A note on goodness-of-fit test of continuation ratio logistic regression models under case–control data

We extend the discussion of Qin and Zhang's [1997. A goodness of fit test for logistic regression models base on case–control data. Biometrika 84, 609–618] goodness-of-fit test of logistic regression under case–control data to continuation ratio logistic regression (CRLR) models. We first showed that the retrospective CRLR model, which is valid for case–control data (the null hypothesis _H0)${\mathrm{H}}_0)$, is equivalent to an I  -sample semiparametric model. Then under _H0${\mathrm{H}}_0$, we find the semiparametric profile empirical likelihood estimators of distributions of the covariate conditioning on each response category and use them to define a Kolmogorov–Smirnov type test for assessing the global fit of CRLR models under case–control data. Unlike prospective CRLR models, retrospective CRLR models cannot be partitioned to a series of retrospective binary logistic regression models studied by Qin and Zhang [1997. A goodness of fit test for logistic regression models base on case–control data. Biometrika 84, 609–618].