[R-sig-ME] reference on unidentifiability of observation-level variance in Bernoulli variables?

[Ben Bolker]

  Hi folks,

  It has often been discussed on this list and other R-help lists that
overdispersion is (broadly speaking) unidentifiable for an ungrouped
Bernoulli response variable (if some kind of grouping can be imposed,
then it becomes identifiable). For example:


https://stat.ethz.ch/pipermail/r-help/2008-February/154058.html
http://finzi.psych.upenn.edu/R/Rhelp02a/archive/91242.html

Peter Dalgaard:

> The point being that you  cannot have a distribution on {0, 1} where >
the variance is anything but  p(1-p) where p is the mean; if you put > a
distribution on p and integrate  it out, you still end up with the >
same variance.

   I am curious if anyone has a *printed* (book or peer-reviewed
article) for this, or even can point to notes that actually go to the
trouble of doing the integration and proving the statement.  I have
looked in some of the usual places (MASS; McCullough, Searle, and
Neuhaus; Zuur et al.) and haven't come across anything.

  Something that showed the computation of the intraclass correlation
and worked out the mean of the logistic-normal-binomial distribution as
a function of the mean and variance of the underlying normal
distribution would be nice too, although I'm guessing it doesn't have a
straightforward analytical solution ...

  I'm hoping I haven't missed anything obvious -- on the other hand, if
I have it will be easy for someone to answer (and please don't be
offended if it was in your book, which was on my shelf all the time and
I forgot to look there ...)

 thanks
  Ben Bolker