Using the half normal distribution to quantify covariate balance in cluster-randomized pragmatic trials

Background: Pragmatic trials often consist of cluster-randomized controlled trials (C-RCTs), where staff of existing clinics or sites deliver interventions and randomization occurs at the site level. Covariate-constrained randomization (CCR) methods are often recommended to minimize imbalance on important site characteristics across intervention and control arms because sizable imbalances can occur by chance in simple randomizations when the number of units to be randomized is relatively small. CCR methods involve multiple random assignments initially, an assessment of balance achieved on site-level covariates from each randomization, and the final selection of an allocation that produces acceptable balance. However, no clear consensus exists on how to assess imbalance or identify allocations with sufficient balance. In this article, we describe an overall imbalance index (I) that is based on the mean of the absolute value of the standardized differences in means on the site characteristics. Methods: We derive the theoretical distribution of I, then conduct simulation studies to examine its empirical properties under the varying covariate distributions and inter-correlations. Results: I has an expected value of 0.798 and, assuming independent site characteristics, a variance of 0.363/k, where k is the number of site characteristics being balanced. Simulations indicated that the properties of I are robust under varying covariate circumstances as long as k is greater than 3 and the covariates are not too highly inter-correlated. Conclusions: We recommend that values of I below the 10th percentile indicate sufficient overall site balance in CCRs. Definitions of acceptable randomizations might also include individual covariate criteria specified in advance, in addition to overall balance criteria.