Note: For a fuller treatment, download our series of lectures Hierarchical Linear Models.
The Empty Model
As a first step, R&B begin with an empty model containing no covariates.
Each school’s intercept, β0j, is then set equal to a grand mean, γ00, and a random error u0j.
Substituting (2) into (1) produces
To estimate this in SPSS, go to Analyze → Mixed Models → Linear…
The Specify Subjects and Repeated menu appears. In this example the grouping variable is id, so it should be placed in the Subjects box.
The Repeated box stays empty. It is only used when the analyst wants to specify a covariance pattern for repeated measures (the R matrix; see A Review of Random Effects ANOVA Models). Click Continue.
A new menu pops up for specifying the variables in the model. The empty model has no independent variables, so place the dependent variable mathach in the appropriate box.
The intercept in the empty model is treated as randomly varying. This is not the default, so click on Random to get the following menu:
Check the Include Intercept option. Also, bring the id variable over to the Combinations box. The Covariance Type is irrelevant when there is only one random effect, in this case the random intercept. Click Continue.
Next, click on Statistics to pull up an additional menu to choose which results get reported in the output.
Choose Parameter Estimates to report estimates for the fixed effects. Click Continue, then OK. A portion of the results is the following:
These results correspond to Table 4.2 in R&B.
The next step is to estimate a means-as-outcomes model.
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