Xu, Man posted on Wednesday, March 06, 2013 - 8:50 am
I am trying to interpret the factors in bi-factor rotated solutions.
Say I get a 4 factor soluation in EFA (f1, f2, f3, f4). Then in an equaivalent bi-factor version of the EFA I still have 4 factors (bf1, bf2, bf3, bf4), but the first factor has become the general factor(bf1), with 3 others (bf2, bf3, bf4), as the "group/specific" factor.
I was wondering if it makes sense at all to try to match the factors from the EFA to the "group/specific" factors from the bi-factor EFA? For example, would bf2 correspond to f2 and bf3 correspond to f3?
Or has the substantive meaning of factors structures and their loadings all changed completely due to rotation methods?
The substantive meaning of the factors has changed with the two rotation methods.
Xu, Man posted on Wednesday, March 06, 2013 - 10:05 pm
If the a priori is to have n factors in data, and EFA indeed indicated a n-factor structure, does this mean that under bi-factor rotation, instead of the n factor, it is the n+1 factor solution that will be most likely to reveal a structure with 1 general factor and n group factors, correspondng to the a priori?
A model with n factors corresponds to a model with 1 general and n-1 specific factors. With EFA these 2 models will have the same fit.
Xu, Man posted on Thursday, March 07, 2013 - 4:06 pm
So, if n-dimension data can be adequately explained under n EFA factors, then under bi-factor rotation, only n-1 groups of items will be able to show up as group factors.
I guess I was a bit confused because I tried to look at the fi-factor EFA structure in the same route as the typical EFA to CFA type of model development. e.g., if I see n factors in EFA, then I will go on to specify a CFA with n factors.
If I see n-1 group factors under bi-factor EFA, would I only specify n-1 group factors, or would I specify n group factors as in typical CFA bi-factor analysis?
One observation is that in some data it may be easier to find an interpretable loading pattern when using a general factor. For instance, EFA with regular rotation may suggest 4 factors while a 5-factor solution has problems. EFA with bi-factor rotation, however, may find that 4 specific factors come out interpretable together with a general factor. This despite fit being the same for regular EFA with 4 factors and bi-factor EFA with only 3 specific and 1 general factor. I have seen this in the classic Holzinger-Swineford data.
If bi-factor EFA points to n-1 specific factors, bi-factor CFA would also specify n-1 specific factors.
Xu, Man posted on Thursday, March 14, 2013 - 12:15 pm
Thank you - and yes - I read your recent teaching slides on bi-factor and saw that 4 factors from regular EFA were sufficient to explain the 24-variable Holzinger-Swineford data. In the bi-factor EFA, 5 factor (instead of 4) were requested as a priori (if I understood correctly), and each of the 5 factors do correspond to the hypothesis - a general factor, and 4 specific factors.
So maybe, if one knows the expected number of dimensions, say n, perhaps one should look at the bi-factor rotation EFA output both under n and n+1 factors?