Varying cluster sizes in trials with clusters on one treatment arm: sample size adjustments when testing treatment effects with linear mixed models.

Trials in which treatments induce clustering of observations in one of two treatment arms, such as when comparing group therapy with pharmacological treatment or with a waiting-list group, are examined with respect to the efficiency loss caused by varying cluster sizes. When observations are (approximately) normally distributed, treatment effects can be estimated and tested through linear mixed model analysis. For maximum likelihood estimation, the asymptotic relative efficiency of unequal versus equal cluster sizes is derived. In an extensive Monte Carlo simulation for small sample sizes, the asymptotic relative efficiency turns out to be accurate for the treatment effect, but less accurate for the random intercept variance. For the treatment effect, the efficiency loss due to varying cluster sizes rarely exceeds 10 per cent, which can be regained by recruiting 11 per cent more clusters for one arm and 11 per cent more persons for the other. For the intercept variance the loss can be 16 per cent, which requires recruiting 19 per cent more clusters for one arm, with no additional recruitment of subjects for the other arm.