Scalar Invariance Testing
Message/Author
 Seth J. Schwartz posted on Friday, May 20, 2011 - 9:33 am
Dear Bengt and Linda:

I have been running metric invariance models (with continuous indicators and latent variables) in Mplus for years, but I'm having trouble with scalar (intercept) invariance.

From what I understand, in a multiple-group model, factor loadings are constrained like this:

f BY y1 y2 y3;
y1 (1);
y2 (2);
y3 (3);

And shouldn't intercepts be constrained like this:

[y1] (4);
[y2] (5);
[y3] (6);

However, that doesn't work. I get the SAME EXACT fit indices with intercept constraints imposed as I do without these constraints. And if I try it this way, I get an unidentifed model:

MODEL WHITE:
[y1];
[y2];
[y3];

MODEL BLACK:
[y1];
[y2];
[y3];

What am I doing wrong?

Thanks very much.

Seth
 Linda K. Muthen posted on Friday, May 20, 2011 - 10:58 am
Please see the inputs under multiple group analysis in the Topic 1 course handout. The default in Mplus is to constrain the intercepts and factor loadings across groups.
 Lisa M. Yarnell posted on Monday, November 14, 2011 - 5:55 pm
Hello,

I learned that when testing for scalar invariance, on top of constraining the intercept to be equal across groups, one should also set the means of errors to be zero in all groups.

Is this true, or is this your practice, Dr. Muthen?

Aren't means of errors ALWAYS zero?

Is this just some programming convention that SEM people do to make sure that the general rule that "means of errors are zero" does indeed hold in estimation?

Have you ever heard of this programming technique?

Thank you,
Lisa
 Lisa M. Yarnell posted on Monday, November 14, 2011 - 9:14 pm
As a follow-up question, Dr. Muthen: Is it even possible to set means of errors for measured variables to zero in Mplus? Would this be some sort of unusual modeling?
 Linda K. Muthen posted on Tuesday, November 15, 2011 - 9:30 am
Means of residuals are not parameters in the model. They cannot be fixed at zero. They are zero.
 Lisa M. Yarnell posted on Tuesday, November 15, 2011 - 10:09 am
Thank you! I agree.
 Sarah Herpertz posted on Saturday, November 29, 2014 - 7:48 pm
Hello,

I try to show scalar invariance across times (Pre-/Posttest) in one group.
I have two factors and 4 item-parcels per factor.

The configural and metric models are ok. Unfortunately there is a problem with the scalar invariance. The model fit seems ok, but the intercepts look really strange.

I fixed the first factor loading to one (the others are free). I fixed also the first intercepts of the fixed loading items to zero.
The problem is that the intercepts in the scalar invariance output are really different to the intercepts in the metric invariance output.

For example:

Y2 intercept (metric) – unstand./stand.: 55.043**/ 4.040**

Y2 intercept (scalar) – unstand./stand.: -52.910 (non-sign.)/ -4.032 (non-sign.)

Is that normal?

Thank you very much.
 Linda K. Muthen posted on Sunday, November 30, 2014 - 9:20 am
The standardized intercept is the unstandardized intercept divided by the standard deviation of the variable. Check the size of the standard deviation of the variable to see if this makes sense.
 Sarah Herpertz posted on Tuesday, December 02, 2014 - 10:55 am
Thank you very much.
 Joy Thompson posted on Sunday, March 20, 2016 - 12:53 pm
Hi, Dr. Muthen! When assessing indices during scalar invariance testing (for which intercepts are constrained), is it useful to attend to indices related to latent means? I guess the same question goes for other invariance tests - should one only attend to indices for imposed constraints, or is there reason that some constraints might affect other estimates?
 Linda K. Muthen posted on Sunday, March 20, 2016 - 12:57 pm
When intercepts are free, factor means must be fixed to zero.
 Jone Aliri posted on Tuesday, July 17, 2018 - 10:07 am
Hello,

I have an ESEM with target rotation (categorical items) and I want to test sex invariance. If I use the "configural metric scalar" syntax I have an error because it cannot calculate the metric model, but if I put only "configural scalar" I can calculate the fit of the configural and the scalar models and calculate the diftest.

Is it ok to do that?

Thank you very much.
 Bengt O. Muthen posted on Wednesday, July 18, 2018 - 6:55 am
I think that is sufficient.
 Tor Neilands posted on Friday, January 18, 2019 - 7:41 pm
I am working with a group assessing measurement invariance of a single-factor 13-item scale with items measured on 5-points. We are treating these items as ordinal and using the WLSMV estimator. Thanks to Mplus's invariance assessment features, the team can readily obtain chi-square statistics comparing configural to metric, configural to scalar, and metric to scalar models. The configural-metric comparison is NS (chi-square=17.52, p=.49), yet the metric-scalar comparison is significant at chi-square(38)=106.64, p<.0001. However, the various descriptive fit statistics are all highly similar across the three models, i.e., RMSEA=.06,.05.04; CFI=.993,.993,.991; SRMR=.024,.024,.026.

I am trying to figure out how to advise the team in weighing the chi-square test evidence, which suggests metric invariance is upheld but not scalar invariance, versus the descriptive fit information, which seems to suggest the three models are approximately equivalent. We have found one article by Cheung & Rensvold, 2002, in the SEM Journal, which address goodness of fit indices for invariance comparisons, but your opinions + suggestions for any additional literature touching on this topic would be much appreciated. Ditto for any alternative approaches available in Mplus we could also consider (e.g., alignment to assess approximate invariance if appropriate for this context) would also be appreciated.

Thank you,

Tor Neilands
 Bengt O. Muthen posted on Saturday, January 19, 2019 - 1:26 pm
With categorical outcomes, we recommend going straight from configural to scalar. Alignment is useful if you have several groups. See various alternatives on our web page:

http://www.statmodel.com/MeasurementInvariance.shtml