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CHAPTER OUTLINE
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12-1 MULTIPLE LINEAR REGRESSION MODEL 12-1.1 Introduction 12-1.2 Least Squares Estimation of the Parameters 12-1.3 Matrix Approach to Multiple Linear Regression 12-1.4 Properties of the Least Squares Estimators 12-2 HYPOTHESIS TESTS IN MULTIPLE LINEAR REGRESSION 12-2.1 Test for Significance of Regression 12-2.2 Tests on Individual Regression Coefficients and Subsets of Coefficients 12-2.3 More About the Extra Sum of Squares Method (CD Only) 12-3 CONFIDENCE INTERVALS IN MULTIPLE LINEAR REGRESSION 12-3.1 Confidence Intervals on Individual Regression Coefficients
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Multiple Linear Regression
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12-3.2 Confidence Interval on the Mean Response
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12-4 PREDICTION OF NEW OBSERVATIONS 12-5 MODEL ADEQUACY CHECKING 12-5.1 Residual Analysis 12-5.2 Influential Observations 12-6 ASPECTS OF MULTIPLE REGRESSION MODELING 12-6.1 Polynomial Regression Models 12-6.2 Categorical Regressors and Indicator Variables 12-6.3 Selection of Variables and Model Building 12-6.4 Multicollinearity 12-6.5 Ridge Regression (CD Only) 12-6.6 Nonlinear Regression Models (CD Only)
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LEARNING OBJECTIVES
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After careful study of this chapter, you should be able to do the following: 1. Use multiple regression techniques to build empirical models to engineering and scientific data 2. Understand how the method of least squares extends to fitting multiple regression models
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12-1 MULTIPLE LINEAR REGRESSION MODEL
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3. Assess regression model adequacy 4. Test hypotheses and construct confidence intervals on the regression coefficients 5. Use the regression model to estimate the mean response and to make predictions and to construct confidence intervals and prediction intervals 6. Build regression models with polynomial terms 7. Use indicator variables to model categorical regressors 8. Use stepwise regression and other model building techniques to select the appropriate set of variables for a regression model CD MATERIAL 9. Understand how ridge regression provides an effective way to estimate model parameters where there is multicollinearity. 10. Understand the basic concepts of fitting a nonlinear regression model. Answers for many odd numbered exercises are at the end of the book. Answers to exercises whose numbers are surrounded by a box can be accessed in the e-Text by clicking on the box. Complete worked solutions to certain exercises are also available in the e-Text. These are indicated in the Answers to Selected Exercises section by a box around the exercise number. Exercises are also available for some of the text sections that appear on CD only. These exercises may be found within the e-Text immediately following the section they accompany.
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12-1 12-1.1
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MULTIPLE LINEAR REGRESSION MODEL Introduction
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Many applications of regression analysis involve situations in which there are more than one regressor variable. A regression model that contains more than one regressor variable 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
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0 1x1 2x2
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(12-1)
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where Y represents the tool life, x1 represents the cutting speed, x2 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. The regression model in Equation 12-1 describes a plane in the three-dimensional space of Y, x1, and x2. Figure 12-1(a) shows this plane for the regression model E1Y 2 50 10x1 7x2
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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 x1 when x2 is held constant, and 2 measures the expected change in Y per unit change in x2 when x1 is held constant. Figure 12-1(b) shows a contour plot of the regression model that is, lines of constant E(Y ) as a function of x1 and x2. Notice that the contour lines in this plot are straight lines.
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