R help - Jun 2007 - icc from GLMM?

Dear R users

I would like to ask a question regarding to icc (intraclass correlation) or many
biologists refer it to as repeatability. It is very useful to get icc for many
reasons and it is easy to do so from linear mixed-effects models and many
packages like psy, psychometric, aod and irr have functions to calculate icc. 

icc = between-group variance/(between-group variance + residual variance) 

*residual variance = within-group variance 

However, I have yet to find a convincing reference or some sort on how to
calculate icc from GLMM. I have found below:

icc = between-group variance/(between-group variance + 1)
*between-group variance = scaled between-group variance????
Or variance obtained from random intercept of GLMM
icc = between-group variance/(between-group variance + pi^2/3)
icc = between-group variance/(between-group variance + pi^2/3*(dispersion
parameter))
for binomial GLMM
icc = between-group variance/(between-group variance + 1/(p(1-p))


I am a little confused which one to trust and use. Or there are no easy formulas
to do this? I am guessing formula would change depending on what distribution
you use and what link function as well? I want to calculate icc from GLMM with
Poisson with log link function and also binomial with logit function. Could
anybody help me please?

Many thanks for your help

Shinichi

-- 
Shinichi Nakagawa
Dept of Animal & Plant Sciences
University of Sheffield
Tel: 0114-222-0113
Fax: 0114-222-0002

Shinichi Nakagawa <S.Nakagawa <at> sheffield.ac.uk> writes:
... 
> I am a little confused which one to trust and use. Or there are no easy
form
> to do this? I am guessing formula would change depending on what
distribution
> you use and what link function as well? I want to calculate icc from GLMM
with
> Poisson with log link function and also binomial with logit function. Could
> anybody help me please?
Yes, you are right that ICC depends on assumed data distribution. While ICC is
very handy in linear models it is not the case in GLMM. I suggest you take a
look at the references bellow. There is also some online material by the same
authors. Additionally, I remember that there were lively discussions about ICC
on "multilevel" list at

http://www.jiscmail.ac.uk/lists/multilevel.html

Best wishes, Gregor

@Article{Goldstein:2002,
  author =       {Goldstein, H. and Browne, W. and Rabash, J.},
  title =        {Partitioning variation in multilevel models},
  journal =      {Understanding Statistics},
  year =         {2002},
  volume =       {1},
  number =       {4},
  pages =        {223--231},
  keywords =     {variance ratio, variance partition coefficient,
                  intra-unit correlation, intra-class correlation, normal
                  models, discrete models, random coefficient models}
}

@Article{Browne:2005,
  author =       {Browne, W. J. and Subramanian, S. V. and Jones, K. and
                  Goldstein, H.},
  title =        {Variance partitioning in multilevel logistic models that
                  exhibit overdispersion},
  journal =      {J. R. Stat. Soc. A Stat. Soc.},
  year =         {2005},
  volume =       {168},
  number =       {3},
  pages =        {599--613},
  doi =          {10.1111/j.1467-985X.2004.00365.x},
  checked =      {[2006-04-16]},
  keywords =     {heritability, ratios, intra-class correlation,
                  intra-unit correlation, simulation, linearization, latent
                  variable approach},
}