Meletios Niros I'm assuming by "SEM causal model" you mean a model in which either your 4 dimensions/factors or the 2nd order factor predict one or more outcome variables, whereas in your CFA, you analyzed only the 4 dimensions/1st order factors with or without the second-order factor.

If this is correct, then the reason for the inferior fit of the SEM (with outcomes) compared to the CFA with 2nd order factor (without outcomes) is likely that your 1st order factors may be differentially related to the outcome(s) so that a model in which you use the 2nd order factor to predict the outcome(s) omits statistically significant (residual) paths from the first-order factors to the outcome(s). You could examine this pretty easily by looking at whether the first-order factors have different correlations with the outcome(s).

My question would be: What do you need the 2nd order factor for? Can you drop it altogether in both the CFA and SEM? From my perspective (shaped by measurement theory), second-order factors are often problematic anyway because they have no clear interpretation. In addition, the assumption of unidimensionality of the first-order factors implied by a second-order factor model is often unrealistic and also unexpected from a measurement perspective unless the first-order factors represent interchangeable facets of a construct. I personally would prefer a model with only first-order factors unless I had a very good reason to use a second-order factor.

Dear Christian,

Thank you for your insights. 2nd order utilized in order to receive dimensions suggested by authors that researched the original subscales. This is something that is usually done according to the papers I have read. The issue is: is it better to use the manipulated 2nd order single construct in the model OR the imputed dimensions? The model fit in the second case is well above thresholds!

As for the original CFA (before 2nd order process) I had 3 dimensions instead of 4 and 2 dimensions were merged in one.

Thank you in advance for any additional comments!

Hi all,

a second order factor (SOF) model IS a causal model. If your assumption is that the core latent variable is the thing you refer to with the SOF and you believe that this thing is not only the relevant causal entity in causing its "primary factors" as well as other external variables, than you have to somehow manage it to measure this thing--either by specifying the whole SOF structure (which solely serves identifying the SOF) or by using/generating reflective indicators.

Of course, it is possible that there are some hybrid structures where a DV is caused by the SOF as well as one of its primary factors but it is a contradiction to first test a SOF structure and then to do a regression with the primary factors as regressors.

HTH

Holger

Dear Holger Steinmetz thank you very much for your valuable insights! Have a nice day!

Model fit will always be lower for the SEM model than for the 2nd order CFA model, which will be lower than for the 1st order CFA model. So this is normal. However, this lower SEM fit must be as close as possible to the fit of your CFA (and above recommended thresholds), showing that the additional restrictions of the structural model only minimally lower model fit. For this see Hair et al., 2019, Multivariate data analysis, 8th ed. (p.710). You can also compare the SEM model with your CFA model by using the ATFI index (see Hair, Babin & Krey, 2017 on CB-SEM).

So if your SEM model seems to have much better fit without the 2nd order factor, maybe there is not enough justification to use the 2nd order? Or maybe there are theoretically justified modifications to 1) the SEM model/2) the factors included in your 2nd order construct which might improve the fit? For example, maybe there is one factor which correlates very differently with your 2nd order construct than the other 3 factors.

Also, I'm not sure you can 'use' a 2nd order construct in EFA, but rather imply its existence. The 1st order constructs should still appear distinct enough in your EFA for discriminant validity (but probably with some cross-loadings which hint to a 2nd order construct).