Applied Statistics and Probability for Engineers

12-5 MODEL ADEQUACY CHECKING 443

Either the relationship between strength and wire length is not linear (requiring that a term

involving x^21 , say, be added to the model), or other regressor variables not presently in the

model affected the response.

In Example 12-9 we used the standardized residuals 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 un-

usual 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

diei 2 ˆ^2

Figure 12-8 Plot of residuals against x 1.

where hiiis the ith diagonal element of the matrix

The Hmatrix is sometimes called the “hat” matrix,since

Thus Htransforms the observed values of yinto a vector of fitted values.

Since each row of the matrix Xcorresponds to a vector, say ,

another way to write the diagonal elements of the hat matrix is

x¿i 3 1, xi 1 , xi 2 ,p, xik 4

yˆ

yˆXˆX ̨ 1 X¿X 2 ^1 ̨X¿yHy

H 1 ¿ 2 ^1 ¿

ri (12-42)

ei

2    ˆ^211 hii 2

i1, 2,p, n

–4 1

–3

–2

0

1

2

3

4

ei

x 1

–1

5

6

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 –4 100 200 300 400 500 600 700

–3

–2

–1

0

1

2

3

4

5

6

ei

x 2

Figure 12-9 Plot of residuals against x 2.

hiix¿i 1 X¿X (^2) (12-43)
 (^1) x
i
Note that apart from 2 , hiiis the variance of the fitted value. The quantities hiiwere used in
the computation of the confidence interval on the mean response in Section 12-3.2.
yˆi
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