The Variance of Intraclass Correlations in Three and Four Level

Intraclass correlations are used to summarize the variance decomposition in popula- tions with multilevel hierarchical structure. There has recently been considerable interest in estimating intraclass correlations from surveys or designed experiments to provide design parameters for planning future large-scale randomized experiments. The large sample distribution of estimates of the intraclass correlation in two-level situations is well known. Contemporary educational and social experiments often involve sampling designs with three or four levels of nesting, leading to three- and four-level intraclass correlation structures. The present article provides expressions for the large sample variances and covariances of estimates of three- and four-level intraclass correlation structures. An appendix describes the multilevel models on which the results of this paper are based and the method used to derive those results. [This paper was published in "Educational and Psychological Measurement," v72 n6 p893-909 Dec 2012 (EJ982773).]