Sample size calculation for one‐armed clinical trials with clustered data and binary outcome

The formula of Fleiss and Cuzick (1979) to estimate the intraclass correlation coefficient is applied to reduce the task of sample size calculation for clustered data with binary outcome. It is demonstrated that this approach reduces the complexity of sample size calculation to the determination of the null and alternative hypothesis and the formulation of the quantitative influence of the belonging to the same cluster on the therapy success probability.

where  denotes the probability that two experimental units from the same cluster have the same response (therapy success yes/no) and  denotes the probability that two experimental units from different clusters have the same response.Given  as the overall probability for a therapy success, it holds that  = (same response | different cluster) = (both experimental units have therapy success) + (both experimental units have therapy failure) The computation of  is slightly more difficult.Assume that within the same cluster, the probability for a therapy success is  1  for  1 ∈ (0, 1∕) if a reference experimental unit from the same cluster has a therapy success and  2  for  2 ∈ (0, 1∕)if a reference experimental unit from the same cluster does not have a therapy success.Then it holds that  = (therapy success) = (therapy success | reference experimental unit has therapy success) + (therapy success | reference experimental unit has therapy failure) It follows that  = (same response | same cluster) = (two successes | same cluster) + (two failures | same cluster) One can deduce that This implies that to compute the  it suffices to estimate the overall probability for a therapy success ( 0 under  0 and  1 for power calculations) and the factor  1 with which the probability of a therapy success increases if a reference experimental unit from the same cluster has a therapy success.Note that applying this formula facilitates the communication between statistician and clinician since the  can be computed by answering the question "How much does the belonging to the same cluster influence the success probability?"which is much easier to answer than a blind guess of the .
In practice, one will estimate  1 and compute the  for  0 and  1 .Taking the maximum of the two resulting values ensures type I error rate control and fulfillment of the power condition.Eventually, one can follow the standard procedure to multiply the unclustered sample size  init by the design effect (Kieser, 2020, chap. 13) that is also applicable to binary data (Donner et al., 1981) to obtain the final sample size  final = (1 + ( − 1) ⋅ ) ⋅  init .
Of note, the presented calculation implicitly assumes that all experimental units within the same cluster are exchangeable.If there were subclusters such as quadrants in dentistry or additional covariates,  1 or  2 might be different for different experimental units within the same cluster.In those more complex cases, one may calculate the sample size based on simulation.
As last point, it should be mentioned that the presented methodology can be applied to a multiarm trial as well by starting with the corresponding sample size for independent data in a multiarm trial instead of the binomial test in the one-arm setting.The crucial point is that the computation of the  by applying the formula of Fleiss and Cuzick reduces to the estimation of the influence of the belonging to the same cluster on the success probability.

A C K N O W L E D G M E N T S
Open access funding enabled and organized by Projekt DEAL.

C O N F L I C T O F I N T E R E S T S TAT E M E N T
The author states that there is no conflict of interest.

D ATA AVA I L A B I L I T Y S TAT E M E N T
Data sharing is not applicable to this article as no new data were created or analyzed in this study.