Combining diagnostic test results to increase accuracy

When multiple diagnostic tests are performed on an individual or multiple disease markers are available it may be possible to combine the information to diagnose disease. We consider how to choose linear combinations of markers in order to optimize diagnostic accuracy. The accuracy index to be maximized is the area or partial area under the receiver operating characteristic (ROC) curve. We propose a distribution-free rank-based approach for optimizing the area under the ROC curve and compare it with logistic regression and with classic linear discriminant analysis (LDA). It has been shown that the latter method optimizes the area under the ROC curve when test results have a multivariate normal distribution for diseased and non-diseased populations. Simulation studies suggest that the proposed non-parametric method is efficient when data are multivariate normal.The distribution-free method is generalized to a smooth distribution-free approach to: (i) accommodate some reasonable smoothness assumptions; (ii) incorporate covariate effects; and (iii) yield optimized partial areas under the ROC curve. This latter feature is particularly important since it allows one to focus on a region of the ROC curve which is of most relevance to clinical practice. Neither logistic regression nor LDA necessarily maximize partial areas. The approaches are illustrated on two cancer datasets, one involving serum antigen markers for pancreatic cancer and the other involving longitudinal prostate specific antigen data.