  In this article, we propose the outcome-adjusted balance measure to perform
model selection for the generalized propensity score (GPS), which serves as an
essential component in estimation of the pairwise average treatment effects
(ATEs) in observational studies with more than two treatment levels. The
primary goal of the balance measure is to identify the GPS model specification
such that the resulting ATE estimator is consistent and efficient. Following
recent empirical and theoretical evidence, we establish that the optimal GPS
model should only include covariates related to the outcomes. Given a
collection of candidate GPS models, the outcome-adjusted balance measure
imputes all baseline covariates by matching on each candidate model, and
selects the model that minimizes a weighted sum of absolute mean differences
between the imputed and original values of the covariates. The weights are
defined to leverage the covariate-outcome relationship, so that GPS models
without optimal variable selection are penalized. Under appropriate
assumptions, we show that the outcome-adjusted balance measure consistently
selects the optimal GPS model, so that the resulting GPS matching estimator is
asymptotically normal and efficient. We compare its finite sample performance
with existing measures in a simulation study. We illustrate an application of
the proposed methodology in the analysis of the Tutoring data.
