Several intraclass correlations (ICC) when modeling heterogeneous variances?

Question

In a nested design (e.g., students in schools, ants in colonies, individuals in a species, leaves in a tree), ICC is defined as the proportion of variance among observations from the same subject (over total variation, i.e., variance within + residual variation). See, e.g., answer here

When in the design there are one or more categorical factors grouping several subjects (between-subjects factors, e.g., type of school, species of ants or trees), possibilities for heterogeneous variances appear. Residual variances can be modeled as heterogeneous in a mixed, random-intercepts model, for example as in (R code, nlme package):

lme(fixed = Y ~ A, random = ~1 | Subject, weights = varIdent(form = ~1 | A))

where there is then a single estimated variation among subjects, but as many residual variations estimated as levels in A.

(1) Is it valid/appropriate to estimate and report several ICCs, one for each level of the fixed factor? This would represent the correlation among observations from the same subject for subjects within each of the groups defined by the factor A.

(2) Moreover, if variances among subjects are modeled as heterogeneous among groups (this is, random intercepts for each level in A come from different normal distributions, see here), the answer to (1) is the same?

Answer 1

There is a very pedagogical paper on models with heterogeneous variance by Donald Hedeker and Robin J. Mermelstein.

You can find it here: http://www.uic.edu/classes/bstt/bstt513/Hedeker_Mermelstein_07.pdf. They report separate ICC:s for different groups. Have look at how they have done.

Answer 2

I have found an example in the Chapter 3 (p. 99) of this book of different ICCs reported and compared when heterogeneous residual variations were estimated for each group.

With two groups (control vs. treatment, two levels of factor A) with heterogeneous residual variations, they should be estimated as expected:

for control: ICC_group = s²_group / (s²_group + s²_{residual-control})

for treatment: ICC_group = s²_group / (s²_group + s²_{residual-treatment})

So I think that makes the situation (1) a valid (or already used) one.

Then I have found an example in this chapter (suggested read by @Erik) of different longitudinal mixed models that estimate separate variations among subjects and/or residual variations within subjects.

The logic is the same. For example, in a random-intercept model with a different (random) variation per group of subjects (two levels of factor A) [and also with a different residual variation estimated within each group], ICCs would be:

for control: ICC_control = s²_{group-control} / (s²_{group-control} + s²_{residual-control})

for treatment: ICC_treatment = s²_{group-treatment} / (s²_{group-treatment} + s²_{residual-treatment})

So, I would say that (2) is also yes: is mostly the same answer.