1 / 46

Models With Two or More Quantitative Variables

Learn about types of regression models with multiple quantitative variables, including models with interaction effects. Explore how the relationship between variables varies based on interactions, and run regressions to analyze these effects.

toddp
Download Presentation

Models With Two or More Quantitative Variables

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related