Evaluating and comparing biomarkers with respect to the area under the receiver operating characteristics curve in two-phase case–control studies

Two-phase sampling design, where biomarkers are subsampled from a phase-one cohort sample representative of the target population, has become the gold standard in biomarker evaluation. Many two-phase case–control studies involve biased sampling of cases and/or controls in the second phase. For example, controls are often frequency-matched to cases with respect to other covariates. Ignoring biased sampling of cases and/or controls can lead to biased inference regarding biomarkers' classification accuracy. Considering the problems of estimating and comparing the area under the receiver operating characteristics curve (AUC) for a binary disease outcome, the impact of biased sampling of cases and/or controls on inference and the strategy to efficiently account for the sampling scheme have not been well studied. In this project, we investigate the inverse-probability-weighted method to adjust for biased sampling in estimating and comparing AUC. Asymptotic properties of the estimator and its inference procedure are developed for both Bernoulli sampling and finite-population stratified sampling. In simulation studies, the weighted estimators provide valid inference for estimation and hypothesis testing, while the standard empirical estimators can generate invalid inference. We demonstrate the use of the analytical variance formula for optimizing sampling schemes in biomarker study design and the application of the proposed AUC estimators to examples in HIV vaccine research and prostate cancer research.