Exact confidence intervals for the average causal effect on a binary outcome

Based on the physical randomization of completely randomized experiments, Rigdon and Hudgens (2015) propose two approaches to obtaining exact confidence intervals for the average causal effect on a binary outcome. They construct the first confidence interval by combining, with the Bonferroni adjustment, the prediction sets for treatment effects among treatment and control groups, and the second one by inverting a series of randomization tests. With sample size $n$, their second approach requires performing $O(n^4)$ randomization tests. We demonstrate that the physical randomization also justifies other ways to constructing exact confidence intervals that are more computationally efficient. By exploiting recent advances in hypergeometric confidence intervals and the stochastic order information of randomization tests, we propose approaches that either do not need to invoke Monte Carlo, or require performing at most $O(n^2)$ randomization tests. We provide technical details and R code in the Supplementary Material.