Covariate-constrained randomization is an effective treatment allocation procedure for controlling imbalance across multiple baseline covariates in randomized trials. Motivated by the GroupPMPlus cluster randomized trial, we introduce the asymptotic theory for a broad class of estimators, known as M-estimators, under covariate-constrained randomization. Here, M-estimators refer to estimators obtained by optimizing an objective function, such as a log-likelihood function, and include commonly used methods such as analysis of covariance and linear mixed models. We show that M-estimators remain consistent in this setting but can exhibit non-Gaussian asymptotic distributions depending on the specification. Using examples of common M-estimators, we delineate conditions under which covariate-constrained randomization can be safely ignored in statistical analysis. Our results extend to stratified covariate-constrained randomization and semiparametric efficient estimators based on data-adaptive machine learning methods. We illustrate these theoretical findings using the GroupPMPlus study to evaluate the causal effect of a psychological treatment on mental health outcomes following a disaster.
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