G Lecture 6 Multilevel Notation; Level 1 and Level 2 Equations

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Presentation transcript:

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 G89.2247 Lecture 6

G89.2247 Lecture 6

G89.2247 Lecture 6

G89.2247 Lecture 6

G89.2247 Lecture 6

Average Regression Line G89.2247 Lecture 6

G89.2247 Lecture 6

G89.2247 Lecture 6

Covariance Matrix of the Level 2 Random Effects G89.2247 Lecture 6

G89.2247 Lecture 6

G89.2247 Lecture 6

Models With Cross-Level Interactions G89.2247 Lecture 6

G89.2247 Lecture 6

Level 2 Equations: 0j = 00 + 01Zj +u0j 1j = 10 + 11Zj + u1j G89.2247 Lecture 6

G89.2247 Lecture 6

G89.2247 Lecture 6

Covariance Matrix of the Residual Level 2 Random Effects G89.2247 Lecture 6

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) G89.2247 Lecture 6

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 G89.2247 Lecture 6

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 G89.2247 Lecture 6