The study on the optimal linear combination of markers based on the partial area under the ROC curve

Abstract

The aim of this work is to construct a composite diagnostic tool based on multiple biomarkers under the criterion of the partial area under a ROC curve (pAUC) for a pre-determined specificity range. Recently several studies are interested in the optimal linear combination maximizing the whole area under a ROC curve (AUC). In this study, we focus on finding the optimal linear combination by a direct maximization of the pAUC under normal assumption. In order to find an analytic solution, the first derivative of the pAUC is derived. The form is so complicated, that a further validation on the Hessian matrix is difficult. In addition, we find that the pAUC maximizer may not be unique and sometimes, local maximizers exist. As a result, the existing algorithms, which depend on the initial-point, are inadequate to serve our needs. We propose a new algorithm by adopting several initial points at one time. In addition, when the population parameters are unknown and only a random sample data set is available, the maximizer of the sample version of the pAUC is shown to be a strong consistent estimator of its theoretical counterpart. We further focus on determining whether a biomarker set, or one specific biomarker has a significant contribution to the disease diagnosis. We propose three statistical tests for the identification of the discriminatory power. The proposed tests are applied to biomarker selection for reducing the variable number in advanced analysis. Numerical studies are performed to validate the proposed algorithm and the proposed statistical procedures. Keywords: Discriminatory power; Hypothesis testing; Optimal linear combination; Partial area under ROC curve; Stepwise biomarker selection; Receiver operating curve; Specificity; Sensitivity.?