Receiver operating characteristic

Receiver operating characteristic

In signal detection theory, a receiver operating characteristic (ROC), or simply ROC curve, is a graphical plot of the sensitivity vs. (1 - specificity) for a binary classifier system as its discrimination threshold is varied. The ROC can also be represented equivalently by plotting the fraction of true positives (TPR = true positive rate) vs. the fraction of false positives (FPR = false positive rate). Also known as a Relative Operating Characteristic curve, because it is a comparison of two operating characteristics (TPR & FPR) as the criterion changes. [Signal detection theory and ROC analysis in psychology and diagnostics : collected papers; Swets, 1996]

ROC analysis provides tools to select possibly optimal models and to discard suboptimal ones independently from (and prior to specifying) the cost context or the class distribution. ROC analysis is related in a direct and natural way to cost/benefit analysis of diagnostic decision making. Widely used in medicine, radiology, psychology and other areas for many decades, it has been introduced relatively recently in other areas like machine learning and data mining.

Basic concept

style="padding-left:1em;" | TPR = 0.63 || style="padding-left:2em;" | TPR = 0.77 || style="padding-left:2em;" | TPR = 0.24 || style="padding-left:2em;" | TPR = 0.88
style="padding-left:1em;" | FPR = 0.28 || style="padding-left:2em;" | FPR = 0.77 || style="padding-left:2em;" | FPR = 0.88 || style="padding-left:2em;" | FPR = 0.24
style="padding-left:1em;" | ACC = 0.68 || style="padding-left:2em;" | ACC = 0.50 || style="padding-left:2em;" | ACC = 0.18 || style="padding-left:2em;" | ACC = 0.82

Plots of the four results above in the ROC space are given in the figure. The result A clearly shows the best among B and C. The result B lies on the random guess line (the diagonal line), and it can be seen in the table that the accuracy of B is 50%. However, when C is mirrored onto the diagonal line, as seen in C', the result is even better than A.

Since this mirrored C method or test simply reverses the predictions of whatever method or test produced the C contingency table, the C method has positive predictive power simply by reversing all of its decisions. When the C method predicts p or n, the C' method would predict n or p, respectively. In this manner, the C' test would perform the best. While the closer a result from a contingency table is to the upper left corner the better it predicts, the distance from the random guess line in either direction is the best indicator of how much predictive power a method has, albeit, if it is below the line, all of its predictions including its more often wrong predictions must be reversed in order to utilize the method's power.

Curves in ROC space

Discrete classifiers, such as decision tree or rule set, yield numerical values or binary label. When a set is given to such classifiers, the result is a single point in the ROC space. For other classifiers, such as naive Bayesian and neural network, they produce probability values representing the degree to which class the instance belongs to. For these methods, setting a threshold value will determine a point in the ROC space. For instance, if probability values below or equal to a threshold value of 0.8 are sent to the positive class, and other values are assigned to the negative class, then a confusion matrix can be calculated. Plotting the ROC point for each possible threshold value results in a curve.

Further interpretations

Sometimes, the ROC is used to generate a summary statistic. Three common versions are:
* the intercept of the ROC curve with the line at 90 degrees to the no-discrimination line
* the area between the ROC curve and the no-discrimination line
* the area under the ROC curve, or "AUC".
* d' (pronounced "d-prime"), the distance between the mean of the distribution of activity in the system under noise-alone conditions and its distribution under signal plus noise conditions, divided by their standard deviation, under the assumption that both these distributions are normal with the same standard deviation. Under these assumptions, it can be proved that the shape of the ROC depends only on d'.

The AUC is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one. [Fawcett, T. (2006). An introduction to ROC analysis. Pattern Recognition Letters, 27, 861-874.] It can be shown that the area under the ROC curve is equivalent to the Mann-Whitney U, which tests for the median difference between scores obtained in the two groups considered if the groups are of continuous data. It is also equivalent to the Wilcoxon test of ranks. The AUC has been found to be related to the Gini coefficient(G) by the following formula [Hand, D.J., & Till, R.J. (2001). A simple generalization of the area under the ROC curve to multiple class classification problems. Machine Learning, 45, 171-186.] G_1 + 1 = 2 x AUC, where:

:G_1 = 1 - sum_{k=1}^{n} (X_{k} - X_{k-1}) (Y_{k} + Y_{k-1})

In this way, it is possible to calculate the AUC by using an average of a number of trapezoidal approximations.

However, any attempt to summarize the ROC curve into a single number loses information about the pattern of tradeoffs of the particular discriminator algorithm.

The machine learning community most often uses the ROC AUC statistic. This measure can be interpreted as the probability that when we randomly pick one positive and one negative example, the classifier will assign a higher score to the positive example than to the negative.In engineering, the area between the ROC curve and the no-discrimination line is often preferred, because of its useful mathematical properties as a non-parametric statistic. This area is often simply known as the discrimination. In psychophysics, d' is the most commonly used measure.

The illustration at the top right of the page shows the use of ROC graphs for the discrimination between the quality of different epitope predicting algorithms. If you wish to discover at least 60% of the epitopes in a virus protein, you can read out of the graph that about 1/3 of the output would be falsely marked as an epitope. The information that is not visible in this graph is that the person that uses the algorithms knows what threshold settings give a certain point in the ROC graph.

Sometimes it can be more useful to look at a specific region of the ROC Curve rather than at the whole curve. It is possible to compute partial AUC. [Cite journal| doi = 10.1177/0272989X8900900307| volume = 9| issue = 3| pages = 190-195| last = McClish| first = Donna Katzman| title = Analyzing a Portion of the ROC Curve| journal = Med Decis Making| accessdate = 2008-09-29| date = 1989-08-01| url =] For example, one could focus on the region of the curve with low false positive rate, which is often of prime interest for population screening tests. [Cite journal| doi = 10.1111/1541-0420.00071| volume = 59| issue = 3| pages = 614-623| last = Dodd| first = Lori E.| coauthors = Margaret S. Pepe| title = Partial AUC Estimation and Regression| journal = Biometrics| accessdate = 2007-12-18| date = 2003| url =]


The ROC curve was first used during the World War II for the analysis of radar signals before it was employed in signal detection theory.cite book|author=D.M. Green and J.M. Swets| title=Signal detection theory and psychophysics| publisher=John Wiley and Sons Inc.| date=1966| location=New York| id=ISBN 0-471-32420-5] Following the attack on Pearl Harbor in 1941, the United States army began new research to increase the prediction of correctly detected Japanese aircraft from their radar signals.

In the 1950s, ROC curves were employed in psychophysics to assess human (and occasionally non-human animal) detection of weak signals. In medicine, ROC analysis has been extensively used in the evaluation of diagnostic tests. [cite journal|author=M.H. Zweig and G. Campbell| journal=Clinical chemistry| pmid=8472349| title=Receiver-operating characteristic (ROC) plots: a fundamental evaluation tool in clinical medicine| volume=39| issue=8| date=1993| pages=561–577] [cite book|author=M.S. Pepe| title=The statistical evaluation of medical tests for classification and prediction| location=New York| publisher=Oxford| date=2003] ROC curves are also used extensively in epidemiology and medical research and are frequently mentioned in conjunction with evidence-based medicine. In radiology, ROC analysis is a common technique to evaluate new radiology techniques. [cite journal|author=N.A. Obuchowski| title=Receiver operating characteristic curves and their use in radiology| pmid=14519861| journal=Radiology| volume=229| issue=1| date=2003| pages=3–8|doi=10.1148/radiol.2291010898] . In the social sciences, ROC analysis is often called the ROC Accuracy Ratio, a common technique for judging the accuracy of default probability models.

ROC curves also proved useful for the evaluation of machine learning techniques. The first application of ROC in machine learning was by Spackman who demonstrated the value of ROC curves in comparing and evaluating different classification algorithms. [cite conference|author=Spackman, K. A.| date=1989| title=Signal detection theory: Valuable tools for evaluating inductive learning| booktitle=Proceedings of the Sixth International Workshop on Machine Learning| location=San Mateo, CA| pages=160–163| publisher=Morgan Kaufman]


ee also

* Constant false alarm rate
* Detection theory
* False alarm

General references


Further reading

* Balakrishnan, N., "Handbook of the Logistic Distribution", Marcel Dekker, Inc., 1991, ISBN-13: 978-0824785871.

* Gonen M., "Analyzing Receiver Operating Characteristic Curves Using SAS", SAS Press, 2007, ISBN: 978-1-59994-298-1.

* Green, William H., "Econometric Analysis", fifth edition, Prentice Hall, 2003, ISBN 0-13-066189-9.

* Hosmer, David W. and Stanley Lemeshow, "Applied Logistic Regression", 2nd ed., New York; Chichester, Wiley, 2000, ISBN 0-471-35632-8.

* Lasko, T. A., J.G. Bhagwat, K.H. Zou and L. Ohno-Machado (Oct. 2005). The use of receiver operating characteristic curves in biomedical informatics. "Journal of Biomedical Informatics" 38(5):404-415. PMID 16198999

* Mason, S. J. and N.E. Graham, Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation. "Q.J.R. Meteorol. Soc." (2002), 128, pp. 2145–2166.

* Pepe, M. S. (2003). "The statistical evaluation of medical tests for classification and prediction". Oxford. ISBN 0198565828

* Stephan, Carsten, Sebastian Wesseling, Tania Schink, and Klaus Jung. Comparison of Eight Computer Programs for Receiver-Operating Characteristic Analysis. Clin. Chem., Mar 2003; 49: 433 - 439. []

* Swets, J.A. (1995). "Signal detection theory and ROC analysis in psychology and diagnostics: Collected papers." Lawrence Erlbaum Associates.

External links

* [ A simple example of a ROC curve]
* [ An introduction to ROC analysis]
* [ A more thorough treatment of ROC curves and signal detection theory]
* [ Diagnostic test evaluation - online calculator]
* [ Kelly H. Zou's Bibliography of ROC Literature and Articles]
* [ Tom Fawcett's ROC Convex Hull: tutorial, program and papers]
* [ Peter Flach's tutorial on ROC analysis in machine learning]
* [ The magnificent ROC] — An explanation and interactive demonstration of the connection of ROCs to archetypal bi-normal test result plots


* [ SAS and R code for ROC curves]
* [ Comparison of Eight Computer Programs for Receiver-Operating Characteristic Analysis, Clinical Chemistry. 2003;49:433-439]
* [ StAR, a software for the statistical comparison of ROC curves]
* [ ROCR, a comprehensive R package for evaluating scoring classifiers] ( [ Introductory article] )
* [ List of ROC analysis software]
* [ ROC package for R (part of the BioConductor suite)]
* [ Standalone PERF program used by the KDD Cup competition]
* [ Web-based calculator of ROC curves from user-supplied data]
* [ ROC curve visualiser]
* [ Analyse-it ROC software]
* [ GeneXproTools ROC Analysis]
* [ ROCTools ROC software]
* [ ROC Curve Tools, m-files for MATLAB, written by Dr. Gavin C. Cawley]
* [ MedCalc software]
* [ AUCCalculator, a Java program for finding AUC-ROC by Jesse Davis and Mark Goadrich]

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