Frequentist inference for cluster randomised trials with multiple primary outcomes

The use of a single primary outcome is generally either recommended or required by many influential randomised trial guidelines to avoid the problem of "multiple testing". Without correction, the probability of rejecting at least one of a set of null hypotheses (the family-wise error rate) is often much greater than the nominal rate of any single test so that statistics like p-values and confidence intervals have no reliable interpretation. Cluster randomised trials though may require multiple outcomes to adequately describe the effects of often complex and multi-faceted interventions. We propose a method for inference for cluster randomised trials with multiple outcomes that ensures a nominal family-wise error rate and produces simultaneous confidence intervals with nominal "family-wise" coverage. We adapt the resampling-based stepdown procedure of Romano and Wolf (2005) using a randomisation-test approach within a generalised linear model framework. We then adapt the Robbins-Monro search procedure for confidence interval limits proposed by Garthwaite and Buckland (1996) to this stepdown process to produce a set of confidence intervals. We show that this procedure has nominal error rates and coverage in a simulation-based study of parallel and stepped-wedge cluster randomised studies and compare results from the analysis of a real-world stepped-wedge trial under both the proposed and more standard analyses.