Confidence bounds for the sensitivity lack of a less specific diagnostic test, without gold standard

We consider the problem of comparing two diagnostic tests based on a sample of paired test results without true state determinations, in cases where the second test can reasonably be assumed to be at least as specific as the first. For such cases, we provide two informative confidence bounds: A lower one for the prevalence times the sensitivity gain of the second test with respect to the first, and an upper one for the sensitivity of the first test. Neither conditional independence of the two tests nor perfectness of any of them needs to be assumed. An application of the proposed confidence bounds to a sample of 256 pairs of laboratory test results for toxigenic Clostridium difficile provides evidence for a dramatic sensitivity gain through first appropriately culturing C. difficile from stool samples before applying an enzyme-immuno-assay.