Cluster-randomized experiments are widely used due to their logistical convenience and policy relevance. To analyze them properly, we must address the fact that the treatment is assigned at the cluster level instead of the individual level. Standard analytic strategies are regressions based on individual data, cluster averages, and cluster totals, which differ when the cluster sizes vary. These methods are often motivated by models with strong and unverifiable assumptions, and the choice among them can be subjective. Without any outcome modeling assumption, we evaluate these regression estimators and the associated robust standard errors from a design-based perspective where only the treatment assignment itself is random and controlled by the experimenter. We demonstrate that regression based on cluster averages targets a weighted average treatment effect, regression based on individual data is suboptimal in terms of efficiency, and regression based on cluster totals is consistent and more efficient with a large number of clusters. We highlight the critical role of covariates in improving estimation efficiency, and illustrate the efficiency gain via both simulation studies and data analysis. Moreover, we show that the robust standard errors are convenient approximations to the true asymptotic standard errors under the design-based perspective. Our theory holds even when the outcome models are misspecified, so it is model-assisted rather than model-based. We also extend the theory to a wider class of weighted average treatment effects.
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