Constructing a confidence interval for the fraction who benefit from treatment, using randomized trial data.
The fraction who benefit from treatment is the proportion of patients whose potential outcome under treatment is better than that under control. Inference on this parameter is challenging since it is only partially identifiable, even in our context of a randomized trial. We propose a new method for constructing a confidence interval for the fraction, when the outcome is ordinal or binary. Our confidence interval procedure is pointwise consistent. It does not require any assumptions about the joint distribution of the potential outcomes, although it has the flexibility to incorporate various user-defined assumptions. Our method is based on a stochastic optimization technique involving a second-order, asymptotic approximation that, to the best of our knowledge, has not been applied to biomedical studies. This approximation leads to statistics that are solutions to quadratic programs, which can be computed efficiently using optimization tools. In simulation, our method attains the nominal coverage probability or higher, and can have narrower average width than competitor methods. We apply it to a trial of a new intervention for stroke.