Area Under Curve - NCSS - area under the curve formula

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• (1) Area of curve
• (2) Area under the curve problems
• (3) Are under the curve
• (4) Desmos area under curve

NCSS Statistical Software NCSS.com
Chapter 390
Area Under Curve
Introduction
Suppose you are studying a drug that influences plasma concentration. A popular method of looking at the
absorption and elimination properties of the drug is to follow the plasma concentration across time. When the
concentration values are plotted on the vertical axis, the time values are plotted on the horizontal axis, and the
points are joined with a line, a curve results. One method of making comparisons among different types of drugs
and different doses of the same drug is to compute the area under the curve (AUC).
AUC is computed by the trapezoidal rule as follows:
AUC = 1 n-1
2 (Ti+1 - Ti )(Ci+1 + Ci - 2B)
i =1
where Ti is the ith time value, Ci is the ith concentration value, n is the number of time values, and B is the baseline
value. The area between the baseline and the curve is computed by this formula.
The AUC should be calculated from zero to a time at which the concentration has returned to its regular levels.
Also, when making comparisons, you should ensure that all AUC's are computed for the same time intervals.
Data Structure
Two different data structures may be used to store data for analysis by this procedure. In the first format, the X
(time) values are stored in one variable and the Y (concentration) values are stored in other variables, one variable
per group. The AUC dataset is in this format.
In the other format, the X values are stored in one variable, group values are stored in a second variable, and the
measurements are stored in a third variable. The data in the AUC dataset was reorganized to be in this format in
the AUC1 dataset.
Format Type 1
AUC dataset
Time P1 P2 P3
05 4 6
1 15 14 17
2 20 16 22
3 21 18 23
4 21 17 25
5 19 15 22
10 15 12 18
50 6 3 7
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Area Under Curve
Format Type 2
AUC1 dataset
Time Subject Concentration
0 P1 5
1 P1 15
2 P1 20
3 P1 21
4 P1 21
5 P1 19
10 P1 15
50 P1 6
0 P2 4
1 P2 14
2 P2 16
3 P2 18
4 P2 17
5 P2 15
10 P2 12
50 P2 3
0 P3 6
1 P3 17
2 P3 22
3 P3 23
4 P3 25
5 P3 22
10 P3 18
50 P3 7
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Area Under Curve
Example 1 - Calculating Area Under the Curve (Multiple Y
Columns)
This section presents a tutorial of calculating the AUC for three individuals labeled P1, P2, and P3. The responses
for each individual are contained in separate columns. The data are contained on the AUC dataset.
Setup
To run this example, complete the following steps:
1 Open the AUC example dataset
? From the File menu of the NCSS Data window, select Open Example Data.
? Select AUC and click OK.
2 Specify the Area Under Curve procedure options
? Find and open the Area Under Curve procedure using the menus or the Procedure Navigator.
? The settings for this example are listed below and are stored in the Example 1 settings template. To load
this template, click Open Example Template in the Help Center or File menu.
Option Value
Variables Tab
Y Variable(s) ........................................... P1-P3
X Variable ............................................... Time
Reports Tab
AUC Report ............................................ Checked
Data Report ............................................ Checked
3 Run the procedure
? Click the Run button to perform the calculations and generate the output.
Area Section
Area Section with Baseline = 0
Area Under Y Time at Time Time
Y Variables Curve Max Max of Y Min Max Count
P1 594 21 3 0 50 8
P2 442 18 3 0 50 8
P3 701 25 4 0 50 8
This report shows the area under curve as well as supporting information.
Y Variables
This gives the value of the Y variable (if multiple Y variables were specified) or the value of the break variable (if
a break variable was specified).
Area Under Curve
The is the area under the curve calculated in the interval between Time Min and Time Max. It is calculated using
the trapezoidal formula presented earlier.
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