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I am using Structural Equation Modeling (SEM) in R with the lavaan package. I would like to compare four groups of children regarding how strongly their processing speed (PS) and their working memory (WM) correlate with their reasoning skills (Gf).

For this, I need to compare the latent factor covariances (PS ~~ Gf, WM ~~Gf and WM ~~PS) between the four groups to test if they differ significantly. Before comparing the covariances, I tested for metric invariance to see if the groups are comparable. After establishing metric invariance, I was going to put equality constraints on the factor covariances and compare the resulting model with the model for metric invariance to see if setting the latent covariances to equality worsens model fit considerably.

But during the process of testing for metric invariance, I was puzzled: Depending on the way I get identification for my model - either by fixing each factor's variance to 1 (UVI) or by fixing the loading of one item per factor to 1 (ULI) - I either get metric invariance, or I don't.

I found some answers about this phenomenon here and here, both of which state that the choice of identification/scaling depends on the type of question one wants to answer. Both say that if one wants to compare factor loadings among groups, UVI makes more sense. But in my case, I'd like to compare latent factor covariances. I'd like to make statements such as "In group 1, the correlation between latent factor PS and Gf was significantly smaller than in group 2".

So my questions are...

  1. What scaling/identification method (fixing the factor variances vs. fixing one factor loading per factor) should I choose if I want to compare latent factor covariances, ideally their standardized version, among four groups?

  2. Is it correct that I have to establish metric invariance (equality of factor loadings across groups) or can I just compare a completely free multigroup model to one with cross-group equality constraints on the factor covariances?

Thank you so much!!!

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  1. To compare factor correlations, I would use identify the configural model by fixing factor variances to 1. Upon constraining loadings to equality, you need to free latent variances in all but 1 reference group. After establishing (at least partial) metric invariance, you can test whether latent variances are equal across groups by fixing them all to 1 again (comparing fit of latent-homoskedasticity model to fit of metric-invariance model). If you can't reject the $H_0$ of latent homoskedasticity, then you can proceed to directly compare latent covariances across groups (which will be correlations because all variances = 1). If not, then you can define "phantom constructs": a higher-order factor has variance = 1, freely estimated loading (interpreted as the lower-order factor's $SD$) in all but the reference group (leave loading fixed to 1 for identification), and fix all lower-order factor (co)variances to 0. Then the factor covariances are estimated only among higher-order "phantom" constructs, so they will be correlations.
  2. Yes, latent covariance-structure parameters are only comparable given (at least partial) metric invariance
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