Dear Sören, Mark, and Jon,
At 12:51 PM -0700 3/25/09, Mark Difford wrote:
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.
This is nicely put. Less concisely, see pages
139-149 of my (under development)
book on psychometric theory using R
(http://personality-project.org/r/book/Chapter6.pdf)
In particular, on page 149:
"Although on the surface, the component model and
factor model appear to very similar
(compare Tables 6.6 and 6.7), they are in fact
very different. One example of this is when an
additional variable is added to the correlation
matrix (Table 6.8). In this case, two additional
variables are added to the correlation matrix.
The factor pattern does not change, but the
component pattern does. Why is this? Because the
components are aimed at accounting for
all of the variance of the matrix, adding new
variables increases the amount of variance to be
explained and changes the previous estimates. But
the common part of the variables (that
which is estimated by factors) is not sensitive
to the presence (or absence) of other variables.
Although a fundamental difference between the two
models, this problem of the additional
variable is most obvious when there are not very
many variables and becomes less of an
empirical problem as the number of variables increases."
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.
For 3 variables, there is only one factor
possible, so rotation is not a problem. (For 1
factor, there are 3 unknown factor loadings and 3
known correlations. The model is just
identified. )
>
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.
If you want to think of what these variables have
in common, use factor analysis, if you want to
summarize them all most efficiently with one
composite, use principal components. These are
very different models.
As Mark said, the difference is that FA accounts
for the covariances (the off diagonal elements)
which reflect what the variables have in common.
PCS accounts for the entire matrix, which in a 3
x3 problem, is primarily the diagonal variances.
Let me know if you need more information.
Bill
>
(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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______________________________________________
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--
William Revelle http://personality-project.org/revelle.html
Professor http://personality-project.org/personality.html
Department of Psychology http://www.wcas.northwestern.edu/psych/
Northwestern University http://www.northwestern.edu/
Attend ISSID/ARP:2009 http://issid.org/issid.2009/
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