Measurement invariance of 2nd-order f... PreviousNext
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 Joseph E. Glass posted on Sunday, August 19, 2012 - 5:41 pm

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;

model g1:
[f1@0 f2@0 f3@0];
model g2:
[f1@0 f2@0 f3@0];
 Linda K. Muthen posted on Monday, August 20, 2012 - 9:02 am
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.
 Mi-young Webb posted on Friday, May 08, 2015 - 8:28 am
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?

Thank you for your help!!!
 Bengt O. Muthen posted on Friday, May 08, 2015 - 8:48 am
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.
 Clark Thompson posted on Sunday, December 20, 2015 - 4:50 am

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:


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);
 Linda K. Muthen posted on Sunday, December 20, 2015 - 11:49 am
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

-same as configural but with equal 2nd order factor loadings

-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).

 Bengt O. Muthen posted on Friday, November 10, 2017 - 4:33 pm
This looks ok. Although for Scalar one could argue for the factor intercepts being fixed@0 for all time points (full scalar intercept invariance).
 Hillary Gorin posted on Friday, March 16, 2018 - 10:11 am

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?

 Bengt O. Muthen posted on Friday, March 16, 2018 - 1:56 pm
We want them invariant over time and it is not clear what it would mean if some factor intercepts are free.
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