I wonder whether accounting for overdispersion in a GLM (Quasi - Poisson instead of Poisson family) has an effect on the model checking plots (plot of residuals against fitted values, a scale–location plot of residuals against fitted values, a normal QQ plot and a plot of residuals against leverages). I tested this and apart from the scale in the plots nothing changes.

Is this hazard or is this always the case?

  • $\begingroup$ It's always the case -- quasi Poisson is Poisson with an additional scale factor. $\endgroup$ – Glen_b May 1 '15 at 10:34
  • $\begingroup$ is the scale factor the variance (which is assumed to be a linear function of the mean for quasi-poisson)? $\endgroup$ – kalakaru May 1 '15 at 10:40
  • $\begingroup$ see the edits to my answer $\endgroup$ – Glen_b May 1 '15 at 10:55

It's always the case -- quasi Poisson is effectively Poisson with an additional scale factor.

The dispersion parameter corresponds to rescaling a Poisson variable by that parameter.

To be more specific, the scale factor I referred above to is the dispersion parameter.

In a quasi-Poisson GLM, $E(Y)=\mu\,$, $Var(Y)=\phi.\mu$.

You can get that by taking a Poisson($\lambda$) and multiplying by $\phi$, such that $\mu=\lambda.\phi$. e.g. $Z\sim \text{Pois}(\lambda)$ and $Y=\phi.Z$; then $Y$ has the required properties for a quasi-Poisson. (To my recollection it's the only member of the natural exponential family that has the right variance function.)

(People using them to model overdispersed counts don't worry too much about that since they look on it as a quasi-likelihood model.)

For example, Clark and Thayer [1] say

The Over-Dispersed Poisson distribution is a generalization of the Poisson, in which the range is a constant $\phi$ times the positive integers. That is, the variable $Y$ can take on values ${0,1\phi, 2\phi, 3\phi, 4\phi,...}$. It has a variance equal to $\phi$ times the mean.

[1]: Clark D.R. and Thayer, C.A. (2004),
A Primer on the Exponential Family of Distributions,
CAS 2004 Discussion Papers 117-148,
Casualty Actuarial Society. Arlington, Virginia



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