Correcting a significance test for clustering

A common mistake in analysis of cluster randomized trials 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 correction to the t statistic that would be computed if clustering were (incorrectly) ignored. The correction is a multiplicative factor depending on the total sample size, the cluster size, and the intraclass correlation p. The corrected t statistic has Student's t distribution with reduced degrees of freedom. The corrected statistic reduces to the t statistic computed by ignoring clustering when p = 0. It reduces to the t statistic computed using cluster means when p = 1. If 0 < p < 1, it lies between these two, and the degrees of freedom are in between those corresponding to these two extremes.