# GLM Model checking Plots - Quasi Poisson - Poisson

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?

• It's always the case -- quasi Poisson is Poisson with an additional scale factor. Commented May 1, 2015 at 10:34
• is the scale factor the variance (which is assumed to be a linear function of the mean for quasi-poisson)? Commented May 1, 2015 at 10:40
• see the edits to my answer Commented May 1, 2015 at 10:55

## 1 Answer

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.)

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.

People using this to model overdispersed counts don't usually worry too much about that since they look on it as a quasi-likelihood model; they're not looking on the scaled Poisson as a specific distributional choice.

[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