We investigate large-sample properties of treatment effect estimators under unknown interference in randomized experiments. The inferential target is a generalization of the average treatment effect estimand that marginalizes over potential spillover effects. We show that estimators commonly used to estimate treatment effects under no interference are consistent for the generalized estimand for several common experimental designs under limited but otherwise arbitrary and unknown interference. The rates of convergence depend on the rate at which the amount of interference grows and the degree to which it aligns with dependencies in treatment assignment. Importantly for practitioners, the results imply that if one erroneously assumes that units do not interfere in a setting with limited, or even moderate, interference, standard estimators are nevertheless likely to be close to an average treatment effect if the sample is sufficiently large. Conventional confidence statements may, however, not be accurate.
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 blockingthat 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.experimental design | blocking | big data | causal inference
The paper introduces a class of experimental designs that allows experimenters to control the robustness and efficiency of their experiments. The designs build on a recently introduced algorithm in discrepancy theory, the Gram-Schmidt walk. We provide a tight analysis of this algorithm, allowing us to prove important properties of the designs it produces. These designs aim to simultaneously balance all linear functions of the covariates, and the variance of an estimator of the average treatment effect is shown to be bounded by a quantity that is proportional to the loss function of a ridge regression of the potential outcomes on the covariates. No regression is actually conducted, and one may see the procedure as regression adjustment by design. The class of designs is parameterized so to give experimenters control over the worse case performance of the treatment effect estimator. Greater covariate balance is attained by allowing for a less robust design in terms of worst case variance. We argue that the trade-off between robustness and efficiency is an inherent aspect of experimental design. Finally, we provide non-asymptotic tail bounds for the treatment effect estimator under the class of designs we describe.
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