Whether an approach is appropriate or not depends in part on what you are trying to do. If all you want to do is to "group the variables" then there is nothing wrong with computing some measure of similarity among the variables and then applying some form of cluster analysis. On the other hand, if you have some model then you have to be more careful.

Factor analysis is a more ambitious approach that attempts to find underlying factors that can be used to help you interpret the pattern of covariation shared among the variables. Factor analysis methods are often used to group variables but that is not the purpose for which the methods were developed. One uses an oblique solution if one does not wish to constrain solutions to those in which the estimated factors are uncorrelated. At least in biology, it is difficult to justify restrictions to only orthogonal factors. Discrimination is a different problem.

For the original request of an oblique factor analysis that penalizes non-zero loadings you might wish to look at: Katz, J. O. and F. J. Rohlf. 1975. Primary product functionplane, an oblique rotation to simple structure. Multivariate Behavioral Research, 10:219‑232. Software for it is not generally available but it will be included along with the better known factor analytic methods in the next version of NTSYSpc.

Jim

-----------------------
F. James Rohlf
State University of New York, Stony Brook, NY 11794-5245
www: http://life.bio.sunysb.edu/ee/rohlf

From: Classification, clustering, and phylogeny estimation [mailto:[log in to unmask]] On Behalf Of Art Kendall
Sent: Monday, September 06, 2004 9:26 AM
To: [log in to unmask]
Subject: Re: Clustering Variables

It is many years since I was current on the factor analysis literature. I have retired and no longer have access to databases of abstracts like DIALOG, ORBIT, or PsychInfo. If you have a friend in a university or a government agency they might be able to do a search for you.

Since most clustering grew up around grouping cases (rows in the original data matrix), how is transposing the data matrix and using the same algorithms problematic in clustering variables (columns)? Just the opposite, one of the oldest methods of clustering cases was to standardize then transpose the data matrix and factor it. (this approach was big in the 1960's & 1970's).

I have a gut feeling (not a thought out opinion) that an oblique solution means that you end up with measures that do not have discriminant validity.

SPSS has had many varieties of factor analysis for many years. It has used 2 kinds of data, 7 kinds of extraction, and 4 kinds of rotation. (56 different "methods"!) Maybe some of those combinations would meet your needs. [For those of us who use methods that other create, it sure would be nice if someone were to use this framework and produce a document advising on when to use the options. ]

to get details like algorithms and lit cites go to
http://support.spss.com/
login as "guest"
password "guest"
<statistics>
<algorithms>
then <catpca> <catreg> <cluster> <discriminant> <factor> <overals> <proximities> <quick cluster> <twostep cluster>

The ANSWERTREE add-on and new TREE procedure in the base module may also be relevant.

kinds of data: SPSS can work on a correlation matrix or a covariance matrix. In Psych, the means of variables are usually arbitrary, so correlations are more common. However, much of the development of factoring was from psych and ed. Perhaps the math psych list would have more current people .
Society for Mathematical Psychology: MPSYCH Listserv

quote from SPSS about the extractions available
Available methods are principal components, unweighted least squares, generalized least squares, maximum likelihood, principal axis factoring, alpha factoring, and image factoring.
end quote.
there are more details in the <help>.

quote from SPSS <help> about the rotations available. These

Varimax Method. An orthogonal rotation method that minimizes the number of variables that have high loadings on each factor. It simplifies the interpretation of the factors.

Direct Oblimin Method. A method for oblique (nonorthogonal) rotation. When delta equals 0 (the default), solutions are most oblique. As delta becomes more negative, the factors become less oblique. To override the default delta of 0, enter a number less than or equal to 0.8.

Quartimax Method. A rotation method that minimizes the number of factors needed to explain each variable. It simplifies the interpretation of the observed variables.

Equamax Method. A rotation method that is a combination of the varimax method, which simplifies the factors, and the quartimax method, which simplifies the variables. The number of variables that load highly on a factor and the number of factors needed to explain a variable are minimized.

Promax Rotation. An oblique rotation, which allows factors to be correlated. It can be calculated more quickly than a direct oblimin rotation, so it is useful for large datasets.

end quote.

Art
[log in to unmask]
Social Research Consultants
University Park, MD USA
(301) 864-5570

Wolfgang M. Hartmann wrote:

Thank you for the nice response,

I kmow that in practice transposing the matrix is a common, but do not think

of it as a very valid approach. (Higher order) Factor analysis with oblique rotation

and restrictions penalizing nonzero loadings would sound good for me. Would

you know of any references for such an approach?

Wolfgang

In SPSS all of the few dozen Proximity (similarity measures) can be applied to variables. (After the data are transformed and transposed) The Proximity matrix can then be read into the variety of cluster procedures. Or the transposed data can be read directly into the CLUSTER, or Quick cluster procedure. I see no reason (given that you want to cluster variables) that the TWOSTEP cluster could not read a transposed data matrix.

Of course there are all of the varieties of factor analysis which are more commonly used to group variables. The CATPCA procedure factors categorical variables.

When the variables are used to classify or differentiate a categorical variable, there are procedures like DISCRIMINANT or the various