Inference in Experiments With Matched Pairs

Abstract

This article studies inference for the average treatment effect in randomized controlled trials where treatment status is determined according to a “matched pairs” design. By a “matched pairs” design, we mean that units are sampled iid from the population of interest, paired according to observed, baseline covariates and finally, within each pair, one unit is selected at random for treatment. This type of design is used routinely throughout the sciences, but fundamental questions about its implications for inference about the average treatment effect remain. The main requirement underlying our analysis is that pairs are formed so that units within pairs are suitably “close” in terms of the baseline covariates, and we develop novel results to ensure that pairs are formed in a way that satisfies this condition. Under this assumption, we show that, for the problem of testing the null hypothesis that the average treatment effect equals a prespecified value in such settings, the commonly used two-sample t-test and “matched pairs” t-test are conservative in the sense that these tests have limiting rejection probability under the null hypothesis no greater than and typically strictly less than the nominal level. We show, however, that a simple adjustment to the standard errors of these tests leads to a test that is asymptotically exact in the sense that its limiting rejection probability under the null hypothesis equals the nominal level. We also study the behavior of randomization tests that arise naturally in these types of settings. When implemented appropriately, we show that this approach also leads to a test that is asymptotically exact in the sense described previously, but additionally has finite-sample rejection probability no greater than the nominal level for certain distributions satisfying the null hypothesis. A simulation study and empirical application confirm the practical relevance of our theoretical results.

Keywords:

• Matched pairs
• Matched pairs t-test
• Permutation test
• Randomized controlled trial
• Treatment assignment
• Two-sample t-test

Supplementary materials

The supplemental materials include (1) a document which provides proofs for all results for the authors’ paper “Inference in Experiments with Matched Pairs,” details for Remark 3.8, as well as some additional simulations and (2) The R code used for simulations in the paper.

Acknowledgments

We thank John Duchi for references to the “blossom” algorithm and Silvia Barbareschi for excellent research assistance. We are additionally grateful to Panos Toulis, two anonymous reviewers, and the associate editor for helpful comments.

Additional information

Funding

The research of the second author is supported by NSF grant MMS-1949845; the research of the third author is supported by NSF grant SES-1530661.