In many theses and examples on the internet I see that people do a factor analysis with principal component analysis as "extraction method". But when you google the two, it is said that they are two different things and that you use either one or the other. Also Stata provides different factor loading values when running PCA and FA.
Still, I keep on seeing in theses that people display a table calling it a "Factor analysis", but then they put "Components" (from PCA) as columns in the tables and then load factor values from factor analysis. The note under the table then states: Extraction method: Principal component analysis. I just don't understand how to combine the two? Because when I google it, it says you can use PCA if you want to reduce the date of instance an FA for underlying variables. But it seems that you can combine the two (extraction method in FA)??
Do they then simply use the PCA to indicate the number of components? Where do you do the rotations for PCA of FA or both? Why do people report the 'components' and insert the factor loadings of FA?
I inserted a screenshot of an example which I see a lot.
Really hope someone can help me with this issue!
Hello Mayumi,
I agree that PCA and common factor analysis (FA) have quite different aims and require a different perspective about what is realistic.
I suspect the answer to your question has multiple parts:
1. Authors (especially of textbooks) tend to follow organizations and presentations used in previous textbooks. It's been said, with only slight facetiousness, that stat texts are not really written for the students so much as they are for the instructors. Who chooses them, after all?
2. Statistical software doesn't always distinguish the two (in fact, SPSS may be singularly more responsible for the conflation than any other analytic package. The default extraction method under the "FACTOR" subprogram is principal components, and has been for 50 years or more).
3. Ignorance on the part of the user, which is not helped by the many published examples (as you note in your query) which conflate the methods.
4. "Loadings" are just correlations, and so may be reported for PCA or FA.
5. Rotations are just re-arrangement of axes in space, and so may be reported for PCA or FA.
Now, for the contrasting perspective. If the number of variables being analyzed is large, PCA results tend to be very close to FA results under the principal axis extraction method.
Good luck with your work.
Your confusion is correct as many people confuse both techniques and use them interchangeably, which is not correct. However, David has responded comprehensively, and I hope you have found your answer.
I will add only one thing, that PCA is a reduction technique where the aim is to reduce the number of items and to find component/dimensions, whereas FA is more of finding dimensions in the data and not reducing the number of items necessarily.
Dear Mayumi,
Your question is very clear and needs some clarifications. Kindly read my input below;
Factor analysis (FA) is usually applied in a data set to provide information on the most meaningful parameters in samples that describe the whole data set allowing data reduction with least loss of information.
Principal component analysis (PCA) generates principal components (PCs) which are sometimes not easily interpreted; therefore it is desirable to rotate the PCs by varimax rotation. Varimax rotation is applied on PCs with eigen value > 1 considered as significant to obtain new group of variables called varimax factors (VF). In FA, the number of VFs produced due to varimax rotation is equal to the number of variables in accordance with common features including latent, hypothetical, and unobserved variables.
Get and read the following published papers online, especially No.(3) for better reference purpose:
1) Source identification and evaluation of surface water quality using factor and discriminant analysis.
2) Spatial assessment of air quality patterns in Malaysia using multivariate analysis.
3) River water quality assessment using environmentric techniques, a case study of River Jakara.
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