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RE: st: PCA and rotation

From   "Nick Cox" <[email protected]>
To   <[email protected]>
Subject   RE: st: PCA and rotation
Date   Mon, 4 Jan 2010 18:58:21 -0000

These recommendations sound tendentious to me, except that no doubt they make much sense in the context of the authors' aims. 

The idea that factor analysis is _uniformly_ superior to principal component analysis is I think only defensible if your aims are limited to those addressed explicitly by factor analysis. 

Indeed, even non-experts on intelligence tests can guess that different intelligence factors, if they exist, are most unlikely to be uncorrelated, but there are plenty of contexts (e.g. in meteorology and oceanography) in which orthogonal components are precisely what is desired from a physical point of view. 

[email protected] 

Diana Kornbrot 28 Dec 2009 

The following intereesting article recommends
1. ALWAYS use factor analysis not principal components, as errors are included in PC anf may differ across replications 
2. ALWAYS use oblique rotation rather than orthogonal rotation, as otherwise you may miss higher order factors
Reeve, C. L., & Blacksmith, N. (2009). Identifying g: A review of current factor analytic practices in the science of mental abilities. Intelligence, 37(5), 487-494. 

Makes sens to me.
Other views?

On 21/12/2009 21:33, "Michael I. Lichter" <[email protected]> wrote:

I recently found that when I extracted components using -pca-, rotated
them using an orthogonal rotation (e.g., -rotate, varimax-), and scored
them using -predict-, the correlations between what I presumed were
uncorrelated factors were actually as high as 0.6. I know that component
scores may be correlated, but this seemed a bit much. Somebody else
noted the same thing a few months ago
( On the
other hand, I found that factor scores (produced with -factor, pcf-) for
the same data remained virtually uncorrelated after orthogonal rotation.

I therefore assumed that the behavior of rotated PCs was a bug. I
contacted Stata. Isabel Canette told me that I was mistaken. She
referred me to "Methods of Multivariate Analysis" by A. Rencher, Second
Edition,Wiley, 2002, page 403, where Rencher says:

  "...If the resulting components do not have satisfactory interpretation,
   they can be further rotated, seeking dimensions in which many of the
  coefficients of the linear combination and near zero to simplify
    However, the new rotated components are correlated, and they do not
  successively account for maximum variance. They are, therefore, no
  longer principal components in the usual sense, and their routine use
  is questionable".

In other words, it's not a bug, it's ... something else. Isabel said
that for this reason Stata discourages the use of rotation after -pca-.

What's odd is that I've seen a number of articles that use varimax
rotations  (with Kaiser normalization) of principal components in scale
development. The authors only use the PCA to guide scale development;
they perform further analysis with Cronbach's alpha and create summative
scales rather than using factor scores. Still, their interpretation of
the components are based on rotated component loadings that, at least
from Rencher's perspective, are "questionable".

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