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Biostatistics Advance Access originally published online on February 27, 2008
Biostatistics 2008 9(3):566-576; doi:10.1093/biostatistics/kxm050
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

ROC analysis with multiple classes and multiple tests: methodology and its application in microarray studies

Jialiang Li

Department of Statistics and Applied Probability, National University of Singapore, Singapore 117546

Jason P. Fine*

Department of Statistics, University of Wisconsin, Madison, WI 53706, USA
fine{at}biostat.wisc.edu

* To whom correspondence should be addressed.

The accuracy of a single diagnostic test for binary outcome can be summarized by the area under the receiver operating characteristic (ROC) curve. Volume under the surface and hypervolume under the manifold have been proposed as extensions for multiple class diagnosis (Scurfield, 1996, 1998). However, the lack of simple inferential procedures for such measures has limited their practical utility. Part of the difficulty is that calculating such quantities may not be straightforward, even with a single test. The decision rule used to generate the ROC surface requires class probability assessments, which are not provided by the tests. We develop a method based on estimating the probabilities via some procedure, for example, multinomial logistic regression. Bootstrap inferences are proposed to account for variability in estimating the probabilities and perform well in simulations. The ROC measures are compared to the correct classification rate, which depends heavily on class prevalences. An example of tumor classification with microarray data demonstrates that this property may lead to substantially different analyses. The ROC-based analysis yields notable decreases in model complexity over previous analyses.

Keywords: Class prevalence; Diagnostic accuracy; Maximum likelihood estimation; Multicategory classification; Multinomial logistic regression

Received March 24, 2006; accepted for publication May 10, 2006.


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