Applied Statistics and Probability for Engineers

12-1 MULTIPLE LINEAR REGRESSION MODEL 413

and the corresponding two-dimensional contour plot. Notice that, although this model is a lin-

ear regression model, the shape of the surface that is generated by the model is not linear. In

general, any regression model that is linear in parameters(the ’s) is a linear regression

model, regardless of the shape of the surface that it generates.

Figure 12-2 provides a nice graphical interpretation of an interaction. Generally, interac-

tion implies that the effect produced by changing one variable (x 1 , say) depends on the level

of the other variable (x 2 ). For example, Fig. 12-2 shows that changing x 1 from 2 to 8 produces

a much smaller change in E(Y) when x 2 2 than when x 2 10. Interaction effects occur fre-

quently in the study and analysis of real-world systems, and regression methods are one of the

techniques that we can use to describe them.

As a final example, consider the second-order model with interaction

(12-6)

If we let x 3 x^21 , x 4 x^22 , x 5 x 1 x 2 ,  3  11 ,  4  22 , and  5  12 , Equation 12-6 can be

written as a multiple linear regression model as follows:

Figure 12-3(a) and (b) show the three-dimensional plot and the corresponding contour plot for

E 1 Y 2  800  10 x 1  7 x 2 8.5x^21  5 x^22  4 x ̨ 1 x 2

Y 0  1 x 1  2 x 2  3 x 3  4 x 4  5 x 5 

Y 0  1 x 1  2 x 2  11 x^21  22 x^22  12 x 1 x 2 

2

4

6

8

10

(^468100)
(^02)
0
200
400
600
800
(a)
x 1
x 2
0
0
246810
(b)
x 1
2
4
6
8
10
x 2
720
586
452
318
117 184
653
519
385
251
(^00)
2
246810
(b)
x 1
2
4
6
8
10
x 2
4
6
8
10
(^468100)
(^02)
0
200
400
800
1000
(a)
x 1
x 2
E(Y)
25
600
100
175
700625550
800750
325250
475400
Figure 12-2 (a) Three-dimensional plot of the regression
model E(Y) 50  10 x 1  7 x 2  5 x 1 x 2. (b) The contour
plot.
Figure 12-3 (a) Three-dimensional plot of the regression
model E(Y) 800  10 x 1  7 x 2 8.5x^21  5 x^22  4 x 1 x 2.
(b) The contour plot.
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