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Efficiency loss because of varying cluster size in cluster randomized trials is smaller than literature suggests

Sample size formulae for cluster randomized trials are based on the assumption of equal cluster sizes, but in practice this assumption will rarely be met. Over the last decade at least 15 publications have appeared that discuss the amount of efficiency loss for treatment evaluation that is due to cluster size variation in cluster randomized trials [1–15]. Eight of these 15 papers appeared in Statistics in Medicine (SiM), of which the last one was in 2010 [14]. Ten of the 15 papers use the same approximation for the efficiency loss, based on an intraclass correlation (ICC) that goes to zero [2–7,9,12,14,15]. In a paper in SiM 2007 we showed this approximation to be very inaccurate and we proposed two simple alternatives [8]. One of these can be combined with the popular inaccurate approximation to get a better one, and the other alternative is quite accurate in itself. In another SiM paper in 2010 [13] we showed this last alternative to apply to binary outcomes as well. The recent appearance of three papers using the inaccurate approximation [12, 14, 15], of which one in SiM, suggests that our alternatives have gone unnoticed. This letter therefore summarizes the approximations and demonstrates their behavior on a realistic example, the distribution of general practice list size in the UK as published in [1]. We hope that this contributes to the abandonment of the inaccurate approach in favor of one of the alternatives that we offer. According to the popular approach, the relative efficiency (RE) of unequal versus equal cluster sizes is: