It is a diagnostic and it is an estimate of a variance ratio.
What then should one do if the diagnostic indicates that
the binomial or poisson variance is not accounting properly
for the variation observed in the data?  Give up doing any
analysis, on the basis of the Bates (!) non-existence axiom?

I take it that you are happy to call the relevant quantity 
an estimate if it comes from a model that has observation 
level random effects?

It is well to note that these two different statistics canvass
just two of a much wider range of ways in which the 
binomial or poisson variance might be inflated.  A recent
paper that canvasses this arena is:

Faddy, Malcolm & Smith, David (2011) Analysis of count data 
with covariate dependence in both mean and variance. 
Journal of Applied Statistics, 38(12), pp.

This is an area where the practice has, for those who want
to be fastidious about such matters, run slightly ahead of the 
mathematical theory.  There are good precedents for this.
Applied people in electromagnetic wave theory and elsewhere 
were working with infinite spikes in their density estimates 
for some time before the mathematics was developed that 
made the details of the mathematical deductions respectable.
Regards
John.

John Maindonald             email: john.maindonald@anu.edu.au
phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
Centre for Mathematics & Its Applications, Room 1194,
John Dedman Mathematical Sciences Building (Building 27)
Australian National University, Canberra ACT 0200.
http://www.maths.anu.edu.au/~johnm

On 01/03/2013, at 2:12 AM, Douglas Bates <bates@stat.wisc.edu> wrote:

> 
> On Wed, Feb 27, 2013 at 6:45 PM, John Maindonald <john.maindonald@anu.edu.au> wrote:
> I think a response is in order!
> 
> Whether one believes in it or not, the variance ratio that the dispersion
> estimate is designed to accommodate is, probably more often than not
> with allegedly binomial or poisson errors, real.  The dispersion estimate
> that can be obtained from glm() fits provides important clues, that users
> ignore at peril to the credibility of the analysis.  In the common
> situation where a dispersion substantially greater than 1 is identified,
> the variance adjustment is a simple ad hoc recourse that generally
> does a good job in allowing calculation of SEs that are plausible.  One
> can set down models for which the dispersion variance correction is
> pretty much right.  Sussing out the details of models that would lead
> to something very close to the dispersion correction would surely be
> a good PhD project.
> 
> You seem to be describing the use of a diagnostic.  I would have no trouble with providing such a value labelled as a diagnostic quantity.  I do have a problem with labeling it as an estimate of a parameter because there is no such parameter to estimate.
> 
> As an aside, I do not myself find what comes out of the magic world of
> negative binomial distributions more plausible than the way that I
> understand the dispersion "correction".
> 
> GLMMs with observation level random effects are another way to
> handle the matter -- albeit the dispersion correction is different.
> I have data where results for most species (there were quite a
> number of species) suggested to me that a GLM type dispersion
> correction did a better job than observation level random effects
> in a GLMM.  This tentative conclusion does though need careful
> checking.  If I live to be 99 with my faculties pretty much intact,
> maybe I will sometime make a stab at investigating further and
> writing it up.
> 
> John Maindonald             email: john.maindonald@anu.edu.au
> phone : +61 2 (6125)3473    fax  : +61 2(6125)5549
> Centre for Mathematics & Its Applications, Room 1194,
> John Dedman Mathematical Sciences Building (Building 27)
> Australian National University, Canberra ACT 0200.
> http://www.maths.anu.edu.au/~johnm
> 
> On 28/02/2013, at 9:57 AM, Douglas Bates <bates@stat.wisc.edu> wrote:
> 
> > On Wed, Feb 27, 2013 at 4:31 PM, Ben Bolker <bbolker@gmail.com> wrote:
> >
> >> <juwb08@...> writes:
> >>
> >>>
> >>> Hi All,
> >>>
> >>> I would like to calculate a dispersion parameter for a glmer with a
> >>> Poisson family (in lme4). I found a function posted on this mailing
> >>> list that had been suggested to use for calculating a dispersion
> >>> parameter for a glmer with a binomial family. (Found on
> >>> https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q1/015383.html )
> >>>
> >>
> >> [snip]
> >>
> >>  Please see slightly more robust code at
> >>
> >> http://glmm.wikidot.com/faq#overdispersion_est
> >>
> >> It should be applicable to GLMMs generally, although please note all the
> >> boldface warnings about this being a crude approximation of the
> >> overdispersion parameter.
> >>
> >
> > As you know, my opinion is that you can approximate it however you want
> > because it doesn't exist :-)
> >
> >       [[alternative HTML version deleted]]
> >
> > _______________________________________________
> > R-sig-mixed-models@r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> 
> 


	[[alternative HTML version deleted]]

