Applied Statistics and Probability for Engineers

412 CHAPTER 12 MULTIPLE LINEAR REGRESSION

In general, the dependent variableor responseYmay be related to kindependentor

regressor variables.The model

(12-2)

is called a multiple linear regression model with kregressor variables. The parameters j,j0,

1,p,k, are called the regression coefficients. This model describes a hyperplane in the k-

dimensional space of the regressor variables {xj}. The parameterjrepresents the expected change

in response Yper unit change in xjwhen all the remaining regressors xi(ij) are held constant.

Multiple linear regression models are often used as approximating functions. That is, the

true functional relationship between Yand x 1 , x 2 ,p, xkis unknown, but over certain ranges of

the independent variables the linear regression model is an adequate approximation.

Models that are more complex in structure than Equation 12-2 may often still be analyzed

by multiple linear regression techniques. For example, consider the cubic polynomial model

in one regressor variable.

(12-3)

If we let x 1 x, x 2 x^2 , x 3 x^3 , Equation 12-3 can be written as

(12-4)

which is a multiple linear regression model with three regressor variables.

Models that include interactioneffects may also be analyzed by multiple linear regres-

sion methods. An interaction between two variables can be represented by a cross-product

term in the model, such as

(12-5)

If we let x 3 x 1 x 2 and  3  12 , Equation 12-5 can be written as

which is a linear regression model.

Figure 12-2(a) and (b) shows the three-dimensional plot of the regression model

Y 50  10 x 1  7 x 2  5 x 1 x 2

Y 0  1 x 1  2 x 2  3 x 3 

Y 0  1 x 1  2 x 2  12 x 1 x 2 

Y 0  1 x 1  2 x 2  3 x 3 

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

Y 0  1 x 1  2 x 2 p ̨kx ̨k

Figure 12-1 (a) The regression plane for the model E(Y) 50  10 x 1  7 x 2. (b) The contour plot.

0

(^20)
246810
(b)
x 1
2
4
6
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10
x 2
4
6
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10
(^468100)
(^02)
0
40
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(a)
x 1
x 2
E(Y)
220
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