Applied Statistics and Probability for Engineers

442 CHAPTER 12 MULTIPLE LINEAR REGRESSION

Figure 12-7 Plot of residuals against yˆ.

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 probabil-

ity plot of these residuals is shown in Fig. 12-6. No severe deviations from normality are ob-

viously apparent, although the two largest residuals (e 15 5.88 and e 17 4.33) do not fall

extremely close to a straight line drawn through the remaining residuals.

The standardized residuals

Figure 12-6 Normal probability plot of residuals.

are often more useful than the ordinary residuals when assessing residual magnitude. The

standardized residuals corresponding to e 15 and e 17 are and

, 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 in Fig. 12-7, and against x 1 and x 2 in Figs. 12-8 and

12-9, respectively.*The two largest residuals, e 15 and e 17 , are apparent. Figure 12-8 gives

some indication that the model underpredicts the pull strength for assemblies with short wire

length and long wire length and overpredicts the strength for assemblies

with intermediate wire length 17 x 1  142. The same impression is obtained from Fig. 12-7.

1 x 1  62 1 x 1  152

yˆ

d 17 4.33 1 4.23521.89

d 15 5.88 1 5.23522.57

*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 dis-

playing the relationship between the response yand each individual regressor.

di (12-41)

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