Applied Statistics and Probability for Engineers

444 CHAPTER 12 MULTIPLE LINEAR REGRESSION

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 eiis

Furthermore, the hiielements must fall in the interval 0hii 1. This implies that the stan-

dardized 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 residu-

als that might be unusually large, observations 15 and 17. The standardized residuals are

Now h15,150.0737 and h17,170.2593, so the studentized residuals are

and

Notice that the studentized residuals are larger than the corresponding standardized residuals.

However, the studentized residuals are still not so large as to cause us serious concern about

possible outliers.

12-5.2 Influential Observations

When using multiple regression, we occasionally find that some subset of the observations is

unusually influential. Sometimes these influential observations are relatively far away from

the vicinity where the rest of the data were collected. A hypothetical situation for two variables

is depicted in Fig. 12-10, where one observation in x-space is remote from the rest of the data.

The disposition of points in the x-space is important in determining the properties of the

model. For example, point (xi 1 , xi 2 ) in Fig. 12-10 may be very influential in determining R^2 , the

estimates of the regression coefficients, and the magnitude of the error mean square.

r 17 

e 17

2 ˆ^211 h17,17 2



4.33

2 5.2352 11 0.2593 2

2.20

r 15 

e 15

2 ˆ^211 h15,15 2



5.88

2 5.2352 11 0.0737 2

2.67

d 15 

e 15

2 ˆ^2



5.88

2 5.2352

2.57 and d 17 

e 17

2 MSE



4.33

2 5.2352

1.89

V ̨ 1 ei 2 ^211 hii 2 , i1, 2,p, n

xi 1 x 1

xi 2

x 2

Region containing

all observations

except the ith

Figure 12-10 A

point that is remote in

x-space.

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