  Data analysis based on information from several sources is common in economic
and biomedical studies. This setting is often referred to as the data fusion
problem, which differs from traditional missing data problems since no complete
data is observed for any subject. We consider a regression analysis when the
outcome variable and some covariates are collected from two different sources.
By leveraging the common variables observed in both data sets, doubly robust
estimation procedures are proposed in the literature to protect against
possible model misspecifications. However, they employ only a single propensity
score model for the data fusion process and a single imputation model for the
covariates available in one data set. It may be questionable to assume that
either model is correctly specified in practice. We therefore propose an
approach that calibrates multiple propensity score and imputation models to
gain more protection based on empirical likelihood methods. The resulting
estimator is consistent when any one of those models is correctly specified and
is robust against extreme values of the fitted propensity scores. We also
establish its asymptotic normality property and discuss the semiparametric
estimation efficiency. Simulation studies show that the proposed estimator has
substantial advantages over existing doubly robust estimators, and an assembled
U.S. household expenditure data example is used for illustration.

Methodology