[R] pca vs. pfa: dimension reduction

Mark Difford mark_difford at yahoo.co.uk
Wed Mar 25 20:51:58 CET 2009

Hi Sören,

>> (1) Is there an easy example, which explains the differences between  
>> pca and pfa? (2) Which R procedure should I use to get what I want?

There are a number of fundamental differences between PCA and FA (Factor
Analysis), which unfortunately are quite widely ignored. FA is explicitly
model-based, whereas PCA does not invoke an explicit model. FA is also
designed to detect structure, whereas PCA focuses on variance, to put things
simply. In more detail, the two methods "attack" the covariance matrix in
different ways: in PCA the focus of decomposition is on the diagonal
elements, whereas in FA the focus is on the off-diagonal elements.

Take a look at Prof. Revelle's psych package (funtion omega &c). Note also
that factanal has a rotation = "none" option.

Regards, Mark.

soeren.vogel wrote:
> Can't make sense of calculated results and hope I'll find help here.
> I've collected answers from about 600 persons concerning three  
> variables. I hypothesise those three variables to be components (or  
> indicators) of one latent factor. In order to reduce data (vars), I  
> had the following idea: Calculate the factor underlying these three  
> vars. Use the loadings and the original var values to construct an new  
> (artificial) var: (B1 * X1) + (B2 * X2) + (B3 * X3) = ArtVar (brackets  
> for readability). Use ArtVar for further analysis of the data, that  
> is, as predictor etc.
> In my (I realise, elementary) psychological statistics readings I was  
> taught to use pca for these problems. Referring to Venables & Ripley  
> (2002, chapter 11), I applied "princomp" to my vars. But the outcome  
> shows 4 components -- which is obviously not what I want. Reading  
> further I found "factanal", which produces loadings on the one  
> specified factor very fine. But since this is a contradiction to  
> theoretical introductions in so many texts I'm completely confused  
> whether I'm right with these calculations.
> (1) Is there an easy example, which explains the differences between  
> pca and pfa? (2) Which R procedure should I use to get what I want?
> Thank you for your help
> Sören
> Refs.:
> Venables, W. N., and Ripley, B. D. (2002). Modern applied statistics  
> with S (4th edition). New York: Springer.
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