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

Instrumental variable (IV) methods are widely used in medical 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 IV is often applied at the cluster level, for example, hospitals' or physicians' preference for a certain treatment where each hospital or physician naturally defines a cluster. This paper proposes to embed such observational IV data into a cluster-randomized encouragement experiment using nonbipartite 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 (PER), 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 PER estimand in that it does not suffer from 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 coronary artery bypass graft surgery on patients' 30-day mortality rate. R package ivdesign implements the proposed method.