Design effect in multicenter studies: gain or loss of power?

Background: In a multicenter trial, responses for subjects belonging to a common center are correlated. Such a clustering is usually assessed through the design effect, defined as a ratio of two variances. The aim of this work was to describe and understand situations where the design effect involves a gain or a loss of power. Methods: We developed a design effect formula for a multicenter study aimed at testing the effect of a binary factor (which thus defines two groups) on a continuous outcome, and explored this design effect for several designs (from individually stratified randomized trials to cluster randomized trials, and for other designs such as matched pair designs or observational multicenter studies). Results: The design effect depends on the intraclass correlation coefficient (ICC) (which assesses the correlation between data for two subjects from the same center) but also on a statistic S, which quantifies the heterogeneity of the group distributions among centers (thus the level of association between the binary factor and the center) and on the degree of global imbalance (the number of subjects are then different) between the two groups. This design effect may induce either a loss or a gain in power, depending on whether the S statistic is respectively higher or lower than 1. Conclusion: We provided a global design effect formula applying for any multicenter study and allowing identifying factors - the ICC and the distribution of the group proportions among centers - that are associated with a gain or a loss of power in such studies.