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Multiple Linear Regression

This analysis employs Multiple Linear Regression (MLR) to assess the significance of adding RANK as an independent variable alongside YRSRANK to predict SALARY. The study calculates key statistics including Total Sum of Squares (SST), Sum of Squares Regression (SSR), and R-Squared values for both models. Results show that the inclusion of RANK substantially increases the explained variation in SALARY, raising R-Squared from approximately 11.67% to 67.74%. A Partial F-test is performed to confirm the significance of this additional contribution to model performance.

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Multiple Linear Regression

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  1. Multiple Linear Regression (MLR) Testing the additional contribution made by adding an independent variable.

  2. Predicting SALARY using YRSRANK

  3. Predicting SALARY using YRSRANK SST = SSY = variation in SALARY

  4. Predicting SALARY using YRSRANK SST = SSY = variation in SALARY

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

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

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

  11. Predicting SALARY using RANK and YRSRANK

  12. Predicting SALARY using RANK and YRSRANK

  13. Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY

  14. Predicting SALARY using RANK and YRSRANK SST = SSY = variation in SALARY

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

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

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

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

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

  24. Predicting SALARY using RANK and YRSRANK

  25. Predicting SALARY using RANK and YRSRANK

  26. 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. 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.

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