Fast computation of exact confidence intervals for randomized experiments with binary outcomes

Given a randomized experiment with binary outcomes, exact confidence intervals for the average causal effect of the treatment can be computed through a series of permutation tests. This approach requires minimal assumptions and is valid for all sample sizes, as it does not rely on large-sample approximations such as those implied by the central limit theorem. We show that these confidence intervals can be found in $O(nlogn)$ permutation tests in the case of balanced designs, where the treatment and control groups have equal sizes, and $O\left(n^2\right)$ permutation tests in the general case. Prior to this work, the most efficient known constructions required $O\left(n^2\right)$ such tests in the balanced case (Li and Ding, 2016), and $O\left(n^4\right)$ tests in the general case (Rigdon and Hudgens, 2015). Our results thus facilitate exact inference as a viable option for randomized experiments far larger than those accessible by previous methods. We also generalize our construction to produce confidence intervals for other causal estimands, including the relative risk ratio and odds ratio, yielding similar computational gains.