The case-control design provides an effective way to collect covariate information conditioning on subjects’ disease status. The standard logistic regression model can be used to model the interaction between two covariates under such a design, but the prospective logistic regression method might not be the most efficient one when certain appropriate constraints can be imposed on the covariate distribution. We develop a hybrid approach for the statistical inference of the interaction under the case-control design.We use a parametric model to characterize the conditional distribution of one covariate given the another covariate in the control population, while leaving the distribution of the later covariate to be fully nonparametric. A maximum hybrid parametric and empirical likelihood method is adopted for the evaluation of all parameters. The estimator and the associated test derived from the proposed semiparametric model are suitable for evaluating the interaction between two covariates of various types (discrete or continuous). Asymptotic results for both the estimators and the test statistics were established, and the advantages of the proposed method over the existing ones are demonstrated through simulation results and a real data example.
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