[R-sig-ME] residual variance in nested mixed model with family=Poisson?

[Douglas Bates]

On Thu, Aug 21, 2008 at 4:50 AM, logodall <logodall at yahoo.fr> wrote:
> Hello,
>    I would like to ask if it is possible (and if so how) to obtained an estimate of the residual variance of a mixed model having a set of nested random factors when using family=Poisson. I have searched in the help files and in the forum and I found no information in this regard. When using family=Gaussian, one obtains by default this variance but not when using Poisson. In the latest verion of lme4, one can now obtain the residuals of a fitted model when family=Poisson, and one can of course calculated the variance of these residuals. However, not being sure of how these residuals are calculated (are they standardised?, raw?, etc), I am unsure of using their variance as an estimate of the residual variance in a nested mixed model.

I don't know how the residual variance of a mixed model would be
defined for family = Poisson.  I formulate the probability model for
GLMMs in terms of the (unconditional) distribution of the random
effects, B, and the conditional distribution of the response, Y, given
the random effects.  The distribution of B is multivariate Gaussian.
The components of Y|B are independent and, for a Poisson GLMM, have a
Poisson distribution.  They are completely determined by the
conditional mean, E[Y|B].  (see slides 9 and 10 of
http://www.stat.wisc.edu/~bates/PotsdamGLMM/GLMMD.pdf).

It is not an oversight that there is no estimate of a residual
variance given for a Poisson GLMM.  Such a parameter doesn't exist in
the way that I formulate the model.