1. © Buddy Freeman, 2015 Multiple Linear Regression (MLR) Testing the additional contribution made by adding an independent variable.

2. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK

3. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY

4. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY

5. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697

6. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression

7. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR = 117,824,722

8. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR = 117,824,722 R Square = SSR/SST ≈.1167 or 11.67%

9. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR = 117,824,722 R Square = SSR/SST ≈.1167 or 11.67%

10. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR = 117,824,722 R Square = SSR/SST ≈.1167 or 11.67% Adding RANK as a second independent variable will explain more of the variation in SALARY, but will it be a significant amount?

11. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK

12. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK

13. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY

14. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY

15. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697

16. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression

17. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR(RANK and YRSRANK) = 683,715,472.1

18. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR(RANK and YRSRANK) = 683,715,472.1 R Square = SSR/SST ≈.6774 or 67.74%

19. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR(RANK and YRSRANK) = 683,715,472.1 R Square = SSR/SST ≈.6774 or 67.74%

20. © Buddy Freeman, 2015 Predicting SALARY using YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR (YRSRANK) = 117,824,722 R Square = SSR/SST ≈.1167 or 11.67% Adding RANK as a second independent variable will explain more of the variation in SALARY, but will it be a significant amount?

21. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression SSR(RANK and YRSRANK) = 683,715,472.1 R Square = SSR/SST ≈.6774 or 67.74% SSR (YRSRANK) = 117,824,722

22. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY variation in SALARY= 1,009,361,697 SSR = variation explained by regression Additional contribution made by adding RANK = SSR(RANK | YRSRANK) = 683,715,472.1 - 117,824,722 = 565,890,750.1 SSR(RANK and YRSRANK) = 683,715,472.1 R Square = SSR/SST ≈.6774 or 67.74% SSR (YRSRANK) = 117,824,722 We may determine if this additional contribution is significant by performing a partial F-test.

23. © Buddy Freeman, 2015 Partial F-test ( α =.05) Additional contribution made by adding RANK = SSR(RANK | YRSRANK) = 683,715,472.1 - 117,824,722 = 565,890,750.1, the numerator.

24. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK

25. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK

26. © Buddy Freeman, 2015 Partial F-test ( α =.05) In Simple Linear Regression, what was the relationship between the F-test and the t-test? The square root of the F ≈ 9.5969, the t value for RANK.

27. © Buddy Freeman, 2015 Predicting SALARY using RANK and YRSRANK The partial F-test and the t-test are equivalent, provided that one is examining the additional contribution of a single independent variable.