Additional Notes: Investigating a Random Slope. When we have fixed level-1 predictors at level 2 we show them like this:

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1 Ron Heck, Summer 01 Seminars 1 Multilevel Regression Models and Their Applications Seminar Additional Notes: Investigating a Random Slope We can begin with Model 3 and add a Random slope parameter. If you remember, we have the slope model for low SES as a fixed parameter. Where there is a random slope, we often refer to the model as a slopes-as-outcome model. When we have fixed level-1 predictors at level we show them like this: If we wish to examine the lowses slope ( 1 ) as randomly varying, we have to add the variance parameter to the model ( u 1 ). So we will now have the following: u The new model will consist of the level-1 model: Y lowses female, i 0 1 i i i the level- intercept model: zenroll _ y schcontext schqualcomp teacheff u, and the level- slope model: u When we substitute them all into one combined equation we will have the complete model with 10 parameters to estimate (7 fixed effects, random effects, and the residual). Yi 00 01zenroll _ y 0schcontext 03schqualcomp 04teacheff lowses female u lowses u 10 i 0 i 1 i 0 i Once again, we note that when we substitute for the random slope ( 1 ) we multiply its average effect ( 10 ) and its random effect ( u 1 ) by the level-1 X predictor (lowses). This results in the following two terms ( 10 lowses i ) and u1 lowsesi appearing in the combined model. Summer 01 Quantitative Methods Series at Portland State University

2 Additional Notes: Investigating a Random Slope The output (Table 1) confirms 10 parameters to estimate (7 fixed effects, random level- effects, and the level-1 residual). Table 1 Next we can look at the fixed effects (Table ). Our primary interest initially, however, is in the variance components. Table The variance components table (Table 3) provides evidence that the student SES-math achievement slope varies across schools (Wald Z = 49.31, p <.003). Keep in mind that because variances cannot be below 0 (i.e., negative), we can use a one-tailed test for their significance (i.e., this means dividing the significance level by ). Note that the default covariance matrix of random effects at level has a variance for the intercept and a variance for the slope, but there is no covariance between them.

3 Additional Notes: Single-Level to ML Models 3 This is referred to as a diagonal covariance matrix, since there is no off-diagonal covariance. You can see that the off-diagonal element (the covariance) is fixed at 0. Intercept 0 0 Slope In the variance components table (Table 3) you can see that the level- variances for the intercept and slope are 6.3 and 49.31, respectively. Both are significant. The results, therefore, suggest we might build a model to explain variance in the slope parameter across schools. Table 3 The fit of Model 3 with a random slope has the following fit indices (Table 4). We can use the AIC to determine whether the model with random slope fits better than Model 3. For Model 3 (9 parameters) the - LL is 176, For the current model it is 176, The difference is distributed as chi square (4.416, 1 df). The required chi-square at p =.05 is Therefore, we can conclude that the model with random low SES slope represents an improved fit to the data over the model with the slope fixed. Table 4

4 Additional Notes: Investigating a Random Slope 4 Building a Model to Explain the Random Slope We would like to build a parallel model to the intercept and then only if variables are not significant, can we drop them out of the final model. The slope model with four predictors added will look like the following: zenroll _ y schcontext schqualcomp teacheff u When we substitute the above equation into the previous combined model with 10 parameters, we will now have 14 parameters to estimate. Y zenroll _ y schcontext schqualcomp teacheff lowses 11lowsesi * zenroll _ y 1lowsesi * schcontext 13lowses i * schqualcomp 14lowses i * teacheff 0 femalei ulowses u i i i 1 i 0 In the combined model, the slope predictors ( ) become cross-level interactions. We will build the interactions in SPSS through opening the Fixed Effect box and clicking on Build Nested Terms. After clicking on that, we will put lowses in the Build Term box using By* (lowses*zenroll_y). Do the same for each of the 4 school predictors in model 3. Click continue. Now we have added four effects to the model. Below is the -LL and other indices (Table 5). Table 5 We can see that the model with 14 parameters (-LL = ) estimated fits better than the model with 10 parameters (-LL = ). We can construct a chi-square test with 4 degrees of freedom ( = ) with 4 df). The required chi-square with 4 df is 9.49 at p =.05. Hence, the current model represents an improved fit. The fixed effects (Table 6) suggest there are two promising effects (lowses*schcontext) and lowses*schqualcomp. The meaning is that in schools 1-SD above the grand mean in terms of

5 Additional Notes: Single-Level to ML Models 5 more challenging student composition (i.e., higher levels of students on free/reduced lunch, receiving languages services), the impact of individual lowses on math outcomes is diminished (4.73, p <.01). So the combined effect in those schools is = Hence there is less of a gap in achievement for low SES students in these schools. This suggests the effect of student composition diminishes the negative effect of low SES on achievement. As we might suspect, in schools where there are high numbers of low SES students, the impact of individual low SES status on achievement is lessened considerably. On the other hand, in schools where very few students require these types of special academic programs, the impact of being lowses on outcomes is exaggerated ( = -19.9). Regarding school quality, the results suggest that the gap between lowses and achievement is actually larger (-1.71), p <.08). This suggests that in higher quality schools in terms of academic focus, being low SES is actually more consequential in terms of achievement ( (-1.71). For example, achievement expectations may be higher in such schools. Table 6 We could decide to remove the variables in the slope model that are not significant (or we can leave them in the model. The variance table (Table 7) there is still considerable variability in school intercepts and the random slope to be explained between schools. Table 7 The last thing we might want to do is examine whether there is a relationship between the

6 Additional Notes: Investigating a Random Slope 6 intercept and the slope. We can obtain an unstructured covariance matrix (UN). It looks like the following: Intercept IS IS Slope. This change will add one random parameter to be estimated in the model. We can see whether the model s fit is improved by adding the random covariance. We can examine the change in the -LL for this model with one additional parameter estimated. For the model with 15 parameters the fit indices are as follows (Table 8): Table 8 The likelihood ratio test (i.e., the difference in model -LL) between two nested models is as follows: 176, , = (1df). The required chi-square coefficient with 1 degree of freedom at p =.05 is We can conclude that the model with unstructured covariance matrix fits the data better than the diagonal covariance matrix. Notice that with the random low SES slope, the two variables from the previous model (i.e., lowses*schcontext, lowses*schqualcomp) are both significant at p <.05 (Table 9).

7 Additional Notes: Single-Level to ML Models 7 Table 9 The covariance results are the following (Table 10): Table 10 The table suggests that the covariance between the slope and intercept is negative and significant (Wald Z = , p <.01). We can also obtain the correlation by specifying (UNR). The fit indices and parameter estimates are the same, but the covariance between the intercept and slope is expressed as a correlation (instead of a covariance). We can see (Table 11) that school achievement is negatively related (r = -.68, p <.01) to the slope describing the impact of low student SES on achievement. Table 11 Examine the variance accounted for by the intercepts at each level.