Binary response models with M-phase case-control data

The present study aimed to characterize the relationship between a binary response and covariates of interest through a more general single-index regression model. With M -phase (M ≥ 2) case-control data supplemented by information on a response and certain covariates, we primarily propose a pseudo likelihood estimation for the index coefficients of this type of semiparametric model. Additionally, our approach can be readily adopted to accommodate case-control sampling with a continuous response and outcome-dependent sampling. With the considered data setting in the receiver operating characteristic curve analysis, an estimation for the accuracy measure is provided and is borrowed to seek an optimal linear predictor in the class of potential linear predictors. To check model correctness, a pseudo least squares approach is further employed as an aid to devising suitable testing procedures. The general theoretical frameworks for the proposed estimators and the bootstrap inference are also developed in this article. Finally, extensive simulations and two empirical applications are used to illustrate the applicability of our methodology.