A goodness of fit test for multiplicative-intercept risk models based on case-control data

Qin and Zhang (1997) considered a goodness-of-fit test for the logisitic regression model under a case-control sampling plan on the basis of a Kolmogorov-Smirnov-type statistic. There, however, does not exist a goodness-of-fit test for the multiplicative-intercept risk model or the odds-linear model described in the literature. By extending the work of Qin and Zhang (1997), and by indicating the equivalence of the multiplicative-intercept risk model and a two-sample semiparametric selection bias model, we propose a Kolmogorov-Smirnov-type statistic to test the validity of the multiplicative-intercept risk model based on case-control data. We also propose a bootstrap procedure for finding the P-values of the proposed test. In addition, we establish some asymptotic results associated with the proposed test statistic and justify the proposed bootstrap procedure. As an application of the proposed test procedure, we consider simulation results and the analysis of two real data sets.