Calibration Curves III: A Different View -

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Calibration Curves III: A Different View

May 1, 2009
By: John W. Dolan
LCGC North America

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John W. Dolan

This is the third installment in a series of "LC Troubleshooting" columns that focus on calibration curves used for liquid chromatography (LC) methods. We started (1) by looking at the issue of whether or not to force a calibration curve through the origin (x = 0, y = 0). Last month (2) we looked at some techniques to determine the limits of detection and quantification, specifically looking at the signal-to-noise ratio (S/N) as a tool in this process. This month we will consider some alternate ways to look at the data so as to determine if they appear to be normal or are trying to tell us that something is amiss. Next month we'll look at some different calibration techniques.

The Conventional Plot

Figure 1

Figure 1 shows a plot of a hypothetical calibration curve comprising five replicate injections of an exponentially diluted standard at concentrations of 1, 2, 5, 10, 20, 50, 100, 200, 500, and 1000 ng/mL. This presentation of the calibration curve looks quite impressive, with the coefficient of determination r² = 0.9999. The y-intercept (0.0906) is less than the standard error of the y-intercept (SEy = 0.4979), so based upon earlier discussion (1), we can justify forcing the curve through the intercept.

Although the curve statistics look impressive for Figure 1, such plots have marginal visual value for problem diagnosis. This is because most of the points are crowded together at the lower end of the curve, where there isn't much detail to be seen.

The %-Error Plot

Another way of plotting the standard curve data is the %-error plot, as shown in Figure 2 for the data set of Figure 1. In this case, both axes are changed. The x-axis (concentration) is plotted on a logarithmic scale instead of a linear one. This spreads out the data points across the graph, allowing more detail to be seen at the low concentrations. Often, we are less interested in the absolute response for a given concentration than how close that response is to the expected value. We can obtain this information by converting the response (y-data) into %-error from the calibration curve. This is quite simple. First, we use the regression equation for the curve to plot the expected response at each concentration. Because we can force the curve through the origin, we will use the y = mx format, which for the present curve is y = 1.0002x. Thus, a concentration of 2 ng/mL is expected to give a response of (1.0002)(2) = 2.0004. One of the 2-ng/mL injections gave a response of 1.8859. This is converted to %-error as:

Figure 2

or 100 (1.8859 – 2.0004)/2.0004 = –5.7% error. This value (x = 2 ng/mL, y = –5.7%) is plotted along with the remaining points to obtain Figure 2.

Figure 2 can give us additional information about the data that are not present in Figure 1. First, notice how the data begin to scatter more around the expected value (0% error) at values less than 50 ng/mL. And as the concentration is decreased, the scatter increases. This is reminiscent of Figure 2 of last month's "LC Troubleshooting" (2), in which we observed a decrease in S/N with decreased concentration. Many of the errors in an LC method increase with decreasing sample concentration, such as weighing, volumetric, and detection errors. These combine to give the expected increase in error at lower concentrations, which ultimately determines the lower limit of quantification and detection of a method (2). If the data are behaving normally, we expect the scatter in the data to be roughly equal above and below the expected values — and this is the case in Figure 2. If the data are distributed normally, we also expect that ≈68% of the values will lie within ±1 standard deviation (SD) of the mean — this would be three or four of the five injections at each concentration. I have plotted dashed lines at the ±1 SD limits in Figure 2, and it can be seen that, once again, the data behave normally.

These data give us assurance that, although there is more error at the lower end of the curve than at higher concentrations, the errors are distributed normally and behave as expected. This means that we can have more confidence in the results. Although the pattern of error — increasing error at lower concentrations — is inevitable, this does not mean that we will be unable to reduce the error at the test concentrations. Often, weighing out more reference standard and using larger dilution volumes will help reduce error, as will the use of volumetric glassware instead of graduated cylinders. Sometimes an internal standard will help to reduce error in sample preparation steps. Larger injections will result in larger peaks, which are easier to integrate, reducing the error in data processing. These, and other method improvements, usually will help to reduce the overall error of the method, which generally is most obvious at the lowest concentrations in the calibration curve.

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