Design and analysis of trials with a partially nested design and a binary outcome measure

Where treatments are administered to groups of patients or by therapists, outcomes for patients in the same group or treated by the same therapist may be more similar, leading to clustering. Trials of such treatments should take account of this effect. Where such a treatment is compared with an un-clustered treatment, the trial has a partially nested design. This paper compares statistical methods for this design where the outcome is binary. Investigation of consistency reveals that a random coefficient model with a random effect for group or therapist is not consistent with other methods for a null treatment effect, and so this model is not recommended for this design. Small sample performance of a cluster adjusted test of proportions, a summary measures test, and logistic generalized estimating equations and random intercept models are investigated through simulation. The expected treatment effect is biased for the logistic models. Empirical test size of two-sided tests is only raised slightly, but there are substantial biases for one-sided tests. Three formulae are proposed for calculating sample size and power based on (i) difference proportions, (ii) the log-odds ratio or (iii) the arc-sine transformation of proportions. Calculated power from the formulae is compared with empirical power from a simulations study. Logistic models appeared to perform better than those based on proportions with the likelihood ratio test performing best in the range of scenarios considered. For these analyses, the log-odds ratio method of calculation of power gave an approximate lower limit for empirical power.