  It is becoming increasingly common for researchers to consider incorporating
external information from large studies to improve the accuracy of statistical
inference instead of relying on a modestly sized dataset collected internally.
With some new predictors only available internally, we aim to build improved
regression models based on individual-level data from an "internal" study while
incorporating summary-level information from "external" models. We propose a
meta-analysis framework along with two weighted estimators as the composite of
empirical Bayes estimators, which combines the estimates from the different
external models. The proposed framework is flexible and robust in the ways that
(i) it is capable of incorporating external models that use a slightly
different set of covariates; (ii) it can identify the most relevant external
information and diminish the influence of information that is less compatible
with the internal data; and (iii) it nicely balances the bias-variance
trade-off while preserving the most efficiency gain. The proposed estimators
are more efficient than the naive analysis of the internal data and other naive
combinations of external estimators.
