[R] Factor Rotation

[Prof Brian D Ripley]

On Wed, 26 Apr 2000, Barry Cooke wrote:

> How does one rotate the loadings from a principal component analysis?
> Help on function prcomp() from package mva mentions rotation:
> 
> Arguments
> 
> retx		a logical value indicating whether the rotated
> 		variables should be returned.
> 
> Values
> 
> rotation	the matrix of variable loadings (i.e., a matrix 
> 		whose olumns contain the eigenvectors).  The 
> 		function princomp returns this in the element 
> 		loadings.
> 
> x		if retx is true the value of the rotated data
> 		(the data multiplied by the rotation matrix) 
> 		is returned.
> 
> but it is unclear to me exactly how the rotation is to be done.

You are being confused by prcomp's S-compatible notation.  All PCAs are a
`rotation' of the coordinate system to a basis in which the first component
has maximum variance, the next has maximal variance subject to
uncorrelatedness with the first etc.

> For instance how would one specify a varimax rotation?  Or are
> loadings, by default, rotated by the varimax method in prcomp()?
> 
> MASS v2 describes the function rotate() on p. 409-410; however R
> complements does not cover this.  A search of CRAN indicates Peter
> Ho <peter at esb.ucp.pt> asked the same question Sept 29, 1999, but
> I found no reply.

There is no code to do this in R that I know of. Rotation in the sense
you are describing it belongs with factor analysis, and that is not
currently implemented in R.  Indeed `loadings' is FA notation applied to
PCA, and this subject is bedevilled with confusion between the two.
In my view most of the rotation criteria only make sense for FA.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272860 (secr)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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