# [S] Factor analysis model

Greg Arnold (G.Arnold@massey.ac.nz)
Tue, 3 Nov 1998 14:01:48 +1200

Factor analysis is something I have often explained to others, but I have
never had a question for which it seemed an appropriate tool. Attempting
the calculations for the first time, S+ produces some numbers I find
puzzling .

These calculations are all performed on:
Version 3.4 Release 1 for Sun SPARC

The model is

x = L f + e

where x is the standardised p dimensional data vector
f an unobserved k dimensional vector of common factors
e a p dimensional vector of factors one unique to each component of x.
As a first step the components of f are uncorrelated with unit variance,
and the components of e are uncorrelated.

Using the standard data set 'testscores' the analysis goes like this:
[The 'scale' function makes no difference, but guarantees to the reader
that x is standardised]

> testscores.no.rot <- factanal(scale(testscores) ,rotation='none', factors=2)

> summary(testscores.no.rot)

[useful stuff with which I have no problem]

My problems lies with the properties of f. I believe these are the
'scores' component of the factanal, but

> var(testscores.no.rot\$scores)
Factor1 Factor2
Factor1 0.99703074 -0.00790189
Factor2 -0.00790189 0.52859672
>

But should not f1, f2 have unit variance?
[var(f1) may see close enough, but other less highly correlated data sets
give all var(fi)<<1]
Just what do the 'scores' really represent?

Greg Arnold
Statistics, Social Science Building
Institute of Information Science and Technology
Massey University
P B 11 222
Palmerston Nth
New Zealand

g.arnold@massey.ac.nz

Phone +64 6 350 4254
Fax +64 6 350 2261

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