C22. Negative-binomial variation

Topic: Standard errors, and models for counts · more questions on this topic
‹ C21

Why can a negative-binomial model handle overdispersed counts that a Poisson model cannot?

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Definitions/notation. The Poisson model has \operatorname{Var}(Y)=\mu, whereas a negative-binomial model with mean \mu>0 and size \nu>0 has \operatorname{Var}(Y)=\mu+\mu^2/\nu. The extra positive term allows greater variation than Poisson at the same mean, as arises from Gamma mixing of Poisson intensities; the Poisson limit is recovered as \nu\to\infty.