First, what is 'constraint': A model of this type relates to in-variance. We want to test if factor loadings, variances of factors, covariances between factors, and variances of the 'error' terms remain the same ACROSS groups - or, if they change, the change is INSIGNIFICANT. We do that by artificially 'keep' constant one of the elements listed above (in the least restricted model, keep constant factor loadings only) or all of them (in the fully restricted model). The computer will do it for you. In AMOS, you should choose 'unstandardized estimates' if you want to see which element is kept constant.
Then we compare estimates of the less restricted model with estimates of the more restricted model, and hope that the change would be insignificant. Again, the computer will do it.
A good cross-group measurement model should have four components like above: Top left: No constraints (two models are run at the same time, chi-square difference is computed at the end). Top right: Constrained factor loadings (the least restricted model - this is minimum requirement). Bottom Left: Constrained structural covariances (the more restricted model). Bottom Right: Constrained measurement residuals (the fully restricted model).
Perfect: All models have good fit
Very good: Except the last model, other three have good fit
Good: The model constraining for factor loadings has good fit.
Poor: The model constraining for factor loadings has poor fit. In this case, we conclude that the groups that are compared do not share the same factors. Two separate measurements (one for men, another for women, for instance) should be used.
The above model has perfect fit. How can I say that?
- All 4 models have RMSEA smaller than 0.06, GFI close to 1, p value is smaller 5%
- Compared with the less constrained model, the more constrained model has a good fit and the 'gap' between 'less constrained' and 'more constrained' is insignificant. In AMOS, look for this at 'Model Comparison' tab.
Assuming model Unconstrained to be correct:
| Model | DF | CMIN | P | NFI Delta-1 |
IFI Delta-2 |
RFI rho-1 |
TLI rho2 |
| Measurement weights | 4 | 2.632 | .621 | .011 | .012 | -.010 | -.012 |
| Structural covariances | 7 | 3.306 | .855 | .014 | .015 | -.023 | -.026 |
| Measurement residuals | 13 | 11.212 | .593 | .048 | .051 | -.011 | -.012 |
- The model constraining for factor loadings is insignificantly different from the unconstrained model.
