The Variance of Intraclass Correlations in Three- and Four-Level Models

Intraclass correlations are used to summarize the variance decomposition in populations 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.