Optimal Linear Combination of Biomarkers by Weighted Youden Index Maximization

In medical research, it is common practice to combine various biomarkers to improve the accuracy of disease diagnosis. The weighted Youden index (WYI), which assigns unequal weights to sensitivity and specificity based on their relative importance, serves as an important and flexible evaluation metric of diagnostic tests. However, no existing methods have been designed specifically to identify the optimal linear combination of biomarkers that maximizes the WYI. In this paper, we propose a novel method to construct an optimal diagnosis score and determine the best cut-off point at the same time. The estimated combination coefficients and cut-off point are shown to have cube root asymptotics, and their joint limiting distribution is established rigorously. Further, the asymptotic normality of the optimal in-sample WYI is established, and out-of-sample inference for score distribution and comparison is investigated. These results provide deep theoretical insights for methods of Youden index maximization for the first time. Computationally, an iterative marginal optimization algorithm, different from the existing literature, is adopted to deal with the objective function that is neither continuous nor smooth. Simulation studies support the theoretical results and demonstrate the superiority of the proposed method. Two real-world examples-coronary disease and Alzheimer's disease diagnosis-are presented for illustration.