Assessing Goodness of Fit of the Experimental Calibration Data and the Range of Linearity
A correlation coefficient, r, of 0.9900 is the goal that achieves “goodness of fit” between instrument response and analyte amount or concentration for both chromatographic and spectroscopic determinative techniques. An interesting question arises. Over how many orders of magnitude in amount or concentration does it take to yield an r? The answer to this question serves to define what is meant by the range of linearity TEQA. This range will differ among determinative techniques. The majority of column chromatographic used in TEQA occurs at analyte concentration levels from low ppb to low ppm. Both linear (or first-order least squares regression) and polynomial (second-order and even third-order least squares regression) calibration covering one or more orders of magnitude above the instrument’s IDL are all useful ways to address the degree of goodness of fit. The least squares slope, called a response factor in some EPA methods, and the uncertainty in the slope are also useful criteria in assessing linearity. The coefficient of determination, r^{2}, is a measure of the amount of variation in the dependent variable (instrument response) that is accounted for by the independent variable (analyte amount or concentration), (pp. П8-120)^{7}
Let us delve into the concept of calibration linearity or lack thereof a bit more. The linear dynamic range of an instrument is defined as that range of concentration from the analyte’s limit of quantitation x on up to where a departure from linearity is observed. This concentration is the called the analyte’s limit of linearity, x _{oi} This concept is best understood graphically as shown in Figure 2.19.
The linear dynamic range for most analytical instruments designed to perform TEQA covers at least two orders of magnitude. For example, a range of analyte concentration from 10 to 1,000 ppb covers two orders of magnitude and is quite suitable for TEQA. Some methods have a linear dynamic range over five or six orders of magnitude. Nonlinear least squares regression analysis does exist, can be found in most contemporary statistical software packages, and can be applicable to TEQA.
Consider the construction of a seven-point calibration for the determination of lead, Pb, in a human blood specimen as shown in Figure 2.20. This is an example of environ-health TEQA. Declines in blood Pb concentration levels in children over the past 20 years represent one of the real triumphs in public health in the U.S. Refer to the discussion in Chapter 1. Figure 2.20 shows
FIGURE 2.20
FIGURE 2.21
a calibration plot for quantitating Pb in human blood using flame atomic absorption spectroscopy (F1AA). FLAA was once the dominant determinative technique to quantitate metals. The desire for much lower IDLs brought the graphite furnace atomic absorption spectroscopy (GFAA) to the public health laboratory. Today, inductively coupled plasma atomic emission spectroscopy (ICP- AES) is more commonly found. For laboratories that quantitate trace metals or ultratrace metals concentration levels of detection, ICP-mass spectrometry (ICP-MS) is preferred. GFAA, ICP- AES and ICP-MS determinative techniques are introduced in Chapter 4. Referring to Figure 2.20, note that the concentration levels are in the ppm range. This range is accommodated nicely by F1AA as the determinative technique of choice (the F1AA determinative technique is introduced in Chapter 4). It becomes almost immediately evident that applying a linear least squares fit to the experimental calibration data is not optimum. Figure 2.21 shows that a quadratic (second-order) least squares regression line is a much better fit. Figure 2.22 shows that a cubic (third-order) least squares regression line makes a slight improvement over the quadratic fit. All three least squares regressed lines or curves are polynomials that can be described in terms of a set of coefficients, as shown below:
Polynomial |
Type of Fit |
у = aO + a lx |
Linear |
у = aO + a lx + alxl |
Quadratic |
v = c/0 + alx + alxl + «3x3 |
Cubic |
Recall from our previous discussion in this chapter that for the linear fit, a_{0} corresponds to the у intercept b while a, corresponds to the slope of the least squares regression line m.
It is instructive to examine the sign of the highest-order coefficients in the linear and quadratic equations introduced above. Consider assigning real numbers to each coefficient and plotting these equations as a function of x,f(x). This can be accomplished easily using a graphics calculator. Figure 2.23 shows three superimposed fix) (a quick review of classic analytic geometry). This result is shown in Figure 2.23. Superimposing curves for the first quadrant for all three polynomials clearly shows just what effect a positive or negative coefficient for x^{2} has on the direction of curves 2 and 3. Most nonlinear regressed quadratic curves plotted in the first quadrant
FIGURE 2.22
for all positive values for .v tend to level off, as shown by curve 3. This curve is quite common for instruments used in TEQA when the linear dynamic range of the detector has been exceeded. A third-order polynomial (not shown in Figure 2.23) tends toward more of a sigmoid or slanted curve shape.
For example, to mathematically conduct a polynomial evaluation of calibration linearity for quantitating Pb in human blood by FI AA, refer to guidelines such as that published by the National Committee for Clinical Laboratory Standards (NCCLS) EP6-A, vol. 21, no. 20. Evaluation of the Linearity of Quantitative Measurement Procedures: A Statistical Approach: Approved Guideline. The author had describes the mathematics for this earlier.^{27}