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Models With Two or More Quantitative Variables

Models With Two or More Quantitative Variables. Types of Regression Models. First-Order Model With 2 Independent Variables. Relationship Between 1 Dependent & 2 Independent Variables Is a Linear Function Assumes No Interaction Between X 1 & X 2

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Models With Two or More Quantitative Variables

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  1. Models With Two or More Quantitative Variables EPI809/Spring 2008

  2. Types of Regression Models EPI809/Spring 2008

  3. First-Order Model With 2 Independent Variables • Relationship Between 1 Dependent & 2 Independent Variables Is a Linear Function • Assumes No Interaction Between X1 & X2 • Effect of X1 on E(Y) Is the Same Regardless of X2 Values EPI809/Spring 2008

  4. First-Order Model With 2 Independent Variables • Relationship Between 1 Dependent & 2 Independent Variables Is a Linear Function • Assumes No Interaction Between X1 & X2 • Effect of X1 on E(Y) Is the Same Regardless of X2 Values • Model EPI809/Spring 2008

  5. No Interaction EPI809/Spring 2008

  6. No Interaction E(Y) E(Y) = 1 + 2X1 + 3X2 12 8 4 0 X1 0 0.5 1 1.5 EPI809/Spring 2008

  7. No Interaction E(Y) E(Y) = 1 + 2X1 + 3X2 12 8 4 E(Y) = 1 + 2X1 + 3(0) = 1 + 2X1 0 X1 0 0.5 1 1.5 EPI809/Spring 2008

  8. No Interaction E(Y) E(Y) = 1 + 2X1 + 3X2 12 8 E(Y) = 1 + 2X1 + 3(1) = 4 + 2X1 4 E(Y) = 1 + 2X1 + 3(0) = 1 + 2X1 0 X1 0 0.5 1 1.5 EPI809/Spring 2008

  9. No Interaction E(Y) E(Y) = 1 + 2X1 + 3X2 12 E(Y) = 1 + 2X1 + 3(2) = 7 + 2X1 8 E(Y) = 1 + 2X1 + 3(1) = 4 + 2X1 4 E(Y) = 1 + 2X1 + 3(0) = 1 + 2X1 0 X1 0 0.5 1 1.5 EPI809/Spring 2008

  10. No Interaction E(Y) E(Y) = 1 + 2X1 + 3X2 E(Y) = 1 + 2X1 + 3(3) = 10 + 2X1 12 E(Y) = 1 + 2X1 + 3(2) = 7 + 2X1 8 E(Y) = 1 + 2X1 + 3(1) = 4 + 2X1 4 E(Y) = 1 + 2X1 + 3(0) = 1 + 2X1 0 X1 0 0.5 1 1.5 EPI809/Spring 2008

  11. No Interaction E(Y) E(Y) = 1 + 2X1 + 3X2 E(Y) = 1 + 2X1 + 3(3) = 10 + 2X1 12 E(Y) = 1 + 2X1 + 3(2) = 7 + 2X1 8 E(Y) = 1 + 2X1 + 3(1) = 4 + 2X1 4 E(Y) = 1 + 2X1 + 3(0) = 1 + 2X1 0 X1 0 0.5 1 1.5 Effect (slope) of X1 on E(Y) does not depend on X2 value EPI809/Spring 2008

  12. First-Order Model Worksheet Run regression with Y, X1, X2 EPI809/Spring 2008

  13. Types of Regression Models EPI809/Spring 2008

  14. Interaction Model With 2 Independent Variables • Hypothesizes Interaction Between Pairs of X Variables • Response to One X Variable Varies at Different Levels of Another X Variable EPI809/Spring 2008

  15. Interaction Model With 2 Independent Variables • Hypothesizes Interaction Between Pairs of X Variables • Response to One X Variable Varies at Different Levels of Another X Variable • Contains Two-Way Cross Product Terms EPI809/Spring 2008

  16. 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 2. Contains Two-Way Cross Product Terms 3. Can Be Combined With Other Models • Example: Dummy-Variable Model EPI809/Spring 2008

  17. Effect of Interaction EPI809/Spring 2008

  18. Effect of Interaction 1. Given: EPI809/Spring 2008

  19. Effect of Interaction 1. Given: 2. Without Interaction Term, Effect of X1 on Y Is Measured by 1 EPI809/Spring 2008

  20. Effect of Interaction 1. Given: 2. Without Interaction Term, Effect of X1 on Y Is Measured by 1 3. With Interaction Term, Effect of X1 onY Is Measured by 1 + 3X2 • Effect changes As X2 changes EPI809/Spring 2008

  21. Interaction Model Relationships EPI809/Spring 2008

  22. Interaction Model Relationships E(Y) = 1 + 2X1 + 3X2 + 4X1X2 E(Y) 12 8 4 0 X1 0 0.5 1 1.5 EPI809/Spring 2008

  23. Interaction Model Relationships E(Y) = 1 + 2X1 + 3X2 + 4X1X2 E(Y) 12 8 E(Y) = 1 + 2X1 + 3(0) + 4X1(0) = 1 + 2X1 4 0 X1 0 0.5 1 1.5 EPI809/Spring 2008

  24. E(Y) 12 8 4 0 X1 0 0.5 1 1.5 Interaction Model Relationships E(Y) = 1 + 2X1 + 3X2 + 4X1X2 E(Y) = 1 + 2X1 + 3(1) + 4X1(1) = 4 + 6X1 E(Y) = 1 + 2X1 + 3(0) + 4X1(0) = 1 + 2X1 EPI809/Spring 2008

  25. E(Y) 12 8 4 0 X1 0 0.5 1 1.5 Interaction Model Relationships E(Y) = 1 + 2X1 + 3X2 + 4X1X2 E(Y) = 1 + 2X1 + 3(1) + 4X1(1) = 4 + 6X1 E(Y) = 1 + 2X1 + 3(0) + 4X1(0) = 1 + 2X1 Effect (slope) of X1 on E(Y) does depend on X2 value EPI809/Spring 2008

  26. Interaction Model Worksheet Multiply X1by X2 to get X1X2. Run regression with Y, X1, X2 , X1X2 EPI809/Spring 2008

  27. Thinking challenge • Assume Y: Milk yield, X1: food intake and X2: weight • Assume the following model with interaction • Interpret the interaction ^ Y = 1 + 2X1 + 3X2 + 4X1X2 EPI809/Spring 2008

  28. Types of Regression Models EPI809/Spring 2008

  29. Second-Order Model With 2 Independent Variables • 1. Relationship Between 1 Dependent & 2 or More Independent Variables Is a Quadratic Function • 2. Useful 1St Model If Non-Linear Relationship Suspected EPI809/Spring 2008

  30. Second-Order Model With 2 Independent Variables • 1. Relationship Between 1 Dependent & 2 or More Independent Variables Is a Quadratic Function • 2. Useful 1St Model If Non-Linear Relationship Suspected • 3. Model EPI809/Spring 2008

  31. Second-Order Model Worksheet Multiply X1by X2 to get X1X2; then X12, X22. Run regression with Y, X1, X2 , X1X2, X12, X22. EPI809/Spring 2008

  32. Models With One Qualitative Independent Variable EPI809/Spring 2008

  33. Types of Regression Models EPI809/Spring 2008

  34. Dummy-Variable Model 1. Involves Categorical X Variable With 2 Levels • 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 • May Be Combined With Quantitative Variable (1st Order or 2nd Order Model) EPI809/Spring 2008

  35. Dummy-Variable Model Worksheet X2 levels: 0 = Group 1; 1 = Group 2. Run regression with Y, X1, X2 EPI809/Spring 2008

  36. Interpreting Dummy-Variable Model Equation EPI809/Spring 2008

  37. Interpreting Dummy-Variable Model Equation     Y     X   X Given: i 0 1 1 i 2 2 i Y  Starting s alary of c ollege gra d' s X  GPA 1 0 i f Male X  2 1 if Female EPI809/Spring 2008

  38. Interpreting Dummy-Variable Model Equation     Y     X   X Given: i 0 1 1 i 2 2 i Y  Starting s alary of c ollege gra d' s X  GPA 1 0 i f Male X  2 1 if Female Males ( X  0 ): 2       Y     X       X (0) i 0 1 1 i 2 0 1 1 i EPI809/Spring 2008

  39. Interpreting Dummy-Variable Model Equation     Y     X   X Given: i 0 1 1 i 2 2 i Y  Starting s alary of c ollege gra d' s X  GPA 1 0 i f Male X  2 1 if Female Same slopes Males ( X  0 ): 2       Y     X       X (0) i 0 1 1 i 2 0 1 1 i Females ( X  1 ): 2        Y     X    (   )   X (1) i 0 1 1 i 2 1 1 i 0 2 EPI809/Spring 2008

  40. Dummy-Variable Model Relationships Y ^ Same Slopes 1 Females ^ ^ 0 + 2 ^ 0 Males 0 X1 0 EPI809/Spring 2008

  41. Dummy-Variable Model Example EPI809/Spring 2008

  42. Dummy-Variable Model Example  Y  3  5 X  7 X Computer O utput: i 1 i 2 i 0 i f Male X  2 1 if Female EPI809/Spring 2008

  43. Dummy-Variable Model Example  Y  3  5 X  7 X Computer O utput: i 1 i 2 i 0 i f Male X  2 1 if Female Males ( X  0 ): 2  Y  3  5 X  7  3  5 X (0) i 1 i 1 i EPI809/Spring 2008

  44. Dummy-Variable Model Example  Y  3  5 X  7 X Computer O utput: i 1 i 2 i 0 i f Male X  2 1 if Female Same slopes Males ( X  0 ): 2  Y  3  5 X  7  3  5 X (0) i 1 i 1 i Females (X  1 ): 2  Y  3  5 X  7  (1) (3 + 7)  5 X i 1 i 1 i EPI809/Spring 2008

  45. Sample SAS codes for fitting linear regressions with interactions and higher order terms PROC GLM data=complex; Class gender; model salary = gpa gendergpa*gender; RUN; EPI809/Spring 2008

  46. Conclusion • Explained the Linear Multiple Regression Model • Tested Overall Significance • Described Various Types of Models • Evaluated Portions of a Regression Model • Interpreted Linear Multiple Regression Computer Output • Described Stepwise Regression • Explained Residual Analysis • Described Regression Pitfalls EPI809/Spring 2008

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