Testing the Goodness of Fit of Multivariate Multiplicative-intercept Risk Models Based on Case-control Data

Abstract

The validity of the multivariate multiplicative-intercept risk model with I +1 categories based on casecontrol data is tested. After reparametrization, the assumed risk model is equivalent to an (I +1) -sample semiparametric model in which the I ratios of two unspecified density functions have known parametric forms. By identifying this (I +1) -sample semiparametric model, which is of intrinsic interest in general (I +1) -sample problems, with an (I +1) -sample semiparametric selection bias model, we propose a weighted Kolmogorov-Smirnov-type statistic to test the validity of the multivariate multiplicativeintercept risk model. Established are some asymptotic results associated with the proposed test statistic, also established is an optimal property for the maximum semiparametric likelihood estimator of the parameters in the (I +1) -sample semiparametric selection bias model. In addition, a bootstrap procedure along with some results on analysis of two real data sets is proposed.