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Re: st: rotated components in pca, order of


From   khigbee@stata.com
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: rotated components in pca, order of
Date   Thu, 05 Jul 2007 09:52:13 -0500

Cathy L. Antonakos <cathya@umich.edu> asks:

> I'm running a principal components analysis in Stata, and am
> puzzled by the order of the components in the rotated solution.
> Prior to rotation, the components are listed as "comp 1" "comp 2"
> "comp 3" "comp 4" in the table of component loadings. After rot
> ation, the order of the components changes to: 3, 2, 1, 4. The
> loadings also change, which I expected.
> 
> I can figure out which of the rotated components is what, based
> on running a correlation between the rotated component scores and
> selected variables that I expect to load highly on particular
> components.
> 
> But I don't know why the order of the components has changed. I
> did look in Statalist and the manual and found references on how
> to generate component scores, but nothing more.
> 
> My question is - are the rotated components reordered because due
> to rotation the percentage of variance explained by a particular
> component has changed? After rotation, in this example, does
> component 3 now explain more variance than component 1 did in the
> unrotated solution?

If you had Stata 10 instead of an earlier version of Stata, the
rotated components would be labeled in the output from 1 to 4
instead of 3, 2, 1, 4.  The issue is only cosmetic--the rotated
components are the same either way.  Rotation is creating new
components that are linear combinations of (all of) the original
components. Afterwards the resulting columns are reordered from
highest to lowest percent of variance explained (just as you
guessed in your question above).

Previous to Stata 10, when we reordered the rotated components,
the column labels ended up reordered at the same time.  There is
no good reason to do that, and so with Stata 10 the columns are
labeled from 1 up.


> I'm also wondering if anyone can explain what the "component
> rotation matrix" is. I've seen this in factor analysis output as
> "factor rotation matrix" but despite quite a bit of searching, I
> haven't been able to find information about what this is.

Maybe the best way to understand the "factor rotation matrix"
(found in the e(r_T) returned matrix result) is by example.
Borrowing an example from "[MV] pca postestimation" we have

    . webuse audiometric
    . pca l* r*, comp(4)
    . rotate

    . mat L = e(L)
    . mat rL = e(r_L)
    . mat rT = e(r_T)

matrix L has the unrotated components, matrix rL has the rotated components,
and rT has the component rotation matrix.  These matrices are related by

    rL = L * rT

You can verify by

    . mat chk = L * rT

    . mat list rL
    . mat list chk


Ken Higbee    khigbee@stata.com
StataCorp     1-800-STATAPC

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