  There are multiple cluster randomised trial designs that vary in when the
clusters cross between control and intervention states, when observations are
made within clusters, and how many observations are made at that time point.
Identifying the most efficient study design is complex though, owing to the
correlation between observations within clusters and over time. In this
article, we present a review of statistical and computational methods for
identifying optimal cluster randomised trial designs. We also adapt methods
from the experimental design literature for experimental designs with
correlated observations to the cluster trial context. We identify three broad
classes of methods: using exact formulae for the treatment effect estimator
variance for specific models to derive algorithms or weights for cluster
sequences; generalised methods for estimating weights for experimental units;
and, combinatorial optimisation algorithms to select an optimal subset of
experimental units. We also discuss methods for rounding weights to whole
numbers of clusters and extensions to non-Gaussian models. We present results
from multiple cluster trial examples that compare the different methods,
including problems involving determining optimal allocation of clusters across
a set of cluster sequences, and selecting the optimal number of single
observations to make in each cluster-period for both Gaussian and non-Gaussian
models, and including exchangeable and exponential decay covariance structures.
