Design and analysis of non-pharmacological treatment trials with multiple therapists per patient

In trials of physical and talking therapies, nesting of patients within therapists has statistical implications analogous to those of cluster randomised trials. Nevertheless, the clustering effect may be more complex, as it interacts with treatment. For some therapies, individual patients may receive care from multiple therapists of the same type, so that patients are no longer strictly nested within therapists, creating a 'multiple-membership' relationship between patients and therapists. This paper considers methods of analysis and sample size estimation for trials with multiple-membership clustering effects. It is motivated by a trial of a psychotherapy for the treatment of adolescent depression with cognitive behavioural therapy. We tested methods and issues in a Monte Carlo simulation study, simulating trials with multiple membership. Results demonstrate satisfactory performance in terms of convergence and give estimates of the intra-cluster correlation coefficient and empirical test size similar to a simple hierarchical design. We derive formulae for sample size and power for multiple-membership trial designs. We then compare estimates of power from this formula with empirical power derived from the simulation study. Finally, we show that we can easily extend formulae for sample size and power to allow consideration of power and sample size for certain types of more complex interventions. These include situations where therapists of different types deliver separate components of the intervention, creating a cross-classified relationship, or where several therapists deliver a group-administered treatment, creating further levels.