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Mplus Discussion > Structural Equation Modeling >
 Student posted on Thursday, February 26, 2015 - 5:44 pm
Hello, I had a conceptual question about testing for measurement and structural invariance. I am interested in examining gender differences in a mediation model. First, I fit the mediation model separately for males and females. Then I tested for configural, metric, scalar, and residual variance invariance.

First question: Do you suggest that fitting the model separately for males and females is necessary?
Second question: Upon reading through the threads, as well as other papers, it seems that testing for residual variance invariance is not entirely necessary and may be overly stringent-- would you recommend to skip this?

After metric invariance is established, I understand that I need to test for structural invariance. I am a bit fuzzy on which steps are absolutely "necessary." Specifically, do you recommend that, similar to establishing measurement invariance, I first compare variances of the factors, then next the covariances of the factors, and lastly regression coefficients of the factors? Or is it possible to just test the regression coefficients only? I am asking because I have read a few papers that do this, and also I am primarily interested in whether there are gender differences in the 3 paths (DV->mediator, DV->IV, mediator->IV).

Thank you for your help.
 Linda K. Muthen posted on Thursday, February 26, 2015 - 6:03 pm
If the same model does not fit well in each group, testing for measurement invariance is not warranted.

Some disciplines do test for the invariance of residual variances. Some do not. This would need to be your choice.

Measurement invariance is necessary if you want the compare structural parameters of factor means, variances, covariances, and regression coefficients. Which of these parameters you compare is your choice.
 Student posted on Thursday, February 26, 2015 - 6:16 pm
Thank you for your response, Dr. Muthen.

Just to make sure, once measurement invariance is established, you can just compare the path coefficients across gender, and do NOT need to initially test for means, variances, and covariances beforehand-- is that a correct understanding?
 Linda K. Muthen posted on Friday, February 27, 2015 - 6:07 am
 Student posted on Friday, February 27, 2015 - 6:23 am
Thank you!
 Student posted on Thursday, April 09, 2015 - 4:19 pm
I had another conceptual question about measurement invariance. If you want to test for measurement invariance across gender for a mediation model (X->M->Y), would you:
A) test for measurement invariance for each latent variable (i.e., X, M, Y) separately in different CFA models, or
B) test measurement invariance in just one model where you have CFAs for all latent variable together?
 Bengt O. Muthen posted on Friday, April 10, 2015 - 6:58 am
There are pros and cons for the 2 approaches. I would lean towards B where the structural part of the model is held saturated, so all paths are in (including X-> Y).
 Student posted on Friday, April 10, 2015 - 7:25 am
That was my inclination as well-- so my follow-up question is whether my only option would be to test each (configural, metric, scalar) model separately, as the "MODEL = CONFIGURAL METRIC SCALAR;" option does not allow "ON" statements.

Another question was whether and why you recommend including covariates when you test for measurement and structural invariance. I've seen arguments both for and against, but without a clear explanation as to why that approach is preferred.

Thank you.
 Bengt O. Muthen posted on Friday, April 10, 2015 - 8:08 am
Q1. You can use WITH among all your latents to make the structural part saturated.

Q2. I would keep it simple and not use covariates, but that's a long story perhaps better suited for SEMNET discussion.
 Student posted on Friday, April 10, 2015 - 8:10 am
Thank you, Dr. Muthen!
 Student posted on Saturday, April 11, 2015 - 10:19 pm
Hi Drs. Muthen,

Following up from above, what do you recommend if you wanted to test for measurement invariance in a mediation model (X->M->Y) where the mediator M is an observed continuous variable? Would you just use WITH between the X and Y latent variables, and leave M (and thus, the X->M and M->Y paths) out of the model?
 Bengt O. Muthen posted on Sunday, April 12, 2015 - 5:05 pm
I would make sure that the X, M,Y covariance matrix is just-identified so off-diagonals are represented by 3*2/2 = 3 parameters, however you want to accomplish that, including all paths or all WITHs.
 Student posted on Sunday, April 12, 2015 - 7:44 pm
I still have a conceptual question about measurement invariance testing, in the context of SEM. The Mplus topic handout, as well as the majority of other resources on measurement invariance tend to focus on CFAs and not necessarily other SEMs (e.g., mediation models). Since measurement invariance just looks at the measurement model, I believe that in a simple mediation model (X->M->Y), it would be fine to:
1) fit separate CFA models for the latent variables (only for X and Y, if M is an observed variable) for group A and group B-- meaning for each group, there is one CFA model for X, a separate CFA model for Y, etc.
2) establish measurement invariance for each latent variable separately
3) fit mediation model for group A and group B
4) test structural invariance (e.g., paths).

Is there a problem with this approach that I'm not understanding? I'd appreciate any input, given that there are very limited resources clarifying how exactly to go about invariance testing beyond just CFA models.
 Bengt O. Muthen posted on Monday, April 13, 2015 - 7:35 am
I think that is a reasonable approach.
 Student posted on Monday, April 13, 2015 - 8:58 am
Thank you, Dr. Muthen.
 Student posted on Tuesday, April 14, 2015 - 7:12 am
After establishing measurement invariance, I am interested in comparing paths across groups and had a few questions:

1) should factor loadings be constrained equal across groups? If so, what if only partial scalar (or metric) invariance was met? Do you recommend that factor loadings be constrained equal anyway when testing for differences in paths or to free the loadings that were not invariant?
2) should factor means be constrained to 0 across groups?
3) Do you recommend first comparing the baseline model (all paths free) to the most restricted model (all paths constrained)? Or to compare the baseline model to separate restricted models (different paths constrained in different models)? I've seen both methods and wondered which may be the preferred approach.

Thank you.
 Bengt O. Muthen posted on Tuesday, April 14, 2015 - 8:23 am
I would try SEMNET for this kind of general analysis strategy questions.
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