Semiparametric Receiver Operating Characteristic Analysis in the Presence of an Imperfect Reference Standard via a Box-Cox Density Ratio Model

Abstract: Receiver operating characteristic (ROC) analysis is commonly used to evaluate the diagnostic accuracy of continuous biomarkers. In practice, the true disease status may be unavailable and only a nominal disease status provided by an imperfect reference standard is observed. Existing nonparametric methods have been developed for ROC analysis in this setting, but may suffer from reduced estimation efficiency, numerical instability, or sensitivity to the choice of biomarker scale. We propose a semiparametric method based on a Box-Cox density ratio model, which links the biomarker distributions of the truly healthy and diseased populations while leaving the baseline distribution unspecified. A key feature of the proposed method is that the transformation parameter is estimated from the data rather than specified in advance, allowing the density-ratio structure to adapt to different transformation scales. We develop an empirical likelihood approach for estimation and an expectation-maximization algorithm for computation. We establish the asymptotic distributions of estimators of the ROC curve, area under the curve, Youden's index, and the sensitivity and specificity at the Youden-optimal cutoff, and develop bootstrap confidence intervals and a goodness-of-fit test. Simulation studies demonstrate that the proposed method provides accurate and numerically stable estimation and inference across a range of distributional settings without requiring the transformation scale to be specified in advance. The proposed method is illustrated using data from a malaria study.