  We consider multi-site randomized controlled trials with a large number of
small sites and imperfect compliance, conducted in non-random convenience
samples in each site. We show that an Empirical-Bayes (EB) estimator can be
used to estimate a lower bound of the variance of intention-to-treat (ITT)
effects across sites. We also propose bounds for the coefficient from a
regression of site-level ITTs on sites' control-group outcome. Turning to local
average treatment effects (LATEs), the EB estimator cannot be used to estimate
their variance, because site-level LATE estimators are biased. Instead, we
propose two testable assumptions under which the LATEs' variance can be written
as a function of sites' ITT and first-stage (FS) effects, thus allowing us to
use an EB estimator leveraging only unbiased ITT and FS estimators. We revisit
Behaghel et al. (2014), who study the effect of counselling programs on job
seekers job-finding rate, in more than 200 job placement agencies in France. We
find considerable ITT heterogeneity, and even more LATE heterogeneity: our
lower bounds on ITTs' (resp. LATEs') standard deviation are more than three
(resp. four) times larger than the average ITT (resp. LATE) across sites. Sites
with a lower job-finding rate in the control group have larger ITT effects.
