Use stepwise regression and other model building techniques to select the appropriate set of vari- ables for a regression model

8. Use stepwise regression and other model building techniques to select the appropriate set of vari- ables for a regression model

  12-1 MULTIPLE LINEAR REGRESSION MODEL 12-1.1 Introduction

  Many applications of regression analysis involve situations in which there are more than one regressor or predictor variable. A regression model that contains more than one regressor vari- able is called a multiple regression model.

  As an example, suppose that the effective life of a cutting tool depends on the cutting speed and the tool angle. A multiple regression model that might describe this relationship is

  Y

  0 1 x 1 2 x 2 (12-1) where Y represents the tool life, x 1 represents the cutting speed, x 2 represents the tool angle,

  and is a random error term. This is a multiple linear regression model with two regressors. The term linear is used because Equation 12-1 is a linear function of the unknown parameters

  0 , 1 , and 2 .

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  12-1 MULTIPLE LINEAR REGRESSION MODEL

  E(Y) 120

  Figure 12-1 (a) The regression plane for the model E(Y )

  50 10x 1 7x 2 . (b) The contour plot.

  The regression model in Equation 12-1 describes a plane in the three-dimensional space

  of Y, x 1 , and x 2 . Figure 12-1(a) shows this plane for the regression model

  E 1Y 2 50 10x 1 7x 2

  where we have assumed that the expected value of the error term is zero; that is E( ) 0. The

  parameter 0 is the intercept of the plane. We sometimes call 1 and 2 partial regression

  coefficients, because 1 measures the expected change in Y per unit change in x 1 when x 2 is held constant, and 2 measures the expected change in Y per unit change in x 2 when x 1 is held

  constant. Figure 12-1(b) shows a contour plot of the regression model— that is, lines of con-

  stant E(Y ) as a function of x 1 and x 2 . Notice that the contour lines in this plot are straight lines.

  In general, the dependent variable or response Y may be related to k independent or regressor variables. The model

  Y

  0 1 x 1 2 x 2 p

  k x ˛ k

  (12-2) is called a multiple linear regression model with k regressor variables. The parameters j ,

  ˛

  j

  0, 1, p , k, are called the regression coefficients. This model describes a hyperplane in

  the k-dimensional space of the regressor variables {x j }. The parameter j represents the

  expected change in response Y per unit change in x j when all the remaining regressors x i (i j)

  are held constant.

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

  true functional relationship between Y and x 1 ,x 2 ,p,x k is 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.

  Y

  0 1 x

  2 x

  3 x 3 (12-3)

  If we let x

  1 x, x 2 x ,x 3 x , Equation 12-3 can be written as

  Y

  0 1 x 1 2 x 2 3 x 3 (12-4)

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

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  CHAPTER 12 MULTIPLE LINEAR REGRESSION

  Models that include interaction effects 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

  Y

  0 1 x 1 2 x 2 12 x 1 x 2 (12-5)