Using a monotonic density ratio model to increase the power of the goodness‐of‐fit test for logistic regression models with case‐control data

Abstract

Logistic regression models are widely used in case‐control data analysis, and testing the goodness‐of‐fit of their parametric model assumption is a fundamental research problem. In this article, we propose to enhance the power of the goodness‐of‐fit test by exploiting a monotonic density ratio model, in which the ratio of case and control densities is assumed to be a monotone function. We show that such a monotonic density ratio model is naturally induced by the retrospective case‐control sampling design under the alternative hypothesis. The pool‐adjacent‐violator algorithm is adapted to solve for the constrained nonparametric maximum likelihood estimator under the alternative hypothesis. By measuring the discrepancy between this estimator and the semiparametric maximum likelihood estimator under the null hypothesis, we develop a new Kolmogorov‐Smirnov‐type statistic to test the goodness‐of‐fit for logistic regression models with case‐control data. A bootstrap resampling procedure is suggested to approximate the ‐value of the proposed test. Simulation results show that the type I error of the proposed test is well controlled and the power improvement is substantial in many cases. Three real data applications are also included for illustration.