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Group-randomized trials are randomized studies that allocate intact groups of individuals to different comparison arms. A frequent practical limitation to adopting such research designs is that only a limited number of groups may be available, and therefore, simple randomization is unable to adequately balance multiple group-level covariates between arms. Therefore, covariate-based constrained randomization was proposed as an allocation technique to achieve balance. Constrained randomization involves generating a large number of possible allocation schemes, calculating a balance score that assesses covariate imbalance, limiting the randomization space to a prespecified percentage of candidate allocations, and randomly selecting one scheme to implement. When the outcome is binary, a number of statistical issues arise regarding the potential advantages of such designs in making inference. In particular, properties found for continuous outcomes may not directly apply, and additional variations on statistical tests are available. Motivated by two recent trials, we conduct a series of Monte Carlo simulations to evaluate the statistical properties of model-based and randomization-based tests under both simple and constrained randomization designs, with varying degrees of analysis-based covariate adjustment. Our results indicate that constrained randomization improves the power of the linearization F-test, the KC-corrected GEE t-test (Kauermann and Carroll, 2001, Journal of the American Statistical Association 96, 1387-1396), and two permutation tests when the prognostic group-level variables are controlled for in the analysis and the size of randomization space is reasonably small. We also demonstrate that constrained randomization reduces power loss from redundant analysis-based adjustment for non-prognostic covariates. Design considerations such as the choice of the balance metric and the size of randomization space are discussed. [ABSTRACT FROM AUTHOR]