Measuring classifier performance: a coherent alternative to the area under the ROC curve Article

First Online: 16 June 2009 Received: 21 August 2008 Revised: 24 March 2009 Accepted: 04 May 2009 DOI :
10.1007/s10994-009-5119-5

Cite this article as: Hand, D.J. Mach Learn (2009) 77: 103. doi:10.1007/s10994-009-5119-5
Abstract The area under the ROC curve (AUC ) is a very widely used measure of performance for classification and diagnostic rules. It has the appealing property of being objective, requiring no subjective input from the user. On the other hand, the AUC has disadvantages, some of which are well known. For example, the AUC can give potentially misleading results if ROC curves cross. However, the AUC also has a much more serious deficiency, and one which appears not to have been previously recognised. This is that it is fundamentally incoherent in terms of misclassification costs: the AUC uses different misclassification cost distributions for different classifiers. This means that using the AUC is equivalent to using different metrics to evaluate different classification rules. It is equivalent to saying that, using one classifier, misclassifying a class 1 point is p times as serious as misclassifying a class 0 point, but, using another classifier, misclassifying a class 1 point is P times as serious, where p ≠P . This is nonsensical because the relative severities of different kinds of misclassifications of individual points is a property of the problem, not the classifiers which happen to have been chosen. This property is explored in detail, and a simple valid alternative to the AUC is proposed.

Keywords ROC curves Classification AUC Specificity Sensitivity Misclassification rate Cost Loss Error rate Editor: Johannes Fürnkranz.

Download to read the full article text

References
Adams, N. M., & Hand, D. J. (1999). Comparing classifiers when the misallocation costs are uncertain.

Pattern Recognition ,

32 , 1139–1147.

CrossRef Google Scholar
Bradley, A. P. (1997). The use of the area under the ROC curve in the evaluation of machine learning algorithms.

Pattern Recognition ,

30 , 1145–1159.

CrossRef Google Scholar
Dodd, L. E., & Pepe, M. S. (2003). Partial AUC estimation and regression.

Biometrics ,

59 , 614–623.

CrossRef MathSciNet Google Scholar
Fawcett, T. (2004).

ROC graphs: notes and practical considerations for researchers . Palo Alto: HP Laboratories.

Google Scholar
Fawcett, T. (2006). An introduction to ROC analysis.

Pattern Recognition Letters ,

27 , 861–874.

CrossRef Google Scholar
Flach, P. A. (2003). The geometry of ROC space: understanding machine learning metrics through isometrics. In Proc. 20th international conference on machine learning (ICML’03) (pp. 194–201).

Hand, D. J. (1997).

Construction and assessment of classification rules . New York: Wiley.

MATH Google Scholar
Hand, D. J. (2004).

Measurement theory and practice: the world through quantification . London: Arnold.

MATH Google Scholar
Hand, D. J. (2005). Good practice in retail credit scorecard assessment.

Journal of the Operational Research Society ,

56 , 1109–1117.

MATH CrossRef Google Scholar
Hand, D. J. (2006). Classifier technology and the illusion of progress (with discussion).

Statistical Science ,

21 , 1–34.

MATH CrossRef MathSciNet Google Scholar
Hand, D. J., & Till, R. J. (2001). A simple generalisation of the area under the ROC curve for multiple class classification problems.

Machine Learning ,

45 , 171–186.

MATH CrossRef Google Scholar
Hanley, J. A. (1989). Receiver operating characteristic (ROC) methodology: the state of the art.

Critical Reviews in Diagnostic Imaging ,

29 , 307–335.

Google Scholar
Hanley, J. A., & McNeil, B. J. (1982). The meaning and use of the area under an ROC curve.

Radiology ,

143 , 29–36.

Google Scholar
Hastie, T., Tibshirani, R., & Friedman, J. (2001).

The elements of statistical learning . New York: Springer.

MATH Google Scholar
Jamain, A., & Hand, D. J. (2008). Mining supervised classification performance studies: a meta-analytic investigation.

Journal of Classification ,

25 , 87–112.

CrossRef Google Scholar
Krzanowski, W. J., & Hand, D. J. (2009).

ROC curves for continuous data . London: Chapman and Hall.

MATH Google Scholar
McClish, D. K. (1989). Analyzing a portion of the ROC curve.

Medical Decision Making ,

9 , 190–195.

CrossRef Google Scholar
Pepe, M. S. (2003).

The statistical evaluation of medical tests for classification and prediction . Oxford: Oxford University Press.

MATH Google Scholar
Provost, F., & Fawcett, T. (1997). Analysis and visualization of classifier performance: Comparison under imprecise class and cost distributions. In KDD-97—third international conference on knowledge discovery and data mining .

Provost, F., Fawcett, T., & Kohavi, R. (1998). The case against accuracy estimation for comparing induction algorithms. In Proceedings of the 15th international conference on machine learning, ICML-98 .

Rudin, W. (1964).

Principles of mathematical analysis (2nd edn.). New York: McGraw-Hill.

MATH Google Scholar
Scott, M. J. J., Niranjan, M., & Prager, R. W. (1998). Parcel: feature subset selection in variable cost domains (Technical Report CUED/F-INFENG/TR. 323). Cambridge University Engineering Department, UK.

Thomas, L. C., Edelman, D. B., & Crook, J. N. (2002).

Credit scoring and its applications . Philadelphia: Society for Industrial and Applied Mathematics.

MATH Google Scholar
Webb, A. (2002).

Statistical pattern recognition (2nd edn.). New York: Wiley.

MATH Google Scholar © Springer Science+Business Media, LLC 2009

Authors and Affiliations 1. Department of Mathematics Imperial College London London UK 2. Institute for Mathematical Sciences Imperial College London London UK