Optimal Multilevel Matching in Clustered Observational Studies: A Case Study of the School Voucher System in Chile∗

A distinctive feature of a clustered observational study is its multilevel or nested data structure arising from the assignment of treatment, in a non-random 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 paper, the optimal strategy is the opposite: first match individuals and, once all possible combinations of matched individuals are known, then match clusters. In this paper we use dynamic and integer programming to implement this strategy and extend optimal matching methods to hierarchical and multilevel settings. In particular, our method attempts to replicate 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 user. Our method 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 method on 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.