  Causal inference analyses often use existing observational data, which in
many cases has some clustering of individuals. In this paper we discuss
propensity score weighting methods in a multilevel setting where within
clusters individuals share unmeasured confounders that are related to treatment
assignment and the potential outcomes. We focus in particular on settings where
models with fixed cluster effects are either not feasible or not useful due to
the presence of a large number of small clusters. We found, both through
numerical experiments and theoretical derivations, that a strategy of grouping
clusters with similar treatment prevalence and estimating propensity scores
within such cluster groups is effective in reducing bias from unmeasured
cluster-level covariates under mild conditions on the outcome model. We apply
our proposed method in evaluating the effectiveness of center-based pre-school
program participation on children's achievement at kindergarten, using the
Early Childhood Longitudinal Study, Kindergarten data.
