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The residuals from the multiple regression model, defined by ei yi yi, play an important role in judging model adequacy just as they do in simple linear regression. As noted in Section 11-7.1, several residual plots are often useful; these are illustrated in Example 12-9. It is also
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2 5 10 20 30 40 50 60 70 80 90 98 99 Probability
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6 5 4 3 2 ei 1 0 1 2 3 4 5 6 4 2 0 2 4 6 7 10 20 30 ^ yi 40 50 60 70
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Figure 12-6
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Figure 12-7
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Plot of residuals against y.
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helpful to plot the residuals against variables not presently in the model that are possible candidates for inclusion. Patterns in these plots may indicate that the model may be improved by adding the candidate variable. EXAMPLE 12-9 The residuals for the model from Example 12-1 are shown in Table 12-3. A normal probability plot of these residuals is shown in Fig. 12-6. No severe deviations from normality are obviously apparent, although the two largest residuals (e15 5.88 and e17 4.33) do not fall extremely close to a straight line drawn through the remaining residuals. The standardized residuals 2MSE ei 2 2 ei
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are often more useful than the ordinary residuals when assessing residual magnitude. The standardized residuals corresponding to e15 and e17 are d15 5.88 15.2352 2.57 and d17 4.33 14.2352 1.89, and they do not seem unusually large. Inspection of the data does not reveal any error in collecting observations 15 and 17, nor does it produce any other reason to discard or modify these two points. The residuals are plotted against y in Fig. 12-7, and against x1 and x2 in Figs. 12-8 and 12-9, respectively.* The two largest residuals, e15 and e17, are apparent. Figure 12-8 gives some indication that the model underpredicts the pull strength for assemblies with short wire length 1x1 62 and long wire length 1x1 152 and overpredicts the strength for assemblies with intermediate wire length 17 x1 142 . The same impression is obtained from Fig. 12-7.
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*There are other methods, described in Montgomery, Peck, and Vining (2001) and Myers (1990), that plot a modified version of the residual, called a partial residual, against each regressor. These partial residual plots are useful in displaying the relationship between the response y and each individual regressor.
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In Example 12-9 we used the standardized residuals di ei 2 2 as a measure of residual magnitude. Some analysts prefer to plot standardized residuals instead of ordinary residuals, because the standardized residuals are scaled so that their standard deviation is approximately unity. Consequently, large residuals (that may indicate possible outliers or unusual observations) will be more obvious from inspection of the residual plots. Many regression computer programs compute other types of scaled residuals. One of the most popular is the studentized residual 2
Either the relationship between strength and wire length is not linear (requiring that a term 2 involving x1, say, be added to the model), or other regressor variables not presently in the model affected the response.
hii 2
1, 2, p , n
where hii is the ith diagonal element of the matrix H 1 2
The H matrix is sometimes called the hat matrix, since
X 1X X2
Thus H transforms the observed values of y into a vector of fitted values y. 31, xi1, xi2, p , xik 4 , Since each row of the matrix X corresponds to a vector, say x i another way to write the diagonal elements of the hat matrix is
x 1X X2 i
Note that apart from 2, hii is the variance of the fitted value yi. The quantities hii were used in the computation of the confidence interval on the mean response in Section 12-3.2.
Under the usual assumptions that the model errors are independently distributed with mean zero and variance 2, we can show that the variance of the ith residual ei is V 1ei 2
hii 2,
1, 2, p , n
Furthermore, the hii elements must fall in the interval 0 hii 1. This implies that the standardized residuals understate the true residual magnitude; thus, the studentized residuals would be a better statistic to examine in evaluating potential outliers. To illustrate, consider the two observations identified in Example 12-9 as having residuals that might be unusually large, observations 15 and 17. The standardized residuals are d15 Now h15,15 2 e15
25.2352 5.88