Applied Statistics and Probability for Engineers

12-6 ASPECTS OF MULTIPLE REGRESSION MODELING 449

In fitting polynomials, we generally like to use the lowest-degree modelconsistent with

the data. In this example, it would seem logical to investigate the possibility of dropping the

quadratic term from the model. That is, we would like to test

The general regression significance test can be used to test this hypothesis. We need to deter-

mine the “extra sum of squares” due to  11 , or

The sum of squares from Table 12-12. To find , we fit a

simple linear regression model to the original data, yielding

It can be easily verified that the regression sum of squares for this model is

Therefore, the extra sum of the squares due to  11 , given that  1 and  0 are in the model, is

The analysis of variance, with the test of H 0 : 11 0 incorporated into the procedure, is

displayed in Table 12-13. Note that the quadratic term contributes significantly to the model.

0.0312

0.52540.4942

SSR 1  110  1 , 02 SSR 1  1 , 110  02 SSR 1  10  02

SSR 1  10  02 0.4942

yˆ1.900363200.00910056x

SSR 1  1 , 110  02 0.5254 SSR 1  10  02

SSR 1  110  1 , 02 SSR 1  1 , 110  02 SSR 1  10  02

H 1 :  11

0

H 0 :  11  0

Table 12-12 Test for Significance of Regression for the Second-Order Model in Example 12-11

Source of Sum of Degrees of Mean

Variation Squares Freedom Square f 0 P-value

Regression 0.5254 2 0.262700 2171.07 5.18E-15

Error 0.0011 9 0.000121

Total 0.5265 11

Table 12-13 Analysis of Variance for Example 12-11, Showing the Test for H 0 :  11  0

Source of Degrees of Mean

Variation Sum of Squares Freedom Square f 0 P-value

Regression 2 0.262700 2171.07 5.18E-15

Linear 1 0.494200 4084.30 1.17E-15

Quadratic 1 0.031200 258.18 5.51E-9

Error 0.0011 9 0.00121

Total 0.5265 11

SSR 1  11 0  0 , 12 0.0312

SSR^1  1 0 ^02 0.4942

SSR^1  1 , 11 0 ^02 0.5254

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