Theory of general balance applied to step wedge designs

A standard idealized step-wedge design satisfies the requirements, in terms of the structure of the observation units, to be considered a balanced design and can be labeled as a criss-cross design (time crossed with cluster) with replication. As such, Nelder's theory of general balance can be used to decompose the analysis of variance into independent strata (grand mean, cluster, time, cluster:time, residuals). If time is considered as a fixed effect, then the treatment effect of interest is estimated solely within the cluster and time:cluster strata; the time effects are estimated solely within the time stratum. This separation leads directly to scalar, rather than matrix, algebraic manipulations to provide closed-form expressions for standard errors of the treatment effect estimate. We use the tools provided by the theory of general balance to obtain an expression for the standard error of the estimated treatment effect in a general case where the assumed covariance structure includes random-effects at the time and time:cluster levels. This provides insights that are helpful for experimental design regarding the assumed correlation within clusters over time, sample size in terms of numbers of clusters and replication within cluster, and components of the standard error for estimated treatment effect.