overdispersion

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When applying the generalized linear model or GLM to the real world, a phenomenon called overdispersion occurs when the observed variance of the data is larger than the predicted variance. This is particularly apparent in the case of a Poisson regression model, where

predicted variance = predicted mean,

predicted variance = predicted mean(1- predicted mean).

A parameter, called the dispersion parameter, $\phi$ , is introducted to the model to lower this overdispersion effect.

The GLM, with the inclusion of this dispersion parameter, has the following density function:

f_{{Y_{i}}}(y_{i}\mid\theta_{i})=\operatorname{exp}[\frac{y\theta_{i}-b(\theta% _{i})}{a(\phi)}+c(y,\phi)]

Dispersion parameters for some of the well known distributions from the exponential family are listed in the following table:

1 J. M. Hilbe, Negative Binomial Regression, Cambridge University Press, Cambridge (2007).
2 A. Agresti, An Introduction to Categorical Data Analysis, Wiley & Sons, New York (1996).
3 P. McCullagh and J. A. Nelder, Generalized Linear Models, Chapman & Hall/CRC, 2nd ed., London (1989).
4 A. J. Dobson, An Introduction to Generalized Linear Models, Chapman & Hall, 2nd ed. (2001).

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dispersion parameter

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Definition

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