# Interpretation of covariance in CFA vs regression in SEM

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.

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.