Applied Statistics and Probability for Engineers

other columns, so the intercept in this model will always be estimated by. The (4 4)

matrix for the four coded variables is the correlation matrix

This matrix contains several large correlation coefficients, and this may indicate significant

multicollinearity. The inverse of is

The variance inflation factors are the main diagonal elements of this matrix. Note that three of

the variance inflation factors exceed 10, a good indication that multicollinearity is present.

Equation S12-1 was solved for various values of , and the results are summarized in

Table S12-3. The ridge trace is shown in Fig. S12-1. The instability of the least squares estimates

is evident from inspection of the ridge trace. It is often difficult to choose a value of

from the ridge trace that simultaneously stabilizes the estimates of all regression coefficients.

Because the bias in the coefficients increases as increases, it is usually best to choose as small

as possible; yet we want to be large enough to provide reasonable stability in the coefficients.

In our example, most of the change in the regression coefficients has occurred when 0.05

0.1. We will choose 0.064, which implies that the regression model is

using. Converting the model to the original variables wj, we have

This is the ridge regression model for the cement data.

yˆ2.99130.8920w 1 0.3483w 2 3.3209w 3 0.0623w 4

ˆ 0 y25.53

yˆ25.5318.0566x 1 17.2202x 2 36.0743x 3 4.7242x 4

j* 1  02

1 ¿ 2 ^1 D

20.769 25.813 0.608 44.042

25.813 74.486 12.597 107.710

0.608 12.597 8.274 18.903

44.042 107.710 18.903 163.620

T

X¿X

X¿XD

1.00000

0.84894

0.91412

0.93367

0.84894

1.00000

0.76899

0.97567

0.91412

0.76899

1.00000

0.86784

0.93367

0.97567

0.86784

1.00000

T

y X¿X

12-4

Table S12-3 Ridge Regression Estimates for the Cement Data

 1 *() * 2 () * 3 () * 4 ()

.000 28.3318 65.9996 64.0479 57.2491

.001 31.0360 57.0244 61.9645 44.0901

.002 32.6441 50.9649 60.3899 35.3088

.004 34.1071 43.2358 58.0266 24.3241

.008 34.3195 35.1426 54.7018 13.3348

.016 31.9710 27.9534 50.0949 4.5489

.032 26.3451 22.0347 43.8309 1.2950

.064 18.0566 17.2202 36.0743 4.7242

.128 9.1786 13.4944 27.9363 6.5914

.256 1.9896 10.9160 20.8028 7.5076

.512 2.4922 9.2014 15.3197 7.7224

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