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What's New in Stata 9: Linear Mixed Models

This section illustrates the use of xtmixed using examples from the paper Using SAS Proc Mixed to Fit Multilevel Models, Hierarchical Models, and Individual Growth Models by Judith Singer and can be downloaded from Professor Singer's web site.

use http://www.ats.ucla.edu/stat/paperexamples/singer/hsb12

Example on page 329.

Level 1: MathAch_ij =  β_0j + r_ij  
Level 2: β_0j =  γ₀₀ + u_0j

Combining the two equations into one by substituting the level 2 equation into the level 1 equation, we have the equation below, with the random effects identified by placing them in square brackets.

MathAch_ij =  γ₀₀ + [ u_0j + r_ij ]

/* unconditional means model */
xtmixed mathach || school: , variance mle
estat ic

Example on page page 331.

Level 1: MathAch_ij =  β_0j + r_ij  
Level 2: β_0j =  γ₀₀ + γ₀₁(MeanSES) + u_0j

Combining the two equations into one by substituting the level 2 equation into the level 1 equation, we have the equation below, with the random effects identified by placing them in square brackets.

MathAch_ij =  γ₀₀ + γ₀₁(MeanSES) + [ u_0j + r_ij ]

/* random intercept -- level 2 predictor */
xtmixed mathach meanses || school: , variance mle
estat ic

Example on page 335.

Here is the model expressed using multiple equations.

Level 1: MathAch_ij =  β_0j + β_1j (cses) + r_ij  
Level 2: β_0j =  γ₀₀  + u_0j
Level 2: β_1j =  γ₁₀  + u_1j

Combining the two equations into one by substituting the level 2 equation to level 1 equation, we have the following with the random effects identified by placing them in square brackets.

MathAch_ij =  γ₀₀  + γ₁₀(cses) +  [ u_1j(cses) + u_0j + r_ij ]   

/* random slopes and random intercept -- level 1 predictor */
xtmixed mathach cses || school: cses, variance cov(un) mle
estat ic

Example on page 337.

We can express this as a multiple equation model like this.

Level 1: MathAch_ij =  β_0j + β_1j (cses) + r_ij  
Level 2: β_0j =  γ₀₀  + γ₀₁(MeanSES) + γ₀₂(Sector) + u_0j
Level 2: β_1j =  γ₁₀  + γ₁₁(MeanSES) + γ₁₂(Sector) + u_1j

Combining the two equations into one by substituting the level 2 equations into the level 1 equation, we have the following equation.  The random effects are identified by placing them in square brackets.

MathAch_ij =   γ₀₀  + γ₀₁(MeanSES) + γ₀₂(Sector) + γ₁₀ (cses) + γ₁₁(MeanSES*cses) +  γ₁₂(Sector*cses) +
     [  u_0j + u_1j(cses) +  r_ij ]

/* random slopes and random intercept -- cross level interactions */
generate msesXcses = meanses*cses
generate secXcses = sector*cses

xtmixed mathach meanses sector cses msesXcses secXcses || school: cses, variance cov(un) mle
estat ic

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