  Numerous publications have now addressed the principles of designing,
analyzing, and reporting the results of, stepped-wedge cluster randomized
trials. In contrast, there is little research available pertaining to the
design and analysis of multi-arm stepped-wedge cluster randomized trials,
utilized to evaluate the effectiveness of multiple experimental interventions.
In this paper, we address this by explaining how the required sample size in
these multi-arm trials can be ascertained when data are to be analyzed using a
linear mixed model. We then go on to describe how the design of such trials can
be optimized to balance between minimizing the cost of the trial, and
minimizing some function of the covariance matrix of the treatment effect
estimates. Using a recently commenced trial that will evaluate the
effectiveness of sensor monitoring in an occupational therapy rehabilitation
program for older persons after hip fracture as an example, we demonstrate that
our designs could reduce the number of observations required for a fixed power
level by up to 58%. Consequently, when logistical constraints permit the
utilization of any one of a range of possible multi-arm stepped-wedge cluster
randomized trial designs, researchers should consider employing our approach to
optimize their trials efficiency.
