There's no correct answer and I guess it depends on exactly what you want to show. 

Before I saw your path diagram, I had envisioned fitting the same structural model to batteries A and B with a covariance between gA and gB. Once I got the models to fit, I would start forcing loadings to be identical and force the covariance between gA and gB to be 1.  This approach would test whether your new battery A had the same factor structure as B and whether the general factor can be shown to be identical across A and B.

If you did that with the existing path model, the first thing you'd notice is that your first-order model is no longer identified because your latent variables would only have one indicator.  I would solve this by using bundles (aka parcels) as indicators. 

I would always avoid adding correlated errors unless you have a good justification and I would try to fit the model without them before I added them.

Morning,

I am not sure I understand you correctly.  You have 2 different questionnaires for IQ. You measure the same underlying constructs but with different items? If the onlyrics thing you need is to show that your new questionnaire is as good as the previous one or even better, than you need to show:

1. That the constructs are working in a similar way (previous constructs and new constructs should be correlated)

2. The relationship that was theorised previously is tested with the new questionnaire better that with the old one.

3. Or you can show that consistency measures and discriminant validity and convergent validity are better than in the old questionnaire.

1: you should make convergent and discriminant validity analysis for all contstructs underlying IQ. Those which measure the same construct (one old vs. one new) should have a low discriminant validity. Those which are measuring different constructs should not be correlated.

2: Let us assume that the theory said that IQ is the main predictor of a person's success. Then you should check the influence the influence of the old IQ and of the new IQ on the success. The IQ which explained the variance in success better, is the better one.

3: just make two different models with all underlying constructs and their underlying items. Show that the new questionnaire proposes a better model fit and reliability and validity indicators.

What you cannot do is running the same CFA for the 2 IQs, because the items are DIFFERENT. You can try to aggregate from items to super variables  (factor analysis, dimension reduction) and then add the aggregated variable instead of all items to the assumably same constructs. Then you should create a dummy variable  (let's say 1 for new, 0 for old  questionnaire). Previously you should have restructured your  questionnaire. 

Then  in AMOS you create three groups: all, new, old. You use the whole dataset for "all"- group,  use your dummy variable as selection variable.  1 for new, 0 for old. Then you run your analyzis and compare two groups new vs. old. This test would show that there is (or there is not) significant difference between two questionnaires. Or, more correctly, yhe resukts in IQ are moderated by the tyoe of questionnaire. Sohnds weird... and the more I write this,  the less I believe that some journal will like this approach... But you can try ;)

Best,

Eugene 

I think Alan's answer is the direction I would go. I would model Test A and Test B and create a higher order factor 'g' with equality constrains imposed. 

I actually began with the model Alan proposed, but then I started wondering whether that was the way to go. 

Thank you all for your answers!

I think both Alan's and Yevgen's suggestions should be considered.  I'd also point out that some recent models of intelligence test data are bifactor models, which are generalizations of the 2nd order model you're considering.  I would also consider them.  See Gignac's recent publications for more on bifactor models of intelligence test data, e.g., Gignac, G. E. (2016).  The higher-order model imposes a proportionality constraint:  That is why the bifactor model tends to fit better.  Intelligence, 55, 57-68.  Good luck in your research.

Dear Michael, thank you for your recommendations. With great interest I have read your ICP paper (2015) on the Bifactor model in personality, as I am conducting research on higher order factors in personality as well. 

Very interesting to see this model being applied to personality!