SMHS ROC

Scientific Methods for Health Sciences - Receiver Operating Characteristic (ROC) Curve

Overview

Receiver operating characteristic (ROC curve) is a graphical plot, which illustrates the performance of a binary classifier system as its discrimination threshold varies. The ROC curve is created by plotting the fraction of true positive out of the total actual positives vs. the fraction of false positives out of the total actual negatives at various threshold settings. In this section, we are going to introduce the ROC curve and illustrate applications of this method with examples.

Motivation

We have talked about the cases with a binary classification where the outcomes are either absent or present and the test results are positive or negative. We have also previously discussed sensitivity and specificity which are associated with the concepts of true positive and true negatives. With ROC curve, we are looking to demonstrate the following aspects:

• To show the tradeoff between sensitivity and specificity;
• The closer the curve follows the left-hand border and top border of ROC space, the more accurate is the test;
• The closer the curve comes to the 45-degree diagonal, the less accurate is the test;
• The slope of the tangent line at a cut-point gives the likelihood ratio for the value of the test;
• The area under the curve is a measure of test accuracy.

Also, see the SMHS Type I and Type II error section.

		Actual condition
		Absent ($H_0$ is true)	Present ($H_1$ is true)
Test Result	Negative (fail to reject $H_0$)	Condition absent + Negative result = True (accurate) Negative (TN, 0.98505)	Condition present + Negative result = False (invalid) Negative (FN, 0.00025)Type II error (β)
	Positive	(reject $H_0$) Condition absent + Positive result = False Positive (FP, 0.00995) Type I error (α)	Condition Present + Positive result = True Positive (TP, 0.00475)
Test Interpretation	Power = 1-FN=1-0.00025 = 0.99975	Specificity: TN/(TN+FP) =0.98505/(0.98505+0.00995) = 0.99	Sensitivity: TP/(TP+FN) =0.00475/(0.00475+ 0.00025)= 0.95

Theory

Review of basic concepts in a binary classification:

		Disease Status		Metrics
		Disease	No Disease	Prevalence=$\frac{\sum Condition\, positive}{\sum Total\, population}$
Screening Test	Positive	a (True positives)	b (False positives)	Positive predictive value (PPV)=$\frac{\sum Ture\, positive}{\sum Test\,positives}$	False discovery rate (FDR)=$\frac{\sum False\, positive}{\sum Test\, positive}$
	Negative	c (False negatives)	d (True negatives)	False omission rate (FOR)=$\frac{\sum False\, negative} {\sum Test\, negative}$	Negative predictive value (NPV)=$\frac{\sum True\, negative}{\sum Test\, negative}$
	Positive Likelihood Ratio=TPR/FBR	True positive rate (TPR)=$\frac{\sum True\, positive} {\sum Condition\, positive}$	False positive rate (FPR)=$\frac{\sum False\, positive}{\sum Condition\, positive}$
	Negative Likelihood Ratio=FNR/TNR	False negative rate (FNR)=$\frac{\sum False\, negative} {\sum condition\, negative}$	True negative rate (TNR)=$\frac{\sum True\, negative}{\sum Condition\, negative}$

• Caution: The ROC curve is a powerful technique as each individual point on the curve represents an improper scoring rule that is optimized by fitting an inappropriate model. ( The area under the ROC curve (AUC) is a linear translation of the Wilcoxon-Mann-Whitney- rank correlation statistics. Using the ROC curve to identify thresholds may be a low-precision and somewhat arbitrary operation, which may not replicate well from one study to the next. A specific potential problem with over-reliance on the ROC curve may be that it tempts analysts to always identify thresholds for binary test-classification even if such hard-thresholds do not really exist. The field of decision theory may provide more reliable protocols for segmenting, clustering of classifying cases into groups or categories.

Introduction of ROC Curve

Sensitivity and specificity are both characteristics of a test but they also depend on the definition of what constitutes an abnormal test. Consider a diagnostic medical test where the cut-points would without doubt influence the test results. We can use the hypothyroidism data from the likelihood ratio to illustrate how these two characteristics change depending on the choice of T4 level that defines hypothyroidism. Recall the data where patients with suspected hypothyroidism are reported.

T4	<1	1-2	2-3	3-4	4-5	5-6	6-7	7-8	8-9	9-10	10-11	11-12	>12
Hypothyroid	2	3	1	8	4	4	3	3	1	0	2	1	0
Euthyroid	0	0	0	0	1	6	11	19	17	20	11	4	4

With the following cut-points, we have the data listed:

T4 value	Hypothyroid	Euthyroid
5 or less	18	1
5.1 - 7	7	17
7.1 - 9	4	36
9 or more	3	39
Totals:	32	93

Suppose that patients with T4 of 5 or less are considered to be hypothyroid, then the data would be displayed as the following and the sensitivity ($TP/(TP+FN) =18/32$) is 0.56 and specificity ($TN/(TN+FP)=92/93$) is 0.99 in this case.

T4 value	Hypothyroid	Euthyroid
5 or less	18	1
More than 5	14	92
Totals:	32	93

Now suppose, we decided to be less stringent on the disease and consider the patients with T4 values of 7 or less to be hypothyroid, then the data would be recorded as the following and the sensitivity in this case would be 0.78 and specificity 0.81:

T4 value	Hypothyroid	Euthyroid
7 or less	25	18
More than 7	7	75
Totals:	32	93

If we move the cut-point for hypothyroidism a little bit higher, say 9, we would have the sensitivity of 0.91 and specificity 0.42:

T4 value	Hypothyroid	Euthyroid
9 or less	29	54
More than 9	3	39
Totals:	32	93

To sum up, we have the pairs of sensitivity an specificity with corresponding cut-points in the following table:

Cut points	Sensitivity	Specificity
5	0.56	0.99
7	0.78	0.81
9	0.91	0.42

From the table above, we observe that the sensitivity improves with increasing cut-point T4 value while specificity increases with decreasing cut-point T4 value. That is a tradeoff between sensitivity and specificity. The table above can also be shown as TP and FP.

Cut points	Rate of True Positives	Rate of False Positives
5	0.56	0.01
7	0.78	0.19
9	0.91	0.58

Plotting the sensitivity vs. (1-specificity), we get the ROC curve:

x<-c(0,0.01,0.19,0.4,0.58,1)
y<-c(0,0.56,0.78,0.81,0.91,1)
plot(x,y,type='o',main='ROC curve for T4',xlab='False positive rate (1-specificity)',ylab='True positive rate (sensitivity)')

install.packages("pROC")
library("pROC")
HypothyroidResponse <- c(2,3,1,8,4,4,3,3,1,0,2,1,0)
PredictorThreshold <- c('poor','good','good','poor','good','poor','poor','poor','good','good','poor','good','good') 
# Syntax (response, predictor)
auc(factor(PredictorThreshold), HypothyroidResponse)
# the AUC=P(random positive example > random negative example)

> Area under the curve: 0.8214

• ROC interpretation: The area under the ROC Curve can be used to classify the accuracy of a diagnostic test according to the following academic point system:

0.90-1: excellent;

0.8-0.9: good;

0.7-0.8: fair;

0.6-0.7: poor;

0.5-0.7:fail.

With our example above, the area under the T4 ROC curve is 0.86, which shows that the accuracy of the test is good in separating hypothyroid from euthyroid patients.

• Computing ROC: The calculation protocol of the area under the ROC curve is important. The area measures discrimination, which is the ability of the test to correctly classify those with and without the disease. The area under the curve is the percentage of randomly drawn pairs for which this is true. Two methods are commonly used to calculate the area of the scope:

non-parametric method based on constructing trapezoids under the curve as approximation of area;

a parametric method using a MLE to fit a smooth curve to data points.

Applications

We can use the smaller summary Alzheimer's disease (AD) dataset of Global Cortical Surface Curvedness morphometry measures to illustrate the concepts of false-positive, false-negative, sensitivity, specificity and power of a test.

Suppose we try to determine the best threshold of the curvedness (a measure of cortical surface smoothness) that can be used as a test for AD. We would combine NC+MCI and explore the effects of varying the threshold of the curvedness on the false-positive and false-negative rates. The table below uses these simple formulas to compute the number of false-positive ($=COUNTIF(CurvednessTable!Curvedness29:Curvedness105, "<="\&A2)$) and false-negative ($=COUNTIF(CurvednessTable!Curvedness2:Curvedness28, ">"\&A2)$) frequencies for each threshold between $35\times 10^4$ and $75\times 10^4$ (in increments of $20,000$).

Threshold	AD_Test_GMV<Threshold_FalsePositives	FalseNegatives
350000	0	26
370000	0	26
390000	1	25
410000	3	23
430000	5	23
450000	7	23
470000	11	20
490000	23	18
510000	39	15
530000	47	13
550000	56	13
570000	64	13
590000	68	12
610000	70	8
630000	71	7
650000	75	7
670000	76	5
690000	76	3
710000	77	2
730000	77	1
750000	77	0

The figure below illustrates graphically the effect of increasing the Curvedness threshold (binary classification of cases into AD group or NC+MCI group) on the false-positive (red) and false-negative (blue) classification rates.

We can dig a bit deeper by plotting the impact of thresholding the global cortical surface shape curvedness on the false-positive (FP) and false-negative (FN) rates. The R-script below generates this figure, where the thresholds ($\times 10^4$) are used as labels for each point in the FP vs. FN scatter plot. If committing FN errors is $R$ times more expensive than committing FP errors, then the best operating point (hence ROC) will be tangent to the FP-FN curve/line with a slope of $-R$. Therefore, if $R=\frac{1}{8}$, we should set the threshold to 53 (as the slope of the green line will be $\frac{13-14}{48-40}=-\frac{1}{8}$) and if $R=2$, the threshold should be 41 (as the slope of the purple line will be $\frac{23-25}{3-2}=-\frac{2}{1}=-2$):

par(mfrow = c(1,1))

leg.txt <- c("FP vs. FN", "Thresholds", "Line Slope=-1/8", "Line Slope = -2")

Threshold <- c(35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75)
FalsePositives <-c(0,0,1,3,5,7,11,23,39,47,56,64,68,70,71,75,76,76,77,77,77)
FalseNegatives <- c(26,26,25,23,23,23,20,18,15,13,13,13,12,8,7,7,5,3,2,1,0)
plot(FalsePositives, FalseNegatives, type="b", pch=21, col="red",  lty=1, 
xlab="False-Positives", ylab="FalseNegatives", 
main = "Impact of Thresholding the Global Cortical Surface Shape Curvedness on the FP vs. FN")

linex1 <- c(40, 48)
liney1 <- c(14, 13)
linez1 <- lm(liney1 ~linex1)
abline (linez1, col="green")
linex2 <- c(3, 2)
liney2 <- c(23, 25)
linez2 <- lm(liney2 ~linex2)
abline (linez2, col="purple")
# lines(linex2, liney2, ol=" purple ",lwd=2.5)

text(FalsePositives, FalseNegatives, Threshold, cex=0.6, pos=1, col="blue")
legend(list(x = 58,y = 29), legend = leg.txt, col=c("red", "blue", "green", "purple"), 
#col = c(1,3,1,2), 
lty = 1, merge = TRUE)

Below is an example using the ROCR R-package to generate the ROC curve and compute the AUC.

#install.packages("randomForest")
#library("randomForest")

# train the random forest model
AD.rf <-randomForest(AD_ClassTrue_1 ~ GMV, ntree=1000, keep.forest=TRUE, importance=TRUE)

# to generate probabilities, instead of class labels, use type="prob"
#randomForest generates probabilities for both the class labels,
# test on new data
# AD.rf.pr = predict(AD.rf,type="prob",newdata=WMV)
AD.rf.pr <- predict(AD.rf, GMV)

#prediction is ROCR function
AD.rf.pred <- prediction(AD.rf.pr, AD_ClassTrue_1)
AD.rf.pr
#performance in terms of true and false positive rates
AD.rf.perf = performance(AD.rf.pred,"tpr","fpr")

#plot the curve
plot(AD.rf.perf,main="ROC Curve for Random Forest Classification of AD using GMV",
col=2,lwd=2)
abline(a=0,b=1,lwd=2,lty=2,col="gray")

#compute area under curve
auc <- performance(AD.rf.pred,"auc")
auc

• This article titled The Meaning And the Use Of The Area Under A Receiver Operating Characteristic (ROC) Curve presented a representation and interpretation of the area under a ROC curve obtained by the ‘rating’ method, or by mathematical predictions based on patient characteristics. It showed that that in such a setting the area represents the probability that a randomly chosen diseased subject is (correctly) rated or ranked with greater suspicion than a randomly chosen non-diseased subject.

• This article illustrated practical experimental techniques for measuring ROC curves and discussed about the issues of case selection and curve-fitting. It also talked about possible generalizations of conventional ROC analysis to account for decision performance in complex diagnostic tasks and showed ROC analysis related in direct and natural way to cost/benefit analysis of diagnostic decision making. This paper developed the concepts of ‘average diagnostic cost’ and ‘average net benefit’ to identify the optimal compromise among various kinds of diagnostic error and suggested ways in ROC analysis to optimize diagnostic strategies.

Software

# With the given example in R:
x<-c(0,0.01,0.19,0.58,1)
y<-c(0,0.56,0.78,0.91,1)
plot(x,y,type='o',main='ROC curve for T4',xlab='False positive rate (specificity)',ylab='True positive rate (sensitivity)')

Problems

6.1) Suppose that a new study is conducted on lung cancer and the following data is collected in identify between two types of lung cancers (say type a and type b). Conduct the ROC curve for this example by varying the cut-points from 2 to 10 by increasing 2 units each time. Calculate the area under the curve and interpret on the accuracy of the test.

measurements	<1	1-2	2-3	3-4	4-5	5-6	6-7	7-8	8-9	9-10	10-11	11-12	>12
Type a	2	1	4	2	8	7	4	3	0	0	1	2	2
Type b	1	3	0	2	2	5	10	23	18	20	15	8	2

6.2) When a serious disease can be treated if it is caught early, it is more important to have a test with high specificity than high sensitivity.

a. True

b. False

6.3) The positive predictive value of a test is calculated by dividing the number of:

(a) True positives in the population

(b) True negatives in the population

(c) People who test positive

(d) People who test negative

6.4) A new screening test has been developed for diabetes. The table below represents the results of the new test compared to the current gold standard.

	Condition positive	Condition negative	Total
Test positive	80	70	150
Test negative	10	240	250
Total	90	310	400

What is the sensitivity of the test?

(a) 77%

(b) 89%

(c) 80%

(d) 53%

6.5) What is the specificity of the test?

(a) 77%

(b) 89%

(c) 80%

(d) 53%

6.6) What is the positive predictive value of the test?

(a) 77%

(b) 89%

(c) 80%

(d) 53%

References

• Introduction to ROC Curves
• Plotting and Intrepretting an ROC Curve
• The Area Under an ROC Curve
• ROC Wikipedia