Exercise 2.1, step 6: Re-specifying the model

This inspection of the modification indices shows that not all factor loadings and intercepts are equal across groups. This indicates that there could be problems with the comparability of the attitude scale. Remarkably, virtually all deviations from full scalar equivalence relate to one item, namely IMSMETN. Apparently, the exact meaning of this specific item concerning immigrants from own social groups differs considerably from one group to another. The parameters for the other two items, however, are stable across groups.

Based on these findings, we re-estimate the model, this time without equality constraints on the problematic item IMSMETN. This newly estimated model implies partial rather than full scalar equivalence, since factor loadings and intercepts are not equal for the whole set, but only for two items.

In order to avoid any confusion about variables names, it is advisable to completely re-specify the model to be estimated. First, save the current project under the name ‘fullequivalence’. Next, open the previously saved AMOS project ‘basemodel’, so that you do not have to create the groups and assign the data again. Repeat steps 3 (Specify the model) and 4 (Estimate the model) as explained before. However, there are two things that need to be done differently.

  1. When assigning the variable names to the indicators (i.e. dragging the variable names into the rectangles), drag IMDFETN into the first rectangle and IMSMETN into the second. By default, AMOS constrains the factor loading of the first item to 1. This item is called the marker item. The constraint on the marker item is a necessary condition to enable the model to be estimated. You can see that the first item is the marker item because the number ‘1’ appears next to the factor loading. Because the factor loading of the marker item is constrained to 1 in all groups, it is impossible to allow factor loadings to vary across groups. Because we want to set free factor loadings for IMSMETN, we have to make sure that this item is not used as the marker item.
  2. Just before clicking the ‘Calculate Estimates’ button, we have to manually remove the equality constraints on IMSMETN. Double-click the ‘Measurement intercepts’ model in the left-hand column. The ‘Manage Models’ window pops up. Here, you get an overview of all parameter constraints that are imposed across groups. The combination of letters and numbers refer to specific parameters in the model - they are also represented in the graphical representation of the model. The letter ‘a’ refer to factor loadings, and the letter ‘i’ to intercepts. A1_1, for example refers to the factor loading of the second item in group 1 (not the factor loading of the first item; the factor loading of the marker item is not a parameter since it is constrained to one). I3_7 refers to the intercept for the third item in group 7. The equality signs indicate that two factor loadings and three intercepts are constrained to be equal across all 51 groups. To remove all constraints on IMSMETN, the first (with the a1s) and fourth (with the i2s) lines need to be deleted. Select these lines, and press delete. The remaining equality constraints refer to the other two items. Close the ‘Manage Models’ window.

 

Figure 2.18: Parameter constraints

Now click the ‘calculate estimates’ button. Once the models are estimated, save the project as ‘partialequivalence’.

Question

Look at the fit of the ‘measurement intercepts’ model (this is the model for which we deleted some equality constraints). What do you see?

 

Figure 2.19: Partial equivalence: Model fit on 'CMIN'

 

Figure 2.20: Partial equivalence: Model fit on 'RMSEA'

Solution

The CFI of the re-specified model equals 0.94, and the RMSEA 0.027. These indices suggest that the model without equality constraints on IMSMETN fits the data quite well. Moreover, the CFI of the partially equivalent model is substantially better than the CFI of the previously estimated fully equivalent model (0.94 vs. the 0.902 we found before).

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