10.1177/0272989X04271046 NOV–DEC MEDICAL DECISION MAKING/NOV–DEC 2004 XIONG, MKEEL, MILLER, MORRIS METHODOLOGY COMBINING CORRELATED DIAGNOSTIC TESTS Combining Correlated Diagnostic Tests: Application to Neuropathologic Diagnosis of Alzheimer’s Disease Chengjie Xiong, PhD, Daniel W. McKeel Jr, MD, J. Philip Miller, John C. Morris, MD This article studies the problem of combining correlated di- agnostic tests to maximize the discriminating power between the diseased population and the healthy population. The au- thors consider all possible linear combinations of multiple di- agnostic tests and search for the one that achieves the largest area under the receiver operating characteristic (ROC) curve. They discuss the statistical estimation of the optimum linear combination test and the associated maximum area under the ROC curve. Their approach is based on the assumption of multivariate normal distribution of the multiple diagnostic tests. They also present the application of the proposed tech- niques to the neuropathologic diagnosis of Alzheimer’s dis- ease based on brain lesions from 5 different brain locations using a data set from the Washington University Alzheimer’s Disease Research Center. Key words: confidence interval es- timate; eigenvalue; eigenvector; maximum likelihood esti- mate; receiver operating characteristic (ROC) curve; z trans- formation. (Med Deci Making 2004;24:659-669) I n medical diagnosis, subjects are assumed to be 1 of 2 basic types: healthy and diseased. Because of the noisy background on which the disease is presented, the presence or absence of the disease may not always be correctly decided. The probabilities of an incorrect decision in the healthy population (1– specificity) and of a correct decision in the disease population (sensi- tivity), respectively, depend on the diagnostic criteria and the decision threshold adopted. When a diagnostic test is based on an observed variable that lies on a con- tinuous or graded scale, an assessment of the test can be made through the use of a receiver operating character- istic (ROC) curve, 1–3 which plots sensitivity against 1– specificity. If a test could perfectly discriminate, it would have a cutoff or threshold result above which the entire diseased population would fall and below which all healthy population would fall, or vice versa. The curve would then pass through point (0, 1) on the grid [0,1] × [0,1]. The closer an ROC curve comes to this ideal point, the better its discriminating ability. A test with no discriminating ability will produce a curve that follows the diagonal of the grid. The area under ROC curve 4 has been a particularly popular index for summarizing the discriminating ability of a diagnostic test. This area represents the probability that results for a randomly selected indi- vidual from the diseased population and a randomly selected individual from the healthy population will be ranked in the correct order. Many authors have stud- ied the statistical estimation/inference and compari- son associated with ROC curves. Both parametric and nonparametric approaches have been proposed in the literature. DeLong and others 5 used a partial likelihood solution to the discrete logistic model to estimate the sensitivity and specificity of a monitoring test. Ma and Hall 6 constructed simultaneous confidence bands for an entire ROC curve based on the maximum likelihood estimates. Dorfman and Alf, 7 Metz, 3 and Swets and Pickett 4 also studied the maximum likelihood estimation of the area under the ROC curve under the binormal model assumption. When multiple diagnostic tests are applied to the same case samples, the resulting ROC curve estimates are correlated. As an example, the neuropathologic di- agnosis for Alzheimer’s disease (AD) is typically based on the densities and distribution of brain lesions such MEDICAL DECISION MAKING/NOV–DEC 2004 659 DOI: 10.1177/0272989X04271046 Received 11 June 2003 from the Division of Biostatistics (CX, JPM), the Departments of Pathology and Immunology (DWM, JCM), and the De- partment of Neurology (JCM), Washington University in St. Louis, Mis- souri. Revision accepted for publication 30 August 2004. Address correspondence and reprint requests to Chengjie Xiong, PhD, Division of Biostatistics, Campus Box 8067, Washington University in St. Louis, St. Louis, MO 63110; phone: 314-362-3635; fax: 314-362- 2693; e-mail: chengjie@wubios.wustl.edu.