A rank-based approach to randomized controlled trials with multiple co-primary endpoints of different scales

Randomized controlled trials with co-primary endpoints refer to trials that are designed to evaluate if the intervention is superior to the control on each endpoint. Data analysis and sample size estimation can be complicated when the endpoints are of different scales. In contrast to trials with multiple primary endpoints, where multiplicity is a concern, multiple co-primary endpoints could cause substantial power/efficiency reduction. We propose a rank-based approach to data analysis and sample size estimation for such studies. For each endpoint, we quantify the treatment effect using the win probability (WinP) that a subject in the treatment group has a better score than (or a win over) a subject in the control group. Inference for the endpoint-specific WinPs is carried out by using multivariate linear mixed models with a unstructured variance-covariance matrix for win fractions, which are derived from (mid)ranks and shown to be asymptotically uncorrelated. We focus on confidence intervals (CIs) for WinPs and testing null hypothesis based on whether all lower limits of the CIs are above 0.50. Sample size formulae are derived with the focus on determining the sample size required to guarantee with a pre-specified assurance probability that the lower limit of CI for each endpoint is above 0.50. Results from a simulation study based on a published trial on ulcerative colitis suggest that our approach performed well in terms of CI coverage and assurance probability. The results also show that baseline adjustments can result in a gain in efficiency, but dichotomizing data can decrease efficiency substantially.