Matching One Sample According to Two Criteria in Observational Studies

Multivariate matching has two goals (i) to construct treated and control groups that have similar distributions of observed covariates, and (ii) to produce matched pairs or sets that are homogeneous in a few key covariates. When there are only a few binary covariates, both goals may be achieved by matching exactly for these few covariates. Commonly, however, there are many covariates, so goals (i) and (ii) come apart, and must be achieved by different means. As is also true in a randomized experiment, similar distributions can be achieved for a high-dimensional covariate, but close pairs can be achieved for only a few covariates. We introduce a new polynomial-time method for achieving both goals that substantially generalizes several existing methods; in particular, it can minimize the earthmover distance between two marginal distributions. The method involves minimum cost flow optimization in a network built around a tripartite graph, unlike the usual network built around a bipartite graph. In the tripartite graph, treated subjects appear twice, on the far left and the far right, with controls sandwiched between them, and efforts to balance covariates are represented on the right, while efforts to find close individual pairs are represented on the left. In this way, the two efforts may be pursued simultaneously without conflict. The method is applied to our on-going study in the Medicare population of the relationship between superior nursing and sepsis mortality. The match2C package in R implements the method. Supplementary materials for this article are available online.