Use of likelihood ratios for comparisons of binary diagnostic tests: Underlying ROC curves
Authors: Bandos Andriy I. , Rockette Howard E. , Gur David 
Published: November 2010

Purpose:When comparing binary test results from two diagnostic systems, superiority in both “sensitivity” and “specificity” also implies differences in all conventional summary indices and locally in the underlying receiver operating characteristics (ROC) curves. However, when one of the two binary tests has higher sensitivity and lower specificity (or vice versa), comparisons of their performance levels are nontrivial and the use of different summary indices may lead to contradictory conclusions. A frequently used approach that is free of subjectivity associated with summary indices is based on the comparison of the underlying ROC curves that requires the collection of rating data using multicategory scales, whether natural or experimentally imposed. However, data for reliable estimation of ROC curves are frequently unavailable. The purpose of this article is to develop an approach of using “diagnostic likelihood ratios,” namely, likelihood ratios of “positive” or “negative” responses, to make simple inferences regarding the underlying ROC curves and associated areas in the absence of reliable rating data or regarding the relative binary characteristics, when these are of primary interest.Methods:For inferences related to underlying curves, the authors exploit the assumption of concavity of the true underlying ROC curve to describe conditions under which these curves have to be different and under which the curves have different areas. For scenarios when the binary characteristics are of primary interest, the authors use characteristics of “chance performance” to demonstrate that the derived conditions provide strong evidence of superiority of one binary test as compared to another. By relating these derived conditions to hypotheses about the true likelihood ratios of two binary diagnostic tests being compared, the authors enable a straightforward statistical procedure for the corresponding inferences.Results:The authors derived simple algebraic and graphical methods for describing the conditions for superiority of one of two diagnostic tests with respect to their binary characteristics, the underlying ROC curves, or the areas under the curves. The graphical regions are useful for identifying potential differences between two systems, which then have to be tested statistically. The simple statistical tests can be performed with well known methods for comparison of diagnostic likelihood ratios. The developed approach offers a solution for some of the more difficult to analyze scenarios, where diagnostic tests do not demonstrate concordant differences in terms of both sensitivity and specificity. In addition, the resulting inferences do not contradict the conclusions that can be obtained using conventional and reasonably defined summary indices.Conclusions:When binary diagnostic tests are of primary interest, the proposed approach offers an objective and powerful method for comparing two binary diagnostic tests. The significant advantage of this method is that it enables objective analyses when one test has higher sensitivity but lower specificity, while ensuring agreement with study conclusions based on other reasonable and widely acceptable summary indices. For truly multicategory diagnostic tests, the proposed method can help in concluding inferiority of one of the diagnostic tests based on binary data, thereby potentially saving the need for conducting a more expensive multicategory ROC study.