Plotting and Intrepretating an ROC Curve

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Introduction to ROC Curves

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The sensitivity and specificity of a diagnostic test depends on more than just the "quality" of the test--they also depend on the definition of what constitutes an abnormal test.  Look at the the idealized graph at right showing the number of patients with and without a disease arranged according to the value of a diagnostic test. This distributions overlap--the test (like most) does not distinguish normal from disease with 100% accuracy. The area of overlap indicates where the test cannot distinguish normal from disease. In practice, we choose a cutpoint (indicated by the vertical black line) above which we consider the test to be abnormal and below which we consider the test to be normal. The position of the cutpoint will determine the number of true positive, true negatives, false positives and false negatives. We may wish to use different cutpoints for different clinical situations if we wish to minimize one of the erroneous types of test results.

We can use the hypothyroidism data from the likelihood ratio section to illustrate how sensitivity and specificity change depending on the choice of T4 level that defines hypothyroidism. Recall the data on patients with suspected hypothyroidism reported by Goldstein and Mushlin (J Gen Intern Med 1987;2:20-24.). The data on T4 values in hypothyroid and euthyroid patients are shown graphically (below left) and in a simplified tabular form (below right).

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 values of 5 or less are considered to be hypothyroid.  The data display then reduces to:
 

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

You should be able to verify that the sensivity is 0.56 and the specificity is 0.99.

Now, suppose we decide to make the definition of hypothyroidism less stringent and now consider patients with T4 values of 7 or less to be hypothyroid.  The data display will now look like this:
 

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

You should be able to verify that the sensivity is 0.78 and the specificity is 0.81.
 
 Lets move the cut point for hypothyroidism one more time:

T4 value	Hypothyroid	Euthyroid
< 9	29	54
9 or more	3	39
Totals:	32	93

You should be able to verify that the sensivity is 0.91 and the specificity is 0.42.
 
Now, take the sensitivity and specificity values above and put them into a table:

Cutpoint	Sensitivity	Specificity
5	0.56	0.99
7	0.78	0.81
9	0.91	0.42

Notice that you can improve the sensitivity by moving to cutpoint to a higher T4 value--that is, you can make the criterion for a positive test less strict.  You can improve the specificity by moving the cutpoint to a lower T4 value--that is, you can make the criterion for a positive test more strict.  Thus, there is a tradeoff between sensitivity and specificity. You can change the definition of a positive test to improve one but the other will decline.

The next section covers how to use the numbers we just calculated to draw and interpret an ROC curve.
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Plotting and Intrepretating an ROC Curve

| Previous Section | Main Menu | Next Section |This section continues the hypothyroidism example started in the the previous section. We showed that the table at left can be summarized by the operating characteristics at right:
 

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

Cutpoint	Sensitivity	Specificity
5	0.56	0.99
7	0.78	0.81
9	0.91	0.42

The operating characteristics (above right) can be reformulated slightly and then presented graphically as shown below to the right:

Cutpoint	True Positives	False Positives
5	0.56	0.01
7	0.78	0.19
9	0.91	0.58

This type of graph is called a Receiver Operating Characteristic curve (or ROC curve.) It is a plot of the true positive rate against the false positive rate for the different possible cutpoints of a diagnostic test.

An ROC curve demonstrates several things:

1. It shows the tradeoff between sensitivity and specificity (any increase in sensitivity will be accompanied by a decrease in specificity).
2. The closer the curve follows the left-hand border and then the top border of the ROC space, the more accurate the test.
3. The closer the curve comes to the 45-degree diagonal of the ROC space, the less accurate the test.
4. The slope of the tangent line at a cutpoint gives the likelihood ratio (LR) for that value of the test. You can check this out on the graph above. Recall that the LR for T4 < 5 is 52. This corresponds to the far left, steep portion of the curve. The LR for T4 > 9 is 0.2. This corresponds to the far right, nearly horizontal portion of the curve.
5. The area under the curve is a measure of text accuracy. This is discussed further in the next section.

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