Semiparametric Binary Response Models with Supplementary 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. Based on case-control data supplemented by information on population totals, we primarily propose a pseudo likelihood estimation for the index coefficients of the considered semiparametric model. Two additional estimation procedures are further provided for different research purposes although the resulting estimators are relatively inefficient compared to the pseudo maximum likelihood one in general situations. Under the assumption of strictly increasing response function, an accuracy measure in the receiver operating characteristic curve analysis is borrowed to build an estimation criterion in which only those subjects with complete covariate information are included in estimation. To check the model correctness, a pseudo least squares estimation is employed as an aid to devising a suitable testing procedure. In addition, the general large sample properties of the proposed estimators and the related bootstrap inference procedures are developed. To assess the finite-sample performance of the estimation and inference procedures, an extensive simulation experiment is conducted. An empirical example from a lowbirthweight study in Auckland is also used to illustrate the applicability of our methodologies.