GraphPad Statistics Guide

Area under the curve

Area under the curve

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Area under the curve

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How to: Area under the curve

The area under the curve is an integrated measurement of a measurable effect or phenomenon. It is used as a cumulative measurement of drug effect in pharmacokinetics and as a means to compare peaks in chromatography.

Note that Prism also computes the area under a Receiver Operator Characteristic (ROC) curve as part of the separate ROC analysis.

Start from a data or results table that represents a curve. Click Analyze and choose Area under the curve from the list of XY analyses.

Interpreting area-under-the-curve results

If your data come from chromatography or spectroscopy, Prism can break the data into separate regions and determine the highest point (peak) of each. Prism can only do this, however, if the regions are clearly defined: the signal, or graphic representation of the effect or phenomenon, must go below the baseline between regions and the peaks cannot overlap.

For each region, Prism shows the area in units of the X axis times units of the Y axis. Prism also shows each region as a fraction of the total area under all regions combined. The area is computed using the trapezoid rule. It simply connects a straight line between every set of adjacent points defining the curve, and sums up the areas beneath these areas.

Next, Prism identifies the peak of each region. This is reported as the X and Y coordinates of the highest point in the region and the two X coordinates that represent the beginning and end of the region.

Prism may identify more regions than you are interested in. In this case, go back to the Parameters dialog box and enter a larger value for the minimum width of a region and/or the minimum height of a peak.

Limitations of this analysis

Note these limitations:

The baseline must be horizontal.

There is no smoothing or curve fitting.

Prism will not separate overlapping peaks. The program will not distinguish two adjacent peaks unless the signal descends all the way to the baseline between those two peaks. Likewise, Prism will not identify a peak within a shoulder of another peak.

If the signal starts (or ends) above the baseline, the first (or last) peak will be incomplete. Prism will report the area under the tails it “sees”.

Prism does not extrapolate back to X=0, if your first X value is greater than zero.

Prism does not extrapolate beyond the highest X value in your data set, so does not extrapolate the curve down to the baseline.

If you enter data with replicate Y values, or as Mean and SD or SEM, Prism only analyzes the mean values.

Prism does not combine the SD values to come up with a confidence interval for the AUC or a SE for the AUC. These calculations have been described by Gagnon (1), but Prism does not yet do them.

Prism no longer insists that the X values be equally spaced. When it sums the areas of the trapezoids, it is fine if some are fatter than others.

How Prism computes area under the curve

Prism computes the area under the curve using the trapezoid rule, illustrated in the figure below.

In Prism, a curve (created by nonlinear regression) is simply a series of connected XY points, with equally spaced X values. Prism can compute area under the curve also for XY tables you enter, and does not insist that the X values be equally spaced. The left part of the figure above shows two of these points and the baseline as a dotted line. The area under that portion of the curve, a trapezoid, is shaded. The middle portion of the figure shows how Prism computes the area. The two triangles in the middle panel have the same area, so the area of the trapezoid on the left is the same as the area of the rectangle on the right (whose area is easier to calculate). The area, therefore, is ΔX*(Y1+Y2)/2. Prism uses this formula repeatedly for each adjacent pair of points defining the curve.

The area is computed between the baseline you specify and the curve, starting from the first X value in your data set and ending at the largest X value. Prism does not extend the curve beyond your data.

What counts as a peak?

By default, Prism only considers points above the baseline to be part of peaks, so only reports peaks that stick above the baseline. You can choose to consider peaks that go below the baseline.

By default, Prism ignores any peaks whose height is less than 10% of the distance from minimum to maximum Y value, but you can change this definition in the area under the curve parameters dialog. You can also tell it to ignore peaks that are very narrow.

Total peak area vs. total area vs. net area

Prism always reports the Total Area, which includes: Positive peaks, negative peaks, peaks that are not high enough to count, and peaks that are too narrow to count. The only choice you make in the analysis dialog that affects the definition of total area is the definition of the baseline.

Prism also reports the Total Peak Area. Here Prism only includes the peaks you ask it to consider. This value is affected by several choices in the analysis dialog: The definition of baseline, your choice about including or ignoring negative peaks, and your definition of peaks too small to count. The total area is not a useful value to report, but it puts the results in context and might help you spot problems or better understanding what Prism is or is not including in the total area.

If you ask Prism to define peaks below the baseline as peaks, then Prism subtracts the area of peaks below the baseline from the area of peaks above the baseline, and reports this difference as the Net Area.

Reference

1. Robert C. Gagnon and John J. Peterson, Estimation of Confidence Intervals for Area Under the Curve from Destructively Obtained Pharmacokinetic Data, Journal of Pharmacokinetics and Pharmacodynamics, 26: 87-102, 1998.