  Cluster-randomized experiments are increasingly used to evaluate
interventions in routine practice conditions, and researchers often adopt
model-based methods with covariate adjustment in the statistical analyses.
However, the validity of model-based covariate adjustment is unclear when the
working models are misspecified, leading to ambiguity of estimands and risk of
bias. In this article, we first adapt two conventional model-based methods,
generalized estimating equations and linear mixed models, with weighted
g-computation to achieve robust inference for cluster-average and
individual-average treatment effects. To further overcome the limitations of
model-based covariate adjustment methods, we propose an efficient estimator for
each estimand that allows for flexible covariate adjustment and additionally
addresses cluster size variation dependent on treatment assignment and other
cluster characteristics. Such cluster size variations often occur
post-randomization and, if ignored, can lead to bias of model-based estimators.
For our proposed efficient covariate-adjusted estimator, we prove that when the
nuisance functions are consistently estimated by machine learning algorithms,
the estimator is consistent, asymptotically normal, and efficient. When the
nuisance functions are estimated via parametric working models, the estimator
is triply-robust. Simulation studies and analyses of three real-world
cluster-randomized experiments demonstrate that the proposed methods are
superior to existing alternatives.
