# 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?

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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.

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