# Cluster Unit Randomized Trials

## 7. Sample Size Assessment

The usual approach to sample size estimation for cluster randomization trials is to multiply formulas found in standard clinical trial textbooks by an estimate of the variance inflation factor

VIF = [1+(m-1) ρ]. For example, the number of subjects required to compare two means in a completely randomized design that allocates clusters of size m to each of two groups is given by

where denotes the magnitude of difference to be detected, σ^{2} denotes the variance of the targeted outcome measure, and Z_{a/2},Z_{β}_{ }denote the critical values of the standard normal distribution corresponding to a two-sided significance test with error rate α and power 1-β, respectively. Equivalently, the required number of clusters per group is given by k=n/m.

The formula above may also be written as

Z_{β}= {km/{[1+(m-1) ρ]2σ^{2}}}^{½}|| – Z_{α}_{/2}

which more directly shows the increase on trial power (corresponding to increasing values of Z_{β}) obtained by varying the values of k and m. This version of the formula makes it clear that while power can be improved indefinitely by increasing the number of clusters randomized k, increasing their size m can only increase power to a certain point, as limited by the values of k and ρ. Indeed, one can show that even if all clusters enrolled are (theoretically) of infinite size, it will be impossible to achieve a power of 80% if the number of randomized clusters is insufficiently large.