1. Lecture 5

2. Linear Models for Correlated Data: Inference

3. 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 Example: Growth of Sitka Trees

4. Weighted-Least Squares Estimation

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

12. Estimation of Mean Model: Weighted Least Squares

13. 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

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. “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? 1.Uniform Correlation Model 2.Balanced Data

37. When can we use OLS and ignore V? (cont’d) 1.(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)

43. (Recall slide) 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 Example: Growth of Sitka Trees

44. Maximum Likelihood Estimation under a Gaussian Assumption

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

48. (Recall slide) 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 Example: Growth of Sitka Trees

49. Restricted Maximum Likelihood Estimation

50. (Recall slide) 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 Example: Growth of Sitka Trees

51. Generalized Least Square Estimator Robust Estimation (unstructured covariance matrix)

52. Robust Estimation of V under a saturated model

53. Robust Estimation of V “restricted ML” – makes estimates unbiased

54. Example

55. Robust Estimation vs. A Parametric Approach

56. Maximum Likelihood Estimation of V

57. Example: Growth of sitka trees

58. Figure 1. Observed data and mean response profiles in each of the four growth chambers for the treatment and control.

59. Figure 2. Observed mean response in each of the four chambers. Season 1 Season 2

61. Example: Growth of sitka trees (cont’d)

62. We first consider the 1998 data.

63. Example: Growth of sitka trees (cont’d) Unstructured covariance matrix

65. Example: Growth of sitka trees (cont’d)

72. Sitka spruce data: Estimated covariance matrix for 1988

73. Sitka spruce data: Estimated covariance matrix for 1989

75. Summary: Unstructured Covariance Matrix

76. Summary: Parametric Models for Covariance Reasons for parametric modelling:

77. Summary: Parametric Models for Covariance (cont’d) Reasons for parametric modelling (cont’d):

78. Summary: Unstructured vs. Parametric Covariance

79. Overall Summary