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

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  • Man-Jen Hsu
  • Huey-Miin Hsueh

Abstract

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. Copyright Springer-Verlag 2013

Suggested Citation

  • Man-Jen Hsu & Huey-Miin Hsueh, 2013. "The linear combinations of biomarkers which maximize the partial area under the ROC curves," Computational Statistics, Springer, vol. 28(2), pages 647-666, April.
    • Handle: RePEc:spr:compst:v:28:y:2013:i:2:p:647-666
    DOI: 10.1007/s00180-012-0321-5
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    References listed on IDEAS

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    1. Lori E. Dodd & Margaret S. Pepe, 2003. "Partial AUC Estimation and Regression," Biometrics, The International Biometric Society, vol. 59(3), pages 614-623, September.
    2. Margaret Sullivan Pepe & Gary Longton & Garnet L. Anderson & Michel Schummer, 2003. "Selecting Differentially Expressed Genes from Microarray Experiments," Biometrics, The International Biometric Society, vol. 59(1), pages 133-142, March.
    3. Tian, Lili, 2010. "Confidence interval estimation of partial area under curve based on combined biomarkers," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 466-472, February.
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    1. Yu, Wenbao & Park, Taesung, 2015. "Two simple algorithms on linear combination of multiple biomarkers to maximize partial area under the ROC curve," Computational Statistics & Data Analysis, Elsevier, vol. 88(C), pages 15-27.
    2. Schneider, Matthew J. & Gorr, Wilpen L., 2015. "ROC-based model estimation for forecasting large changes in demand," International Journal of Forecasting, Elsevier, vol. 31(2), pages 253-262.

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