 ### Description

The Receiver Operating Curve (ROC) originated during World War II with the use of radar in signal detection. This was extended to the use of diagnostic tests for identifying disease states, using plots of sensitivity versus specificity for different test results. The area under a ROC curve serves as a measure of the diagnostic accuracy (discrimination performance) for a test.

To generate a receiver operating curve it is first necessary to determine the sensitivity and specificity for each test result in the diagnosis of the disorder in question.

The x axis ranges from 0 to 1, or 0% to 100%, and can be either the

• false positive rate (1 - (specificity)), or

• true negative rate (specificity)

The false positive rate is the one typically used.

The y axis ranges from 0 to 1, or 0% to 100%

• true positive rate (sensitivity), with range 0 to 1 (or 0 to 100%)

When the x-axis is the false positive rate (1 - (specificity)), the curve starts at (0,0) and increases towards (1,1). When the x-axis is the true negative rate (specificity), the curve starts at (0, 1) and drops towards (1, 0). The endpoints for the curve will run to these points.

Area under Curve

One way of measuring the area under a curve is by measuring subcomponent trapezoids. Data points can be connected by straight lines defined by:

y = ((slope) * x) + intercept

The area under each line can be determined by integration of (y * dx) over the interval of x1 to x2:

area = (((slope) / 2) * ((x2 ^ 2) - (x1 ^2))) + ((intercept) * (x2 - x1))

By summating the areas under each segment, an approximation of the area under the entire curve can be reached. However, the trapezoidal method tends to underestimate areas (Hanley, 1983), so that other techniques for measuring area should be used if greater accuracy is required.

The maximum area under ROC curve is 1 and is seen with the ideal test. The closer the area under the ROC curve is to 1, the better (more accurate) the test.

Comparison of Two Methods

Two methods can be compared by the area under their respective ROC curves. The method with the larger area under the ROC curve is preferable over one with a smaller area, allowing for variability, as being more accurate.