  The clustered observational study (COS) design is the observational study
counterpart to the clustered randomized trial. In a COS, a treatment is
assigned to intact groups, and all units within the group are exposed to the
treatment. However, the treatment is non-randomly assigned. COSs are common in
both education and health services research. In education, treatments may be
given to all students within some schools but withheld from all students in
other schools. In health studies, treatments may be applied to clusters such as
hospitals or groups of patients treated by the same physician. In this
manuscript, we study the identification of causal effects in clustered
observational study designs. We focus on the prospect of differential selection
of units to clusters, which occurs when the units' cluster selections depend on
the clusters' treatment assignments. Extant work on COSs has made an implicit
assumption that rules out the presence of differential selection. We derive the
identification results for designs with differential selection and that
contexts with differential cluster selection require different adjustment sets
than standard designs. We outline estimators for designs with and without
differential selection. Using a series of simulations, we outline the magnitude
of the bias that can occur with differential selection. We then present two
empirical applications focusing on the likelihood of differential selection.
