The linear combinations of biomarkers which maximize the partial area under the ROC curves

As biotechnology has made remarkable progress nowadays, there has also been a great improvement on data collection with lower cost and higher quality outcomes. More often than not investigators can obtain the measurements of many disease-related features simultaneously. When multiple potential biomarkers are available for constructing a diagnostic tool of a disease, an effective approach is to combine these biomarkers to build one single indicator. For continuous-scaled variables, the use of linear combinations is popular due to its easy interpretation. Su and Liu (J Ame Stat Assoc 88(424):1350-1355, 1993) derived the best linear combination under the criterion of the area under the receiver operating characteristic (ROC) curve, when the joint normality of biomarkers is assumed. However, in many investigations, the emphases are placed only on a limited extent of clinical relevancy, instead of the whole ROC curve. The goal of this study is to find the linear combination that maximizes the partial area under a ROC curve (pAUC) for a pre-specified range. In order to find an analytic solution, the first derivative of the pAUC under normal assumption is derived. The explicit form is so complicated, that a further validation on the Hessian matrix is difficult. On the other hand, we find that the pAUC maximizer may not be unique and local maximizers do exist in some cases. Consequently, the existing algorithms find an initial-point dependent solution and are inadequate to serve our needs. Hence, we propose a new algorithm by adopting several initial points at one time. Intensive numerical studies have been performed to show the adequacy of the proposed algorithm. Real examples are also provided for illustration.