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Lecture 5

Lecture 5. Linear Models for Correlated Data: Inference. Inference. Estimation Methods Weighted Least Squares (WLS) (V i known) Maximum Likelihood (V i unknown) Restricted Maximum Likelihood (V i unknown) Robust Estimation (V i unknown) Hypothesis Testing

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Lecture 5

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  1. Lecture 5

  2. Linear Models for Correlated Data: Inference

  3. Inference • Estimation Methods • Weighted Least Squares (WLS)(Vi known) • Maximum Likelihood (Vi unknown) • Restricted Maximum Likelihood (Vi unknown) • Robust Estimation (Vi unknown) • Hypothesis Testing • Example: Growth of Sitka Trees

  4. Weighted-Least Squares Estimation

  5. Weighted-Least Squares Estimation (cont’d)

  6. Weighted-Least Squares Estimation (cont’d)

  7. Weighted-Least Squares Estimation (cont’d)

  8. Weighted-Least Squares Estimation (cont’d)

  9. Weighted-Least Squares Estimation (cont’d)

  10. Weighted-Least Squares Estimation (cont’d)

  11. Weighted-Least Squares Estimation (cont’d)

  12. Estimation of Mean Model: Weighted Least Squares

  13. Estimation of Mean Model: Weighted Least Squares (cont’d)

  14. Estimation of Mean Model: Weighted Least Squares (cont’d)

  15. Note that we can re-write the WRRS as:

  16. What does this equation say?Examples…

  17. Examples: V diagonal

  18. Examples: V diagonal (cont’d)

  19. Examples: V not diagonal

  20. Examples: AR-1 (V not diagonal)

  21. Examples: AR-1 (V not diagonal) (cont’d)

  22. Weighted Least Squares Estimation:Summary

  23. Pigs – “WLS” Fit “WLS” Model results

  24. Pigs – “WLS” Fit

  25. Pigs – “WLS” Fit

  26. Pigs – “WLS” Fit

  27. Pigs – “WLS” Fit

  28. Pigs – OLS fit . regress weight time Source | SS df MS Number of obs = 432 -------------+------------------------------ F( 1, 430) = 5757.41 Model | 111060.882 1 111060.882 Prob > F = 0.0000 Residual | 8294.72677 430 19.2900622 R-squared = 0.9305 -------------+------------------------------ Adj R-squared = 0.9303 Total | 119355.609 431 276.927167 Root MSE = 4.392 ------------------------------------------------------------------------------ weight | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- time | 6.209896 .0818409 75.88 0.000 6.049038 6.370754 _cons | 19.35561 .4605447 42.03 0.000 18.45041 20.26081 ------------------------------------------------------------------------------ OLS results

  29. Pigs – “WLS” Fit

  30. Pigs – “WLS” Fit “WLS” Model results

  31. Pigs – OLS fit . regress weight time Source | SS df MS Number of obs = 432 -------------+------------------------------ F( 1, 430) = 5757.41 Model | 111060.882 1 111060.882 Prob > F = 0.0000 Residual | 8294.72677 430 19.2900622 R-squared = 0.9305 -------------+------------------------------ Adj R-squared = 0.9303 Total | 119355.609 431 276.927167 Root MSE = 4.392 ------------------------------------------------------------------------------ weight | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- time | 6.209896 .0818409 75.88 0.000 6.049038 6.370754 _cons | 19.35561 .4605447 42.03 0.000 18.45041 20.26081 ------------------------------------------------------------------------------ OLS results

  32. Efficiency

  33. Efficiency (cont’d)

  34. Example

  35. Example (cont’d)

  36. When can we use OLS and ignore V? • Uniform Correlation Model • Balanced Data

  37. When can we use OLS and ignore V? (cont’d) • (Uniform Correlation) With a common correlation between any two equally-spaced measurements on the same unit, there is no reason to weight measurements differently. 2. (Balanced Data) This would not be true if the number of measurements varied between units because, with >0, units with more measurements would then convey more information per unit than units with fewer measurements.

  38. When can we use OLS and ignore V? (cont’d) In many circumstances where there is a balanced design, the OLS estimator is perfectly satisfactory for point estimation.

  39. Example: Two-treatment crossover design

  40. Example: Two-treatment crossover design (cont’d)

  41. Example: Two-treatment crossover design (cont’d)

  42. Example: Two-treatment crossover design (cont’d)

  43. (Recall slide) Inference • Estimation Methods • Weighted Least Squares (WLS)(Vi known) • Maximum Likelihood (Vi unknown) • Restricted Maximum Likelihood (Vi unknown) • Robust Estimation (Vi unknown) • Hypothesis Testing • Example: Growth of Sitka Trees

  44. Maximum Likelihood Estimation under a Gaussian Assumption

  45. Maximum Likelihood Estimation under a Gaussian Assumption (cont’d)

  46. Maximum Likelihood Estimation under a Gaussian Assumption (cont’d)

  47. Maximum Likelihood Estimation under a Gaussian Assumption (cont’d)

  48. (Recall slide) Inference • Estimation Methods • Weighted Least Squares (WLS)(Vi known) • Maximum Likelihood (Vi unknown) • Restricted Maximum Likelihood (Vi unknown) • Robust Estimation (Vi unknown) • Hypothesis Testing • Example: Growth of Sitka Trees

  49. Restricted Maximum Likelihood Estimation

  50. (Recall slide) Inference • Estimation Methods • Weighted Least Squares (WLS)(Vi known) • Maximum Likelihood (Vi unknown) • Restricted Maximum Likelihood (Vi unknown) • Robust Estimation (Vi unknown) • Hypothesis Testing • Example: Growth of Sitka Trees

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