1. 12b - 1 © 2000 Prentice-Hall, Inc. Statistics Multiple Regression and Model Building Chapter 12 part II

2. 12b - 2 © 2000 Prentice-Hall, Inc. Learning Objectives 1.Describe Curvilinear Regression Models 2.Summarize Interaction Models 3.Explain Models with Qualitative Variables 4.Evaluate Portions of Regression Models 5.Describe Stepwise Regression Analysis

3. 12b - 3 © 2000 Prentice-Hall, Inc. Types of Regression Models

4. 12b - 4 © 2000 Prentice-Hall, Inc. Models With a Single Quantitative Variable

5. 12b - 5 © 2000 Prentice-Hall, Inc. Types of Regression Models

6. 12b - 6 © 2000 Prentice-Hall, Inc. First-Order Model With 1 Independent Variable

7. 12b - 7 © 2000 Prentice-Hall, Inc. First-Order Model With 1 Independent Variable 1.Relationship Between 1 Dependent & 1 Independent Variable Is Linear

8. 12b - 8 © 2000 Prentice-Hall, Inc. First-Order Model With 1 Independent Variable 1.Relationship Between 1 Dependent & 1 Independent Variable Is Linear 2.Used When Expected Rate of Change in Y Per Unit Change in X Is Stable

9. 12b - 9 © 2000 Prentice-Hall, Inc. First-Order Model With 1 Independent Variable 1.Relationship Between 1 Dependent & 1 Independent Variable Is Linear 2.Used When Expected Rate of Change in Y Per Unit Change in X Is Stable 3.Used With Curvilinear Relationships If Relevant Range Is Linear

10. 12b - 10 © 2000 Prentice-Hall, Inc. First-Order Model Relationships  1 < 0  1 > 0 Y X 1 Y X 1

11. 12b - 11 © 2000 Prentice-Hall, Inc. First-Order Model Worksheet Run regression with Y, X 1

12. 12b - 12 © 2000 Prentice-Hall, Inc. Types of Regression Models

13. 12b - 13 © 2000 Prentice-Hall, Inc. Second-Order Model With 1 Independent Variable 1.Relationship Between 1 Dependent & 1 Independent Variables Is a Quadratic Function 2.Useful 1 St Model If Non-Linear Relationship Suspected

14. 12b - 14 © 2000 Prentice-Hall, Inc. Second-Order Model With 1 Independent Variable 1.Relationship Between 1 Dependent & 1 Independent Variables Is a Quadratic Function 2.Useful 1 St Model If Non-Linear Relationship Suspected 3.Model Linear effect Curvilinear effect

15. 12b - 15 © 2000 Prentice-Hall, Inc. Second-Order Model Relationships  2 > 0  2 < 0

16. 12b - 16 © 2000 Prentice-Hall, Inc. Second-Order Model Worksheet Create X 1 2 column. Run regression with Y, X 1, X 1 2.

17. 12b - 17 © 2000 Prentice-Hall, Inc. Types of Regression Models

18. 12b - 18 © 2000 Prentice-Hall, Inc. Third-Order Model With 1 Independent Variable 1.Relationship Between 1 Dependent & 1 Independent Variable Has a ‘Wave’ 2.Used If 1 Reversal in Curvature

19. 12b - 19 © 2000 Prentice-Hall, Inc. Third-Order Model With 1 Independent Variable 1.Relationship Between 1 Dependent & 1 Independent Variable Has a ‘Wave’ 2.Used If 1 Reversal in Curvature 3.Model Linear effect Curvilinear effects

20. 12b - 20 © 2000 Prentice-Hall, Inc. Third-Order Model Relationships  3 < 0  3 > 0

21. 12b - 21 © 2000 Prentice-Hall, Inc. Third-Order Model Worksheet Multiply X 1 by X 1 to get X 1 2. Multiply X 1 by X 1 by X 1 to get X 1 3. Run regression with Y, X 1, X 1 2, X 1 3.

22. 12b - 22 © 2000 Prentice-Hall, Inc. Models With Two or More Quantitative Variables

23. 12b - 23 © 2000 Prentice-Hall, Inc. Types of Regression Models

24. 12b - 24 © 2000 Prentice-Hall, Inc. First-Order Model With 2 Independent Variables 1.Relationship Between 1 Dependent & 2 Independent Variables Is a Linear Function 2.Assumes No Interaction Between X 1 & X 2 Effect of X 1 on E(Y) Is the Same Regardless of X 2 Values Effect of X 1 on E(Y) Is the Same Regardless of X 2 Values

25. 12b - 25 © 2000 Prentice-Hall, Inc. First-Order Model With 2 Independent Variables 1.Relationship Between 1 Dependent & 2 Independent Variables Is a Linear Function 2.Assumes No Interaction Between X 1 & X 2 Effect of X 1 on E(Y) Is the Same Regardless of X 2 Values Effect of X 1 on E(Y) Is the Same Regardless of X 2 Values 3.Model

26. 12b - 26 © 2000 Prentice-Hall, Inc. No Interaction

27. 12b - 27 © 2000 Prentice-Hall, Inc. No Interaction E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3X 2

28. 12b - 28 © 2000 Prentice-Hall, Inc. No Interaction E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3X 2 E(Y) = 1 + 2X 1 + 3(0) = 1 + 2X 1

29. 12b - 29 © 2000 Prentice-Hall, Inc. No Interaction E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3X 2 E(Y) = 1 + 2X 1 + 3(1) = 4 + 2X 1 E(Y) = 1 + 2X 1 + 3(0) = 1 + 2X 1

30. 12b - 30 © 2000 Prentice-Hall, Inc. No Interaction E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3(2) = 7 + 2X 1 E(Y) = 1 + 2X 1 + 3X 2 E(Y) = 1 + 2X 1 + 3(1) = 4 + 2X 1 E(Y) = 1 + 2X 1 + 3(0) = 1 + 2X 1

31. 12b - 31 © 2000 Prentice-Hall, Inc. No Interaction E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3(2) = 7 + 2X 1 E(Y) = 1 + 2X 1 + 3X 2 E(Y) = 1 + 2X 1 + 3(1) = 4 + 2X 1 E(Y) = 1 + 2X 1 + 3(0) = 1 + 2X 1 E(Y) = 1 + 2X 1 + 3(3) = 10 + 2X 1

32. 12b - 32 © 2000 Prentice-Hall, Inc. No Interaction Effect (slope) of X 1 on E(Y) does not depend on X 2 value E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3(2) = 7 + 2X 1 E(Y) = 1 + 2X 1 + 3X 2 E(Y) = 1 + 2X 1 + 3(1) = 4 + 2X 1 E(Y) = 1 + 2X 1 + 3(0) = 1 + 2X 1 E(Y) = 1 + 2X 1 + 3(3) = 10 + 2X 1

33. 12b - 33 © 2000 Prentice-Hall, Inc. First-Order Model Relationships

34. 12b - 34 © 2000 Prentice-Hall, Inc. First-Order Model Worksheet Run regression with Y, X 1, X 2

35. 12b - 35 © 2000 Prentice-Hall, Inc. Types of Regression Models

36. 12b - 36 © 2000 Prentice-Hall, Inc. Interaction Model With 2 Independent Variables 1.Hypothesizes Interaction Between Pairs of X Variables Response to One X Variable Varies at Different Levels of Another X Variable Response to One X Variable Varies at Different Levels of Another X Variable

37. 12b - 37 © 2000 Prentice-Hall, Inc. Interaction Model With 2 Independent Variables 1.Hypothesizes Interaction Between Pairs of X Variables Response to One X Variable Varies at Different Levels of Another X Variable Response to One X Variable Varies at Different Levels of Another X Variable 2.Contains Two-Way Cross Product Terms

38. 12b - 38 © 2000 Prentice-Hall, Inc. 1.Hypothesizes Interaction Between Pairs of X Variables Response to One X Variable Varies at Different Levels of Another X Variable Response to One X Variable Varies at Different Levels of Another X Variable 2.Contains Two-Way Cross Product Terms 3.Can Be Combined With Other Models Example: Dummy-Variable Model Example: Dummy-Variable Model Interaction Model With 2 Independent Variables

39. 12b - 39 © 2000 Prentice-Hall, Inc. Effect of Interaction

40. 12b - 40 © 2000 Prentice-Hall, Inc. Effect of Interaction 1.Given:

41. 12b - 41 © 2000 Prentice-Hall, Inc. Effect of Interaction 1.Given: 2.Without Interaction Term, Effect of X 1 on Y Is Measured by  1

42. 12b - 42 © 2000 Prentice-Hall, Inc. Effect of Interaction 1.Given: 2.Without Interaction Term, Effect of X 1 on Y Is Measured by  1 3.With Interaction Term, Effect of X 1 on Y Is Measured by  1 +  3 X 2 Effect Increases As X 2i Increases Effect Increases As X 2i Increases

43. 12b - 43 © 2000 Prentice-Hall, Inc. Interaction Model Relationships

44. 12b - 44 © 2000 Prentice-Hall, Inc. Interaction Model Relationships E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3X 2 + 4X 1 X 2

45. 12b - 45 © 2000 Prentice-Hall, Inc. Interaction Model Relationships E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3X 2 + 4X 1 X 2 E(Y) = 1 + 2X 1 + 3(0) + 4X 1 (0) = 1 + 2X 1

46. 12b - 46 © 2000 Prentice-Hall, Inc. Interaction Model Relationships E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3X 2 + 4X 1 X 2 E(Y) = 1 + 2X 1 + 3(1) + 4X 1 (1) = 4 + 6X 1 E(Y) = 1 + 2X 1 + 3(0) + 4X 1 (0) = 1 + 2X 1

47. 12b - 47 © 2000 Prentice-Hall, Inc. Interaction Model Relationships Effect (slope) of X 1 on E(Y) does depend on X 2 value E(Y) X1X1X1X1 4 8 12 0 010.51.5 E(Y) = 1 + 2X 1 + 3X 2 + 4X 1 X 2 E(Y) = 1 + 2X 1 + 3(1) + 4X 1 (1) = 4 + 6X 1 E(Y) = 1 + 2X 1 + 3(0) + 4X 1 (0) = 1 + 2X 1

48. 12b - 48 © 2000 Prentice-Hall, Inc. Interaction Model Worksheet Multiply X 1 by X 2 to get X 1 X 2. Run regression with Y, X 1, X 2, X 1 X 2

49. 12b - 49 © 2000 Prentice-Hall, Inc. Types of Regression Models

50. 12b - 50 © 2000 Prentice-Hall, Inc. Second-Order Model With 2 Independent Variables 1.Relationship Between 1 Dependent & 2 or More Independent Variables Is a Quadratic Function 2.Useful 1 St Model If Non-Linear Relationship Suspected

51. 12b - 51 © 2000 Prentice-Hall, Inc. Second-Order Model With 2 Independent Variables 1.Relationship Between 1 Dependent & 2 or More Independent Variables Is a Quadratic Function 2.Useful 1 St Model If Non-Linear Relationship Suspected 3.Model

52. 12b - 52 © 2000 Prentice-Hall, Inc. Second-Order Model Relationships  4 +  5 > 0  4 +  5 < 0  3 2 > 4  4  5

53. 12b - 53 © 2000 Prentice-Hall, Inc. Second-Order Model Worksheet Multiply X 1 by X 2 to get X 1 X 2 ; then X 1 2, X 2 2. Run regression with Y, X 1, X 2, X 1 X 2, X 1 2, X 2 2.

54. 12b - 54 © 2000 Prentice-Hall, Inc. Testing Model Portions

55. 12b - 55 © 2000 Prentice-Hall, Inc. 1.Tests the Contribution of a Set of X Variables to the Relationship With Y 2.Null Hypothesis H 0 :  g+1 =... =  k = 0 Variables in Set Do Not Improve Significantly the Model When All Other Variables Are Included Variables in Set Do Not Improve Significantly the Model When All Other Variables Are Included 3.Used in Selecting X Variables or Models 4.Part of Most Computer Programs Testing Model Portions

56. 12b - 56 © 2000 Prentice-Hall, Inc. Models With One Qualitative Independent Variable

57. 12b - 57 © 2000 Prentice-Hall, Inc. Types of Regression Models

58. 12b - 58 © 2000 Prentice-Hall, Inc. Dummy-Variable Model 1.Involves Categorical X Variable With 2 Levels e.g., Male-Female; College-No College e.g., Male-Female; College-No College 2.Variable Levels Coded 0 & 1 3.Number of Dummy Variables Is 1 Less Than Number of Levels of Variable 4.May Be Combined With Quantitative Variable (1 st Order or 2 nd Order Model)

59. 12b - 59 © 2000 Prentice-Hall, Inc. Dummy-Variable Model Worksheet X 2 levels: 0 = Group 1; 1 = Group 2. Run regression with Y, X 1, X 2

60. 12b - 60 © 2000 Prentice-Hall, Inc. Interpreting Dummy- Variable Model Equation

61. 12b - 61 © 2000 Prentice-Hall, Inc. Interpreting Dummy- Variable Model Equation Given: Starting s alary of c ollege gra d' s s GPA i i if Female f Male         Y Y X X X X Y Y X X X X i i i i i i             R R S S T T       0 0 1 1 1 1 2 2 2 2 1 1 2 2 0 0 1 1

62. 12b - 62 © 2000 Prentice-Hall, Inc. Interpreting Dummy- Variable Model Equation Given: Starting s alary of c ollege gra d' s s GPA i i if Female Males ( f Male                     ): Y Y X X X X Y Y X X X X Y Y X X X X i i i i i i i i i i i i X X             R R S S T T                             0 0 1 1 1 1 2 2 2 2 1 1 2 2 0 0 1 1 1 1 2 2 0 0 1 1 1 1 0 0 1 1 0 0 2 2 0 0 a a f f

63. 12b - 63 © 2000 Prentice-Hall, Inc. Interpreting Dummy- Variable Model Equation Same slopes

64. 12b - 64 © 2000 Prentice-Hall, Inc. Dummy-Variable Model Relationships Y X1X1X1X1 0 0 Same Slopes  1 0000  0 +  2 ^ ^ ^ ^ Females Males

65. 12b - 65 © 2000 Prentice-Hall, Inc. Dummy-Variable Model Example

66. 12b - 66 © 2000 Prentice-Hall, Inc. Dummy-Variable Model Example Computer O utput: f Male if Female i i   Y Y X X X X X X i i i i i i         R R S S T T 3 3 5 5 7 7 0 0 1 1 1 1 2 2 2 2

67. 12b - 67 © 2000 Prentice-Hall, Inc. Dummy-Variable Model Example Computer O utput: f Male if Female Males ( i i     ): Y Y X X X X X X Y Y X X X X i i i i i i i i i i i i X X         R R S S T T             3 3 5 5 7 7 0 0 1 1 3 3 5 5 7 7 0 0 3 3 5 5 1 1 2 2 2 2 1 1 1 1 2 2 0 0 a a f f

68. 12b - 68 © 2000 Prentice-Hall, Inc. Dummy-Variable Model Example Same slopes

69. 12b - 69 © 2000 Prentice-Hall, Inc. Selecting Variables in Model Building

70. 12b - 70 © 2000 Prentice-Hall, Inc. Selecting Variables in Model Building A Butterfly Flaps its Wings in Japan, Which Causes It to Rain in Nebraska. -- Anonymous Use Theory Only! Use Computer Search!

71. 12b - 71 © 2000 Prentice-Hall, Inc. Model Building with Computer Searches 1.Rule: Use as Few X Variables As Possible 2.Stepwise Regression Computer Selects X Variable Most Highly Correlated With Y Computer Selects X Variable Most Highly Correlated With Y Continues to Add or Remove Variables Depending on SSE Continues to Add or Remove Variables Depending on SSE 3.Best Subset Approach Computer Examines All Possible Sets Computer Examines All Possible Sets

72. 12b - 72 © 2000 Prentice-Hall, Inc. Conclusion 1.Described Curvilinear Regression Models 2.Summarized Interaction Models 3.Explained Models with Qualitative Variables 4.Evaluated Portions of Regression Models 5.Described Stepwise Regression Analysis

73. End of Chapter Any blank slides that follow are blank intentionally.