Confidence intervals for the common intraclass correlation in the analysis of clustered binary responses

In cluster randomized trials, it is often of interest to estimate the common intraclass correlation at the design stage for sample size and power calculations, which are greatly affected by the value of a common intraclass correlation. In this article, we construct confidence intervals (CIs) for the common intraclass correlation coefficient of several treatment groups. We consider the profile likelihood (PL)-based approach using the beta-binomial models and the approach based on the concept of generalized pivots using the ANOVA estimator and its asymptotic variance. We compare both approaches with a number of large sample procedures as well as both parametric and nonparametric bootstrap procedures in terms of coverage and expected CI length through a simulation study, and illustrate the methodology with two examples from biomedical fields. The results support the use of the PL-based CI as it holds the preassigned confidence level very well and overall gives a very competitive length.