## Linear Portions of the Curves:

Many scientific instruments will only yield a linear response function over a certain concentration range; beyond this region, the calibration curve will be nonlinear. It is therefore important to choose the correct region for linear regression analysis in order to minimise any errors due to nonlinearity.

A quick and convenient way to accomplish this is to visually inspect the graph and select only those data points that appear to lie on a straight line, as well as a few points on either side of the linear portion. Consider, for example, the fluorescence calibration data from the previous example but over an extended concentration range:

 C (pg/mL): Intensity: 0 2 4 6 8 10 12 14 16 18 20 2.1 5 9 12.6 17.3 21 24.7 28.4 31 32.9 33.9

Extended fluorescence calibration curve showing curvature at high concentrations

### Finding the Linear Portion:

To find the linear portion, we can calculate R for successive sets of data points, and observe the range for which there is a noticeable decrease in the R-value. A relatively quick way to do this is to use the trend line feature in Excel™. To do this, first create a table of the calibration data within a new worksheet, and produce an XY Scatter Plot of the data using the Chart function as described previously. Next, add the trend line, being sure to set the options to show the R2 value on the chart. By default, this should include all the points in the calibration dataset.

Note how the source data is framed when the chart is selected. If you move the cursor over the box in the lower left corner of each column, you can change the selection of points used to produced the chart.

Use this feature to omit the highest concentration–intensity point from the chart and obtain a new R2 value. Repeat this procedure several times, and compile a table of the corresponding ranges and R2 values.

### Showing the Linear Portion:

The following table shows the R2 values for different ranges (and, therefore, different values of n), together with the resulting calibration plot:

Concentration Rangen R2
0 to 20110.98478
0 to 18100.99374
0 to 1690.99759
0 to 1480.99848
0 to 1270.99776

Download a sample Excel file for this exercise

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To overlay your trend line with the full calibration dataset, right-click (or control-click) on the chart, select Source Data..., and choose the Series tab. Use the Add button, and fill in the cell locations for the x and y values.

As you can see from table, the correlation coefficient drops off after the 9th data point. The plot shows the correlation coefficent for points 1–8, with the full set of calibration points superposed as an added data series. This shows that the 9th point (C = 16 pg/mL) does actually lie below the line. We can reasonably conclude, therefore, that the linear portion of the calibration curve lies between 0 and 14 pg/mL.

You can use the equation for the trend line to calculate the y-residuals from the regression line – that is, the distance of each actual y value from the straight line – in order to corroborate your choice. A more rigorous method to test the validity of your choice is to use a t-test, which is discussed in a later section.

Continue to Regression Equation...

Important: you should always show the complete set of calibration points on your final plot, and indicate which were included in the regression analysis

Download the final Excel file for this exercise