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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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**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**Pigs – “WLS” Fit**“WLS” Model results**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**Pigs – “WLS” Fit**“WLS” Model results**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**When can we use OLS and ignore V?**• Uniform Correlation Model • Balanced Data**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.**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.**(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**Maximum Likelihood Estimation under a Gaussian Assumption**(cont’d)**Maximum Likelihood Estimation under a Gaussian Assumption**(cont’d)**Maximum Likelihood Estimation under a Gaussian Assumption**(cont’d)**(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**(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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