Inference for Matched Tuples and Fully Blocked Factorial Designs

This paper studies inference in randomized controlled trials with multiple treatments, where treatment status is determined according to a "matched tuples" design. Here, by a matched tuples design, we mean an experimental design where units are sampled i.i.d. from the population of interest, grouped into "homogeneous" blocks with cardinality equal to the number of treatments, and finally, within each block, each treatment is assigned exactly once uniformly at random. We first study estimation and inference for matched tuples designs in the general setting where the parameter of interest is a vector of linear contrasts over the collection of average potential outcomes for each treatment. Parameters of this form include but are not limited to standard average treatment effects used to compare one treatment relative to another. We first establish conditions under which a sample analogue estimator is asymptotically normal and construct a consistent estimator of its corresponding asymptotic variance. Combining these results establishes the asymptotic validity of tests based on these estimators. In contrast, we show that a common testing procedure based on a linear regression with block fixed effects and the usual heteroskedasticity-robust variance estimator is invalid in the sense that the resulting test may have a limiting rejection probability under the null hypothesis strictly greater than the nominal level. We then apply our results to study the asymptotic properties of what we call "fully-blocked" $2^K$ factorial designs, which are simply matched tuples designs applied to a full factorial experiment. Leveraging our previous results, we establish that our estimator achieves a lower asymptotic variance under the fully-blocked design than that under any stratified factorial design. A simulation study and empirical application illustrate the practical relevance of our results.