  In cluster-randomized experiments, there is emerging interest in exploring
the causal mechanism in which a cluster-level treatment affects the outcome
through an intermediate outcome. Despite an extensive development of causal
mediation methods in the past decade, only a few exceptions have been
considered in assessing causal mediation in cluster-randomized studies, all of
which depend on parametric model-based estimators. In this article, we develop
the formal semiparametric efficiency theory to motivate several doubly-robust
methods for addressing several mediation effect estimands corresponding to both
the cluster-average and the individual-level treatment effects in
cluster-randomized experiments--the natural indirect effect, natural direct
effect, and spillover mediation effect. We derive the efficient influence
function for each mediation effect, and carefully parameterize each efficient
influence function to motivate practical strategies for operationalizing each
estimator. We consider both parametric working models and data-adaptive machine
learners to estimate the nuisance functions, and obtain semiparametric
efficient causal mediation estimators in the latter case. Our methods are
illustrated via extensive simulations and two completed cluster-randomized
experiments.
