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, $\varphi$, is introducted to the model to lower this overdispersion effect.

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

$$f_{Y_i}(y_i\mid {\theta }_i)=\exp [\frac{y{\theta }_i-b({\theta }_i)}{a(\varphi )}+c(y,\varphi )]$$

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