# Research paper claims to have used PCA, but it sounds like factor analysis

I'm currently reading a research paper entitled Relationships between national economic culture, institutions, and accounting: Implications for IFRS and I am looking at trying to replicate the analysis of the paper (just for my understanding).

The paper has a couple of data sets which are combined. The authors then run a "principal component analysis utilizing a varimax rotation" to evaluate the separate constructs and confirm validity of their constructs. They check the following assumptions for the analysis: sphericity, sampling adequacy, and low communalities.

Paraphrasing the paper results:

The six variables loaded on the first component, which accounted for x% of the variance, while the other variables loaded on the second component, which accounted for an additional X% of the variance. This validated discriminant validity.

My question is, are these not the assumptions and results of exploratory factor analysis, and not of principal component analysis?

My understanding is that PCA is a data reduction technique with the end result being uncorrelated principal components. The assumptions of PCA are linear relationship between all variables, sampling adequacy, variables are somewhat correlated and no significant outliers. Whereas exploratory factor analysis is used as a method to evaluate construct validity which seems to be happening here.

I could include the original paragraph but the paper sits behind a paywall so I am not sure if its allowed.

• This thread on Is PCA followed by a rotation (e.g. varimax) still PCA? will probably clear things for you. Oct 18, 2015 at 10:40
• Hi @usεr11852, thank you for the link. Based on the description this is in essence a "PCA followed by a varimax rotation". Does this mean that the assumptions of PCA should also be checked before beginning the analysis? Oct 18, 2015 at 11:04
• I managed to get through the paywall and see that the actual quote is: a factor analysis [...] was conducted using principal component analysis utilizing a varimax rotation. So they did PCA+varimax, but they view, approach, and interpret it as a way of doing factor analysis. Oct 18, 2015 at 15:23
• Hi @amoeba, does this mean that the pca assumptions should have been checked? Oct 18, 2015 at 16:03
• Are you asking if the "PCA assumptions" should be checked in addition (or instead) to the "FA assumptions"? But are those any different? What would e.g. be an example of a "PCA assumption" that you would check for PCA, but not for FA? Oct 18, 2015 at 16:08

## 1 Answer

Broadly speaking, factor analysis decomposes the total variance in a data matrix, S, into two new matrices: one containing the common factor structure, W, and a second consisting of the unique errors, U. PCA and CFA differ most importantly in that PCA considers the total variance, CFA only analyses W.

In addition, unrotated PCA provides an optimal and mathematically unique solution. Once any factor matrix is rotated, it is no longer either optimal or unique. So, rotated PCA is still PCA but it's no longer an optimal or unique solution.