# Receiver operating characteristic

ROC curve of three epitope predictors.

In signal detection theory, a receiver operating characteristic (ROC), or simply ROC curve, is a graphical plot of the sensitivity, or true positives, vs. (1 − specificity), or false positives, 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.[1]

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. The ROC curve was first developed by electrical engineers and radar engineers during World War II for detecting enemy objects in battle fields, also known as the signal detection theory, and was soon introduced in psychology to account for perceptual detection of signals. ROC analysis since then has been used in medicine, radiology, and other areas for many decades, and it has been introduced relatively recently in other areas like machine learning and data mining.

## Basic concept

 true positive (TP) eqv. with hit true negative (TN) eqv. with correct rejection false positive (FP) eqv. with false alarm, Type I error false negative (FN) eqv. with miss, Type II error sensitivity or true positive rate (TPR) eqv. with hit rate, recall ${\displaystyle TPR=TP/P=TP/(TP+FN)}$ false positive rate (FPR) eqv. with fall-out ${\displaystyle FPR=FP/N=FP/(FP+TN)}$ accuracy (ACC) ${\displaystyle ACC=(TP+TN)/(P+N)}$ specificity (SPC) or True Negative Rate ${\displaystyle SPC=TN/N=TN/(FP+TN)=1-FPR}$ positive predictive value (PPV) eqv. with precision ${\displaystyle PPV=TP/(TP+FP)}$ negative predictive value (NPV) ${\displaystyle NPV=TN/(TN+FN)}$ false discovery rate (FDR) ${\displaystyle FDR=FP/(FP+TP)}$ Matthews correlation coefficient (MCC) ${\displaystyle MCC=(TP*TN-FP*FN)/{\sqrt {PNP'N'}}}$ F1 score ${\displaystyle F1=2TP/(P+P')}$ Source: Fawcett (2006).

A classification model (classifier or diagnosis) is a mapping of instances into a certain class/group. The classifier or diagnosis result can be in a real value (continuous output) in which the classifier boundary between classes must be determined by a threshold value, for instance to determine whether a person has hypertension based on blood pressure measure, or it can be in a discrete class label indicating one of the classes.

Let us consider a two-class prediction problem (binary classification), in which the outcomes are labeled either as positive (p) or negative (n) class. There are four possible outcomes from a binary classifier. If the outcome from a prediction is p and the actual value is also p, then it is called a true positive (TP); however if the actual value is n then it is said to be a false positive (FP). Conversely, a true negative has occurred when both the prediction outcome and the actual value are n, and false negative is when the prediction outcome is n while the actual value is p.

To get an appropriate example in a real-world problem, consider a diagnostic test that seeks to determine whether a person has a certain disease. A false positive in this case occurs when the person tests positive, but actually does not have the disease. A false negative, on the other hand, occurs when the person tests negative, suggesting they are healthy, when they actually do have the disease.

Let us define an experiment from P positive instances and N negative instances. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows:

actual value
p n total
prediction
outcome
p' True
Positive
False
Positive
P'
n' False
Negative
True
Negative
N'
total P N

## ROC space

The ROC space and plots of the four prediction examples.

The contingency table can derive several evaluation "metrics" (see infobox). To draw an ROC curve, only the true positive rate (TPR) and false positive rate (FPR) are needed. TPR determines a classifier or a diagnostic test performance on classifying positive instances correctly among all positive samples available during the test. FPR, on the other hand, defines how many incorrect positive results occur among all negative samples available during the test.

An ROC space is defined by FPR and TPR as x and y axes respectively, which depicts relative trade-offs between true positive (benefits) and false positive (costs). Since TPR is equivalent with sensitivity and FPR is equal to 1 - specificity, the ROC graph is sometimes called the sensitivity vs (1 - specificity) plot. Each prediction result or one instance of a confusion matrix represents one point in the ROC space.

The best possible prediction method would yield a point in the upper left corner or coordinate (0,1) of the ROC space, representing 100% sensitivity (no false negatives) and 100% specificity (no false positives). The (0,1) point is also called a perfect classification. A completely random guess would give a point along a diagonal line (the so-called line of no-discrimination) from the left bottom to the top right corners. An intuitive example of random guessing is a decision by flipping coins (head or tail).

The diagonal divides the ROC space. Points above the diagonal represent good classification results, points below the line poor results. Note that the output of a poor predictor could simply be inverted to obtain points above the line.

Let us look into four prediction results from 100 positive and 100 negative instances:

A B C C'
 TP=63 FP=28 91 FN=37 TN=72 109 100 100 200
 TP=77 FP=77 154 FN=23 TN=23 46 100 100 200
 TP=24 FP=88 112 FN=76 TN=12 88 100 100 200
 TP=76 FP=12 88 FN=24 TN=88 112 100 100 200
TPR = 0.63 TPR = 0.77 TPR = 0.24 TPR = 0.76
FPR = 0.28 FPR = 0.77 FPR = 0.88 FPR = 0.12
ACC = 0.68 ACC = 0.50 ACC = 0.18 ACC = 0.82

Plots of the four results above in the ROC space are given in the figure. The result of method A clearly shows the best predictive power among A, B, and C. The result of 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 across the center point (0.5,0.5), the resulting method C' is even better than A. This mirrored method simply reverses the predictions of whatever method or test produced the C contingency table. Although the original C method has negative predictive power, simply reversing its decisions leads to a new predictive method C' which has positive predictive power. 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. The closer a result from a contingency table is to the upper left corner, the better it predicts, but the distance from the random guess line in either direction is the best indicator of how much predictive power a method has. If the result is below the line (i.e. the method is worse than a random guess), all of the method's predictions must be reversed in order to utilize the its power, thereby moving the result above the random guess line.

## Curves in ROC space

Discrete classifiers, such as decision tree or rule set, yield discrete numerical values or a binary label. Applied to a set of instances, such classifiers yield only a single point in the ROC space. Other classifiers, such as naive Bayesian, Logistic regression or neural network, produce scores (which may, in fact, be an estimate of a probability values or may not depending on the method) 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

How an ROC curve can be interpreted

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[citation needed]
• the area under the ROC curve, or "AUC", or A' (pronounced "a-prime") [2]
• 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-alone 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.[3] It can be shown that the area under the ROC curve is closely related to the Mann–Whitney U, which tests whether positives are ranked higher than negatives. It is also equivalent to the Wilcoxon test of ranks. The AUC is related to the Gini coefficient (${\displaystyle G_{1}}$) by the following formula[4] ${\displaystyle G_{1}+1=2AUC}$, where:

${\displaystyle G_{1}=1-\sum _{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})}$

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 for model comparison.[5] 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[citation needed]. 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.[6] 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.[7] Another common approach for classification problems in which P << N (common in bioinformatics applications) is to use a logarithmic scale for the x-axis.[8]

## History

The ROC curve was first used during World War II for the analysis of radar signals before it was employed in signal detection theory.[9] 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.[9] In medicine, ROC analysis has been extensively used in the evaluation of diagnostic tests.[10][11] 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.[12]. 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.[13]

## References

1. ^ Signal detection theory and ROC analysis in psychology and diagnostics : collected papers; Swets, 1996
2. ^ J. Fogarty, R. Baker, S. Hudson (2005). "Case studies in the use of ROC curve analysis for sensor-based estimates in human computer interaction". ACM International Conference Proceeding Series, Proceedings of Graphics Interface 2005. Waterloo, Ontario, Canada: Canadian Human-Computer Communications Society.CS1 maint: multiple names: authors list (link)
3. ^ Fawcett, T. (2006). An introduction to ROC analysis. Pattern Recognition Letters, 27, 861–874.
4. ^ 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.
5. ^ Hanley, JA (1983-09-01). "A method of comparing the areas under receiver operating characteristic curves derived from the same cases". Radiology. 148 (3): 839–843. PMID 6878708. Retrieved 2008-12-03. Unknown parameter |coauthors= ignored (|author= suggested) (help)
6. ^ McClish, Donna Katzman (1989-08-01). "Analyzing a Portion of the ROC Curve". Med Decis Making. 9 (3): 190–195. doi:10.1177/0272989X8900900307. PMID 2668680. Retrieved 2008-09-29.
7. ^ Dodd, Lori E. (2003). "Partial AUC Estimation and Regression". Biometrics. 59 (3): 614–623. doi:10.1111/1541-0420.00071. PMID 14601762. Retrieved 2007-12-18. Unknown parameter |coauthors= ignored (|author= suggested) (help)
8. ^ [1]
9. ^ a b D.M. Green and J.M. Swets (1966). Signal detection theory and psychophysics. New York: John Wiley and Sons Inc. ISBN 0-471-32420-5.
10. ^ M.H. Zweig and G. Campbell (1993). "Receiver-operating characteristic (ROC) plots: a fundamental evaluation tool in clinical medicine". Clinical chemistry. 39 (8): 561–577. PMID 8472349.
11. ^ M.S. Pepe (2003). The statistical evaluation of medical tests for classification and prediction. New York: Oxford.
12. ^ N.A. Obuchowski (2003). "Receiver operating characteristic curves and their use in radiology". Radiology. 229 (1): 3–8. doi:10.1148/radiol.2291010898. PMID 14519861.
13. ^ Spackman, K. A. (1989). "Signal detection theory: Valuable tools for evaluating inductive learning". Proceedings of the Sixth International Workshop on Machine Learning. San Mateo, CA: Morgan Kaufmann. pp. 160–163.

### General references

• X. H. Zhou, N. A. Obuchowski, and D. M. McClish (2002). New York, USA: Wiley & Sons. ISBN 9780471347729. Missing or empty |title= (help)CS1 maint: multiple names: authors list (link)

• Zou, K.H., O'Malley, A.J., Mauri, L. (2007). Receiver-operating characteristic analysis for evaluating diagnostic tests and predictive models. Circulation, 6;115(5):654–7.
• Lasko, T.A., J.G. Bhagwat, K.H. Zou and Ohno-Machado, L. (2005). The use of receiver operating characteristic curves in biomedical informatics. Journal of Biomedical Informatics, 38(5):404–415.
• Balakrishnan, N., (1991) Handbook of the Logistic Distribution, Marcel Dekker, Inc., ISBN 978-0-8247-8587-1.
• Gonen M., (2007) Analyzing Receiver Operating Characteristic Curves Using SAS, SAS Press, ISBN 978-1-59994-298-1.
• Green, W.H., (2003) Econometric Analysis, fifth edition, Prentice Hall, ISBN 0-13-066189-9.
• Heagerty, P.J., Lumley, T., Pepe, M. S. (2000) Time-dependent ROC Curves for Censored Survival Data and a Diagnostic Marker Biometrics, 56:337 – 344
• Hosmer, D.W. and Lemeshow, S., (2000) Applied Logistic Regression, 2nd ed., New York; Chichester, Wiley, ISBN 0-471-35632-8.
• Brown, C.D., and Davis, H.T. (2006) Receiver operating characteristic curves and related decision measures: a tutorial, Chemometrics and Intelligent Laboratory Systems, 80:24–38
• Mason, S.J. and Graham, N.E. (2002) Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation. Q.J.R. Meteorol. Soc., 128:2145–2166.
• Pepe, M.S. (2003). The statistical evaluation of medical tests for classification and prediction. Oxford. ISBN 0198565828.
• Carsten, S. Wesseling, S., Schink, T., and Jung, K. (2003) Comparison of Eight Computer Programs for Receiver-Operating Characteristic Analysis. Clinical Chemistry, 49:433–439
• Swets, J.A. (1995). Signal detection theory and ROC analysis in psychology and diagnostics: Collected papers. Lawrence Erlbaum Associates.
• Swets, J.A., Dawes, R., and Monahan, J. (2000) Better Decisions through Science. Scientific American, October, pages 82–87.