  In cluster randomized experiments, units are often recruited after the random
cluster assignment, and data are only available for the recruited sample.
Post-randomization recruitment can lead to selection bias, inducing systematic
differences between the overall and the recruited populations, and between the
recruited intervention and control arms. In this setting, we define causal
estimands for the overall and the recruited populations. We first show that if
units select their cluster independently of the treatment assignment, cluster
randomization implies individual randomization in the overall population. We
then prove that under the assumption of ignorable recruitment, the average
treatment effect on the recruited population can be consistently estimated from
the recruited sample using inverse probability weighting. Generally we cannot
identify the average treatment effect on the overall population. Nonetheless,
we show, via a principal stratification formulation, that one can use weighting
of the recruited sample to identify treatment effects on two meaningful
subpopulations of the overall population: units who would be recruited into the
study regardless of the assignment, and units who would be recruited in the
study under treatment but not under control. We develop a corresponding
estimation strategy and a sensitivity analysis method for checking the
ignorable recruitment assumption.
