  In a cluster-randomized experiment, treatment is assigned to clusters of
individual units of interest--households, classrooms, villages, etc.--instead
of the units themselves. The number of clusters sampled and the number of units
sampled within each cluster is typically restricted by a budget constraint.
Previous analysis of cluster randomized experiments under the Neyman-Rubin
potential outcomes model of response have assumed a simple random sample of
clusters. Estimators of the population average treatment effect (PATE) under
this assumption are often either biased or not invariant to location shifts of
potential outcomes. We demonstrate that, by sampling clusters with probability
proportional to the number of units within a cluster, the Horvitz-Thompson
estimator (HT) is invariant to location shifts and unbiasedly estimates PATE.
We derive standard errors of HT and discuss how to estimate these standard
errors. We also show that results hold for stratified random samples when
samples are drawn proportionally to cluster size within each stratum. We
demonstrate the efficacy of this sampling scheme using a simulation based on
data from an experiment measuring the efficacy of the National Solidarity
Programme in Afghanistan.
