Multiple Linear Regression

1. 12 Multiple Linear Regression CHAPTER OUTLINE 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 & subsets of coefficients 12-3 Confidence Intervals in Multiple Linear Regression 12-4.1 Use of t-tests 12-3.2 Confidence interval on the mean response 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 & indicator variables 12-6.3 Selection of variables & model building 12-6.4 Multicollinearity

2. Learning Objectives for Chapter 12 After careful study of this chapter, you should be able to do the following: • Use multiple regression techniques to build empirical models to engineering and scientific data. • Understand how the method of least squares extends to fitting multiple regression models. • Assess regression model adequacy. • Test hypotheses and construct confidence intervals on the regression coefficients. • Use the regression model to estimate the mean response, and to make predictions and to construct confidence intervals and prediction intervals. • Build regression models with polynomial terms. • Use indicator variables to model categorical regressors. • Use stepwise regression and other model building techniques to select the appropriate set of variables for a regression model.

3. 12-1: Multiple Linear Regression Models 12-1.1 Introduction • 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.

4. 12-1: Multiple Linear Regression Models 12-1.1 Introduction • For example, suppose that the effective life of a cutting tool depends on the cutting speed and the tool angle. A possible multiple regression model could be where Y – tool life x1 – cutting speed x2 – tool angle

5. 12-1: Multiple Linear Regression Models 12-1.1 Introduction Figure 12-1(a)The regression plane for the model E(Y) = 50 + 10x1 + 7x2. (b) The contour plot

6. 12-1: Multiple Linear Regression Models 12-1.1 Introduction

7. 12-1: Multiple Linear Regression Models 12-1.1 Introduction Figure 12-2 (a) Three-dimensional plot of the regression model E(Y) = 50 + 10x1 + 7x2 + 5x1x2. (b) The contour plot

8. 12-1: Multiple Linear Regression Models 12-1.1 Introduction Figure 12-3 (a) Three-dimensional plot of the regression model E(Y) = 800 + 10x1 + 7x2 – 8.5x12 – 5x22 + 4x1x2. (b) The contour plot

9. 12-1: Multiple Linear Regression Models 12-1.2 Least Squares Estimation of the Parameters

10. 12-1: Multiple Linear Regression Models 12-1.2 Least Squares Estimation of the Parameters • The least squares function is given by • The least squares estimates must satisfy

11. 12-1: Multiple Linear Regression Models 12-1.2 Least Squares Estimation of the Parameters • The least squares normal Equations are • The solution to the normal Equations are the least squares estimators of the regression coefficients.

12. 12-1: Multiple Linear Regression Models Example 12-1

13. 12-1: Multiple Linear Regression Models Example 12-1

14. 12-1: Multiple Linear Regression Models Figure 12-4 Matrix of scatter plots (from Minitab) for the wire bond pull strength data in Table 12-2.

15. 12-1: Multiple Linear Regression Models Example 12-1

16. 12-1: Multiple Linear Regression Models Example 12-1

17. 12-1: Multiple Linear Regression Models Example 12-1

18. 12-1: Multiple Linear Regression Models 12-1.3 Matrix Approach to Multiple Linear Regression Suppose the model relating the regressors to the response is In matrix notation this model can be written as

19. 12-1: Multiple Linear Regression Models 12-1.3 Matrix Approach to Multiple Linear Regression where

20. 12-1: Multiple Linear Regression Models 12-1.3 Matrix Approach to Multiple Linear Regression We wish to find the vector of least squares estimators that minimizes: The resulting least squares estimate is

21. 12-1: Multiple Linear Regression Models 12-1.3 Matrix Approach to Multiple Linear Regression

22. 12-1: Multiple Linear Regression Models Example 12-2

23. Example 12-2

24. 12-1: Multiple Linear Regression Models Example 12-2

25. 12-1: Multiple Linear Regression Models Example 12-2

26. 12-1: Multiple Linear Regression Models Example 12-2

27. 12-1: Multiple Linear Regression Models Example 12-2

29. 12-1: Multiple Linear Regression Models Estimating 2 An unbiased estimator of 2 is

30. 12-1: Multiple Linear Regression Models 12-1.4 Properties of the Least Squares Estimators Unbiased estimators: Covariance Matrix:

31. 12-1: Multiple Linear Regression Models 12-1.4 Properties of the Least Squares Estimators Individual variances and covariances: In general,

32. 12-2: Hypothesis Tests in Multiple Linear Regression 12-2.1 Test for Significance of Regression The appropriate hypotheses are The test statistic is

33. 12-2: Hypothesis Tests in Multiple Linear Regression 12-2.1 Test for Significance of Regression

34. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-3

35. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-3

36. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-3

37. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-3

38. 12-2: Hypothesis Tests in Multiple Linear Regression R2 and Adjusted R2 The coefficient of multiple determination • For the wire bond pull strength data, we find that R2 = SSR/SST = 5990.7712/6105.9447 = 0.9811. • Thus, the model accounts for about 98% of the variability in the pull strength response.

39. 12-2: Hypothesis Tests in Multiple Linear Regression R2 and Adjusted R2 The adjusted R2 is • The adjusted R2 statistic penalizes the analyst for adding terms to the model. • It can help guard against overfitting (including regressors that are not really useful)

40. 12-2: Hypothesis Tests in Multiple Linear Regression 12-2.2 Tests on Individual Regression Coefficients and Subsets of Coefficients The hypotheses for testing the significance of any individual regression coefficient:

41. 12-2: Hypothesis Tests in Multiple Linear Regression 12-2.2 Tests on Individual Regression Coefficients and Subsets of Coefficients The test statistic is • Reject H0 if |t0| > t/2,n-p. • This is called a partial or marginal test

42. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-4

43. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-4

44. 12-2: Hypothesis Tests in Multiple Linear Regression The general regression significance test or the extra sum of squares method: We wish to test the hypotheses:

45. 12-2: Hypothesis Tests in Multiple Linear Regression A general form of the model can be written: where X1 represents the columns of X associated with 1 and X2 represents the columns of X associated with 2

46. 12-2: Hypothesis Tests in Multiple Linear Regression For the full model: If H0 is true, the reduced model is

47. 12-2: Hypothesis Tests in Multiple Linear Regression The test statistic is: Reject H0 if f0 > f,r,n-p The test in Equation (12-32) is often referred to as a partial F-test

48. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-6

49. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-6

50. 12-2: Hypothesis Tests in Multiple Linear Regression Example 12-6