Estimating HLM Models Using SPSS Menus: Part 2

Jeremy Albright

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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 > Linear…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:

HLM SPSS Results

This corresponds to Table 4.3 in R&B.

The next step is to estimate a random coefficient model.

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