Adjusting a significance test for clustering in designs with two levels of nesting

A common mistake in analysis of cluster randomized experiments is to ignore the effect of clustering and analyze the data as if each treatment group were a simple random sample. This typically leads to an overstatement of the precision of results and anticonservative conclusions about precision and statistical significance of treatment effects. This article gives a simple adjustment to the t statistic that would be computed if clustering were (incorrectly) ignored in an experiment with two levels of nesting (e.g., classrooms and schools) where treatment assignment is made at the highest (e.g., school) level. The adjustment is a multiplicative factor depending on the number of clusters and subclusters, the cluster and subcluster sample sizes, and the cluster and subcluster intraclass correlations ρS and ρC. The adjusted t statistic has Student's t distribution with reduced degrees of freedom. The adjusted statistic reduces to the t statistic computed by ignoring clustering when ρS = ρC = 0. It reduces to the t statistic computed using cluster means when ρS = 1. If ρS and ρC are between 0 and 1, the adjusted t statistic lies between these two and the degrees of freedom are in between those corresponding to these two extremes.