D. Harvey - Modern Analytical Chemistry (794078), страница 39
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When this assumption is false, asshown in Figure 5.11b, the variance associated with each value of y must beincluded when estimating β0 and β1. In this case the predicted slope and intercept areb1 =n ∑ w i x i y i – ∑ w i x i ∑ w i yin ∑ w i x 2i – (∑ w i x i )25.221400-CH05 9/8/99 3:59 PM Page 125Chapter 5 Calibrations, Standardizations, and Blank Correctionsandb0 =∑ w i yi – b1 ∑ w i x in5.23where wi is a weighting factor accounting for the variance in measuring yi. Values ofwi are calculated using equation 5.24.ns –i 2wi =5.24∑ s –i 2where si is the standard deviation associated with yi. The use of a weighting factorensures that the contribution of each pair of xy values to the regression line is proportional to the precision with which yi is measured.EXAMPLE 5.13The following data were recorded during the preparation of a calibration curve,–where Smeas and s are the mean and standard deviation, respectively, for threereplicate measurements of the signal.CA–Smeass0.0000.1000.2000.3000.4000.5000.0012.3624.8335.9148.7960.420.020.020.070.130.220.33–Determine the relationship between Smeas and CA using a weighted linearregression model.SOLUTIONOnce again, as you work through this example, remember that x represents theconcentration of analyte in the standards (C S ), and y corresponds to the–average signal (Smeas).
We begin by setting up a table to aid in the calculation ofthe weighting factor.si–2xiyisi0.0000.1000.2000.3000.4000.5000.0012.3624.8335.9148.7960.420.020.020.070.130.220.332500.002500.00204.0859.1720.669.18wi2.83392.83390.23130.06710.02340.0104Adding together the values in the forth column gives∑ s –i 2= 5293.09which is used to calculate the weights in the last column. As a check on thecalculation, the sum of the weights in the last column should equal the numberof calibration standards, n. In this caseΣwi = 6.00001251400-CH05 9/8/99 3:59 PM Page 126Modern Analytical ChemistryAfter the individual weights have been calculated, a second table is used to aidin calculating the four summation terms in equations 5.22 and 5.23.xiyiwiwixiwiyiwixi2wixiyi0.0000.1000.2000.3000.4000.5000.0012.3624.8335.9148.7960.422.83392.83390.23130.06710.02340.01040.00000.28340.04630.02010.00940.00520.000035.02705.74322.40961.14170.62840.00000.02830.00930.00600.00370.00260.00003.50271.14860.72290.45670.3142Adding the values in the last four columns givesΣwixi = 0.3644 Σwiyi = 44.9499 Σwixi2= 0.0499 Σwixiyi = 6.1451Substituting these values into the equations 5.22 and 5.23 gives the estimatedslopeb1 =(6)(6.1451) – (0.3644)(44.9499)= 122.985(6)(0.0499) – (0.3644)2and the estimated y-interceptb0 =44.9499 – (122.985)(0.3644)= 0.02246The relationship between the signal and the concentration of the analyte,therefore, is–Smeas = 122.98 × CA + 0.02with the calibration curve shown in Figure 5.12.8060Smeas126402000.00.10.20.3CA0.40.50.6Figure 5.12Weighted normal calibration curve for the data in Example 5.13.
Thelines through the data points show the standard deviation of thesignal for the standards. These lines have been scaled by a factor of50 so that they can be seen on the same scale as the calibrationcurve.1400-CH05 9/8/99 3:59 PM Page 127Chapter 5 Calibrations, Standardizations, and Blank CorrectionsEquations for calculating confidence intervals for the slope, the y-intercept,and the concentration of analyte when using a weighted linear regression arenot as easily defined as for an unweighted linear regression.9 The confidence interval for the concentration of an analyte, however, will be at its optimum valuewhen the analyte’s signal is near the weighted centroid, –y, of the calibrationcurvey =1n∑ wi y i5C.4 Weighted Linear Regression with Errors in Both x and yIf we remove the assumption that the indeterminate errors affecting a calibrationcurve are found only in the signal (y), then indeterminate errors affecting thepreparation of standards containing known amounts of analyte (x) must be factored into the regression model.
The solution for the resulting regression line iscomputationally more involved than that for either the unweighted or weightedregression lines, and is not presented in this text. The suggested readings at theend of the chapter list several papers discussing algorithms for this regressionmethod.5C.5 Curvilinear and Multivariate RegressionRegression models based on a straight line, despite their apparent complexity, usethe simplest functional relationship between two variables.
In many cases, calibration curves show a pronounced curvature at high concentrations of analyte (see Figure 5.3b). One approach to constructing a calibration curve when curvature exists isto seek a transformation function that will make the data linear. Logarithms, exponentials, reciprocals, square roots, and trigonometric functions have all been usedin this capacity. A plot of y versus log x is a typical example.
Such transformationsare not without complications. Perhaps the most obvious is that data that originallyhas a uniform variance for the y values will not maintain that uniform variancewhen the variable is transformed.A more rigorous approach to developing a regression model for a nonlinearcalibration curve is to fit a polynomial equation such as y = a + bx + cx2 to the data.Equations for calculating the parameters a, b, and c are derived in the same manneras that described earlier for the straight-line model.10 When a single polynomialequation cannot be fitted to the calibration data, it may be possible to fit separatepolynomial equations to short segments of the calibration curve.
The result is a single continuous calibration curve known as a spline function.The regression models considered earlier apply only to functions containing asingle independent variable. Analytical methods, however, are frequently subject todeterminate sources of error due to interferents that contribute to the measured signal. In the presence of a single interferent, equations 5.1 and 5.2 becomeSmeas = kAnA + kInI + SreagSmeas = kACA + kICI + Sreagwhere kI is the interferent’s sensitivity, nI is the moles of interferent, and CI is the interferent’s concentration. Multivariate calibration curves can be prepared usingstandards that contain known amounts of analyte and interferent.111271400-CH05 9/8/99 3:59 PM Page 128128Modern Analytical Chemistry5D Blank CorrectionsIn discussing ways to standardize a method, we assumed that an appropriatereagent blank had been used to correct Smeas for signals originating from sourcesother than the analyte.
At that time we did not ask an important question—“What constitutes an appropriate reagent blank?” Surprisingly, the answer is notintuitively obvious.In one study,12 analytical chemists were asked to evaluate a data set consisting of a normal calibration curve, three samples of different size but drawn fromthe same source, and an analyte-free sample (Table 5.3). At least four differentapproaches for correcting the signals were used by the participants: (1) ignorethe correction entirely, which clearly is incorrect; (2) use the y-intercept of thecalibration curve as a calibration blank, CB; (3) use the analyte-free sample as areagent blank, RB; and (4) use both the calibration and reagent blanks.
Equations for calculating the concentration of analyte using each approach are shownin Table 5.4, along with the resulting concentration for the analyte in each of thethree samples.That all four methods give a different result for the concentration of analyteunderscores the importance of choosing a proper blank but does not tell uswhich of the methods is correct.
In fact, the variation within each method for thereported concentration of analyte indicates that none of these four methods hasadequately corrected for the blank. Since the three samples were drawn from thesame source, they must have the same true concentration of analyte. Since allfour methods predict concentrations of analyte that are dependent on the size ofthe sample, we can conclude that none of these blank corrections has accountedfor an underlying constant source of determinate error.To correct for all constant method errors, a blank must account for signalsdue to the reagents and solvent used in the analysis and any bias due to interac-Table 5.3Hypothetical Data Used to Study Proceduresfor Method BlanksWsaSstand1.66675.00008.33339.550711.666718.160019.93330.25000.50000.75000.84131.00001.48701.6200SampleNumber123Wxb62.474682.7915103.1085analyte-freecSsamp0.80001.00001.20000.1000Calibration equation: Sstand = 0.0750 × Ws + 0.1250Source: Modified from Cardone, M.
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