A note on statistical power in multi-site randomized trials with multiple treatments at each site

We derive the statistical power functions in multi-site randomized trials with multiple treatments at each site, using multi-level modelling. An F statistic is used to test multiple parameters in the multi-level model instead of the Wald chi square test as suggested in the current literature. The F statistic is shown to be more conservative than the Wald statistic in testing any overall treatment effect among the multiple study conditions. In addition, we improvise an easy way to estimate the non-centrality parameters for the means comparison t-tests and the F test, using Helmert contrast coding in the multi-level model. The variance of treatment means, which is difficult to fathom but necessary for power analysis, is decomposed into intuitive simple effect sizes in the contrast tests. The method is exemplified by a multi-site evaluation study of the behavioural interventions for cannabis dependence.