# How does Prism compute "area under the curve" (AUC)? What are its units?

Prism's calculations of area under the curve are very simple, as explained here.

**Area under what curve?**

This page is about Prism's analysis named "Area under the curve". Prism has a separate analysis to create a Receiver Operator Characteristic (ROC) curve and this analysis also computes the area under the the ROC curve. Read this explanation.

The input to the area-under-the-curve analysis can be either a data table or a curve. The analysis sees a fitted curve as simply a series of connected XY points, so the calculations are the same either way. If you analyze a data table, Prism simply computes the area under the "curve" defined by connecting the points by straight lines.

**How the calculations work**

Prism computes area-under-the-curve by the trapezoidal method. The area under each connecting segment describes a trapezoid, as shown below (left). Prism calculates the area of each trapezoid by calculating the area of the equivalent rectangle (below, right). The area under the curve is the sum of areas of all the rectangles.

The area is computed using the baseline you specify and the curve between two X values. Which X values?

- If all your data points are larger than the baseline, the AUC calculations start at the lowest X value in your data set and end at the largest X value. Note that Prism does not extend the curve beyond the X range of your data.
- If the Y values at the lowest X values are below your baseline: Prism finds the smallest X value in your data associated with a Y value greater than the baseline. It draws a line between that point and the point with the next smallest X value in your data set. It then uses linear interpolation to find where that line crosses the baseline and uses that interpolated value as the first X value to compute the AUC.
- If the Y values at the largest X values are below your baseline: Prism finds the largest X value in your data associated with a Y value greater than the baseline. It draws a line between that point and the point with the next largest X value in your data set. It then uses linear interpolation to find where that line crosses the baseline and uses that interpolated value as the last X value to compute the AUC.

**Notes**

- The units of the AUC are the units of the Y axis times units of the X axis. For example, if your Y axis measures concentration in mmol/L and the X axis measures time in minutes, then the area is expressed in units of (mmol/L) x minutes.
- 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.
- Prism does not extrapolate the curve down to the baseline (except for ROC curves which are extrapolated down to 0,0 and up to 100,100).
- Prism does not require equally spaced X values. When it sums the areas of the trapezoids, it is fine if some are fatter than others.
- If you enter data with replicate Y values, or as Mean and SD or SEM, Prism 7 uses those values to compute a confidence interval (and SE) for the AUC or for the AUC. These calculations have been described by Gagnon (1).

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.