Comparing latent means PreviousNext
Mplus Discussion > Confirmatory Factor Analysis >
 Marlies Maes posted on Monday, September 16, 2013 - 7:38 am
Dear all,

I established measurement invariance for 4 groups, for a model including 4 factors with 3 indicators each.

As a next step, I would like to compare the latent means across those four groups. By default, the first factor mean is constrained to zero.

(a) Is is possible to compare latent means across groups, without constraining any of those latent means to zero?
Or: how can I obtain the latent means for each group? (without one being set to zero)

(b) Is it correct that when I would like to see which groups differ from each other, I need to examine this per pair of groups?

Many thanks for your help.

Best wishes,
 Linda K. Muthen posted on Monday, September 16, 2013 - 11:14 am
To compare all factor means, use a model with factor means zero in all groups versus a model with factor means zero in one group and free in the other groups.

To compare specific pairs of means if you have, for example, three groups and the factor means are fixed to zero in the first group, the z-test of the factor mean in group 2 tests the difference between groups 1 and 2. The z-test of the factor mean in group 2 tests he difference between groups 1 and 3. Use MODEL TEST to test the difference between groups 2 and 3.
 Marlies Maes posted on Tuesday, September 17, 2013 - 1:20 am
Many thanks for your quick answer!

Is it also possible to just see the factor means for each group?

(In other words, is it possible to see the latent mean of each factor for each group, without one being set to zero?)

Many thanks for your help!
 Linda K. Muthen posted on Tuesday, September 17, 2013 - 7:11 am
No. One must be set to zero for identification.
 Ebrahim Hamedi posted on Tuesday, November 05, 2013 - 6:22 pm
On page 433, you mention three possible steps for measurement invariance testing:

excerpt from the manual:
"1. Intercepts, factor loadings, and residual variances free across groups;
factor means fixed at zero in all groups
2. Factor loadings constrained to be equal across groups; intercepts and
residual variances free; factor means fixed at zero in all groups
3. Intercepts and factors loadings constrained to be equal across
groups; residual variances free; factor means zero in one group and
free in the others (the Mplus default)"

I have a question: Given that, in step 3, latent means in one group are fixed to zero, and freely estimated in other groups, I believe, in addition to providing a test of scalar invariance, estimates of step 3 can be used for investigating differences in latent means across groups. That is, the latent means in other groups can be tested for significance relative to the latent means in the reference group. Do you agree with my understanding? Or do you suggest extra modeling for latent mean analysis following scalar invariance testing?

many thanks in advance,
 Bengt O. Muthen posted on Wednesday, November 06, 2013 - 8:36 am
I agree with your understanding.
 mdehne posted on Saturday, July 29, 2017 - 8:45 am
Dear all,

I want to compare the means of a true change model with two measurements (where t1 is the first latent variable second variable is t2-t1 with loadings from t2; see input for clarification below). Thus, I am interested whether the difference variable/true change significantly differs from t1.

To my understanding, the displayed significance for my difference variable just shows the significant difference from zero. How can I conduct mean comparison?

Many thanks for your help!

y1 by x11@1 x21@1;

diff2_1 by x12@1 x22@1;

 Bengt O. Muthen posted on Sunday, July 30, 2017 - 5:04 pm
Don't complicate it - just have

f1 by time 1 indicators

f2 by time 2 indicators

and then fix the f1 mean


and free the f2 mean


The significance (significantly different from zero, i.e. the f1 mean) of the f2 mean tests whether there is a difference in the 2 factor means.
 Milica Lazic posted on Tuesday, September 12, 2017 - 2:13 am
Dear Professor Muthen,

We compared latent mean across gender and age using the parameters of the bifactor-ESEM model. To estimate the measurement invariance we used model=configural, scalar option in analyses section. As a next step, we would like to compare the latent means across groups. Do we can report means from scalar section, or we need to rewrite syntax? By default, in scalar section, the first factor mean is constrained to zero, while the mean of the other group is freely estimated. Is it enough, or we need to fixed mean to zero in syntax?

Thank you in advance
 Linda K. Muthen posted on Tuesday, September 12, 2017 - 6:04 am
See Example 5.27 in the user's guide.
 Tino Wulff posted on Monday, October 30, 2017 - 8:53 am
Dear all,

I'm doing a latent mean comparison as well.
At the level of scalar invariance i have a good fit except of srmr which is around 0.1

Unfortunately, literature is not giving me good advice with that. All I know is that srmr is somehow sensitive to sample size which is about 750. Is it necessary to look at covariance residuals? And if yes, how do I interpret them?
I'm a beginner at this topic and I'm writing a thesis so I'm very grateful for help.

Sorry for bad english :P
 Bengt O. Muthen posted on Monday, October 30, 2017 - 4:23 pm
If most other fit indices are good, you can trust the model despite the smrm.

You can also ask such a general analysis question on SEMNET.
 Tino Wulff posted on Sunday, November 05, 2017 - 1:22 am
Thank you. That helps a lot.

When I'm running a CFA for each group initially and I have a significant ChiČ-value , the mod-indices show me some high with-statements. So I'm adding them to the model to get a better fit. In the other group these with-commands are not needed for a good fit.
How does the programming looks like in the next step for form invariance? Do I have to add these with-commands only for the group where I ran this CFA (with bad fit) or for both groups for comparability?

Thanks in advance.
 Carl Arnold posted on Sunday, November 05, 2017 - 11:00 am
MySQL assumptions:
2 groups
1 latent variable
X manifest variables
Mean difference exists

Can such a constellation be associated with invariant intercepts of the manifest variables as long as the mean values of the latent variable in both groups are fixed to zero? The situation may be different when one of the groups is unrestricted by the averaging restriction. Only, how can you then perform an invariance test? Without this restriction, the model (restricted intercepts and a free mean value) is no longer nested compared to the basic model (freely estimated intercepts).
 Bengt O. Muthen posted on Sunday, November 05, 2017 - 1:38 pm
Answer to Wulff and Arnold:

These general analysis questions are suitable for SEMNET.
 Joe Wasserman posted on Friday, February 16, 2018 - 4:31 pm
I am trying to combine the effects-coding method of scaling latent variables (Little, Slegers, & Card, 2006) with multi-group analysis.

1) Am I incorrect that by constraining item loadings and item intercepts within each group, it should be possible to free the estimation of factor means in all groups?

2) If I'm not incorrect, is there any way to override Mplus's default of fixing factor means in the first group to 1? I can't seem to find one.

A simplified 2-group version of my syntax for the configural model:

f1 BY v1-v4 * (L1-L4);
f2 BY v5-v8 * (L5-L8);
[v1-v8] (T1-T8);

MODEL grp2:
f1 BY v1-v4 * (g2L1-g2L4);
f2 BY v5-v8 * (g2L5-g2L8);
[v1-v8] (g2T1-g2T8);

L1 = 4 - L2 - L3 - L4;
L5 = 4 - L6 - L7 - L8;
T1 = 0 - T2 - T3 - T4;
T5 = 0 - T6 - T7 - T8;
g2L1 = 4 - g2L2 - g2L3 - g2L4;
g2L5 = 4 - g2L6 - g2L7 - g2L8;
g2T1 = 0 - g2T2 - g2T3 - g2T4;
g2T5 = 0 - g2T6 - g2T7 - g2T8;
 Bengt O. Muthen posted on Friday, February 16, 2018 - 6:12 pm
I am not familiar with that way of identifying the model but there are certainly parameterizations with restriction on the intercepts such that factor means can be identified for all groups. Just say
[f1 f2];
in the MODEL: part of your input.
 Joe Wasserman posted on Friday, February 16, 2018 - 6:21 pm
Thank you for the feedback, Bengt. I tried your suggestion to include [f1 f2] in the model but for some reason it is still fixing the factor means to 0 in the first group.
 Bengt O. Muthen posted on Saturday, February 17, 2018 - 11:54 am
Put this statement in the first group.
 Georgios Sideridis posted on Wednesday, July 17, 2019 - 1:01 pm
I am looking at your output example: Table 8 9-24 Schmidt bayestowlevel random slide 71 model 200 draws biter=5000.out
and that output includes a section labeled:

What are the commands to obtain in the output these between level factor score comparisons? I use the following syntax which is identical to the above but with more items but am not getting the comparisons.
VARIABLE: NAMES ARE id Region xgrp q1-q10;
usevariables are q1-q10;
ANALYSIS: estimator=Bayes; type=twolevel random; process = 2;
biterations = (5000);
s1-s10 | fw BY q1-q10;

q1-q10 s1-s10;
[s1-s10*1] (p1-p10);
fb BY q1-q10*1 (p1-p10); !eqn 19 SMR paper
fpsi BY s1-s10*1 (p1-p10); !eqn 20
fb fpsi;

output: tech1 tech3 tech8;
 Bengt O. Muthen posted on Wednesday, July 17, 2019 - 5:23 pm
Add Fscomparison to the Output command and add

Type = Plot3;
Factors = All;
 Georgios Sideridis posted on Friday, July 19, 2019 - 4:10 am
Thank you, very much appreciated! In your 9.30 user's guide example (two level time series...) you have a between level factor score comparisons section. Is there any way we can get tests of significance on these as with latent means? Thank you.
 Bengt O. Muthen posted on Friday, July 19, 2019 - 1:48 pm
 Irene Dias posted on Friday, August 02, 2019 - 3:50 pm
Dear Dr Muthen. I am running a latent means comparison. I have 3 factors and 3 groups. I know how to compare the means of group1 versus groups 2 and 3, but not sure how to compare means of groups 2 and 3. I tried to use MODEL TEST to do it but without success. Could you please take a look at my syntax and tell me what is wrong?

F1 BY Q9* Q10 Q3 Q12 Q6;
F2 BY Q7* Q8 Q1;
F3 BY Q13* Q14;

Q10 WITH Q3 (1);

Model group1:


Model group2:

[F1] (A1);
[F2] (A2);
[F3] (A3);

Model group3:

[F1] (A3);
[F2] (A4);
[F3] (A5);

Model test:
 Bengt O. Muthen posted on Saturday, August 03, 2019 - 11:45 am
For Group 3 you should use the labels A4, A5, A6.

Then Model test for testing that all 3 factor means are equal across groups 2 and 3 is:

0 = A1 - A4;
0 = A2 - A5;
0 = A3 - A6;
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