Applied Statistics and Probability for Engineers

452 CHAPTER 12 MULTIPLE LINEAR REGRESSION

The analysis of variance for this model is shown in Table 12-15. Note that the hypothesis

(significance of regression) would be rejected at any reasonable level of

significance because the P-value is very small. This table also contains the sums of squares

so a test of the hypothesis can be made. Since this hypothesis is also rejected, we

conclude that tool type has an effect on surface finish.

It is also possible to use indicator variables to investigate whether tool type affects both

the slope and intercept. Let the model be

where x 2 is the indicator variable. Now if tool type 302 is used, x 2  0, and the model is

If tool type 416 is used, x 2 1, and the model becomes

Note that  2 is the change in the intercept and that  3 is the change in slope produced by a

change in tool type.

Another method of analyzing these data is to fit separate regression models to the data

for each tool type. However, the indicator variable approach has several advantages. First,

only one regression model must be fit. Second, by pooling the data on both tool types,

more degrees of freedom for error are obtained. Third, tests of both hypotheses on the

parameters  2 and  3 are just special cases of the extra sum of squares method.

12-6.3 Selection of Variables and Model Building

An important problem in many applications of regression analysis involves selecting the set of

regressor variables to be used in the model. Sometimes previous experience or underlying

theoretical considerations can help the analyst specify the set or regressor variables to use in a

particular situation. Usually, however, the problem consists of selecting an appropriate set of

 1  0  22  1  1  32 x 1 

Y 0  1 x 1  2  3 x 1 

Y 0  1 x 1 

Y 0  1 x 1  2 x 2  3 x 1 x 2 

H 0 :  2  0

SSR 1  10  02 SSR 1  20  1 , 02

SSRSSR 1  1 , 20  02

H 0 :  1  2  0

Table 12-15 Analysis of Variance of Example 12-12

Source of Degrees of Mean

Variation Sum of Squares Freedom Square f 0 P-value

Regression 1012.0595 2 506.0297 1103.69 1.02E-18

130.6091 1 130.6091 284.87 4.70E-12

881.4504 1 881.4504 1922.52 6.24E-19

Error 7.7943 17 0.4508

Total 1019.8538 19

SSR 1  2 0  1 , 02

SSR 1  1 0  02

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