Calibrated regression estimation using empirical likelihood under data fusion

Data analysis based on information from different sources, typically known as the data fusion problem, is common in economic and biomedical studies. An interesting question concerns the regression of an outcome variable on certain covariates when combining two distinct datasets. These datasets consist of a primary sample containing the outcome and a subset of the covariates, and a supplemental sample comprising information only on the full set of covariates. Previous methods have proposed doubly robust estimation procedures that employ a single propensity score model for the data fusion process and a single imputation model for the covariates available only in the supplemental dataset. However, it may be questionable to assume that either model is correctly specified due to an unknown data generating process. To address this issue, an empirical likelihood based approach that calibrates multiple propensity scores and imputation models is introduced. The resulting estimator is consistent when any one of the models is correctly specified and is robust against extreme values of the fitted propensity scores. The asymptotic normality property and the estimation efficiency are also discussed. Simulation studies show that the proposed estimator has substantial advantages over existing estimators, and an assembled U.S. household expenditure data example is used for illustration.