  A fundamental aspect of statistics is the integration of data from different
sources. Classically, Fisher and others were focused on how to integrate
homogeneous (or only mildly heterogeneous) sets of data. More recently, as data
are becoming more accessible, the question of if data sets from different
sources should be integrated is becoming more relevant. The current literature
treats this as a question with only two answers: integrate or don't. Here we
take a different approach, motivated by information-sharing principles coming
from the shrinkage estimation literature. In particular, we deviate from the
do/don't perspective and propose a dial parameter that controls the extent to
which two data sources are integrated. How far this dial parameter should be
turned is shown to depend, for example, on the informativeness of the different
data sources as measured by Fisher information. In the context of generalized
linear models, this more nuanced data integration framework leads to relatively
simple parameter estimates and valid tests/confidence intervals. Moreover, we
demonstrate both theoretically and empirically that setting the dial parameter
according to our recommendation leads to more efficient estimation compared to
other binary data integration schemes.
