  Causal or unconfounded descriptive comparisons between multiple groups are
common in observational studies. Motivated from a racial disparity study in
health services research, we propose a unified propensity score weighting
framework, the balancing weights, for estimating causal effects with multiple
treatments. These weights incorporate the generalized propensity scores to
balance the weighted covariate distribution of each treatment group, all
weighted toward a common pre-specified target population. The class of
balancing weights include several existing approaches such as the inverse
probability weights and trimming weights as special cases. Within this
framework, we propose a set of target estimands based on linear contrasts. We
further develop the generalized overlap weights, constructed as the product of
the inverse probability weights and the harmonic mean of the generalized
propensity scores. The generalized overlap weighting scheme corresponds to the
target population with the most overlap in covariates across the multiple
treatments. These weights are bounded and thus bypass the problem of extreme
propensities. We show that the generalized overlap weights minimize the total
asymptotic variance of the moment weighting estimators for the pairwise
contrasts within the class of balancing weights. We consider two balance check
criteria and propose a new sandwich variance estimator for estimating the
causal effects with generalized overlap weights. We apply these methods to
study the racial disparities in medical expenditure between several racial
groups using the 2009 Medical Expenditure Panel Survey (MEPS) data. Simulations
were carried out to compare with existing methods.
