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# 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:

distribution | notation | dispersion parameter $\phi$ |
---|---|---|

Normal | $N(\mu,\sigma^{2})$ | $\sigma^{2}$ |

Poisson | $Poisson(\mu)$ | 1 |

Binomial | $Bin(m,\pi)$ | $\displaystyle{\frac{1}{m}}$ |

Gamma | $Gamma(\alpha,\lambda)$ | $\displaystyle{\frac{1}{\alpha}}$ |

# References

- 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).

## Mathematics Subject Classification

62J12*no label found*

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