Kelly, T‐L, Pratt, N. A note on sample size calculations for cluster randomised crossover trials with a fixed number of clusters. Statistics in Medicine. 2019; 38: 3342– 3345. doi: 10.1002/sim.8191

Medical interventions are applied frequently within centres such as aged care facilities, hospitals or intensive care units. When conducting a randomised controlled trial within these centres, known as clusters, it is often more convenient to apply the same intervention to all individuals in the whole cluster, rather than to randomise individuals separately. For example, in a busy intensive care unit, it may be impractical or impossible to apply different interventions to different patients. A trial with randomisation allocated at the cluster level is known as a cluster randomised trial. If the trial is conducted in parallel, each cluster experiences only one intervention. However, it may be possible to reduce the sample size of a parallel cluster randomised trial by allowing the intervention in each cluster to switch or “crossover” after a treatment period, so that each cluster serves as its own control. Such trials are called cluster randomised crossover trials. In the simplest case of two interventions A and B and two treatment periods, the clusters are randomised 1:1 to the treatment sequence AB or BA in the first and second periods.
Individuals within clusters may be correlated both within clusters in the same period and between periods in the same cluster. These correlations are measured by the intraclass and interperiod correlation coefficients, respectively. Sample size calculations for cluster randomised crossover trials must account for these correlations by incorporating a “design effect” or variance inflation factor, which increases or “inflates” the sample size relative to an individually randomised trial. The total sample size thus depends on the intraclass and interperiod correlation coefficients and the individually randomised sample size. One further piece of information is needed, either the number of individuals per period in each cluster, known as the cluster size, or the number of clusters which need to be recruited.
Girardeau, Ravaud and Donner in 2008 derived a formula for sample size calculations for cluster randomised crossover trials that requires the cluster size to be specified in advance. However, in many randomised trials, the number of clusters is fixed in some way, but the cluster size is not. It is often more convenient to calculate the cluster size using a known number of clusters, rather than the other way around. A version of the Girardeau formula, which can be used when the number of clusters is fixed, is presented in a forthcoming article in Statistics in Medicine. To help in the design, planning and recruitment of cluster randomised crossover trials, formulae are given for the minimum feasible number of clusters, the maximum cluster size and the relationship between the correlation coefficients when there are constraints on both the number of clusters and the cluster size. Practical examples of how to use the formulae are given, including a worked example of a trial in Australia and New Zealand with a maximum of 37 intensive care units available.