Dispersion parameter

Prometheus

Member
I have the understanding that the dispersion parameter, $$\phi$$, in generalised linear models needs to be known for deviance to meaningful or otherwise it can be estimated from the model, but no analysis of deviance can be carried out.

Why is it then that in binomial and poisson models in R/S+ it is always assumed $$\phi = 1$$? If the assumption is not true then the analysis in meaningless, no?

Dason

If that assumption isn't true then you don't actually have binomial or poisson distributed data. The binomial and poisson models are interesting in that once you specify what the mean of the response is that you automatically know what the variance is. This is why we can fix the dispersion parameter beforehand - you don't need another parameter to talk about the variance since once you specify the mean you know the variance.

ted00

New Member
Dason's is a good reply. I'll just add by directing you to the wikipedia pages on binomial and Poisson where you should examine the mathematical forms of the distribution functions, and the page on generalized linear models. Binomial and Poisson don't have dispersion parameters, that's why the software sets default to the trivial value of 1.

Prometheus

Member
Thanks guys. I can see what you mean with regard to the poisson, but can't see it for the binomial. When i start from the pdf of a binomial and put it in the form of the generalised exponential family i always end up with $$\phi = 1/n$$. Am i missing something about weights - never did understand that?

Dason

typically we use bernoulli outcomes so n = 1. If you have binomial outcomes you can think of that as the sum of n bernoullis.

Prometheus

Member
Still not sure - but i haven't looked at the bernoulli distribution in the context of GLMs yet so i'll do that.

Prometheus

Member
Ok, i see what you guys are saying. Makes sense. But then i must be wrong in my calculations when i get $$\phi = 1/n$$? Which is surprising in that they accord with what my lecturer gave.