Document Zbl 1431.62502

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Probab. Stat.</a> <a href="/?q=in%3A430050" title="Articles in this Issue">2019, Article ID 8953530, 13 p. (2019)</a>. </div> <div class="abstract">Summary: <span class="zbmathjax-textit">Background</span>. Evaluation of diagnostic assays and predictive performance of biomarkers based on the receiver operating characteristic (ROC) curve and the area under the ROC curve (AUC) are vital in diagnostic and targeted medicine. The partial area under the curve (pAUC) is an alternative metric focusing on a range of practical and clinical relevance of the diagnostic assay. In this article, we adopt and extend the min-max method to the estimation of the pAUC when multiple continuous scaled biomarkers are available and compare the performances of our proposed approach with existing approaches via simulations. <span class="zbmathjax-textit">Methods</span>. We conducted extensive simulation studies to investigate the performance of different methods for the combination of biomarkers based on their abilities to produce the largest pAUC estimates. Data were generated from different multivariate distributions with equal and unequal variance-covariance matrices. Different shapes of the ROC curves, false positive fraction ranges, and sample size configurations were considered. We obtained the mean and standard deviation of the pAUC estimates through re-substitution and leave-one-pair-out cross-validation. <span class="zbmathjax-textit">Results</span>. Our results demonstrate that the proposed method provides the largest pAUC estimates under the following three important practical scenarios: (1) multivariate normally distributed data for nondiseased and diseased participants have unequal variance-covariance matrices; or (2) the ROC curves generated from individual biomarker are relative close regardless of the latent normality distributional assumption; or (3) the ROC curves generated from individual biomarker have straight-line shapes. <span class="zbmathjax-textit">Conclusions</span>. The proposed method is robust and investigators are encouraged to use this approach in the estimation of the pAUC for many practical scenarios.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A62P10" title="MSC2020">62P10</a> </td> <td class="space"> Applications of statistics to biology and medical sciences; meta analysis </td> </tr></table> </div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1431.62502" data-ciurl="/ci/07039898" data-biburl="/bibtex/07039898.bib" data-amsurl="/amsrefs/07039898.bib" data-xmlurl="/xml/07039898.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07039898.pdf" title="Zbl 1431.62502 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1155/2019/8953530" aria-label="DOI for “On the use of min-max combination of biomarkers to maximize the partial area under the ROC curve”" title="10.1155/2019/8953530">DOI</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Hanley, J. A.; McNeil, B. J., The meaning and use of the area under a receiver operating characteristic (ROC) curve, Radiology, 143, 1, 29-36 (1982) · <a href="https://doi.org/10.1148/radiology.143.1.7063747" class="nowrap">doi:10.1148/radiology.143.1.7063747</a></td> </tr><tr> <td>[2]</td> <td class="space">Zhou, X.-H.; Obuchowski, N. A.; McClish, D. K., Statistical Methods in Diagnostic Medicine. Statistical Methods in Diagnostic Medicine, Wiley Series in Probability and Statistics (2002), New York, NY, USA: Wiley-Interscience [John Wiley & Sons], New York, NY, USA · <a href="/1007.62092" class="nowrap">Zbl 1007.62092</a> · <a href="https://doi.org/10.1002/9780470317082" class="nowrap">doi:10.1002/9780470317082</a></td> </tr><tr> <td>[3]</td> <td class="space">Pepe, M. S., The Statistical Evaluation of Medical Tests for Classification and Prediction. The Statistical Evaluation of Medical Tests for Classification and Prediction, Oxford Statistical Science Series, 28 (2003), Oxford, UK: Oxford University Press, Oxford, UK · <a href="/1039.62105" class="nowrap">Zbl 1039.62105</a></td> </tr><tr> <td>[4]</td> <td class="space">McClish, D. K., Analyzing a Portion of the ROC Curve, Medical Decision Making, 9, 3, 190-195 (1989) · <a href="https://doi.org/10.1177/0272989X8900900307" class="nowrap">doi:10.1177/0272989X8900900307</a></td> </tr><tr> <td>[5]</td> <td class="space">Jiang, Y.; Metz, C. E.; Nishikawa, R. M., A receiver operating characteristic partial area index for highly sensitive diagnostic tests, Radiology, 201, 3, 745-750 (1996) · <a href="https://doi.org/10.1148/radiology.201.3.8939225" class="nowrap">doi:10.1148/radiology.201.3.8939225</a></td> </tr><tr> <td>[6]</td> <td class="space">Dodd, L. E.; Pepe, M. S., Partial AUC estimation and regression, Biometrics: Journal of the International Biometric Society, 59, 3, 614-623 (2003) · <a href="/1210.62152" class="nowrap">Zbl 1210.62152</a> · <a href="https://doi.org/10.1111/1541-0420.00071" class="nowrap">doi:10.1111/1541-0420.00071</a></td> </tr><tr> <td>[7]</td> <td class="space">He, Y.; Escobar, M., Nonparametric statistical inference method for partial areas under receiver operating characteristic curves, with application to genomic studies, Statistics in Medicine, 27, 25, 5291-5308 (2008) · <a href="https://doi.org/10.1002/sim.3335" class="nowrap">doi:10.1002/sim.3335</a></td> </tr><tr> <td>[8]</td> <td class="space">Zhang, D. D.; Zhou, X.-H.; Freeman, D. H.; Freeman, J. L., A non-parametric method for the comparison of partial areas under ROC curves and its application to large health care data sets, Statistics in Medicine, 21, 5, 701-715 (2002) · <a href="https://doi.org/10.1002/sim.1011" class="nowrap">doi:10.1002/sim.1011</a></td> </tr><tr> <td>[9]</td> <td class="space">Ma, H.; Bandos, A. I.; Rockette, H. E.; Gur, D., On use of partial area under the ROC curve for evaluation of diagnostic performance, Statistics in Medicine, 32, 20, 3449-3458 (2013) · <a href="https://doi.org/10.1002/sim.5777" class="nowrap">doi:10.1002/sim.5777</a></td> </tr><tr> <td>[10]</td> <td class="space">Ma, H.; Bandos, A. I.; Gur, D., On the use of partial area under the ROC curve for comparison of two diagnostic tests, Biometrical Journal, 57, 2, 304-310 (2015) · <a href="/1310.62116" class="nowrap">Zbl 1310.62116</a> · <a href="https://doi.org/10.1002/bimj.201400023" class="nowrap">doi:10.1002/bimj.201400023</a></td> </tr><tr> <td>[11]</td> <td class="space">Su, J. Q.; Liu, J. S., Linear combinations of multiple diagnostic markers, Journal of the American Statistical Association, 88, 424, 1350-1355 (1993) · <a href="/0792.62099" class="nowrap">Zbl 0792.62099</a> · <a href="https://doi.org/10.1080/01621459.1993.10476417" class="nowrap">doi:10.1080/01621459.1993.10476417</a></td> </tr><tr> <td>[12]</td> <td class="space">Liu, A.; Schisterman, E. F.; Zhu, Y., On linear combinations of biomarkers to improve diagnostic accuracy, Statistics in Medicine, 24, 1, 37-47 (2005) · <a href="https://doi.org/10.1002/sim.1922" class="nowrap">doi:10.1002/sim.1922</a></td> </tr><tr> <td>[13]</td> <td class="space">Jin, H.; Lu, Y., The optimal linear combination of multiple predictors under the generalized linear models, Statistics & Probability Letters, 79, 22, 2321-2327 (2009) · <a href="/1176.62109" class="nowrap">Zbl 1176.62109</a> · <a href="https://doi.org/10.1016/j.spl.2009.08.002" class="nowrap">doi:10.1016/j.spl.2009.08.002</a></td> </tr><tr> <td>[14]</td> <td class="space">Kang, L.; Liu, A.; Tian, L., Linear combination methods to improve diagnostic/prognostic accuracy on future observations, Statistical Methods in Medical Research, 25, 4, 1359-1380 (2013) · <a href="https://doi.org/10.1177/0962280213481053" class="nowrap">doi:10.1177/0962280213481053</a></td> </tr><tr> <td>[15]</td> <td class="space">Allez, R.; Bouchaud, J., Eigenvector dynamics: General theory and some applications, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 86, 4 (2012) · <a href="https://doi.org/10.1103/PhysRevE.86.046202" class="nowrap">doi:10.1103/PhysRevE.86.046202</a></td> </tr><tr> <td>[16]</td> <td class="space">Pepe, M. S.; Thompson, M. L., Combining diagnostic test results to increase accuracy, Biostatistics, 1, 2, 123-140 (2000) · <a href="/0959.62109" class="nowrap">Zbl 0959.62109</a></td> </tr><tr> <td>[17]</td> <td class="space">Pepe, M. S.; Cai, T.; Longton, G., Combining predictors for classification using the area under the receiver operating characteristic curve, Biometrics: Journal of the International Biometric Society, 62, 1, 221-229, 319 (2006) · <a href="/1091.62125" class="nowrap">Zbl 1091.62125</a> · <a href="https://doi.org/10.1111/j.1541-0420.2005.00420.x" class="nowrap">doi:10.1111/j.1541-0420.2005.00420.x</a></td> </tr><tr> <td>[18]</td> <td class="space">Hsu, M.-J.; Hsueh, H.-M., The linear combinations of biomarkers which maximize the partial area under the ROC curves, Computational Statistics, 28, 2, 647-666 (2013) · <a href="/1305.65045" class="nowrap">Zbl 1305.65045</a> · <a href="https://doi.org/10.1007/s00180-012-0321-5" class="nowrap">doi:10.1007/s00180-012-0321-5</a></td> </tr><tr> <td>[19]</td> <td class="space">Yu, W.; Park, T., Two simple algorithms on linear combination of multiple biomarkers to maximize partial area under the ROC curve, Computational Statistics & Data Analysis, 88, 15-27 (2015) · <a href="/1470.62024" class="nowrap">Zbl 1470.62024</a> · <a href="https://doi.org/10.1016/j.csda.2014.12.002" class="nowrap">doi:10.1016/j.csda.2014.12.002</a></td> </tr><tr> <td>[20]</td> <td class="space">Liu, C.; Liu, A.; Halabi, S., A min-max combination of biomarkers to improve diagnostic accuracy, Statistics in Medicine, 30, 16, 2005-2014 (2011) · <a href="https://doi.org/10.1002/sim.4238" class="nowrap">doi:10.1002/sim.4238</a></td> </tr><tr> <td>[21]</td> <td class="space">Heagerty, P. J.; Pepe, M. S., Semiparametric estimation of regression quantiles with application to standardizing weight for height and age in US children, Journal of the Royal Statistical Society: Series C (Applied Statistics), 48, 4, 533-551 (1999) · <a href="/1058.62520" class="nowrap">Zbl 1058.62520</a> · <a href="https://doi.org/10.1111/1467-9876.00170" class="nowrap">doi:10.1111/1467-9876.00170</a></td> </tr><tr> <td>[22]</td> <td class="space">Huang, X.; Qin, G.; Fang, Y., Optimal combinations of diagnostic tests based on AUC, Biometrics: Journal of the International Biometric Society, 67, 2, 568-576 (2011) · <a href="/1217.62169" class="nowrap">Zbl 1217.62169</a> · <a href="https://doi.org/10.1111/j.1541-0420.2010.01450.x" class="nowrap">doi:10.1111/j.1541-0420.2010.01450.x</a></td> </tr><tr> <td>[23]</td> <td class="space">Egan, J. P., Signal Detection Theory and ROC Analysis (1975), New York, NY, USA: Academic Press, New York, NY, USA</td> </tr><tr> <td>[24]</td> <td class="space">Dorfman, D. D.; Berbaum, K. S.; Metz, C. E.; Lenth, R. V.; Hanley, J. A.; Dagga, H. A., Proper receiver operating characteristic analysis: the bigamma model, Academic Radiology, 4, 2, 138-149 (1997) · <a href="https://doi.org/10.1016/S1076-6332(97)80013-X" class="nowrap">doi:10.1016/S1076-6332(97)80013-X</a></td> </tr><tr> <td>[25]</td> <td class="space">Faraggi, D.; Reiser, B.; Schisterman, E. F., ROC curve analysis for biomarkers based on pooled assessments, Statistics in Medicine, 22, 15, 2515-2527 (2003) · <a href="https://doi.org/10.1002/sim.1418" class="nowrap">doi:10.1002/sim.1418</a></td> </tr><tr> <td>[26]</td> <td class="space">Huang, Y.; Pepe, M. S., A parametric ROC model-based approach for evaluating the predictiveness of continuous markers in case-control studies, Biometrics: Journal of the International Biometric Society, 65, 4, 1133-1144 (2009) · <a href="/1181.62181" class="nowrap">Zbl 1181.62181</a> · <a href="https://doi.org/10.1111/j.1541-0420.2009.01201.x" class="nowrap">doi:10.1111/j.1541-0420.2009.01201.x</a></td> </tr><tr> <td>[27]</td> <td class="space">Hanley, J. A., Receiver operating characteristic (ROC) methodology: the state of the art, Critical Reviews in Computed Tomography, 29, 3, 307-35 (1989)</td> </tr><tr> <td>[28]</td> <td class="space">Nelsen, R. B., An Introduction to Copulas. An Introduction to Copulas, Lecture Notes in Statistics, 139 (1999), Berlin, Germany: Springer, Berlin, Germany · <a href="/0909.62052" class="nowrap">Zbl 0909.62052</a> · <a href="https://doi.org/10.1007/978-1-4757-3076-0" class="nowrap">doi:10.1007/978-1-4757-3076-0</a></td> </tr><tr> <td>[29]</td> <td class="space">Rini, B. I.; Halabi, S.; Rosenberg, J. E.; Stadler, W. M.; Vaena, D. A.; Ou, S.-S.; Archer, L.; Atkins, J. N.; Picus, J.; Czaykowski, P.; Dutcher, J.; Small, E. J., Bevacizumab plus interferon alfa compared with interferon alfa monotherapy in patients with metastatic renal cell carcinoma: CALGB 90206, Journal of Clinical Oncology, 26, 33, 5422-5428 (2008) · <a href="https://doi.org/10.1200/jco.2008.16.9847" class="nowrap">doi:10.1200/jco.2008.16.9847</a></td> </tr><tr> <td>[30]</td> <td class="space">Rini, B. I.; Halabi, S.; Rosenberg, J. E.; Stadler, W. M.; Vaena, D. A.; Archer, L.; Atkins, J. N.; Picus, J.; Czaykowski, P.; Dutcher, J.; Small, E. J., Phase III trial of bevacizumab plus interferon alfa versus interferon alfa monotherapy in patients with metastatic renal cell carcinoma: final results of CALGB 90206, Journal of Clinical Oncology, 28, 13, 2137-2143 (2010) · <a href="https://doi.org/10.1200/JCO.2009.26.5561" class="nowrap">doi:10.1200/JCO.2009.26.5561</a></td> </tr><tr> <td>[31]</td> <td class="space">Nixon, A. B.; Halabi, S.; Shterev, I., Identification of predictive biomarkers of overall survival (OS) in patients (pts) with advanced renal cell carcinoma (RCC) treated with interferon alpha (I) +/- bevacizumab (B): Results from CALGB 90206 (Alliance), Journal of Clinical Oncology, 31, article no. 4520 (2013)</td> </tr><tr> <td>[32]</td> <td class="space">Nixon, A. B.; Pang, H.; Starr, M. D.; Friedman, P. N.; Bertagnolli, M. M.; Kindler, H. L.; Goldberg, R. M.; Venook, A. P.; Hurwitz, H. I., Prognostic and predictive blood-based biomarkers in patients with advanced pancreatic cancer: Results from CALGB80303 (alliance), Clinical Cancer Research, 19, 24, 6957-6966 (2013) · <a href="https://doi.org/10.1158/1078-0432.CCR-13-0926" class="nowrap">doi:10.1158/1078-0432.CCR-13-0926</a></td> </tr><tr> <td>[33]</td> <td class="space">Cox, L. H.; Johnson, M. M.; Kafadar, K., Exposition of statistical graphics technology, Proceedings of the ASA Proceedings of the Statistical Computation Section</td> </tr><tr> <td>[34]</td> <td class="space">Copas, J. B.; Corbett, P., Overestimation of the receiver operating characteristic curve for logistic regression, Biometrika, 89, 2, 315-331 (2002) · <a href="/1017.62062" class="nowrap">Zbl 1017.62062</a> · <a href="https://doi.org/10.1093/biomet/89.2.315" class="nowrap">doi:10.1093/biomet/89.2.315</a></td> </tr><tr> <td>[35]</td> <td class="space">Erho, N.; Crisan, A.; Vergara, I. A.; Mitra, A. P.; Ghadessi, M.; Buerki, C.; Bergstralh, E. J.; Kollmeyer, T.; Fink, S.; Haddad, Z.; Zimmermann, B.; Sierocinski, T.; Ballman, K. V.; Triche, T. J.; Black, P. C.; Karnes, R. J.; Klee, G.; Davicioni, E.; Jenkins, R. B., Discovery and validation of a prostate cancer genomic classifier that predicts early metastasis following radical prostatectomy, PLoS ONE, 8, 6 (2013) · <a href="https://doi.org/10.1371/journal.pone.0066855" class="nowrap">doi:10.1371/journal.pone.0066855</a></td> </tr><tr> <td>[36]</td> <td class="space">Sparano, J. A.; Gray, R. J.; Makower, D. F.; Pritchard, K. I.; Albain, K. S.; Hayes, D. F.; Geyer, C. E.; Dees, E. C.; Perez, E. A.; Olson, J. A.; Zujewski, J. A.; Lively, T.; Badve, S. S.; Saphner, T. J.; Wagner, L. I.; Whelan, T. J.; Ellis, M. J.; Paik, S.; Wood, W. C.; Ravdin, P.; Keane, M. M.; Gomez Moreno, H. L.; Reddy, P. S.; Goggins, T. F.; Mayer, I. A.; Brufsky, A. M.; Toppmeyer, D. L.; Kaklamani, V. G.; Atkins, J. N.; Berenberg, J. L.; Sledge, G. W., Prospective validation of a 21-gene expression assay in breast cancer, The New England Journal of Medicine, 373, 21, 2005-2014 (2015) · <a href="https://doi.org/10.1056/nejmoa1510764" class="nowrap">doi:10.1056/nejmoa1510764</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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