### Note: For a fuller treatment, download our series of lectures Hierarchical Linear Models.

#### 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 Equation (1):

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

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

\[ \begin{equation}\beta_{0j} = \gamma_{00} + \gamma_{01}(MEAN \ SES_j)+ u_{0j}\tag{4}\end{equation} \]

Substituting (4) into (1) yields

\[ \begin{equation}Y_{0j} = \gamma_{00} + \gamma_{01}(MEAN \ SES_j)+ u_{0j} + e_{ij} \tag{5}\end{equation} \]

To estimate this in SPSS, again go to **Analyze > Mixed Models > Linearâ€¦**The **Specify Subjects and Repeated Menu** appears again. Place *schid* in the **Subjects** box and leave the **Repeated** box empty.

Click **Continue**. In the next menu, one specifies the dependent and independent variables. The dependent variable will be *mathach*, and the single covariate will be *meanses*.

The *meanses* variable is entered as a fixed effect, so click on the **Fixed** button to pull up the **Fixed Effects** menu. Bring the *meanses* variable into the **Model** box and make sure **Include Intercept** is checked.

Click **Continue**. Next, click on **Random** to open the **Random Effects** menu. Check **Include Intercept** to specify the intercept as random, and place the grouping variable *schid* in the **Combinations** box. Do NOT place *meanses* in teh **Model** box. It will be treated as a fized effect only. The **Covariance Type** is again irrelevant because there is only one random effect, the random intercept.

Click **Continue**. Finally, click on **Statstics** to choose what gets reported in the output. Put a check next to **Parameter Estimates**.

Click **Continue**, then click **OK**. A portion of the output is the following:

This corresponds to Table 4.3 in R&B.

The next step is to estimate a random coefficient model.

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