Help with SEM interpretation (multigroup and measurement invariance)

[Emma Corley]

Hello everyone, 

I'm really stuck trying to make sense of my model and I would appreciate even the smallest bit of help! 

I'm running a multi-group moderated mediation model with education as a grouping variable (variables are as follows): 

Y= Latent variable of IQ

X= Genes 

M= Brain Vol

Z= Latent Environmental Variable

Across the two educational groups these were the relationships observed:  

Question 1: Interpreting multi-group SEM results

For differences in the individual model paths across groups I compared these using a Wald Chi-Squared? Is that the correct way to do so?  For this, the relationship between Brain Vol and IQ (x2 (1)= 9.604, p= 0.0019) and the relationship between environment and Brain vol (x2 (1)= 7.967, p= 0.0048) were both significant. Would I be right in saying then that for those that haven't been to college/uni, the association between brain vol and IQ and between environment and brain vol is stronger?

Question 2: Measurement Invariance

I've here and on other SEM papers about testing for measurement invariance. For this, I fitted the model again with group.equal= "loadings" and again with group.equal = c("intercepts", "loadings")). From this I then used the function lavTestLRT(multi.sem.group, multi.sem.group1, multi.sem.group2) to access measurement invariance. Is this the correct way to do so? 

These were the results 

I'm guessing based on these results that invariance did not hold up across groups? But I also looked at differences in SRMR and RMSEA (ΔSRMR = .005; ΔRMSEA = .008) and (ΔCFI = 5421.4.; ΔSRMR = .005; ΔRMSEA = .008) respectively. These appear to hold up across the groups. My sample size is very large +20,000, so would CFI differences even hold with such a large sample size?

If you've made it this far, thank you so much for reading! Honestly, any bit of help with this would really really be appreciated! 

Thanks again, 

Emma

[Terrence Jorgensen]

For differences in the individual model paths across groups I compared these using a Wald Chi-Squared? Is that the correct way to do so? 

That is an option.  But it is not scale-invariant, so you will get different results depending how you identify your common factors (IQ, Z).  The LRT does not suffer that limitation, so you might want to fit models that represent your null hypotheses and compare to the unrestricted model.

Would I be right in saying then that for those that haven't been to college/uni, the association between brain vol and IQ and between environment and brain vol is stronger?

That could depend on how you scaled IQ (reference indicator's loading = 1 vs. fixing residual variance to 1).  Slopes are only comparable across groups if you impose measurement-invariance across groups, in which case your interpretation looks correct.

 

Is this the correct way to do so? 

You should only test scalar invariance once you have a defensible (partial) metric-invariance model.

invariance did not hold up across groups?

Doesn't look like it.  

My sample size is very large +20,000, so would CFI differences even hold with such a large sample size?

Large samples are the ones for which delta-CFI has no power.  Only in small samples does it have inflated Type I error rates.

https://doi.org/10.1037/met0000152

Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam
http://www.uva.nl/profile/t.d.jorgensen