  Case-control sampling is a commonly used retrospective sampling design to
alleviate imbalanced structure of binary data. When fitting the logistic
regression model with case-control data, although the slope parameter of the
model can be consistently estimated, the intercept parameter is not
identifiable, and the marginal case proportion is not estimatable, either. We
consider the situations in which besides the case-control data from the main
study, called internal study, there also exists summary-level information from
related external studies. An empirical likelihood based approach is proposed to
make inference for the logistic model by incorporating the internal
case-control data and external information. We show that the intercept
parameter is identifiable with the help of external information, and then all
the regression parameters as well as the marginal case proportion can be
estimated consistently. The proposed method also accounts for the possible
variability in external studies. The resultant estimators are shown to be
asymptotically normally distributed. The asymptotic variance-covariance matrix
can be consistently estimated by the case-control data. The optimal way to
utilized external information is discussed. Simulation studies are conducted to
verify the theoretical findings. A real data set is analyzed for illustration.
