Proposed design effects for SW-CRTs with unequal cluster sizes

By multiplying the sample size for an individually randomised trial by the standard DE for CRTs, and assuming equal cluster sizes, the sample size for an individually randomised trial is adjusted for the effect of clustering. The adjusted DEs make additional adjustments for the effect of an imbalance in cluster sizes. A ‘correction term’ can then be found by subtracting the standard DE from each adjusted DE. This gives the component of the DE that adjusts for the effect of an inequality in cluster size. By adding these correction terms to the standard DE for a SW-CRT, the sample size for an individually randomised trial can be adjusted for the effect of an inequality in cluster size, in addition to the effects of the clustering and stepped-wedge design:

where DE^CRT is an adjusted DE for a CRT and is an adjusted DE for a SW-CRT.

Using the cluster and minimum variance adjusted weights DEs, given previously, we propose two adjusted DEs for SW-CRTs with unequal cluster sizes. One uses the CV in cluster size, whereas for the other, the size of each cluster must be specified. The number of subjects in each cluster in the unadjusted DE is replaced by the average cluster size, m¯. The cluster weights adjusted DE is:

and the minimum variance weights adjusted DE is:

where ρ is the ICC, k is the number of steps, t is the number of measurements taken after each step, m¯ is the average cluster size, b is the number of measurements taken at baseline, CV is the coefficient of variation in cluster size, I is the number of cluster and m _i is the size of the i^th cluster. The sample size for a SW-CRT with unequal cluster sizes can then be found by multiplying the required sample size for an individually randomised trial by one of the adjusted DEs.