This paper studies inference in a randomized controlled trial with multiple treatments, where treatment status is determined according to a ‘matched tuple’ design. Here, a matched tuple plan means that units are sampled by iid from the population of interest, grouped into “homogeneous” blocks with cardinality equal to the number of treatments, and finally assigned to each treatment within each block. means experimental design. just uniformly random. We first study the estimation and inference of the matched tuple design in the general setting where the parameter of interest is the vector of linear contrasts against the collection of average potential outcomes for each treatment. Parameters in this form include, but are not limited to, standard mean treatment effects used to compare one treatment to another. First, establish conditions under which the sample analog estimator is asymptotically normal, and construct a consistent estimator of the corresponding asymptotic variance. Combining these results establishes the asymptotic validity of tests based on these estimators. In contrast, general testing procedures based on linear regression with block fixed effects and the usual heteroskedasticity robust variance estimator allow the resulting test to have strictly large marginal rejection probabilities under the null hypothesis. Nominal level indicating that it is disabled in the sense that it is We then apply the results to examine the asymptotic properties of what is called a “fully blocked” $2^K$ factorial design. This is a simple matched tuple design applied to a full factorial experiment. Leveraging previous results, we establish that the estimator achieves lower asymptotic variances in fully blocked designs than in stratified factorial designs. Simulation studies and empirical applications demonstrate the practical relevance of our results.
