  In this paper, we focus on estimating the average treatment effect (ATE) of a
target population when individual-level data from a source population and
summary-level data (e.g., first or second moments of certain covariates) from
the target population are available. In the presence of heterogeneous treatment
effect, the ATE of the target population can be different from that of the
source population when distributions of treatment effect modifiers are
dissimilar in these two populations, a phenomenon also known as covariate
shift. Many methods have been developed to adjust for covariate shift, but most
require individual covariates from a representative target sample. We develop a
weighting approach based on summary-level information from the target sample to
adjust for possible covariate shift in effect modifiers. In particular, weights
of the treated and control groups within a source sample are calibrated by the
summary-level information of the target sample. Our approach also seeks
additional covariate balance between the treated and control groups in the
source sample. We study the asymptotic behavior of the corresponding weighted
estimator for the target population ATE under a wide range of conditions. The
theoretical implications are confirmed in simulation studies and a real data
application.
