Interpreting multigroup SEM constraining regression coefficients

[Dana Linnell Wanzer]

Hi all,

I am running a multi-group SEM constraining the unstandardized loadings, lv.variances, and regressions to be equal across groups. However, I am having a hard time wrapping my head around how to interpret the standardized (std.all) coefficients between my groups given that the unstandardized values, z-scores, p-values, and std.lv are the same across groups but the std.all is different across groups. I am most interested in comparing the std.all across groups but beyond saying that one group's std.all is slightly larger than another (e.g., group 1 = .202, group 2 = .141, p < .001) what am I missing? Conceptually, I understand why the two have the same p-values but is there a better method for comparing the groups' standardized regression coefficients if I assume structural variance across the groups? 

-Dana

[Terrence Jorgensen]

given that the unstandardized values, z-scores, p-values, and std.lv are the same across groups but the std.all is different across groups

If all parameters are equal across groups, then the standardized values should be the same too. Can you show your summary() output?  If you left residual variances different across groups (common practice since strict invariance is not necessary to compare latent distributions across groups), then that would account for differences in loadings, even if factor variances are equal.  But standardized latent slopes should be equal if unstandardized latent variances and slopes are equal.

Terrence D. Jorgensen

Postdoctoral Researcher, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

UvA web page: http://www.uva.nl/profile/t.d.jorgensen

[Dana Linnell Wanzer]

​Yes, I didn't worry about strict invariance, so I only have the ​loadings, lv.variances, and regressions constrained. I put the summary() output side-by-side in an Excel spreadsheet, which you can find here: https://www.dropbox.com/s/58yc2rk769gzwn9/summary%28%29%20output%20for%20multigroup%20SEM.xlsx?dl=0

My main reason for doing the multigroup SEM is to see whether the regression pathways are similar for the two groups; if they're not similar, then I want to know which pathways are stronger for whom. So I look at the regressions and everything but the std.all is the same (which sounds like it makes sense since I did not worry about strict invariance) but then the only ideas I have for comparing the groups is just talking about the magnitude of the difference between the two groups (e.g., FullGPA ~ GrwthMn is .141 for Group1 and .202 for Group2). 

However, I've seen another paper that was unfortunately not clear in their methods that showed differences in statistical significance across the regression pathways in their multigroup SEM. I am wondering if I am not supposed to interpret the results constraining regressions but rather to go back to my weak invariance model (only loadings constrained) and compare the regression pathways there. 

Best,

Dana Linnell Wanzer, M.A.

Doctoral Student | Evaluation and Applied Research Methods

Claremont Graduate University | School of Social Sciences, Policy, and Evaluation

danal...@gmail.com | (760) 835-3604

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[Terrence Jorgensen]

I didn't worry about strict invariance, so I only have the ​loadings, lv.variances, and regressions constrained.

Then that is why the std.all solutions differ.  If the  residual variances differ, than so do the total variances.  Thus the explained variance implied by the equal slopes translates to an unequal proportion of the total variance of the outcome.

I've seen another paper that was unfortunately not clear in their methods that showed differences in statistical significance across the regression pathways in their multigroup SEM. I am wondering if I am not supposed to interpret the results constraining regressions but rather to go back to my weak invariance model (only loadings constrained) and compare the regression pathways there. 

Testing their difference statistically is a matter of model comparison, just like testing weak invariance.  You compare the fit of a model with the slopes constrained across group to a model without those equality constraints (but other constraints held in place: loadings and latent variances), using the anova() function. 

[Dana Linnell Wanzer]

Okay, I understand now why the std.all would be different since I do not constrain the residual variances.

For the second point, I did do model comparison, find the model chi-square difference to be statistically significant (as well as large enough decreases in CFI/RMSEA), but I'm trying to figure out the best solution for determining where those differences are. As I mentioned, I found one paper that did multi-group comparisons and then showed the structural (regression) pathway differences for the different groups, but there are large differences and the statistical significance changes across groups. I cannot have that be the case since I am constraining loadings and lv.variances to be the same, so the z-scores and p-values are the same regardless of the std.all. 

Essentially, I am showing that there is a decrease in model fit from weak invariance to structural invariance through model fit, but now I want to show that the pathways are different. Do I just compare the std.all for the structural invariance model and just discuss in terms of magnitude of the difference between groups? 

Best,

Dana Linnell Wanzer, M.A.

Doctoral Student | Evaluation and Applied Research Methods

Claremont Graduate University | School of Social Sciences, Policy, and Evaluation

danal...@gmail.com | (760) 835-3604

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[Terrence Jorgensen]

I am showing that there is a decrease in model fit from weak invariance to structural invariance through model fit, but now I want to show that the pathways are different. Do I just compare the std.all for the structural invariance model and just discuss in terms of magnitude of the difference between groups? 

If the constrained model does not fit as well, then it is not valid to compare those (standardized) estimates anyway.  Your statistical test already shows that structural invariance does not hold, so you can compare estimates in your weak-invariance model to describe why it does not hold.  

You can also consider the weak-versus-structural invariance test as an omnibus test, then follow it up with tests of individual parameters.  If you label the regressions (differently in each group) in the weak-invariance model, you can specify each each constraint and test them one at a time using lavTestWald() (see the ?lavTestWald help page). 

[Dana Linnell Wanzer]

Thank you!! I had a feeling I was supposed to be looking at the Weak invariance model, but despite hours searching the literature multigroup analysis discussions are very vague on these procedures. The lavTestWald() looks very interesting and I will check it out. Thank you for your help, Terrence!

Best,

Dana Linnell Wanzer, M.A.

Doctoral Student | Evaluation and Applied Research Methods

Claremont Graduate University | School of Social Sciences, Policy, and Evaluation

danal...@gmail.com | (760) 835-3604