1 In addition, tests and regression diagnostics available for the normal regression special case carry over, at least in look and feel (Cook and Weisberg: chapter 22). Poisson regression applications have been published by a number of respected criminologists (Paternoster and Brame, 1997; Sampson and Laub, 1997; Osgood, 2000). The negative binomial distribution has been suggested by some as an alternative to the Poisson when there is evidence of “overdisperson” (Pa- ternoster and Brame, 1997; Osgood, 2000). Stated loosely for the moment, “overdispersion” implies that there is more variability around the model’s fitted values than is consistent with a Poisson formulation. The negative binomial is proposed as a means to correct for this problem, and some say that it automatically does so (Osgood, 2000). There is a parameter for the negative binomial distribution whose estimated value inflates the Poisson dispersion as needed. The negative binomial variant of Poisson-based regression model is now a conventional way to address apparent overdispersion whether at the level of individual offenders, criminal justice agencies, neighborhoods, or even larger units (Paternoster and Brame, 1997; Paternoster et al., 1997; Sampson and Laub, 1997; Duwe et al., 2002; Piquero et al., 2002; Braga, 2003; Parker, 2004; Stucky, 2003; Lattimore et al. 2004; Bottcher and Ezzell, 2005). A search in sociological abstracts for peer-reviewed journal articles that mention the negative binomial model as a “key word” yielded 47 with substantial...
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This note was uploaded on 02/15/2012 for the course GEO 6938 taught by Professor Staff during the Summer '08 term at University of Florida.
- Summer '08