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Overdispersion and Poisson Regression

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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 conventional 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.

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Notes

  1. The canonical link function is the log of the expected value of the response variable.

  2. A wide variety of other examples could serve just as well. For example, the observational units could be neighborhoods, the response variable could be the number of burglaries in a given month, and among the predictors could be median household income and the proportion of the male population between 15 and 25 years of age.

  3. The Poisson processes are different because they have different values for λ, conditional on age.

  4. The variance of a sum of random variables is a linear combination of their variances and covariances (Freund 1971, p. 195). If those covariances are positive, the variance of the sum is inflated. If those covariances are negative, the variance of the sum is deflated. The Poisson distribution assumes that all of the covariances are zero.

  5. This follows from the assumptions about ɛ i .

  6. Without normalization, the model’s constant would not be identified.

  7. As θ → ∞, one is back to the Poisson distribution. If θ <  0, there is underdispersion.

  8. In order to have sufficient observations to get a good reading on the conditional variance, ranges of conditional means are used.

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Acknowledgments

We are indebted to David Freedman, David McDowall, and the anonymous reviewers for their helpful suggestions. All errors and omissions remain those of the authors.

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Authors and Affiliations

  1. Department of Criminology, University of Pennsylvania, McNeil Building, Suite 483, 3718 Locust Walk, Philadelphia, PA, 19104-6286, USA

    Richard Berk & John M. MacDonald

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  1. Richard Berk
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  2. John M. MacDonald
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Correspondence to Richard Berk.

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Berk, R., MacDonald, J.M. Overdispersion and Poisson Regression. J Quant Criminol 24, 269–284 (2008). https://doi.org/10.1007/s10940-008-9048-4

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