### 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.

\[ \begin{equation}Y_{ij} = \beta_{0j} + e_{ij}\tag{1}\end{equation} \]

Each school’s intercept, \(\beta_{0j}\), is then set equal to a grand mean, \(\gamma_{00}\), and a random error \(u_{0j}\).

\[ \begin{equation}\beta_{0j} = \gamma_{00} + u_{0j}\tag{2}\end{equation} \]

Substituting (2) into (1) produces

\[ \begin{equation}Y_{ij} = \gamma_{00} + u_{0j} + e_{ij}\tag{3}\end{equation} \]

To estimate this in SPSS, go to **Analyze > Mixed Models > Linear…**

The **Specify Subjects and Repeated** menu appears. In this example, grouping variable is *schid*, 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:

Click the **Include Intercept** option. Also, bring the *schid* 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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