Syntax for second-order and bifactor CFA PreviousNext
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 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;

ADHD BY inatt hyper;

ADHD@1;
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;

ADHD BY inat1pc inat2pc
inat3pc inat4pc inat5pc inat6pc inat7pc inat8pc inat9pc
hyper1pc hyper2pc hyper3pc hyper4pc hyper5pc
hyper6pc hyper7pc hyper8pc hyper9pc;

ADHD@1;
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?

Thanks in advance!!
 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.

Thank you very much for your answer!!!
 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!
 ZHIYAO YI posted on Friday, July 19, 2019 - 4:03 pm
Hi Dr. Muthen,

I specified a bifactor model as below:

model:
inattent by adhd1* adhd3 adhd5 adhd7 adhd9 adhd11 adhd13 adhd15 adhd17;

hyper by adhd2* adhd4 adhd6 adhd8 adhd10 adhd12 adhd14 adhd16 adhd18;

adhd by adhd1* adhd3 adhd5 adhd7 adhd9
adhd11 adhd13 adhd15 adhd17
adhd2 adhd4 adhd6 adhd8 adhd10
adhd12 adhd14 adhd16 adhd18;

adhd with inattent@0;
adhd with hyper@0;
inattent with hyper@0;

adhd@1;
inattent@1;
hyper@1;

The results show that some items are positively loaded on inattent(one specific factor) but some are negatively loaded. Same for hyper(another specific factor). But all the items are positively loaded on ADHD(general factor).

I found the results is really strange, since all the other models(one and two factor models) show that the items should positively loaded on the corresponding factors.
I am wondering is this because I specified the bifactor model incorrectly or because of the nature of the bifactor model?
 Bengt O. Muthen posted on Saturday, July 20, 2019 - 1:10 pm
You have to think of the specific factors as picking up residual correlations that are not well fitted by the general factor alone. So say that you have a pair of specific factor indicators, one with a negative loading and one with a positive one. This implies that the sample correlation for this pair of variables is smaller than what is predicted by the general factor alone.
 ZHIYAO YI posted on Saturday, July 20, 2019 - 9:50 pm
Thank you so much for your response!

I have another question about the second-order model.

If there are only two specific factors (I used the same data above), can I still construct a second-order model. I tried many ways, the model was always not identified. Thank you in advance.
 Bengt O. Muthen posted on Sunday, July 21, 2019 - 4:46 pm
No, 3 first-order factor are needed.
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