Interpretation of covariance in CFA vs regression in SEM

Question

I have conducted a confirmatory factor analysis in lavaan (in the context of a group comparison). Moving on to structural equation modelling I realised that my hypothesized structural model is identical to the measurement model, i.e. it has the same (good) fit indices and apart from the structural aspects (see output below) all estimated parameters are identical. At the time I interpreted this to mean that my model or perhaps any structural model without any mediating relationships(?) cannot really be verified beyond the fit of the measurement model. So I decided to stick with just CFA.

The CFA group invariance tests revealed some potentially interesting differences in the significance of latent covariances between the two groups. (As the data does not support invariant factor variances I cannot meaningfully compare the covariance structure according to Brown (2015) so I thought the significance levels might give me a grasp to better interpret some of the group differences)

Out of curiosity I also conducted the same group comparison with the SEM model. Here not just the strength of the relationships, but especially the p values are different. I am now wondering, how I can meaningfully interpret my results. What are the respective meanings of regression and covariance strength as well as significance?

below are the estimated structural relationships in one of the groups (all other aspects of the output are identical for SEM and CFA)

CFA model:

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  nat ~~                                                                
    pat               0.153    0.034    4.545    0.000    0.479    0.479
    ausl              0.254    0.041    6.238    0.000    0.518    0.518
  pat ~~                                                                
    ausl             -0.059    0.040   -1.493    0.135   -0.122   -0.12

SEM model (same model with paths between "nat" and "ausl" and "pat" and "ausl"):

Regressions:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  ausl ~                                                                
    nat               1.140    0.195    5.860    0.000    0.748    0.748
    pat              -0.738    0.186   -3.978    0.000   -0.480   -0.480

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  nat ~~                                                                
    pat               0.153    0.034    4.545    0.000    0.479    0.479

Brown, T. A. (2015). Confirmatory factor analysis for applied research, Second edition, New York ; London: The Guilford Press.

Answer 1

The CFA and SEM are statistically equivalent because you aren't placing any restrictions on the structural model. It is just reparameterizing the 3 covariances as regression slopes, which do have different interpretations despite reproducing the observed covariance matrix equally well. The principal is even more apparent if you used composites to represent the constructs, because you would have 2 observed variables predicting a 3rd observed variable, and that model would have df=0.

I also conducted the same group comparison with the SEM model. Here not just the strength of the relationships, but especially the p values are different.

Compare parameters, not p values. You can only compare regression slopes (or correlations) across groups if metric invariance holds (i.e., equal factor loadings across groups). If that holds, then you could test the $H_0$ of an equivalent slope across groups by setting them equal with identical labels and conducting a LRT.

https://lavaan.ugent.be/tutorial/groups.html

Answer 2

yes this is a common issue. Technically speaking the models are equivalent. In these cases you really need strong theory and/or empirical methods for parameter identification that would allow differentiating them. The interesting thing about equivalent models is that they abound, and you can understand any given model as being merely one of a possibly infinite set. For insights see:

Raykov, T., & Marcoulides, G. A. (2001). Can there be infinitely many models equivalent to a given covariance structure model?. Structural Equation Modeling, 8(1), 142-149.