D-optimality of unequal versus equal cluster sizes for mixed effects linear regression analysis of randomized trials with clusters in one treatment arm

The efficiency loss due to varying cluster sizes in trials where treatments induce clustering of observations in one of the two treatment arms is examined. Such designs may arise when comparing group therapy to a condition with only medication or a condition not involving any kind of treatment. For maximum likelihood estimation in a mixed effects linear regression, asymptotic relative efficiencies (RE) of unequal versus equal cluster sizes in terms of the D-criterion and D(s)-criteria are derived. A Monte Carlo simulation for small sample sizes shows these asymptotic REs to be very accurate for the D(s)-criterion of the fixed effects and rather accurate for the D-criterion. Taylor approximations of the asymptotic REs turn out to be accurate and can be used to predict the efficiency loss when planning a trial. The RE usually will be more than 0.94 and, when planning sample sizes, multiplying both the number of clusters in one arm and the number of persons in the other arm by 1/RE is the most cost-efficient way of regaining the efficiency loss.