For most of this century, randomization has been a cornerstone of scientific experimentation, especially when dealing with humans as experimental units. In practice, however, noncompliance is relatively common with human subjects, complicating traditional theories of inference that require adherence to the random treatment assignment. In this paper we present Bayesian inferential methods for causal estimands in the presence of noncompliance, when the binary treatment assignment is random and hence ignorable, but the binary treatment received is not ignorable. We assume that both the treatment assigned and the treatment received are observed. We describe posterior estimation using EM and data augmentation algorithms. Also, we investigate the role of two assumptions often made in econometric instrumental variables analyses, the exclusion restriction and the monotonicity assumption, without which the likelihood functions generally have substantial regions of maxima. We apply our procedures to real and artificial data, thereby demonstrating the technology and showing that our new methods can yield valid inferences that differ in practically important ways from those based on previous methods for analysis in the presence of noncompliance, including intention-to-treat analyses and analyses based on econometric instrumental variables techniques. Finally, we perform a simulation to investigate the operating characteristics of the competing procedures in a simple setting, which indicates relatively dramatic improvements in frequency operating characteristics attainable using our Bayesian procedures.
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