I have identified a second-order CFA model with ordinal indicators and WLSMV (see below). I would like to test the second order measurement parameters for loading and intercept invariance. Can you point me to reference materials on this topic or instruct me on how to do this? Thank you.
f1 by var1 var2 var3 var4; f2 by var5 var6 var7 var8; f3 by var9 var10 var11 var2; f4 by f1 f2 f3;
The model you show above would be your constrained model. In the second model, you would free the factor loadings and thresholds, fix the mean of f4 to zero, and fix all scale factors at one. See multiple group analysis in the Topic 2 course handout for example inputs for how to do this.
What is the best way to test measurement invariance of a second-order factor model with binary data?
For continuous data, measurement invariance tests goes in a sequence: 1. Configural invariance 2. First-order factor loadings invariance 3. Second-order factor loadings invariance 4. Intercepts of first-order factor indicators (observed variables) invariance 5. Intercepts of second-order factor indicators invariance 6. Disturbances of first-order factors invariance 7. Residual variance of first-order factor indicators or observed variables invariance
For measurement invariance with binary data, I have been advised to test loadings and intercepts invariance together. If so, should I test first-order factor loadings and indicator intercepts invariance test and proceeds to second-order factor loadings and intercepts?
You should not mention the first factor indicator in the group-specific part of the MODEL command. It is no longer fixed to one when you do this.
Margarita posted on Thursday, November 09, 2017 - 3:01 am
Dear Dr. Muthen,
I'd like to confirm if possible whether the below steps are correct for testing 2nd order measurement invariance across time with THETA and WLSMV after the invariance of the 1st-order factors has been established. These are based on UG keeping in mind the continuous nature of the latent variables. I think the most tricky part is the means of the 1s-order factors that become intercepts for the 2nd order factors, and whether the 1st-order factor disturbances should be free at all stages or not.
1.CONFIGURAL (having established 1st-order factor invariance): -equal 1st-order loadings and thresholds -item residual variances @1 in Time1 and free in other times -free factor disturbances -free 2nd-order factor loadings -1st-order factor intercepts @0 in Time1 and free in other times -2nd-order factor means @0 in all times
2.METRIC -same as configural but with equal 2nd order factor loadings
3.SCALAR -same as metric but: -2nd-order factor means @0 in Time1 and free in the others -1st-order factor intercepts equal (except those of Time1 which are fixed @0 - I found it doesn't work otherwise).
I'd like to confirm the below steps are correct for testing MG second-order inv. with WLSMV
Configural Factor loadings free, but one per factor is fixed to 1. Thresholds free across groups. Residual var. fixed at 1 in all gr. Factor var. free across gr. Factor means fixed at 0 in all gr. Factor loadings (second-order factor) free, but one per factor is fixed to 1. Latent mean (s-o factor) fixed to 0 in all gr.
First-order metric Factor load. constrained to be equal across groups. The first thresh. of each item (and second for metric item) is held eq across gr. Resid. var. fixed at 1 in one gr. and free in the other. Factor var. free across gr. Factor means fixed at 0 in all gr. Factor loadings (s-o factor) free, but one per factor is fixed to 1. Latent mean (s-o factor) as configural.
First- and second-order metric Same as first-order metric, exept factor loadings (s-o factor) set eq across gr. and one per factor is fix to 1.
First-order scalar Factor load. and thresholds constr. to be equal across gr. Res. var. fix at 1 in one gr. and free in the other gr. Factor var. are free. Factor means free, but fix to 0 in one group. Factor load. (s-o factor) eq across gr. and one per f. fixed to 1. Latent mean (s-o factor) as conf.
First- and second-order scalar Same as previously, exept: Latent means (for s-o) free, but fix to 0 in one gr.
Configural looks correct. For first- and second-order scalar case, when you say latent means for s-o free, are you referring to the intercepts of the s-o factors in their regression on (measurement of) the second-order factor? If so, I don't think you want those free but fixed at zero and let the means of the second-order factor be free in all but one group.
So the optimal series of hierarchical steps should be: 1) Configural As in the previous post. 2) First order scalar First order factor loadings (and tresholds) constrained to be equal across groups. Residual variances fixed at one in one group and free in the other groups. If the metric of a factor is set by fixing a factor loading to one, factor variances are free across groups. First order factor intercepts free, but fixed to 0 in one group. Second order factor loadings equal across groups. Mean of the second order factor fixed to 0 in all groups. 3) Second-order scalar First order factor loadings tresholds, residual variances, factor variances as before. First order factor intercepts fixed to 0 in all group. Second order factor loadings equal across groups. Mean of the second order factor free, but fixed to 0 in one group.