  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: in typical applications, where the intracluster correlation is not
perfect, it is best to 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. 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.
