Invariance constraints on factor means
Message/Author
 Roger E. Millsap posted on Saturday, November 25, 2000 - 10:21 pm
In a multiple-group latent growth model,
I would like to constrain the factor means
for the intercept and linear factors to
invariance across groups. Unfortunately,
I cannot find a syntax expression that
will accomplish this. I have tried something
like:

Model: ....[f1*](1);
[f2*] (2);
Model: grp2 ....

This doesn't work; grp2 ends up with zero
values for the factor means, while the other
group is nonzero. Note that I don't want
to fix the means to zero, but rather to
estimate common (possibly) nonzero values
for the means. Also, the two factor means
should be allowed to differ within group.
Hope this is clear. How is it done? thanks
 Linda K. Muthen posted on Monday, November 27, 2000 - 9:20 am
In multiple group analysis, the factor means are fixed to zero in the first group by default and free in the other groups by default. You must explicitly refer to the factor means in group 1 to override the default. If you change group 2 to group 1, it should work:

MODEL:

[f1] (1);

MODEL g2:

[f1] (1);
 Amber Watts posted on Friday, March 11, 2011 - 9:07 am
I think I have done what is suggested here, but it still estimates factor mean@0 for group 1 and free for group two. What am I doing wrong?

MODEL: leisure BY walktri strendi sportdi;
home BY yargar houswk homrep care;
[leisure] (1);
[home](2);

Model wave2:
[leisure] (1);
[home](2);
 Linda K. Muthen posted on Friday, March 11, 2011 - 9:17 am
In multiple group analysis, the test of factor mean differences consists of one model where factor means are zero in all groups and another model where factor means are zero in one group and free in the other groups. This is described in the Topic 1 course handout under multiple group analysis. See specifically slides 223 and 224.
 Amber Watts posted on Friday, March 11, 2011 - 10:29 am
So if I write the following code and the results fit equally well, can I assume invariance of factor means?

MODEL:
leisure BY walktri strendi sportdi;
home BY yargar houswk homrep care;
[leisure@0];
[home@0];

Thank you
 Linda K. Muthen posted on Friday, March 11, 2011 - 10:54 am
You would need to do a difference test to determine whether the means are the same across groups. See page 434-435 of the user's guide to see how to do this.
 pauline posted on Monday, May 14, 2012 - 11:31 pm
In am doing a multigroup latent growth model with a quadratic growth form. There are two groups. I would like to freely estimate the intercepts, factor loadings, and residual variances across groups while fixing factor means at zero in all groups. I am unclear as to how to write the group specific model to achieve this. Thank you

Model:
i s q | y@0 y@1 y@2 y@3 y@4 y@5;
 Linda K. Muthen posted on Tuesday, May 15, 2012 - 10:32 am
With growth modeling, testing for measurement invariance of the growth factors cannot be done in the traditional way because the factor loadings must be fixed to specify the growth model. In this case, the first step is to find a well-fitting growth model in each group. If the same growth model does not fit in each group, the groups cannot be compared on the growth factor means, variances, and covariances.
 pauline posted on Tuesday, May 15, 2012 - 4:49 pm
Thank you. Of course, that absolutely makes sense. There is a well fitting quadratic model for each group. Just to clarify, the Grouping option for MG analysis by default looks at equality of factor loadings which in this case of MG LGM is already be fixed for I S and Q anyways. The chi square stat for this MG LGM will then become the baseline for testing further invariance?

I would then proceed to constrain factor means. Am I correct to specify the constraints for grp 2 instead of grp 1 for the group specific model?

Model:
i s q | y@0 y@1 y@2 y@3 y@4 y@5;
[i] (1);
[s] (2);
[q](3);
Model grp 2:
[i](1);
[s](2);
[q](3);

Then factor variances:
i (4);
s (5);
q (6);
And covariances :
i with s (7);
i with q (8);
s with q (9);

Many thanks
 Bengt O. Muthen posted on Tuesday, May 15, 2012 - 6:37 pm

Constraints across groups are only needed in your Model statement. You can check in the output that you get what you want. See also the UG, chapter 14.
 Chang Liu posted on Tuesday, May 07, 2019 - 6:27 pm
Dear Dr. Muthen,

I'm currently doing a Second-Order Latent Growth Models with multiple indicators at each time of assessment. My model is a linear growth curve model and the intercept (I guess by default) is 0. Is it possible to estimate the "true" intercept?

In addition, I plan to test the cohort differences in this growth curve modeling (e.g., differences in the intercept and slope). Do I have to compare the model with cohort 2 freely estimated intercept vs. cohort 2 constrained the intercept to 0? Many thanks!
 Bengt O. Muthen posted on Wednesday, May 08, 2019 - 2:20 pm
Q1: There is no "true" intercept. You can't free it because it is not identified. You can artificially obtain a non-zero estimate if you instead fix say the first measurement intercept at zero - but this is simply moving parameters around in the model and no new information has been gained.

Q2: Right.