Relative efficiency of unequal cluster sizes for variance component estimation in cluster randomized and multicentre trials

Cluster randomized and multicentre trials evaluate the effect of a treatment oil persons nested within clusters, for instance patients within clinics or Pupils within schools. Although equal sample sizes per cluster are generally optimal for parameter estimation, they, are rarely feasible. This paper addresses the relative efficiency (RE) of unequal versus equal cluster sizes for estimating variance components in cluster randomized trials and in multicentre trials with person randomization within centres, assuming a quantitative outcome. Starting from maximum likelihood estimation, the RE is investigated numerically, for a range of cluster size distributions. An approximate formula is presented for computing the RE as a function of the mean and variance of cluster sizes and the intraclass correlation. The accuracy of this approximation is checked and found to be good. It is concluded that the loss of efficiency, for variance component estimation due to variation of cluster sizes rarely exceeds 20% and call be compensated by sampling 25% more clusters