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

[Highland Statistics Ltd]

Message: 2
Date: Tue, 15 Nov 2011 15:53:07 -0500
From: Ben Bolker<bbolker at gmail.com>
To: r-sig-mixed-models<r-sig-mixed-models at r-project.org>
Subject: [R-sig-ME] reference on unidentifiability of
	observation-level variance in Bernoulli variables?
Message-ID:<4EC2D133.6040004 at gmail.com>
Content-Type: text/plain; charset=ISO-8859-1


   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 ...




Ben,

An expression for the intraclass correlation in a Poisson or binomial GLMM can be found in:


Goldstein et al. (2002)
Browne et al. (2005)

For a Poisson GLMM the calculations are simple. For a Binomial GLMM you need to do approximations or simulations.
For a 2-way nested ZIP model the intraclass correlation is given in Chapter 4 of:
Zero Inflated Models and Generalized Linear Mixed Models with WinBUGS and R (2012). Zuur et al.


Alain

Refs:
Goldstein H, Browne W, Rasbash J (2002). Partitioning Variation in Multilevel Models. Understand-ing Statistics. Understanding Statistics, 1(4), 223-231

Browne WJ, Subramanian SV, Jones K (2005). Variance partitioning in multilevel logistic models that exhibit overdispersion. J. R. Statist. Soc. A (2005) 168, Part 3, pp. 599–613









   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