1. Receiver Operating Characteristic (ROC) Curves
Assessing the predictive properties of a test statistic – Decision Theory

2. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion

3. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion

4. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion
TP 
TP = True Positive

5. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion

6. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion
FP 
FP = False Positive

7. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion

8. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion
FN 
FN = False Negative

9. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion

10. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion
TN 
TN = True Negative

11. Binary Prediction Problem Conceptual Framework
Suppose we have a test statistic for predicting the presence or absence of disease.
True Disease Status
Pos
Neg
Test
Criterion TP FP FN TN P N
P+ N

12. Binary Prediction Problem Conceptual Framework

13. Binary Prediction Problem Test Properties
True Disease Status
Pos
Neg
Test
Criterion TP FP FN TN P N
P+ N
Accuracy
= Probability that the test yields a correct result.
= (TP+TN) / (P+N)

14. Binary Prediction Problem Test Properties
True Disease Status
Pos
Neg
Test
Criterion TP FP FN TN P N
P+ N
Sensitivity
= Probability that a true case will test positive
= TP / P
Also referred to as True Positive Rate (TPR) or True Positive Fraction (TPF).

15. Binary Prediction Problem Test Properties
True Disease Status
Pos
Neg
Test
Criterion TP FP FN TN P N
P+ N
Specificity
= Probability that a true negative will test negative
= TN / N
Also referred to as True Negative Rate (TNR) or True Negative Fraction (TNF).

16. Binary Prediction Problem Test Properties
True Disease Status
Pos
Neg
Test
Criterion TP FP FN TN P N
P+ N
1-Specificity
= Prob that a true negative will test positive
= FP / N
Also referred to as False Positive Rate (FPR) or False Positive Fraction (FPF).

17. Binary Prediction Problem Test Properties
True Disease Status
Pos
Neg
Test
Criterion TP FP FN TN P N
P+ N
Positive Predictive Value (PPV)
= Probability that a positive test will truly have disease
= TP / (TP+FP)

18. Binary Prediction Problem Test Properties
True Disease Status
Pos
Neg
Test
Criterion TP FP FN TN P N
P+ N
Negative Predictive Value (NPV)
= Probability that a negative test will truly be disease free
= TN / (TN+FN)

19. Binary Prediction Problem Example
True Disease Status
Pos
Neg
Test
Criterion 27
173
200 73
727
800
100
900
1000
Se =
27/100 = .27
Acc =
(27+727)/1000 = .75
Sp =
727/900 = .81
PPV =
27/200 = .14
FPF = 1- Sp =
.19
NPV =
727/800 = .91

20. Binary Prediction Problem Test Properties
Of these properties, only Se and Sp (and hence FPR) are considered invariant test characteristics.
Accuracy, PPV, and NPV will vary according to the underlying prevalence of disease.
Se and Sp are thus “fundamental” test properties and hence are the most useful measures for comparing different test criteria, even though PPV and NPV are probably the most clinically relevant properties.

21. ROC Curves
Now assume that our test statistic is no longer binary, but takes on a series of values (for instance how many of five distinct risk factors a person exhibits).
Clinically we make a rule that says the test is positive if the number of risk factors meets or exceeds some threshold (#RF > x)
Suppose our previous table resulted from using x = 4.
Let’s see what happens as we vary x.

22. ROC Curves Impact of using a threshold of 3 or more RFs
True Disease Status
Pos
Neg
Test
Criterion 45
200
245 55
700
755
100
900
1000
200
800
.27
.75
Se =
27/100 = .45
Acc =
(27+727)/1000 = .75
.81
.14
Sp =
727/900 = .78
PPV =
27/200 = .18
.91
FPF = 1- Sp =
.22
NPV =
727/800 = .93
Se , Sp , and interestingly both PPV and NPV 

23. ROC Curves Summary of all possible options
Threshold
TPR
FPR 6
0.00 5
0.10
0.11 4
0.27
0.19 3
0.45
0.22 2
0.73 1
0.98
0.80
1.00
As we relax our threshold for defining “disease,” our true positive rate (sensitivity) increases, but so does the false positive rate (FPR).
The ROC curve is a way to visually display this information.

24. ROC Curves Summary of all possible options
x=5
x=4
x=2
The diagonal line shows what we would expect from simple guessing (i.e., pure chance).
Threshold
TPR
FPR 6
0.00 5
0.10
0.11 4
0.27
0.19 3
0.45
0.22 2
0.73 1
0.98
0.80
1.00
What might an even better ROC curve look like?

25. ROC Curves Summary of a more optimal curve
Threshold
TPR
FPR 6
0.00 5
0.10
0.01 4
0.77
0.02 3
0.90
0.03 2
0.95
0.04 1
0.99
0.40
1.00
Note the immediate sharp rise in sensitivity. Perfect accuracy is represented by upper left corner.

26. ROC Curves Use and interpretation
The ROC curve allows us to see, in a simple visual display, how sensitivity and specificity vary as our threshold varies.
The shape of the curve also gives us some visual clues about the overall strength of association between the underlying test statistic (in this case #RFs that are present) and disease status.

27. ROC Curves Use and interpretation
The ROC methodology easily generalizes to test statistics that are continuous (such as lung function or a blood gas). We simply fit a smoothed ROC curve through all observed data points.

28. ROC Curves Use and interpretation
See demo from

29. ROC Curves Area under the curve (AUC)
The total area of the grid represented by an ROC curve is 1, since both TPR and FPR range from 0 to 1.
The portion of this total area that falls below the ROC curve is known as the area under the curve, or AUC.

30. Area Under the Curve (AUC) Interpretation
The AUC serves as a quantitative summary of the strength of association between the underlying test statistic and disease status.
An AUC of 1.0 would mean that the test statistic could be used to perfectly discriminate between cases and controls.
An AUC of 0.5 (reflected by the diagonal 45° line) is equivalent to simply guessing.

31. Area Under the Curve (AUC) Interpretation
The AUC can be shown to equal the Mann-Whitney U statistic, or equivalently the Wilcoxon rank statistic, for testing whether the test measure differs for individuals with and without disease.
It also equals the probability that the value of our test measure would be higher for a randomly chosen case than for a randomly chosen control.

32. Area Under the Curve (AUC) Interpretation
FPR
TPR 1
ROC Curve
AUC   ~
0.540
controls
cases

33. Area Under the Curve (AUC) Interpretation~
.95
TPR 1
FPR
ROC Curve
controls
cases

34. Area Under the Curve (AUC) Interpretation
What defines a “good” AUC?
Opinions vary
Probably context specific
What may be a good AUC for predicting COPD may be very different than what is a good AUC for predicting prostate cancer

35. Area Under the Curve (AUC) Interpretation
= excellent
= good
= fair
= poor
= fail
Remember that <.50 is worse than guessing!

36. Area Under the Curve (AUC) Interpretation
= excellent
= very good
= good
= fair

37. ROC Curves Comparing multiple ROC curves
Suppose we have two candidate test statistics to use to create a binary decision rule. Can we use ROC curves to choose an optimal one?

38. ROC Curves Comparing multiple ROC curves
Adapted from curves at:

39. ROC Curves Comparing multiple ROC curves

40. ROC Curves Comparing multiple ROC curves
We can formally compare AUCs for two competing test statistics, but does this answer our question?
AUC speaks to which measure, as a continuous variable, best discriminates between cases and controls?
It does not tell us which specific cutpoint to use, or even which test statistic will ultimately provide the “best” cutpoint.

41. ROC Curves Choosing an optimal cutpoint
The choice of a particular Se and Sp should reflect the relative costs of FP and FN results.
What if a positive test triggers an invasive procedure?
What if the disease is life threatening and I have an inexpensive and effective treatment?
How do you balance these and other competing factors?
See excellent discussion of these issues at

42. ROC Curves Generalizations
These techniques can be applied to any binary outcome. It doesn’t have to be disease status.
In fact, the use of ROC curves was first introduced during WWII in response to the challenge of how to accurately identify enemy planes on radar screens.

43. ROC Curves Final cautionary notes
We assume throughout the existence of a gold standard for measuring “disease,” when in practice no such gold standard exists.
COPD, asthma, even cancer (can we truly rule out the absence of cancer in a given patient?)
As a result, even Se and Sp may not be inherently stable test characteristics, but may vary depending on how we define disease and the clinical context in which it is measured.
Are we evaluating the test in the general population or only among patients referred to a specialty clinic?
Incorrect specification of P and N will vary in these two settings.