# Estimating HLM Models Using SPSS Menus: Part 3

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

#### Random Coefficient Model

Next, R&B present a model in which student-level SES is included instead of average SES, and they treat the slope of student SES as random. One complication is that R&B present results after group-mean centering student SES. Group-mean centering means that the average SES for each student’s school is subtracted from each student’s individual SES. Unfortunately the *meanses* variable is coded -1, 0, 1 and is therefore only a rough indicator of each school’s average. To get a better estimate of the school average one can take advantage of the **Aggregate** command in SPSS.

The first step to group-mean center a variable is to find the average for each cluster. Go to **Data → Aggregate**

The **Aggregate Data** menu appears. The variable representing each group is known as a “break” variable; put *id* in the **Break Variable(s)** box. The goal is to get the average student SES score from each school, so bring the *ses* variable over to the **Summaries of Variable(s)** box. By default, SPSS assumes the user is interested in getting the mean for each group, so there is no need to change the function. Finally, make sure the **Add aggregated variables to active dataset**radio button is selected.

A new variable *ses_mean* (not to be confused with the trichotomous *meanses*) is now added to the data. To complete the group-mean centering, subtract *ses_mean* from each *ses* variable. Go to **Transform → Compute Variable**.

In the menu that appears, create a **Target Variable** named *grp_ses* that is equal to *ses* minus *ses_mean*.

Click **OK**. The group-centered SES variable can now be used.

The level-1 equation is the following:

The intercept *β _{0j}* can be modelled as a grand mean

*γ*plus random error,

_{00}*u*. Similarly the slope

_{0j}*β*can be modelled as having a grand mean

_{1j}*γ*plus random error

_{10}*u*.

_{1j}Combining (7) and (8) into (6) produces:

To estimate (9) in SPSS go to **Analyze → Mixed Models → Linear**. The **Specify Subjects and Repeated** menu appears again. As before, place *id* 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 single covariate will be *grp_ses*.

To specify the model’s fixed effects click on **Fixed**. In the **Fixed Effects** menu, bring the *grp_ses* variable over to the **Model** box and make sure **Include Intercept** is checked.

Click **Continue**, then on **Random**.

In the **Random Effects** menu, place the grouping variable *id* in the **Combinations** box. Also, because *grp_ses* will have a random slope it is necessary to place it in the **Model** box. Next, make sure that **Include Intercept** is checked so that the intercept is also allowed to vary randomly. Finally, the presence of two random effects means that the dimensions of the covariance matrix **G** are now 2×2. The default in SPSS is to assume a variance components structure, which implies that there is no covariance between the random intercept and random slope (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 **OK**. A portion of the results is the following:

These results correspond to Table 4.4 in R&B. See also the variance-covariance components at the bottom of their page 77.

The final model R&B present is an intercept- and slopes-as-outomes model.

Still have questions? Contact us!