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Instrumental variable methods are widely used in medical and social scienceresearch to draw causal conclusions when the treatment and outcome areconfounded by unmeasured confounding variables. One important feature of suchstudies is that the instrumental variable is often applied at the clusterlevel, e.g., hospitals' or physicians' preference for a certain treatment whereeach hospital or physician naturally defines a cluster. This paper proposes toembed such observational instrumental variable data into a cluster-randomizedencouragement experiment using statistical matching. Potential outcomes andcausal assumptions underpinning the design are formalized and examined. Testingprocedures for two commonly-used estimands, Fisher's sharp null hypothesis andthe pooled effect ratio, are extended to the current setting. We then introducea novel cluster-heterogeneous proportional treatment effect model and therelevant estimand: the average cluster effect ratio. This new estimand isadvantageous over the structural parameter in a constant proportional treatmenteffect model in that it allows treatment heterogeneity, and is advantageousover the pooled effect ratio estimand in that it is immune to Simpson'sparadox. We develop an asymptotically valid randomization-based testingprocedure for this new estimand based on solving a mixed integerquadratically-constrained optimization problem. The proposed design andinferential methods are applied to a study of the effect of usingtransesophageal echocardiography during CABG surgery on patients' 30-daymortality rate.

Bridging preference-based instrumental variable studies and  cluster-randomized encouragement experiments: study design, noncompliance,  and average cluster effect ratio