[R] icc from GLMM?

[Shinichi Nakagawa]

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