1. G89.2247 Lecture 6 Multilevel Notation; Level 1 and Level 2 Equations
Multilevel Models with Random Intercept and Slope
Models Incorporating Cross-Level Interactions
Estimation Issues
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6. Average Regression Line
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9. Covariance Matrix of the Level 2 Random Effects
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12. Models With Cross-Level Interactions
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14. Level 2 Equations: 0j = 00 + 01Zj +u0j 1j = 10 + 11Zj + u1j
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17. Covariance Matrix of the Residual Level 2 Random Effects
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18. Estimation Issues
Unlike standard regression methods, random regression must take into account two kinds of variability
Within macro-level (within person)
Between person random effects
The estimation process requires both estimates of the regression coefficients (fixed effects) and the the variance terms (random effects)
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19. ML and REML Overview
Proc Mixed and similar programs use iterative methods
Step 1: estimate fixed effects, assuming a first guess for the variances
Step 2: use residuals to estimate variances
Step 3 use estimated variances to estimate fixed effects again
Step 4 use residuals to estimate variances again
Continue until values don’t change
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20. Estimating Variances
Maximum likelihood estimation gives consistent estimates of variances
Estimates are biased, however
For sample variances, we fix bias by using (N-1) rather than N
Restricted Maximum likelihood is a general approach to fix the bias
Gives same point estimates of fixed effects, but gives better estimates of the variances
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