Counting Defiers in Health Care with a Design-Based Likelihood for the Joint Distribution of Potential Outcomes

We present a design-based model of a randomized experiment in which the observed outcomes are informative about the joint distribution of potential outcomes within the experimental sample. We derive a likelihood function that maintains curvature with respect to the joint distribution of potential outcomes, even when holding the marginal distributions of potential outcomes constant -- curvature that is not maintained in a sampling-based likelihood that imposes a large sample assumption. Our proposed decision rule guesses the joint distribution of potential outcomes in the sample as the distribution that maximizes the likelihood. We show that this decision rule is Bayes optimal under a uniform prior. Our optimal decision rule differs from and significantly outperforms a ``monotonicity'' decision rule that assumes no defiers or no compliers. In sample sizes ranging from 2 to 40, we show that the Bayes expected utility of the optimal rule increases relative to the monotonicity rule as the sample size increases. In two experiments in health care, we show that the joint distribution of potential outcomes that maximizes the likelihood need not include compliers even when the average outcome in the intervention group exceeds the average outcome in the control group, and that the maximizer of the likelihood may include both compliers and defiers, even when the average intervention effect is large and statistically significant.