# Understanding factor analysis

Can I understand factor analysis in the following way?

Assume I have 5 independent variables (A,B,C,D,E)

Factor analysis allows me to make (D,E) to be dependent variables and allow me to make them to be linear combinations of (A,B,C).

Therefore I will only need to carry (A,B,C) data and the $\Lambda$ matrix, then I can recreate data (D,E) by data(A,B,C) and the $\Lambda$ matrix .

It does data reduction only. Am I right?

• I upvoted this question because it's reasonable. Well-posed questions that expose misunderstandings should be welcomed (and upvoted to indicate that), not disparaged with downvotes, because they can prompt great responses like the one @StasK has already offered here. – whuber Feb 8 '12 at 22:37

No. In factor analysis, all variables are dependent variables, and they depend on latent factors (and also contain measurement errors). While factor scores are often used in place of the original variables, which may seem like a data reduction issue, this is precisely what factor analysis is aimed at. In other words, rather than saying, "Wow, I've got a lot of data that I cannot really process and understand; can I come up with a trick to have fewer variables?", factor analysis is usually performed in the situation "I cannot measure a thing directly, so I will try different approaches to it; I know I will have a lot of data, but this would be related data of known structure, and I shall be able to exploit that structure to learn about that thing that I could not measure directly".

What you described qualifies either as multivariate regression (don't confuse with multiple regression, which encompasses one dependent variable and many explanatory variables; multivariate regression has many dependent variables and the same set of explanatory variables in each individual regression), or canonical correlations (with some stretch of imagination though), or a multiple indicators and multiple causes structural equation model, may be. But no, this is not factor analysis.

to add to @StasK's excellent response, i will clarify further by saying that this problem does fall under the general umbrella of structural equation modeling (SEM). SEM is a technique that can be employed to model covariance structures and, while typically used with unobserved or latent variables, it can also be applied to models with only observed or manifest variables. applying SEM methodology and terminology to your problem, D and E would be considered endogenous variables while A, B, and C are exogenous variables. endogeny suggests that variance in the particular variable is explained by another variable while exogeny suggests that variance is not explained by another variable, latent or manifest.

werner wothke provides some good slides introducing SEM using SAS here.

also look for ed rigdon's site discussing a variety of SEM issues (too new, can't link!).

getting back to basics, if your goal is to understand factor analysis, i would suggest starting with an applied text like brown's confirmatory factor analysis for applied research.

• Ed Rigdon's page is www2.gsu.edu/~mkteer. To make this answer even more complete, I would add that SEM with observed variables only have been first studied in econometrics in 1950s or so under the title of simultaneous equation models. Ken Bollen's excellent book (amazon.com/Structural-Equations-Latent-Variables-Kenneth/dp/…) is actually one of the few books on SEM that covers it in enough detail to learn something useful. – StasK Feb 10 '12 at 3:48