Using the cross-match test to appraise covariate balance in matched Pairs

Having created a tentative matched design for an observational study, diagnostic checks are performed to see whether observed covariates exhibit reasonable balance, or alternatively whether further effort is required to improve the match. We illustrate the use of the cross-match test as an aid to appraising balance on high-dimensional covariates, and we discuss its close logical connections to the techniques used to construct matched samples. In particular, in addition to a significance level, the cross-match test provides an interpretable measure of high-dimensional covariate balance, specifically a measure defined in terms of the propensity score. An example from the economics of education is used to illustrate. In the example, imbalances in an initial match guide the construction of a better match. The better match uses a recently proposed technique, optimal tapered matching, that leaves certain possibly innocuous covariates imbalanced in one match but not in another, and yields a test of whether the imbalances are actually innocuous.