Assessing Goodness-of-Fit of Generalized Logit Models Based on Case-Control Data

We consider testing the validity of the generalized logit model with I+1 categories based on case-control data. After reparametrization, the assumed logit model is equivalent to an (I+1)-sample semiparametric model in which the I log ratios of two unspecified density functions are linear in data. By identifying this (I+1)-sample semiparametric model, which is of intrinsic interest in general (I+1)-sample problems, with a biased sampling model, we propose a weighted Kolmogorov-Smirnov-type statistic to test the validity of the generalized logit model. We establish some asymptotic results associated with the proposed test statistic. We also propose a bootstrap procedure along with some results on simulation and on analysis of three real data sets.