  Cluster-randomized trials (CRTs) involve randomizing entire groups of
participants -- called clusters -- to treatment arms but are often comprised of
a limited or fixed number of available clusters. While covariate adjustment can
account for chance imbalances between treatment arms and increase statistical
efficiency in individually-randomized trials, analytical methods for
individual-level covariate adjustment in small CRTs have received little
attention to date. In this paper, we systematically investigate, through
extensive simulations, the operating characteristics of propensity score
weighting and multivariable regression as two individual-level covariate
adjustment strategies for estimating the participant-average causal effect in
small CRTs with a rare binary outcome and identify scenarios where each
adjustment strategy has a relative efficiency advantage over the other to make
practical recommendations. We also examine the finite-sample performance of the
bias-corrected sandwich variance estimators associated with propensity score
weighting and multivariable regression for quantifying the uncertainty in
estimating the participant-average treatment effect. To illustrate the methods
for individual-level covariate adjustment, we reanalyze a recent CRT testing a
sedation protocol in 31 pediatric intensive care units.
