  Instrumental variable methods are widely used in medical and social science
research to draw causal conclusions when the treatment and outcome are
confounded by unmeasured confounding variables. One important feature of such
studies is that the instrumental variable is often applied at the cluster
level, e.g., hospitals' or physicians' preference for a certain treatment where
each hospital or physician naturally defines a cluster. This paper proposes to
embed such observational instrumental variable data into a cluster-randomized
encouragement experiment using statistical matching. Potential outcomes and
causal assumptions underpinning the design are formalized and examined. Testing
procedures for two commonly-used estimands, Fisher's sharp null hypothesis and
the pooled effect ratio, are extended to the current setting. We then introduce
a novel cluster-heterogeneous proportional treatment effect model and the
relevant estimand: the average cluster effect ratio. This new estimand is
advantageous over the structural parameter in a constant proportional treatment
effect model in that it allows treatment heterogeneity, and is advantageous
over the pooled effect ratio estimand in that it is immune to Simpson's
paradox. We develop an asymptotically valid randomization-based testing
procedure for this new estimand based on solving a mixed integer
quadratically-constrained optimization problem. The proposed design and
inferential methods are applied to a study of the effect of using
transesophageal echocardiography during CABG surgery on patients' 30-day
mortality rate.
