# overdispersion

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,

or the binary logistic regression model, where

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:

Title overdispersion Overdispersion 2013-03-22 14:30:34 2013-03-22 14:30:34 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 62J12 dispersion parameter