  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 assumd.
  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
Clostridium difficile from stool samples before applying an
enzyme-immuno-assay.
