Estimating HLM Models Using Stata: Part 2

Jeremy Albright

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ANOVA HLM Stata

Estimating HLM Models Using Stata: Part 2

The Means-as-Outcomes Model

After estimating the empty model, R&B develop a Means-as-Outcomes model in which a school-level variable, meanses, is added to the model for the intercept. This variable reflects the average student SES level in each school. Recall that, when fitting the empty model, we began with the following equation:

\[ Y_{ij} = \beta_{0j} + e_{ij} \]

Now the intercept can be modeled as a grand mean, \(\gamma_{00}\), plus the effect of the average SES score, \(\gamma_{01}\), plus a random error, \(u_{0j}\).

\[ \beta_{0j} = \gamma_{00} + \gamma_{01}(\text{Mean SES}_j) + u_{0j} \]

Substituting,

\[ Y_{ij} = \gamma_{00} + \gamma_{01}(\text{Mean SES}_j) + u_{0j} + e_{ij} \]

The following will estimate the model.

mixed mathach meanses || schid: , reml
Performing EM optimization: 

Performing gradient-based optimization: 

Iteration 0:   log restricted-likelihood = -23480.642  
Iteration 1:   log restricted-likelihood = -23480.642  

Computing standard errors:

Mixed-effects REML regression                   Number of obs     =      7,185
Group variable: schid                           Number of groups  =        160

                                                Obs per group:
                                                              min =         14
                                                              avg =       44.9
                                                              max =         67

                                                Wald chi2(1)      =     263.15
Log restricted-likelihood = -23480.642          Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     mathach |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     meanses |   5.863538   .3614574    16.22   0.000     5.155095    6.571982
       _cons |   12.64944   .1492799    84.74   0.000     12.35685    12.94202
------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
schid: Identity              |
                  var(_cons) |   2.638696    .404337      1.954144    3.563051
-----------------------------+------------------------------------------------
               var(Residual) |   39.15709   .6608017      37.88312    40.47389
------------------------------------------------------------------------------
LR test vs. linear model: chibar2(01) = 239.95        Prob >= chibar2 = 0.0000

The only difference from the empty model is that we’ve now included meanses as a fixed effect. The estimate of 5.864 tells us how much on average across all schools we would expect math achievement to increase for each unit increase in the school-level mean SES. This effect is statistically significant, \(p < 0.001\).

The next step is to fit a random coefficient model.