Multi-group modeling
Message/Author
 Daniel posted on Tuesday, April 06, 2004 - 9:22 am
I am making a multi-group comparison between two levels of sport participation. I found a significant effect for race, but there is a negative variance in my categorical dependent variable. I used theta parameterization and found that the negative variance is not significant. Continuing with theta parameterization, I used WLS to test for chi-square difference between the two groups. There was a significant difference. Now, do I report my findings from this analysis or from the analysis without theta parameterization? Since it is not possible to constrain the scales to zero, not using theta parameterization would mean that I am modeling with a negative variance. What is the correct way to finish my project?
 Linda K. Muthen posted on Tuesday, April 06, 2004 - 6:04 pm
You would adjust the lambda associated with the variable with the small negative residual variance in the Delta parameterization. You would fix the lambda at a smaller and smaller value until the negative residual variance disappears.
 Laura posted on Wednesday, April 07, 2004 - 9:45 am
I am conducting a multi-group path analysis and have a brief question regarding the output. In the section of the output entitled R-SQUARE, is the information reported under the subheading "R-Square" within each group the proportion of variance accounted for in each dependent variable by all other variables that have direct paths to that dependent variable? Thanks!
 bmuthen posted on Wednesday, April 07, 2004 - 10:20 am
Yes.
 Anonymous posted on Thursday, July 01, 2004 - 1:51 pm
I am running a path analysis on imputed data. The results from the full sample are fine, but when I ran the multiple group analysis, the program didn't seem to do anything. What did I do wrong?

INPUT INSTRUCTIONS

TITLE: path analysis for substance use
DATA: file is c:\impute.dat;
type=imputation;
VARIABLE: NAMES ARE v1-v48;
CATEGORICAL ARE v37 v39 v45;
USEVARIABLES ARE v36 v38 v44 v32 v42 v37 v39 v45 ;
grouping is v20 (1=male 2=female);
MODEL: v37 v39 v45 ON v32 v42;
v37 ON v36;
v39 ON v38;
v45 ON v44;
OUTPUT:;

path analysis for substance use
SUMMARY OF ANALYSIS

Number of groups 2
Average number of observations
Group MALE 1014
Group FEMALE 1021

Number of replications
Requested 10
Completed 0

In a related question, how do I get the chi-squares from the following imputed analysis results?

TESTS OF MODEL FIT
Number of Free Parameters 12
Chi-Square Test of Model Fit
Number of successful computations 10

Proportions Percentiles
Expected Observed Expected Observed
0.990 1.000 0.554 27.857
0.980 1.000 0.752 27.857
0.950 1.000 1.145 27.857
0.900 1.000 1.610 27.857
0.800 1.000 2.343 27.857
0.700 1.000 3.000 29.674
0.500 1.000 4.351 30.634
0.300 1.000 6.064 32.937
0.200 1.000 7.289 32.611
0.100 1.000 9.236 39.154
0.050 1.000 11.070 39.154
0.020 1.000 13.388 39.154
0.010 1.000 15.086 39.154

Thanks very much!
 Linda K. Muthen posted on Thursday, July 01, 2004 - 2:25 pm
You need to send output and data to support@statmodel.com so that I can see what is happening.
 Anonymous posted on Monday, October 04, 2004 - 2:36 pm
Dear Dr. Muthen,
I am doing a multiple group CFA with categorical factor indicators. The factor indicators include some dichotomous, 3 point and 5-point likert type items. In the Example 5.16 in the m-plus manual(p. 65), there's just an example on how to model dichotomous threshold and sacle factors. Would please give us more example on how to specify threshhold for ordinal factor indicators other than binary variables. Also, would you please show how to fix the related scale factors to one for identification purpose for multiple group CFA for ordinal factor indicators?
 Linda K. Muthen posted on Monday, October 04, 2004 - 5:25 pm
Let's say that u3 is measured on a 3-point scale, then Example 5.16 would be

[u3\$1 u3\$2];
{u3@1};

The number of thresholds is equal to the number of categories minus one. There is only one scale factor for an item no matter how many thresholds it has.
 Anonymous posted on Tuesday, October 05, 2004 - 10:23 am
Dear Dr. Muthen,
Thanks alot for your response. Besides having categorical factor indicators, I also have some background categroical variables which I use as predictors for some factors in structural part of my model. In the multiple group analysis, I wonder :
1)if I should specify threshholds and scale factors for them or not?
2) should I constrain their respected thresholds to be freely estimated across groups?
3) Finally, should I constrain their respected scale factors to be fixed@1 across groups?
 Linda K. Muthen posted on Tuesday, October 05, 2004 - 10:34 am
You do not need to anything to binary covariates. Do not include them on the CATEGORICAL list.
 Anonymous posted on Thursday, October 07, 2004 - 9:23 am
Dear Dr. Muthen,

In the multiple CFA analysis, I specified the thresholds for categrocial observed indicators in the group specific model, say blacks, to be freely estimated. I followed the recommendation that the number of thresholds should be equal to the numbert of categories minus 1. Also, for identification purpose, I constrained the the scale factors for the categorical indicators whose factor loadings freely estimated to be 1. In the output, however, the model was NOT identified. Could you please exlain what might has been wrong?

p.s. Identification problem was solved when I removed the the thresholds specifications from the group specific model (let the thresholds to be equal across groups).
 Linda K. Muthen posted on Tuesday, October 12, 2004 - 5:25 pm
If the thresholds are free, the factor means must be zero in all groups.
 Anonymous posted on Thursday, October 14, 2004 - 8:23 am
Dear Dr. Muthen,
The factor means are fixed at 0. Still the model is not identified. However, when I fix all scale factors across groups to be constrained at 1, the model is identified. Am I doing the right thing? Could you please let me know what are the implications of fixing sacle factors at 1?
 Linda K. Muthen posted on Thursday, October 14, 2004 - 10:30 am
When thresholds and factor loadings are free, factor means must be fixed to zero in all groups and scale factors must be fixed to one in all groups. These are required for identification purposes. See Web Note 4 for further information.
 Anonymous posted on Friday, October 15, 2004 - 9:43 am
Dear Linda,

Thanks alot for your time and patience!
 Linda K. Muthen posted on Friday, October 15, 2004 - 10:21 am
For the invariant item, the thresholds are also held invariant, and the scale factor for that item is one in the first group and free to be estimated in the other groups.

For the non-invariant item, the thresholds are non-invariant as well as the loading and the scale factor for that item is fixed to one in all groups. See Example 5.16 of the Mplus User's Guide for an illustration of this.
 Anonymous posted on Friday, November 05, 2004 - 8:13 am
Dear Linda,
Thanks alot for your response.Just to follow up: do you mean to test "factor loading" invariance, in the baseline model, factor laodings and thresholds are freely estimated, while in the nested model, both factor loadings and thresholds are held invariant? Does the chi-squared diff test show the invariance of factor laodings or both the factor loadings and thresholds? I'm only interested in "factor loading invariance".
 LMuthen posted on Friday, November 05, 2004 - 8:38 am
Factor loadings and thresholds need to be considered together with categorical outcomes.
 Anonymous posted on Friday, November 05, 2004 - 10:56 am
Thanks alot. In the nested model, where both factor loadings and thresholds are held invariant, should the the scale factors be freely estimated?
 LMuthen posted on Friday, November 05, 2004 - 2:22 pm
When the factor loadings and threhsolds are held equal, the scale factors should be fixed to one in the first group and be free in the others. Factor means should be zero in the first group and free in the others.
 Anonymous posted on Tuesday, November 09, 2004 - 2:11 pm
Thanks for your response. Inthis case, can I use WLSM for chi square difference test?
 bmuthen posted on Sunday, November 14, 2004 - 12:50 pm
Yes.
 Jeremy Miles posted on Sunday, November 21, 2004 - 8:30 am
Hello,

I am doing a multiple group, comparing the factor structure and factor means of two groups. When I constrain the means of the factors of the two groups to be equal, I get an error:

THE STANDARD ERRORS OF THE MODEL PARAMETER ESTIMATES COULD NOT BE
COMPUTED. THE MODEL MAY NOT BE IDENTIFIED. CHECK YOUR MODEL.
PROBLEM INVOLVING PARAMETER 26.

(Parameter 26 is the mean of the factors, in both groups).

However, when the means are estimated separately, the mean of the first group is 0, and the second is estimated.

So, here's my question:

Is it legitimate to fix the mean of the second group to 0? And then doing the nested model test?

Thanks,
 Linda K. Muthen posted on Sunday, November 21, 2004 - 9:24 am
Yes, this is how equality of the factor means is specified. Because they must be zero in the first class (or one class) for identification purposes, holding them equal is the same thing as fixing them to zero.
 Anonymous posted on Tuesday, November 30, 2004 - 3:04 pm
Dear Dr. Muthen,

When examining partial invariance, I free the factor loading and the threshold in the nested model. My question is should I still fix the factor means in both groups at zero while I free the parameters in the nested model?
 Linda K. Muthen posted on Tuesday, November 30, 2004 - 5:23 pm
See Examples 5.16 and 5.17. They show partial measurement invariance for the Delta and Theta parameterizations. For Delta, you must also fix the scale factor to one in all groups. You do not have to fix the factor means to zero.
 Anonymous posted on Friday, December 03, 2004 - 10:02 am
Dear Dr. Muthen,
Follow up: In this case, in the baseline model factor means were set to zero in both groups for identification purpose, while in the nested model, based on your previous comment, factor means do not have to be set zero. My question: Are these two models, then, nested within each other? In other words, can we consider a nested model whose factor means are freely estimated in one group to be nested within a basline model in which both group factor means were set to zero?
 Linda K. Muthen posted on Friday, December 03, 2004 - 4:13 pm
Yes.
 Anonymous posted on Friday, December 03, 2004 - 9:53 pm
Dear Dr.Muthen,

Thanks a lot for your response. I have summarized the impression I have got so far from the previous discussion. Please correct my understandings if they are not right.
In order to do the multiple group CFA to examine the metric invariance and/or partial metric invariance we need go through 3 steps:
1-Form a baseline model in which factor loadings and thresholds are freely estimated, except that the first factor laoding is fixed at 1 for identification and respected thresholds are held invariant.For identification purpose, factor means and scale factors set to zero and one, respecively, for the items whose factor loadinsg and thresholds are freely estimated.
2-To check the metric invariance, in the nested model, all factor loadings and thresholds are held invariant across groups. In this model, factor means and scale factors for the second and thirds etc group are freely estimated across groups.
3-If full metric invariance in chi square diff test failed, then we step by step free a factor loading and respected threshold. In doing so, we fix the respected scale factor at one and and also fix the respected "factor means" to zero in both groups in the nested model. Here, I should not fix ALL factor means to zero in the nested model and only the ones whose related factor loadings are freed. Is my impression right?

Thanks a lot again!
 Linda K. Muthen posted on Saturday, December 04, 2004 - 6:53 am
Regarding 3, it is not necessary to fix the factor means at zero when you free the factor loading and threshold and fix the scale factor to one as long as the model is identified. See Example 5.16 in the Mplus User's Guide where this is illustrated.
 Anonymous posted on Tuesday, December 07, 2004 - 11:39 am
Dear Dr. Muthen,

I realize that my last question was a bit ambiguous. To clarify: In our baseline model(A), the factor means were set to zero in both groups for identification purposes, while in the constrained model (B), factor means were set to zero for only the first group (although factor loadings and intercepts were constrained to be equal across groups). My question: Under these circumstances, is the constrained model (B) still considered to be nested within the baseline model (A)? More specifically, can we consider a constrained model (B) whose factor means are freely estimated in one group to be nested within a baseline model (A) in which both group factor means were set to zero? The two models are described below.

Model A ( "baseline" model): A model in which factor loadings and thresholds are freely estimated across both groups. Factor means are fixed at zero and scale factors are fixed at one in both groups.
Model B ("constrained" model): A model in which factor loadings and thresholds are held invariant in both groups. Factor means and scale factor are freely estimated in the second group.

Thanks very much. We just want to make sure that we can use a chi-square difference test to test for differences between out two models.
 Linda K. Muthen posted on Tuesday, December 07, 2004 - 2:16 pm
Yes, I would consider these models to be nested.
 millsap posted on Sunday, April 17, 2005 - 8:30 pm
Can Mplus test for invariance of logistic
regression weights in a multiple group
context? It does not appear to be so from
the manual that I have, but my manual
is out of date. Thanks.
 bmuthen posted on Tuesday, April 19, 2005 - 12:21 pm
You can do this in the type=mixture context using the Version 3 KNOWNCLASS option. This treats group as a latent class variable with known (observed) class membership. If you only have earlier versions, you can do the same using the "training data" option.
 Sven D. Klingemann posted on Wednesday, May 02, 2007 - 2:04 pm
Hi Dr. Muthen,
just a quick question with respect to partial measurement invariance: If I am interested in comparing group means and only partial measurement invariance is achieved would you advise to drop these items all together? In my case, some of the items have select cross-loadings that are not invariant, while others are.
 Linda K. Muthen posted on Wednesday, May 02, 2007 - 6:24 pm
I would only drop the items if they don't have strong loadings. Otherwise, I would model the invariance.
 Sven D. Klingemann posted on Thursday, May 03, 2007 - 6:51 am
Thanks Linda.
Doesn't modeling the invariance still affect the mean comparison? I am a bit confused here because I thought that the whole idea of testing for invariance of factor loadings and thresholds was to eventually insure that the differences in means would not be affected by group-specific attributes that are unrelated to the construct(s).
Sven
 Linda K. Muthen posted on Thursday, May 03, 2007 - 8:29 am
If you ignore the measurement non-invariance, it is a problem. But if you allow for it in the model, it should be approximately the same as deleting the item. By modeling it, I mean including a direct effect in a MIMIC model or allowing an intercept or factor loading to vary across groups in a multiple group model.
 Sven D. Klingemann posted on Thursday, May 03, 2007 - 8:31 am
Thank you!!
S.
 Leanne Whiteside-Mansell posted on Tuesday, May 08, 2007 - 3:06 pm
I am examining a CFA with 2 latent factors (6 and 7 likert 1-5 indicators respectively). The constrained model fits poorly compared to the model with parameters freed (some snytax below.

Quesitons:
1) do I have the syntax correct to test these two extremes?
2) the Mod Index suggests that the poor fit is related to [b1p103b]. what syntax would free just this parameter? Assuming this improves the fit, what would this mean regarding the invariance across the groups?

grouping is racelang (
11 = WhiteE
21 = BlackE
31 = Hispanic/English
32 = Hispanic/Spanish
);
ANALYSIS:
TYPE=MISSING h1;
MODEL:
psi14_11 BY b1p103f ...b1p103l;
psi14_12 BY b1p103c ....b1p103j;

PROGRAM to Constrain:

MODEL:
psi14_11 BY b1p103f ... b1p103l;
psi14_12 BY b1p103c...b1p103j;
[psi14_11@0];
[psi14_12@0];
MODEL BlackE:
psi14_11 BY b1p103h ... b1p103l;
psi14_12 BY b1p103b ... b1p103j;
[b1p103f\$1];
{b1p103f@1};
[b1p103h\$1];
{b1p103h@1};
[b1p103i\$1];
{b1p103i@1};
... etc
 Linda K. Muthen posted on Tuesday, May 08, 2007 - 3:20 pm
It looks generally correct but I would have to see the full input to say. Why don't you try it and if you have problems, send the input, data, output, and your license number to support@statmodel.com.
 Rohini Sen posted on Monday, November 15, 2010 - 8:00 am
Dear Dr.Muthen,

I am trying to run a multi-group (male/female)Exploratory SEM on my data set that has 3 factors and 27 items. I'm basically following your example 5.27 verbatim & have placed parameter constraints according to the example mentioned but get the following error:

*** FATAL ERROR
IMPROPER PARAMETER CONSTRAINT FOR EFA MEASUREMENT SPECIFICATION.
(Error Code: 1021)

My input was:

TITLE: Multiple-group EFA where eq. of
factor variances & factor covariances is
imposed in addition to meas.inv. of incpts
DATA: FILE IS creativity3.dat;
VARIABLE:
NAMES ARE y1-y27 sex program phase yrinprog inter fgoal;
USEVARIABLES ARE y1-y27 sex;
MISSING ARE sex(99);
GROUPING IS sex (1 = male 2 = female);
MODEL: f1-f3 BY y1-y27 (*1);!f1-f3 are EFA factors which have the same factor indicators.
f1 with f2(1);
f2 with f3(1);
f1 with f3(1);
f1-f3@1;
[f1-f3@0];

OUTPUT: TECH1 SAMPSTAT;

I'm unable to determine what the problem is here - any directions or thoughts on this is much appreciated.

Thank you!
 Linda K. Muthen posted on Monday, November 15, 2010 - 10:47 am
I think the message is because of the equalities for the factor covariances.
 Rohini Sen posted on Wednesday, December 01, 2010 - 11:00 am
Hi Linda,

Thanks - I removed the equalities of factor covariances but I get this new error .

*** FATAL ERROR
IMPROPER PARAMETER CONSTRAINT FOR EFA MEASUREMENT SPECIFICATION.
(Error Code: 1001)

Thanks,

Rohini
 Linda K. Muthen posted on Wednesday, December 01, 2010 - 11:37 am
 Selahadin Ibrahim posted on Monday, November 19, 2012 - 10:17 am
Hi Linda,
in multi-group analysis should parameter estimates and chi-square contributions for each group be exactly equal when estimated separately and when estimated in multi-group framework with all parameters unconstrained ?

thanks,
 Linda K. Muthen posted on Monday, November 19, 2012 - 1:44 pm
Yes, the results of a totally unconstrained multiple group analysis will be the same as analyzing the two groups separately.
 Selahadin Ibrahim posted on Wednesday, November 21, 2012 - 10:19 am
thanks Linda.
does it matter if several of the mediators are categorical (dichotomous variables)?

 Bengt O. Muthen posted on Wednesday, November 21, 2012 - 4:13 pm
No.
 Yuyu Fan posted on Monday, March 24, 2014 - 8:43 pm
Dear Muthen,

I am testing the measurement invariance of the covariance structure for a common factor model with 5 binary indicators using the delta parameterization (we are not interested in the mean structure) across two groups. may I ask two questions?
The first question is whether threshold can be freely estimated if I only care the covariance structure, as thresholds are analogous to intercepts for CFA with continuous indicators?
The second questions is on the procedures of testing measurement invariance. As the two steps pointed out in the user's guide for measurement invariance of the categorical variables: from the least restrictive to the most restrictive models, the scale factors are first constrained to be equal then freed in one group, then there must be equivalent models when we equalize factor loadings and free scale factors. Specifically, I first constrain the scale factor as 1 for both groups, and tested the configural invarinace with free factor loadings;then, I constrain the factor loading to be equal, and fix scale factors to be 1 in the reference group and free the scale factors of the other group. The two models are actually equivalent models with identical model fit indices. I am confused about the procedures.
In addition, if the residual variances are correlated for some of the indicators, should I just constrain them to be equal in the last step?
Thank you very much!
 Linda K. Muthen posted on Tuesday, March 25, 2014 - 10:10 am
Please see pages 4-11 of the Version 7.1 Mplus Language Addendum which is on the website with the user's guide. The models for testing for measurement invariance are described in more detail there. New options for automatically testing for measurement invariance are also described.