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 Josue Almansa Ortiz posted on Tuesday, April 15, 2008 - 8:01 am
Dear Muthen,
I've a sample of 21 countries, but most of them with sample size less than 40. A CFA model is adjusted for a large set of ordinal items (21).

1. Because of the small sample size per country, total measurement invariance (by country) it is not possible to test, isn't it?

2. Added Country as covariate the M.I. do not show the necessity of direct effects, and factor loadings and variances remains almost equal. I think this can be a "partial measurement invariance" demonstration.
But, although factor loadings did not change, thresholds of some variables changed a lot if country is included as covariate. ¿This change in the thresholds when covariate country is included is meaningful in any sense? (can be an explanation for that? is a symptom of some measurement non-invariance?)
 Linda K. Muthen posted on Tuesday, April 15, 2008 - 10:23 am
I think your sample size is too small to test for measurement invariance across countries. Note that with a MIMIC model, measurement invariance can be tested for the thresholds only. This is done by looking at the direct effect of the covariates on the factor indicators. A significant direct effect signifies measurement non-invariance. It is not possible in a MIMIC model to assess measurement invariance of factor loadings.
 Josue Almansa Ortiz posted on Monday, April 21, 2008 - 10:49 am
Then, once adjusted a model with covariates (country in this case), without direct effects, if there is no significant M.I. for the direct effects (item-responses on the covariates), Is this sufficient to say that thresholds are equal across countries?

 Linda K. Muthen posted on Monday, April 21, 2008 - 11:06 am
Yes. But be sure you are getting the modification indices. You need to add the regression of the items responses on the covariates fixed to zero for the modification indices for these parameters to be printed, for example,

y1-y10 ON x1-x5@0;
 Josue Almansa Ortiz posted on Tuesday, April 22, 2008 - 7:40 am
I didn't know that.

Lots of thanks!!
 Dan Feaster posted on Tuesday, September 23, 2008 - 1:34 pm
I am trying to test invariant intercepts without simultaneously constraining all groups means to zero. My first approach was to compare a model with all 3 groups factor means set to zero and free intercepts to a model in which groups 2 & 3 factor means were freed and all intercepts equated across the 3 groups. However, this is not a nested test. I wanted to try the Marker-Variable Method of identifying means and variances across groups (Little, Slegers & Card, SEM, 2001). In this method you constrain a single loading per latent factor =1 AND constrain that items intercept=0. Unfortunately, I have been unable to override the Mplus default of setting the first group's factor means to zero. I have included: [factor mean name]; statements in the overall model and and the 2 subgroup models, but still get zero estimates for the first group's means. Am I missing something? Thanks, Dan
 Linda K. Muthen posted on Tuesday, September 23, 2008 - 1:58 pm
You are probably not specifying that the means are free in the first group. You need to do this to override the default.
 Bengt O. Muthen posted on Tuesday, September 23, 2008 - 4:55 pm
To get the first group's factor means free you should say

[factor mean name];

in the first group.
 James W Griffith posted on Saturday, January 16, 2010 - 2:25 pm
Dear Dr Muthen and Dr Muthen

We are interested in testing metric invariance across two groups, using a Rasch model. All our items are dichotomous. Have we set up our input correctly?

We saw in your postings and examples how to set up the Rasch model, but in various postings about group invariance, the slope of the first indicator is fixed at 1.0 and the factor variance is left to vary.

We constrained the item slopes to be equal and fixed the variance to 1.0 (in both groups), as follows:

TITLE: Rasch across two groups


dv BY item1* item2-item10 (is);

g item1-item10;

USEVARIABLES item1-item10;


CATEGORICAL ARE item1-item10;

GROUPING IS g (0=g1 1=g2);




 Bengt O. Muthen posted on Saturday, January 16, 2010 - 2:43 pm
With Rasch modeling you typically want to use Estimator = ML (so drop Parameterization=Theta). Also, you probably want to allow for possible factor variance differences across groups which you can accomplish by saying

Model g2:


so that in group 2 the dv factor variance can be freely estimated and different from 1.

You can also test for threshold (difficulty) invariance across the groups.
 Artemis Koukounari posted on Wednesday, February 23, 2011 - 9:02 am

Is it possible to assess if there is partial measurement invariance in an LCA using the KNOWNCLASS option? Or we do we do this by looking at the direct effect of the covariates on the item response probabilities?

In this case, do we need to add the regression of the items responses on the covariates fixed to zero?

Many thanks!
 Linda K. Muthen posted on Wednesday, February 23, 2011 - 12:28 pm
Because in LCA, the only measurement parameters are intercepts or thresholds of the latent class indicators, looking at the direct effects is sufficient. Multiple group analysis allows the invariance of other parameters to be considered but in LCA this is not necessary.

You can ask for MODINDICES (ALL).
 Artemis Koukounari posted on Thursday, February 24, 2011 - 8:22 am
Thanks so much for the explanation. I shall then look at the direct effects. When I tried MODINDICES (ALL) I got the following message: 'This is not available for TYPE=MIXTURE with more than 1 categorical latent variable. Request for MODINDICES is ingored'.
 Linda K. Muthen posted on Thursday, February 24, 2011 - 8:31 am
Then I would add one direct effect at a time where one latent class indicator is regressed on all covariates.
 Jon Gruda posted on Monday, March 21, 2016 - 1:33 pm

How would I test for measurement invariance not between groups (i.e., a categorical variable) but rather as a function of a continuous variable (and in fact, more precisely, as a function of two continuous variables and their interaction). In other words, I would not be comparing factor between groups but along a set of continuous variables and their interaction.

Is there any paper or examples that can point me in the right direction?

Thank you so much!
 Linda K. Muthen posted on Monday, March 21, 2016 - 1:39 pm
I don't know of any such paper. You should post this on a general discussion forum like SEMNET.
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