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Causal inference analyses often use existing observational data, which inmany cases has some clustering of individuals. In this paper we discusspropensity score weighting methods in a multilevel setting where withinclusters individuals share unmeasured confounders that are related to treatmentassignment and the potential outcomes. We focus in particular on settings wheremodels with fixed cluster effects are either not feasible or not useful due tothe presence of a large number of small clusters. We found, both throughnumerical experiments and theoretical derivations, that a strategy of groupingclusters with similar treatment prevalence and estimating propensity scoreswithin such cluster groups is effective in reducing bias from unmeasuredcluster-level covariates under mild conditions on the outcome model. We applyour proposed method in evaluating the effectiveness of center-based pre-schoolprogram participation on children's achievement at kindergarten, using theEarly Childhood Longitudinal Study, Kindergarten data.

Partially Pooled Propensity Score Models for Average Treatment Effect Estimation with Multilevel Data