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In many studies comparing a new 'target treatment' with a control target treatment, the received treatment does not always agree with assigned treatment-that is, the compliance is imperfect. An obvious example arises when ethical or practical constraints prevent even the randomized assignment of receipt of the new target treatment but allow the randomized assignment of the encouragement to receive this treatment. In fact, many randomized experiments where compliance is not enforced by the experimenter (e.g. with non-blinded assignment) may be more accurately thought of as randomized encouragement designs. Moreover, often the assignment of encouragement is at the level of clusters (e.g. doctors) where the compliance with the assignment varies across the units (e.g. patients) within clusters. We refer to such studies as 'clustered encouragement designs' (CEDs) and they arise relatively frequently (e.g. Sommer and Zeger, 1991; McDonald et al., 1992; Dexter et al., 1998) Here, we propose Bayesian methodology for causal inference for the effect of the new target treatment versus the control target treatment in the randomized CED with all-or-none compliance at the unit level, which generalizes the approach of Hirano et al. (2000) in important and surprisingly subtle ways, to account for the clustering, which is necessary for statistical validity. We illustrate our methods using data from a recent study exploring the role of physician consulting in increasing patients' completion of Advance Directive forms.

Clustered Encouragement Designs with Individual Noncompliance: Bayesian Inference with Randomization, and Application to Advance Directive Forms