Improving massive experiments with threshold blocking

Inferences from randomized experiments can be improved by blocking: assigning treatment in fixed proportions within groups of similar units. However, the use of the method is limited by the difficulty in deriving these groups. Current blocking methods are restricted to special cases or run in exponential time; are not sensitive to clustering of data points; and are often heuristic, providing an unsatisfactory solution in many common instances. We present an algorithm that implements a widely applicable class of blocking-threshold blocking-that solves these problems. Given a minimum required group size and a distance metric, we study the blocking problem of minimizing the maximum distance between any two units within the same group. We prove this is a nondeterministic polynomial-time hard problem and derive an approximation algorithm that yields a blocking where the maximum distance is guaranteed to be, at most, four times the optimal value. This algorithm runs in O(n log n) time with O(n) space complexity. This makes it, to our knowledge, the first blocking method with an ensured level of performance that works in massive experiments. Whereas many commonly used algorithms form pairs of units, our algorithm constructs the groups flexibly for any chosen minimum size. This facilitates complex experiments with several treatment arms and clustered data. A simulation study demonstrates the efficiency and efficacy of the algorithm; tens of millions of units can be blocked using a desktop computer in a few minutes.