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Chapter 5 Calibrations, Standardizations, and Blank Corrections 127

Equations for calculating confidence intervals for the slope, the y-intercept,
and the concentration of analyte when using a weighted linear regression are
9
not as easily defined as for an unweighted linear regression. The confidence in-
terval for the concentration of an analyte, however, will be at its optimum value
–
when the analyte’s signal is near the weighted centroid, y, of the calibration
curve

1
y = å w ii y
n

5C.4 Weighted Linear Regression with Errors in Both x and y

If we remove the assumption that the indeterminate errors affecting a calibration
curve are found only in the signal (y), then indeterminate errors affecting the
preparation of standards containing known amounts of analyte (x) must be fac-
tored into the regression model. The solution for the resulting regression line is
computationally more involved than that for either the unweighted or weighted
regression lines, and is not presented in this text. The suggested readings at the
end of the chapter list several papers discussing algorithms for this regression
method.

5 5 Curvilinear and Multivariate Regression
C.
Regression models based on a straight line, despite their apparent complexity, use
the simplest functional relationship between two variables. In many cases, calibra-
tion curves show a pronounced curvature at high concentrations of analyte (see Fig-
ure 5.3b). One approach to constructing a calibration curve when curvature exists is
to seek a transformation function that will make the data linear. Logarithms, expo-
nentials, reciprocals, square roots, and trigonometric functions have all been used
in this capacity. A plot of y versus log x is a typical example. Such transformations
are not without complications. Perhaps the most obvious is that data that originally
has a uniform variance for the y values will not maintain that uniform variance
when the variable is transformed.
A more rigorous approach to developing a regression model for a nonlinear
2
calibration curve is to fit a polynomial equation such as y = a + bx + cx to the data.
Equations for calculating the parameters a, b, and c are derived in the same manner
as that described earlier for the straight-line model. 10 When a single polynomial
equation cannot be fitted to the calibration data, it may be possible to fit separate
polynomial equations to short segments of the calibration curve. The result is a sin-
gle continuous calibration curve known as a spline function.
The regression models considered earlier apply only to functions containing a
single independent variable. Analytical methods, however, are frequently subject to
determinate sources of error due to interferents that contribute to the measured sig-
nal. In the presence of a single interferent, equations 5.1 and 5.2 become

S meas = k A n A + k I n I + S reag

S meas = k A C A + k I C I + S reag
where k I is the interferent’s sensitivity, n I is the moles of interferent, and C I is the in-
terferent’s concentration. Multivariate calibration curves can be prepared using
standards that contain known amounts of analyte and interferent. 11