  Multivariate outcomes are common in pragmatic cluster randomized trials.
While sample size calculation procedures for multivariate outcomes exist under
parallel assignment, none have been developed for a stepped wedge design. In
this article, we present computationally efficient power and sample size
procedures for stepped wedge cluster randomized trials (SW-CRTs) with
multivariate outcomes that differentiate the within-period and between-period
intracluster correlation coefficients (ICCs). Under a multivariate linear mixed
model, we derive the joint distribution of the intervention test statistics
which can be used for determining power under different hypotheses and provide
an example using the commonly utilized intersection-union test for co-primary
outcomes. Simplifications under a common treatment effect and common ICCs
across endpoints and an extension to closed cohort designs are also provided.
Finally, under the common ICC across endpoints assumption, we formally prove
that the multivariate linear mixed model leads to a more efficient treatment
effect estimator compared to the univariate linear mixed model, providing a
rigorous justification on the use of the former with multivariate outcomes. We
illustrate application of the proposed methods using data from an existing
SW-CRT and present extensive simulations to validate the methods.
