Comparing the accuracy of screening tests with verification of disease status restricted to test positives

As a fundamental component of health care, disease screening is of highly importance. Oftentimes, two screening tests for a specific disease are compared in order to determine an optimal screening policy, for example, the digital rectal examination (DRE) and serum prostate specific antigen (PSA) level for screening prostate cancer. Ideally, if a gold standard test is given to each individual being screened to establish their true disease status, the difference in accuracy measures between two tests can be evaluated. In practice, however, it is common that only individuals who test positive on at least one screening test are to receive gold standard tests, which are often invasive and cannot be applied to those with negative results on both tests due to ethical reasons. Under such circumstances, estimates of the differences in accuracy measures between two tests cannot be determined, thus the inference problem within this framework is challenging. In this article, using sensitivity and specificity as measures of test accuracy, we show that their difference between two tests is interval-identified, as bounded by estimable sharp bounds. Here, we develop the asymptotic normality for the estimators of the bounds and construct confidence intervals for the difference by utilizing the method for solving inference problem for partially identified parameters. The performance of constructed confidence intervals for the difference and their sharp bounds are evaluated via simulation studies. We also apply the proposed method to the prostate cancer example to compare the accuracy of DRE and PSA.