Message/Author 


Hello, I have identified a secondorder 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; model g1: [f1@0 f2@0 f3@0]; model g2: [f1@0 f2@0 f3@0]; 


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 secondorder factor model with binary data? For continuous data, measurement invariance tests goes in a sequence: 1. Configural invariance 2. Firstorder factor loadings invariance 3. Secondorder factor loadings invariance 4. Intercepts of firstorder factor indicators (observed variables) invariance 5. Intercepts of secondorder factor indicators invariance 6. Disturbances of firstorder factors invariance 7. Residual variance of firstorder 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 firstorder factor loadings and indicator intercepts invariance test and proceeds to secondorder factor loadings and intercepts? Thank you for your help!!! 


Regarding 5. for continuous items, don't you always have those fixed at zero? Regarding binary items, yes, I would recommend doing loadings and thresholds together. See my answer to Eiko Freid today. 


Hi, I am trying to establish configural measurement invariance for a second order factor model across two groups. However, when I try to free the factor loadings, I get the following error message: THE STANDARD ERRORS OF THE MODEL PARAMETER ESTIMATES COULD NOT BE COMPUTED. THE MODEL MAY NOT BE IDENTIFIED. Is this a problem of nonidentification or did I misspecify my model? Below you can see my input for one of my groups. Thanks a lot in advance for your support! group 1: RE_1 by z1 z2 (l1); RE_2 by u1 u2 (l2); RE_3 by p1 p2 (l3); RE_4 by t1 t2 (l4); REI by RE_1 RE_2 (l5) RE_3 (l6) RE_4 (l7); [z1] (i1); [z2] (i2); [u1] (i3); [u2] (i4); [p1] (i5); [p2] (i6); [t1] (i7); [t2] (i8); [RE_1] (i9); [RE_2] (i10); [RE_3] (i11); [RE_4] (i12); 


You should not mention the first factor indicator in the groupspecific 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 1storder 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 1sorder factors that become intercepts for the 2nd order factors, and whether the 1storder factor disturbances should be free at all stages or not. 1.CONFIGURAL (having established 1storder factor invariance): equal 1storder loadings and thresholds item residual variances @1 in Time1 and free in other times free factor disturbances free 2ndorder factor loadings 1storder factor intercepts @0 in Time1 and free in other times 2ndorder factor means @0 in all times 2.METRIC same as configural but with equal 2nd order factor loadings 3.SCALAR same as metric but: 2ndorder factor means @0 in Time1 and free in the others 1storder factor intercepts equal (except those of Time1 which are fixed @0  I found it doesn't work otherwise). Thanks! 


This looks ok. Although for Scalar one could argue for the factor intercepts being fixed@0 for all time points (full scalar intercept invariance). 


Hello, In reference to your response to Margarita on November 10, 2017, for Scalar, why do you recommend fixing factor intercepts @0 for all time points? Thanks! Hillary 


We want them invariant over time and it is not clear what it would mean if some factor intercepts are free. 


I'd like to confirm the below steps are correct for testing MG secondorder 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 (secondorder factor) free, but one per factor is fixed to 1. Latent mean (so factor) fixed to 0 in all gr. Firstorder 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 (so factor) free, but one per factor is fixed to 1. Latent mean (so factor) as configural. First and secondorder metric Same as firstorder metric, exept factor loadings (so factor) set eq across gr. and one per factor is fix to 1. Firstorder 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. (so factor) eq across gr. and one per f. fixed to 1. Latent mean (so factor) as conf. First and secondorder scalar Same as previously, exept: Latent means (for so) free, but fix to 0 in one gr. 


Configural looks correct. For first and secondorder scalar case, when you say latent means for so free, are you referring to the intercepts of the so factors in their regression on (measurement of) the secondorder factor? If so, I don't think you want those free but fixed at zero and let the means of the secondorder factor be free in all but one group. Metric we don't recommend. 


Thank you very much for your answer. 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) Secondorder 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. 


Right 

Back to top 