Bayesian design and analysis of two-arm cluster randomised trials using assurance: Extension to binary outcomes and...

Bayesian designs for clinical trials using assurance to choose the sample size have been proposed in various trial contexts. Assurance allows for the incorporation of uncertainty on both the treatment effect and nuisance parameters into the sample size calculation. In the case of two-arm cluster randomised trials with continuous outcomes, assurance has been proposed with both a frequentist analysis (hybrid designs) and a Bayesian analysis (fully Bayesian designs). A Bayesian analysis in this context ensures a consistent treatment of probability throughout the design and analysis of the trial. In the fully Bayesian design, inference has been achieved via Markov chain Monte Carlo sampling, and since assurance itself is evaluated via simulation, the result is a computationally intensive and often slow-to-run approach. In the case of two-arm cluster randomised trials with binary outcomes, assurance has not yet been explored to specify sample sizes, either in the hybrid or fully Bayesian case.

This article considers fully Bayesian designs for two-arm cluster randomised trials with continuous and binary outcomes. For the analysis of the trial, we use a (generalised) linear mixed-effects model. We summarise the inference for the treatment effect based on quantiles of the posterior distribution. We use assurance to choose the sample size. In the continuous case, we investigate Integrated Nested Laplace Approximations for inference to speed up calculation of the assurance and compare Integrated Nested Laplace Approximations in computation time and accuracy to Markov chain Monte Carlo. In the binary case, we develop the first fully Bayesian design for cluster randomised trials and conduct a similar comparison between Integrated Nested Laplace Approximations and Markov chain Monte Carlo. We demonstrate our novel approach using assurance to choose sample sizes for the SPEEDY cluster randomised trial, based on the results of a formal prior elicitation exercise with two clinical experts.

We report comparisons of Integrated Nested Laplace Approximations and Markov chain Monte Carlo for a range of different scenarios for cluster randomised controlled trials (RCTs), to determine when each inference scheme should be used, balancing the computational cost in terms of speed and accuracy. Overall Markov chain Monte Carlo with a very large number of samples produces very accurate inference but does not scale well in terms of computational speed compared to Integrated Nested Laplace Approximations. Based on our simulation study, we recommend that Integrated Nested Laplace Approximations is used for inference in cluster trials with binary outcomes and large (n> 500) cluster trials with continuous outcomes, and that Markov chain Monte Carlo is used in smaller (n≤500) cluster trials with continuous outcomes. Our case study demonstrated how to incorporate the uncertainty of trial clinicians into the sample size calculation to give an overall assessment of the likelihood of success of the trial.

A fully Bayesian design can be used for two-arm cluster trials with both continuous and binary outcomes. Integrated Nested Laplace Approximations can allow for more efficient assessment of the assurance for cluster trials with binary outcomes and large cluster trials with continuous outcomes, without loss of accuracy in inference. A fully Bayesian design of a cluster randomised trial provides a coherent design and analysis framework and incorporates uncertainty in model parameters when choosing the sample size.