Cluster randomized controlled trials (cRCTs) are designed to evaluate interventions delivered to groups of individuals. A practical limitation of such designs is that the number of available clusters may be small, resulting in an increased risk of baseline imbalance under simple randomization. Constrained randomization overcomes this issue by restricting the allocation to a subset of randomization schemes where sufficient overall covariate balance across comparison arms is achieved with respect to a pre-specified balance metric. However, several aspects of constrained randomization for the design and analysis of multi-arm cRCTs have not been fully investigated. Motivated by an ongoing multi-arm cRCT, we provide a comprehensive evaluation of the statistical properties of model-based and randomization-based tests under both simple and constrained randomization designs in multi-arm cRCTs, with varying combinations of design and analysis-based covariate adjustment strategies. In particular, as randomization-based tests have not been extensively studied in multi-arm cRCTs, we additionally develop most-powerful permutation tests under the linear mixed model framework for our comparisons. Our results indicate that under constrained randomization, both model-based and randomization-based analyses could gain power while preserving nominal type I error rate, given proper analysis-based adjustment for the baseline covariates. The choice of balance metrics and candidate set size and their implications on the testing of the pairwise and global hypotheses are also discussed. Finally, we caution against the design and analysis of multi-arm cRCTs with an extremely small number of clusters, due to insufficient degrees of freedom and the tendency to obtain an overly restricted randomization space.
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