Comparing two medical tests when results of reference standard are unavailable for those negative via both tests

In studies for comparing the diagnostic accuracy of two qualitative tests, very often the reference standard to confirm the disease status is not applied to all study subjects. We considered a situation when all subjects with a positive result by at least one of the tests had a verified disease status and none of the subjects with both tests negative results had a verification of disease status. In this paper, we discuss whether the information about the ratio of true positive rates and the ratio of false positive rates of two qualitative tests, TNew and TOld, is sufficient to draw a conclusion about effectiveness of the TNew. We show that if there is a statistically significant increase in true positive rates and the increase in true positive rates is statistically larger than increase in false positive rates, then a conclusion about effectiveness of test TNew can be made and this does not require application of the reference standard to the subjects with negative results by both tests. An application of ratio of true positive rates and ratio of false positive rates to post-market studies is also presented.