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A distinctive feature of a clustered observational study is its multilevel ornested data structure arising from the assignment of treatment, in a non-randommanner, to groups or clusters of units or individuals. Examples are ubiquitousin the health and social sciences including patients in hospitals, employees infirms, and students in schools. What is the optimal matching strategy in aclustered observational study? At first thought, one might start by matchingclusters of individuals and then, within matched clusters, continue by matchingindividuals. But as we discuss in this paper, the optimal strategy is theopposite: in typical applications, where the intracluster correlation is notperfect, it is best to first match individuals and, once all possiblecombinations of matched individuals are known, then match clusters. In thispaper we use dynamic and integer programming to implement this strategy andextend optimal matching methods to hierarchical and multilevel settings. Amongother matched designs, our strategy can approximate a paired clusteredrandomized study by finding the largest sample of matched pairs of treated andcontrol individuals within matched pairs of treated and control clusters thatis balanced according to specifications given by the investigator. Thisstrategy directly balances covariates both at the cluster and individual levelsand does not require estimating the propensity score, although the propensityscore can be balanced as an additional covariate. We illustrate our resultswith a case study of the comparative effectiveness of public versus privatevoucher schools in Chile, a question of intense policy debate in the country atthe present.

Optimal Multilevel Matching in Clustered Observational Studies: A Case  Study of the Effectiveness of Private Schools Under a Large-Scale Voucher  System