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

A distinctive feature of a clustered observational study is its multilevel or nested data structure arising from the assignment of treatment, in a nonrandom manner, to groups or clusters of units or individuals. Examples are ubiquitous in the health and social sciences including patients in hospitals, employees in firms, and students in schools. What is the optimal matching strategy in a clustered observational study? At first thought, one might start by matching clusters of individuals and then, within matched clusters, continue by matching individuals. But as we discuss in this article, the optimal strategy is the opposite: in typical applications, where the intracluster correlation is not one, it is best to first match individuals and, once all possible combinations of matched individuals are known, then match clusters. In this article, we use dynamic and integer programming to implement this strategy and extend optimal matching methods to hierarchical and multilevel settings. Among other matched designs, our strategy can approximate a paired clustered randomized study by finding the largest sample of matched pairs of treated and control individuals within matched pairs of treated and control clusters that is balanced according to specifications given by the investigator. This strategy directly balances covariates both at the cluster and individual levels and does not require estimating the propensity score, although the propensity score can be balanced as an additional covariate. We illustrate our results with a case study of the comparative effectiveness of public versus private voucher schools in Chile, a question of intense policy debate in the country at the present.