  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.
