Dear all,
I calculated the intraclass correlations at two nested levels in a binomial mixed model from the random effect variances, and tried to calculate the 95% confidence limits on them from the confidence limits on the variance components, obtained by profiling the likelihood:
(I have 2 groups and ca 10 subgroups per group)

  fit = glmer(trait ~ 1 + (1|group/subgroup), family = binomial(link="logit"),data=mydata)
  vcomps = as.numeric(VarCorr(fit))
  V.link = (pi^2)/3
  V.tot= vcomps [1] + vcomps [2] + V.link
  ICC.group = results[2]/V.tot
  ICC.subgroup = results[1]/V.tot
 icc=c(ICC.group,ICC.subgroup)

  pr=profile(fit,signames=F)
  prclims=(confint(pr,level=level)^2)/V.tot
  low=prclims[2:1,1] # upper and lower CIs on the ICCs
  up=prclims[2:1,2]
  results = round(data.frame(icc, low, up),3)
  row.names(results) = c("group", "subgroup")
  results
#                 icc        low    up
# group    0.000    0.000   Inf
# subgroup 0.095 0.006 0.303

My question is whether this approach would be OK? I only consider the uncertainty in the estimation of the among-group and among-subgroup variances here, keeping the total variance constant. Is that OK? And what is the implicit assumption in calculating the confidence limits this way - e.g. what sources of uncertainty are considered then? (I also came across this reference, https://stuiterproxy.kuleuven.be/doi/abs/10.1080/,DanaInfo=www.tandfonline.com+03610910903324834, about calculating the conf lims on the ICC using likelihood profiling, not sure if that is relevant here).

I also tried calculating the confidence limits using parametric bootstrapping (using bootMer), but there I ran into problems with non convergence in the fits, owing to the small size of my dataset. And then I tried nonparametric cluster bootstrapping, whereby I resampled over subgroups (resampling the same nr of subgroups pers groups as I had initially), but I am not sure that that is entirely correct either (e.g. it doesn't take into account the binomial scatter per subgroup cluster, and also resampling within clusters biases my estimate, so I can't do that). Or does anyone perhaps happen to have other ideas about how to calculate confidence limits on my estimated intraclass correlation?

Cheers,
Tom


_______________________________________________________________________________________

Prof. Tom Wenseleers
*      Lab. of Socioecology and Social Evolution
           Dept. of Biology
           Zoological Institute
           K.U.Leuven
           Naamsestraat 59, box 2466
           B-3000 Leuven
           Belgium
* +32 (0)16 32 39 64 / +32 (0)472 40 45 96
* tom.wenseleers@bio.kuleuven.be
http://bio.kuleuven.be/ento/wenseleers/twenseleers.htm



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