Date: Mon, 19 Sep 2005 20:54:50 -0300
Reply-To: Hector Maletta <hmaletta@fibertel.com.ar>
Sender: "SPSSX(r) Discussion" <SPSSX-L@LISTSERV.UGA.EDU>
From: Hector Maletta <hmaletta@fibertel.com.ar>
Subject: Re: PCA and Rotation
In-Reply-To: <6250203B042D8349A920AA61038309364B526D@UTHEVS3.mail.uthouston.edu>
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Some clarification:
PCA is designed to maximize the first underlying factor, maximizing its
contribution, and (besides) when it extracts all possible factors (number of
factor equal to the number of variables) it accounts for the entire variance
of the variables. Other methods of extraction do not maximize the first, and
besides they try to explain only the COMMON variance, leaving aside a
portion of variance that is regarded as unique to each observed variable.
Historically, PCA was used first to identify a single factor underlying a
set of related measures, supposedly measuring all the same trait
(intelligence, as it was), while Principal Axes was used based on the theory
that instead of a single Intelligence factor there were underlying
"intelligences" for varios "faculties of the mind" such as linguistic,
graphic or mathematical ability.
In BOTH cases, the position of the coordinate system on which the underlying
factors are measured is essentially arbitrary. By rotation, the analyst may
be able to put one factor closer to one set of observed variables, and far
from others, and the opposite for another factor. This make happen by chance
in the initial extraction, but ordinarily doesn't, so rotation is used in
order to get that nice characteristic of a factor linked to sets of
interrelated variables. For instance, using the same IQ example, by rotation
one may get one factor strongly associated with several linguistic tests,
and only weakly related to other tests, while another factor is strongly
related to mathematical tests and weakly to other tests, so one intuitively
calls the first factor "linguistic" and the second "mathematical".
These different factors may be independe from each other, or correlated. It
is perfectly possible that being good in language implies being good also in
math, and so at least to some degree, some correlation between linguistic
and math factors is only to be expected.
Initial patterns of extraction ordinarily extract factors that are
orthogonal or uncorrelated to each other, because each successive factor is
extracted on the unexplained residuals left by the preceding ones, but
rotation can position the factor axes in ways that imply they are correlated
to each other.
Rotation preserving the independence of factors is called orthogonal
rotation. Rotation allowing them to be correlated to each other is called
oblique rotation.
So in Paul Swank responde there is a (probably involuntary) confusion at the
end: oblique rotation yields correlated factors (though it does not FORCE
them to be correlated) while orthogonal rotation methods FORCE rotated
factors to be uncorrelated to each other.
Hector
> -----Original Message-----
> From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU]
> On Behalf Of Swank, Paul R
> Sent: Monday, September 19, 2005 8:26 PM
> To: SPSSX-L@LISTSERV.UGA.EDU
> Subject: Re: PCA and Rotation
>
> Principal components is a data reduction procedure, not a way
> to identify interpretable factors. To do the latter, use
> principal axes analysis or some other factor algorithm that
> targets common factors. It makes sense to rotate these since
> you are intreseted in inpretable factors. However, I suggest
> an oblique rotation to ensure that the factors are no
> correlated before forcing them to be.
>
>
> Paul R. Swank, Ph.D.
> Professor, Developmental Pediatrics
> Director of Research, Center for Improving the Readiness of
> Children for Learning and Education (C.I.R.C.L.E.) Medical
> School UT Health Science Center at Houston
>
> -----Original Message-----
> From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU]
> On Behalf Of Luis O.
> Sent: Monday, September 19, 2005 5:14 PM
> To: SPSSX-L@LISTSERV.UGA.EDU
> Subject: PCA and Rotation
>
> Dear List Members,
>
> I am new in the list and want to ask a very basic question
> regarding the principal component analysis. I was doing some
> analysis by using principal component method and VArimax
> rotation. However, one of my friends told me that the stat
> book says that we should not rotate principal components.
> That is, the principal component analysis should not rotate
> the soluations, because, by theory, it produces a unique
> soluation. On the other hand, when I read some SPSS manuals,
> they usually tell you to use the principal component method
> with some rotation method. Which is correct?
>
> Luis O. Benavent
> Benavnet Talps Research
> Gandía, Spain
>
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