# Exploratory factor analysis: the same eigenvalue for the one factor and two factor solution

I did an exploratory factor analysis of four measures. First, I constrained the four measures to load on a single factor(eigenvalue: 2.14, % of variance: 53.4 ). Second, the four measures were allowed to load on two factors (eigenvalue and % of variance for the first factor respectively: 2.14, 53.4; and the second factor 1.018, 25.451). Surprisingly, these two methods yielded the same eigenvalue and % of variance for the first factor. However, when I read the Amstrong & Kats (2010) paper, I found that they used the same methods, and had different eigenvalue and % of variance for the one factor and two factor method. Just wondering are my results normal, or I missed something? I appreciate any clue about this. Thank you in advance!

the paper I mentioned: Armstrong, T., & Katz, C. (2010). Further evidence on the discriminant validity of perceptual incivilities measures.

• What software are you using? – Placidia Jan 3 '15 at 3:57
• Are you doing Principal components or Factor analysis? – ttnphns Jan 3 '15 at 9:55

(For others reading, the paper referred to can be found here. http://www.tandfonline.com/doi/abs/10.1080/07418820802506198#.VKdrldXF9Uc )

You should get the same eigenvalue for the first factor, regardless of the number of factors that you extract. The first factor is still the first factor. The first factor explains as much variance as it can. The next factor explains what is left over.

However, when you rotate the factor solution, the eigenvalues will change. Part of the purpose of rotating is to balance the eigenvalues. That paper puts the eigenvalues in the paper but (as far as I can tell) never mentions them again.

Incidentally, the eigenvalue for a factor is the sum of the squared (standardized) loadings. That's how I knew they'd been rotated - the loadings change, and I checked the eigenvalues.

It's best not to take as absolute truth anything which is written in a paper about statistics, if it's not written in a methodology/statistics journal the statistics were probably not reviewed especially thoroughly.*

*Which is how I find myself in my current position.

• the eigenvalue for a factor is the sum of the squared (standardized) loadings. Well, the term "eigenvalue" is from PCA method. If the FA extraction method was other than PCA or principal axis I doubt it is right to use the word. "Sum of squared loadings after the extraction" - is the right one. – ttnphns Jan 3 '15 at 9:52
• @ttnphns - yes, thanks. I get annoyed by people mixing components and factors, and then I do it. – Jeremy Miles Jan 5 '15 at 4:53

First of all you are not doing EFA. You are doing PCA

You might want to read about the difference between EFA and PCA here

http://www2.sas.com/proceedings/sugi30/203-30.pdf

or here

http://en.wikipedia.org/wiki/Factor_analysis

BTW, eigenvalues are not used much in EFA - mostly for starting values and a rough guess on how many factors there are (count >1)

Eigenvalues are DEFINED as the eigenvalues of the sample correlation matrix. By definition those numbers are the same, i.e., if you use 2 components PCA the first component is the same as the first component of the 1 component PCA.

Incidentally

53.4% = 2.14 / 4

and

25.451% = 1.018 / 4

Also in EFA you can't say percentage of variance explained - it is different for each observed variable. Also in EFA it doesn't work that way - the first factor of 2 factor EFA is NOT the same as the first factor of the 1 factor EFA.