Derivation of Sample Size Formula for Cluster Randomized Trials with Binary Responses Using a General Continuity Correction Factor and Identification of Optimal Settings for Small Event Rates

Trials for comparing interventions where cluster of subjects, rather than individuals, are randomized, are commonly called cluster randomized trials (CRTs). For comparison of binary outcomes in a CRT, although there are a few published formulations for sample size computation, the most commonly used is the one developed by Donner, Birkett, and Buck (Am J Epidemiol, 1981) probably due to its incorporation in the text book by Fleiss, Levin, and Paik (Wiley, 2003). In this paper, we derive a new 2 approximation formula with a general continuity correction factor (c) and show that specially for the scenarios of small event rates (< 0:01), the new formulation recommends lower number of clusters than the Donner et al. formulation thereby providing better eciency. All known formulations can be shown to be special cases at specic value of the general correction factor (e.g., Donner formulation is equivalent to the new formulation for c = 1). Statistical simulation is presented with data on comparative ecacy of the available methods identifying correction factors that are optimal for rare event rates. Table of sample size recommendation for variety of rare event rates along with code in\R" language for easy computation of sample size in other settings is also provided. Sample size calculations for a published CRT (\Pathways to Health study" that evaluates the value of intervention for smoking cessation) are computed for various correction factors to illustrate that with an optimal choice of the correction factor, the study could have maintained the same power with a 20% less sample size.