Syntax for second-order and bifactor CFA
Message/Author
 Michelle Martel posted on Wednesday, November 23, 2011 - 5:14 pm
Hi,

I was hoping you could let me know if the following syntax look correct.

Second-order factor model:

MODEL:
inatt BY inat1pc inat2pc inat3pc inat4pc inat5pc inat6pc
inat7pc inat8pc inat9pc;

hyper BY hyper1pc hyper2pc hyper3pc hyper4pc hyper5pc
hyper6pc hyper7pc hyper8pc hyper9pc;

inatt@1;
hyper@1;

Bifactor model:

MODEL:
inatt BY inat1pc inat2pc inat3pc inat4pc inat5pc inat6pc
inat7pc inat8pc inat9pc;

hyper BY hyper1pc hyper2pc hyper3pc hyper4pc hyper5pc
hyper6pc hyper7pc hyper8pc hyper9pc;

inat3pc inat4pc inat5pc inat6pc inat7pc inat8pc inat9pc
hyper1pc hyper2pc hyper3pc hyper4pc hyper5pc
hyper6pc hyper7pc hyper8pc hyper9pc;

inatt@1;
hyper@1;
inatt WITH hyper@0;

Thanks so much!
 Linda K. Muthen posted on Wednesday, November 23, 2011 - 5:53 pm
A second-order factor model with two first-order factors is not identified. When you fix the metric by setting the factor variance to one, you must free the first factor loading which is set to one as the default.

See Slide 159 of the Topic 1 course handout for an example of a bi-factor input.
 Katerina Gk posted on Wednesday, October 30, 2013 - 10:46 am
Hi,

Is it possible to test if there is any impact only to the general factor and not to the rest factors of the bi-factor model?

If yes can I include to the input the general factor only?

 Bengt O. Muthen posted on Wednesday, October 30, 2013 - 8:31 pm
Maybe you are asking if the general factor can be predicted by or can be a predictor of other variables - and the specific factors not. If so, yes.
 Katerina Gk posted on Thursday, October 31, 2013 - 3:37 am
Yes this is what I am asking.

 Bengt O. Muthen posted on Thursday, October 31, 2013 - 8:59 pm
For application, search for a Gustafsson-Balke article in Multiv Behav Research from the 90's.
 Ads posted on Tuesday, August 04, 2015 - 2:40 pm
I had a question regarding the order of variance extraction in bifactor CFA. Many articles/texts on confirmatory bifactor models (e.g., Reise, 2012) indicate that first, the variance in indicators is explained by G, and then subfactors explain the residual variance that remains in each indicator (i.e., equivalent to a Schmid-Leiman transformed second-order model).

However, in syntax for a bifactor model the indicators appear to load on both G and the subfactors simultaneously, which would seem more like simultaneous entry in regression. In contrast, G claiming variance first and then subfactors claiming the residual variance would be more like hierarchical entry in regression, and there is no hierarchical-like specification in bifactor model syntax.

Thus, I am concerned when looking at output that I can say G first has claimed X% of variance and then subfactors claim an incremental X% of variance in indicators (as there could be multicollinearity among the predicting factors). Is there a way to get a hierarchical interpretation of variance explained with bifactor models?

As an empirical test, I ran a bifactor model and then ran the same model again with all item loadings on subfactors fixed to 0. Amount of item variance explained by the G factor was similar but slightly different between the models (difference in R2 between the models ranged from 1% to 4% when looking at a few items). Many thanks for your help.
 Bengt O. Muthen posted on Tuesday, August 04, 2015 - 6:47 pm
I think the Reise type of explanation is conceptual, explaining how to interpret the model, rather than computational. Early factor analysis did bi-factor analysis computationally in the implied two steps, but today the estimation is done in one single step.
 Ads posted on Wednesday, August 05, 2015 - 7:46 am
Thank you very much Dr. Muthen for this explanation and background. I wonder then, given the contemporary computational/estimation methods for bifactor models you mentioned, what is the order of variance allocation to each factor (i.e., G and subfactors, given that it appears in bifactor syntax that items load onto multiple factors simultaneously)?

Pardon my ignorance as I must be missing something; it seems then that bifactor loadings differ in interpretation from semipartial correlations of items with factors (i.e., coeffecients don't reflect just variance above and beyond variance accounted for by other "predictor" factors).
 Bengt O. Muthen posted on Wednesday, August 05, 2015 - 6:48 pm
There is no order of variance allocation. The model is fitted to a covariance/correlation matrix so that the parameters values are chosen to maximize this fit.

The loadings are interpreted as regular partial regression coefficients, regressing each item on several factors.
 Ads posted on Wednesday, August 05, 2015 - 7:45 pm
Thank you very much!