  We consider the optimal decision-making problem in a primary sample of
interest with multiple auxiliary sources available. The outcome of interest is
limited in the sense that it is only observed in the primary sample. In
reality, such multiple data sources may belong to heterogeneous studies and
thus cannot be combined directly. This paper proposes a new framework to handle
heterogeneous samples and address the limited outcome simultaneously through a
novel calibrated optimal decision-making method, by leveraging the common
intermediate outcomes in multiple data sources. Specifically, our method allows
the baseline covariates across different samples to have either homogeneous or
heterogeneous distributions. Under the equal conditional means of intermediate
outcomes in different samples given baseline covariates and the treatment
information, we show that the proposed estimator of the conditional mean
outcome is asymptotically normal and more efficient than using the primary
sample solely. Extensive experiments on simulated datasets demonstrate
empirical validity and improved efficiency using our approach, followed by a
real application to electronic health records.
