Chapter 5 - Analytical Sciences Digital Library

196 Analytical Chemistry 2.0
> inverse.predict(model, sample, alpha = 0.05)
$Prediction
[1] 0.2412597
$`Standard Error`
[1] 0.002363588
$Confidence
[1] 0.006562373
Figure 5.24 Output from R’s command for predicting the analyte’s
concentration, C A , from the sample’s signal, S samp .
$`Confidence Limits`
[1] 0.2346974 0.2478221
You may have noticed that this way of
defining weights is different than that
shown in equation 5.28. In deriving equations
for a weighted linear regression, you
can choose to normalize the sum of the
weights to equal the number of points, or
you can choose not to—the algorithm in
R does not normalize the weights.
Practice Exercise 5.7
Use Excel to complete the
regression analysis in Practice
Exercise 5.4.
Click here to review your answer
to this exercise.
Let’s use this command to complete Example 5.12. First, we need to create
an object containing the weights, which in R are the reciprocals of the standard
deviations in y, (s y i )–2 . Using the data from Example 5.12, we enter
> syi=c(0.02, 0.02, 0.07, 0.13, 0.22, 0.33)
> w=1/syi^2
to create the object containing the weights. The commands
> modelw = lm(signal ~ conc, weights = w)
> summary(modelw)
generate the output shown in Figure 5.25. Any difference between the
results shown here and the results shown in Example 5.12 are the result of
round-off errors in our earlier calculations.
> modelw=lm(signal~conc, weights = w)
> summary(modelw)
Call:
lm(formula = signal ~ conc, weights = w)
Residuals:
1 2 3 4 5 6
-2.223 2.571 3.676 -7.129 -1.413 -2.864
Figure 5.25 The summary of R’s regression analysis for
a weighted linear regression. The types of information
shown here is identical to that for the unweighted linear
regression in Figure 5.22.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.04446 0.08542 0.52 0.63
conc 122.64111 0.93590 131.04 2.03e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 4.639 on 4 degrees of freedom
Multiple R-Squared: 0.9998, Adjusted R-squared: 0.9997
F-statistic: 1.717e+04 on 1 and 4 DF, p-value: 2.034e-08