Statistical inference for case-control logistic regression via integrating external summary data

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 a 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 optimal way to utilize external information is discussed. The resultant estimators are shown to be asymptotically normally distributed. Simulation results show favorable evidence for the theoretical findings and reveal that the integration of external information may increase the estimation efficiency and predictive accuracy. Two real data sets are analyzed for illustration.