So I'm not quite sure about the terminology but what ever it's called the actual scenario is straight-forward. I'm dealing with a two-stage cluster sample. It's also stratified but let's set that to the side. So within a particularly strata, enumeration areas were selected with probability proportionate to size, a listing exercise was conducted for selected enumeration areas, then n households were selected from each enumeration area (cluster).
Now each household essentially functions as another cluster since data was solicited for each household member. So let's say we have 20 households within a cluster but 36 observations for variable x from that cluster because we have data for multiple household members. Of course, observations from the same family are not independent observations.
How do I calculate the ICC for x for the entire strata?
It seems to me that the ICC for each cluster is simply a case of single stage cluster sampling. Once I had the ICC for each cluster though I'm not sure how to aggregate that up to the strata.
I'm using the survey package in R so specific help with this application would be appreciated. Thanks!