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

#### Intercepts- and Slopes-as-Outcomes Model

R&B present a final model that includes one further generalization of the random coefficients model. They begin again with the level-1 model:

\[ \begin{equation}Y_{ij} = \beta_{0j} + \beta_{1j}(SES)+ e_{ij} \tag{6}\end{equation} \]

The intercept \(\beta_{0j}\) is now modeled as a function of the average SES level of the school and whether or not the school is public or private. The slope \(\beta_{1j}\) is modeled in similar fashion.

\[ \begin{equation}\beta_{0j} = \gamma_{00} + \gamma_{01}(MEAN\ SES)+ \gamma_{02}(SECTOR)+ u_{0j} \tag{10}\end{equation} \]

\[ \begin{equation}\beta_{1j} = \gamma_{10} + \gamma_{11}(MEAN\ SES)+ \gamma_{12}(SECTOR)+ u_{1j} \tag{11}\end{equation} \]

Substituting (10) and (11) into (6) leads to the combined model:

\[ \begin{equation}Y_{ij} = \gamma_{00} + \gamma_{01}(MEAN\ SES)+ \gamma_{02}(SECTOR) \\ + \gamma_{10}(SES) + \gamma_{11}(MEAN\ SES)(SES) \\ +\gamma_{12}(SECTOR)(SES) +u_{0j} + u_{1j}(SES)+ e_{ij} \tag{12}\end{equation} \]

To estimate (12) in SPSS go to **Analyze > Mixed Models > Linear**. The **Specify Subjects and Repeated** menu appears again. As before, place *schid* in the **Subjects** box and leave **Repeated** blank.

Click **Continue**. In the next menu one specifies the dependent and independent variables. The dependent variable is *mathach*, and the covariates are *grp_ses*, *meanses*, and *schtype*.

To specify the model’s fixed effects click on **Fixed**. The model includes both main effects for all three variables as well as one interaction between *grp_ses* and *meanses* and another between *grp_ses* and *schtype*. Choose **Main Effects** from the drop-down menu in the center. Then bring all three variables over to the **Model** box.

To specify the interaction, change **Main Effects** to **Interaction**. Click on *grp_ses*, then hold down the Ctrl key and click on *meanses*. Click the **Add** button to bring the interaction to the **Model** box.

Next, click on *grp_ses*, hold down the Ctrl key, and click on *schtype*. Click the **Add** button to bring the second interaction to the **Model** box.

Make sure **Include Intercept** is checked and click **Continue**. Next, click on **Random** to specify the random effects in the model.

In the **Random Effects** menu, the grouping variable *schid* should once again appear in the **Combinations** box. The level-1 variable *grp_ses* will have a random slope, so it is necessary to place it in the **Model** box. The **Include Intercept** should also be checked to specify a random intercept. Finally, the presence of two random effects means that the dimensions of the covariance matrix **G** are 2?2. The default in SPSS is to assume a variance components structure, (see the table of covariance structures in A Review of Random Effects ANOVA Models). This assumption can be loosened so that the covariances are free parameters to be estimated from the data. Specify **Unstructured** for the **Covariance Type**.

Click **Continue**. Then click **Statistics** to specify what appears in the output. Check **Parameter Estimates** to get results for the fixed effects.

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

These results correspond to Table 4.5 in R&B and the variance-covariance components on page 83. While most estimates are identical, there are some slight differences in the random effects (for example, R&B report a level-1 variance component of 36.68 whereas SPSS reports the estimate to be 36.72). In general, results will vary somewhat across software packages for more complicated models. Thus, researchers should name the software they have used when reporting results for mixed models.

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