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Organization: SAS Institute Inc. Cary NC
Keywords: scaling
Newsgroups: bit.listserv.stat-l
Date:         Mon, 4 Jan 1993 14:46:09 GMT
Comments:     Warning -- original Sender: tag was NETNEWS@VM.SAS.COM
From:         "Warren F. Kuhfeld" <saswfk@UNX.SAS.COM>
Subject:      Re: qualitative principal components

In article <8289@news.duke.edu>, Frank Harrell <feh@DUKE.EDU> writes:
|> What is the state of the art in scaling techniques for combinations
|> of continuous, ordinal, polytomous, and binary variables? To me,
|> qualitative principal components using the alternating least squares-type
|> techniques in SAS PROC PRINQUAL look very promising, but we have had
|> tremendous convergence problems using this procedure.

As I see it, the problem with using PROC PRINQUAL on many data sets is
simply that at best there are too few observations for the number of
parameters, and at worst, there are more parameters than observations.
Consider 20 categorical variables with 10 categories each.  There are
20 * (10 - 1) parameter estimates required for the optimal scoring.

Another problem is that the algorithms often work "too well", finding
an uninteresting or silly solution that is in fact the optimal
solution.  Consider the two-dimensional point cloud:

                                                    X  X  X
                                                 X    X   X  X
                                               X  X X  X X  X
                                            A  X   X   XX  X
                                                  X  X X  X
                                                    X X  X

If given the freedom to do so, PROC PRINQUAL could transform this to:

A                                                     X

It tries to collapse all the X's into one point.  Often it does not
quite succeed and the X's get *almost* the same scores.  If the
original mean and variance are restored, "A" will get extreme scores.
This problem is most acute when ordinary (period) missing values are
optimally scored.  The example in the SAS manual showing how to use
PRINQUAL to estimate missing data, I now believe, is not very useful.
That technique too frequently leads to optimal but uninteresting
solutions.

In my (unfortunately unpublished) dissertation, I concluded that the
best way to compute principal components of ordered categorical data
was to first perform a rank transformation, and then perform an
ordinary PCA of the ranks.

In the ALS approach, binary variables can be treated as nominal,
ordinary, or interval.  It does not matter; the results will be the
same.

If the total number of categories in all of the nominal variables is
small relative to the total number of observations, consider optimally
scoring them with PRINQUAL.  If the ((degree plus the number of knots)
times the number of spline variables) is small relative to the total
number of observations, consider splines.  However, if either of these
numbers is large, you may get uninteresting results.  If there are
multivariate outliers, you may also get uninteresting results.

The 6.07 release of PRINQUAL for MVS, CMS, VMS has a REITERATE option.
It allows you to output the results, change the model, and start
iterating again using the previous results as a starting point.  It
also allows you to specify random initial scores.  Perhaps this might
help.

--
----------------------------------------------------------------------
Warren F. Kuhfeld          Statistical R & D      (919) 677-8000 x7922
saswfk@unx.sas.com         SAS Institute Inc.     (919) 677-8123 (Fax)
                           Cary, NC 27513-2414