Simple confidence interval and region formulas for comparing diagnostic likelihood ratios under a paired design

A population-based paired design is often used for comparing the diagnostic likelihood ratios of two binary diagnostic tests. However, a case-control paired design, which involves the application of both diagnostic tests to two independent samples, is a good alternative study design especially when the disease is rare. Existing methods for comparing two diagnostic likelihood ratios have been mainly focused on the population-based paired design with little attention paid to the case-control paired design. In this paper, we derive a confidence interval formula for the relative diagnostic likelihood ratio (the ratio of two diagnostic likelihood ratios), which can be used for the comparison of two positive or negative diagnostic likelihood ratios separately. We also derive a confidence region formula for the two relative positive and negative diagnostic likelihood ratios, which allows simultaneous comparison of two positive and negative diagnostic likelihood ratios. The proposed confidence interval and region formulas are simple to compute and can be used for both population-based paired design and case-control paired designs. Simulation studies are used to assess the finite sample performance of the confidence interval and region formulas. The proposed methods are applied to a real data set on coronary artery disease and two diagnostic tests.