Art, et al,
I believe PREFSCAL is a program devised by Willem Heiser and one of
his present or former students (can't recall student's name at
the moment) which modifies their PROXSCAL program for metric and
nonmetric two- and three-way multidimensional scaling (MDS) by
optimizing a STRESS-like criterion to fit a "multidimensional
unfolding" model (a la Coombs, Bennett and Hayes) to off-diagonal
conditional proximity data (where, in the preferential data analysis
context, nonsymmetric rectangular proximities are taken as measuring,
usually up to a non-increasing monotone function, the distances
between stimuli-- corresponding say to rows of the data matrix) and
subjects' ideal points (corresponding to columns of the same matrix),
where the "conditional" nature of the data implies that these data
are comparable only WITHIN rows, but NOT between rows.
I have never been associated with a program named "PREFSCAL". Joe
Kruskal and I were the first to point out, (in the 1969 Kruskal and
Carroll paper "Geometrical models and badness-of-fit functions", in
P. R. Krishnaiah (Ed.), Multivariate Analysis, Vol. 2, pp.
639-671. New York: Academic Press), that a nonmetric MDS program
such as KYST can be used for unfolding analysis of such data if a
slightly different version of STRESS, called STRESS2, is used, in
which the normalizing factor in the denominator is proportional to
VARIANCE of the obtained distances (in this case between stimuli and
subjects' ideal points) rather than to the SUMS OF SQUARES of
distances as in the standard case of MDS of symmetric UNconditional
proximities, the more common form of proximities analyzed via MDS
procedures-- and if the data are in the form of an off-diagonal
conditional proximity matrix of the kind described by Coombs et
al. The data for unfolding analysis ARE proximity data, but, as
described above, of a very special kind-- based on Commbs's unfolding
theory that postulates an individual's preferences are inversely
monotonically related to distance between the stimulus and that
individual's postulated "ideal point"-- which is presumed to
represent that subject's "most preferred" stimulus-- so that, the
closer a stimulus is to that subject's ideal point the greater is
that subject's preference for that stimulus. Kruskal modified later
versions of his MDSCAL program, and all versions of KYST, so that an
option was available, by providing the kind of proximity data
discussed above, using STRESS2 instead of STRESS1 (which is the
original version of STRESS defined by him in his two original
Psychometrika papers on nonmetric MDS), and fitting the data
conditionally by row, so that unfolding analysis of this type of data
was possible (although a problem of degeneracies and
quasi-degeneracies makes the model extremely difficult to fit adequately).
Dominance data, as exemplified by "Paired Comparisons" data, the most
common variety, are data in which, e.g., pairs of stimuli or other
objects are presented to each subject in a paired fashion and the
subject asked to give a binary judgment indicating which of each pair
s/he prefers. (There are many other kinds of dominance data, but
paired comparisons comprise by far the most frequently used.) While
superficially resembling a square two-way proximity data, these
paired comparisons data are usually binary, as mentioned, and tend to
be SKEW-SYMMETRIC rather than symmetric, as standard proximity data
usually are, since, if A is preferred to B, almost by definition B is
DISpreferred to A-- although in certain experimental situations
(e.g., involving certain acoustical stimuli) the two questions ("Is A
preferred to B?" and "Is B preferred to A?") can be asked
independently, and may result in inconsistent responses. There are
many ways to analyze such dominance data, but the one I'm most
closely associated with (in collaboration with Jih-Jie Chang)is
called "MDPREF", in which such data (usually for two or more
subjects-- thus comprising a THREE-WAY matrix of dominance data) are
analyzed via a model in which preference orders for different
subjects are modeled by a "vector model" in which a multidimensional
array of stimuli are projected onto vectors-- or directed line
segments-- in a common R-dimensional space, with an individual's
order or scale value of preference assumed to be modeled via the
order or scale value of projections of these stimuli onto that
individual's vector. MDPREF can also take as input a subjects by
stimuli matrix of preference scale values (or rankings of preferences
by individuals), which actually can be viewed as an off-diagonal
conditional proximity data, except that the preference DATA are
assumed DIRECTLY rather than inversely monotonically related to
actual subject preferences. The metric version of MDPREF (by far
the most frequently used-- nonmetric versions exist but often yield
solutions less desirable via several criteria to the metric version--
even when there's reason to believe the individual preferential
choice data are measured at most on an ordinal scale), can be fit to
these data via some preprocessing (whether of a set of paired
comparisons matrices or of a subject by stimuli preference scale or
rank order matrix) followed by an SVD (singular valued decomposition)
of the resulting rectangular matrix into a product of a stimulus
matrix and a matrix of termini of subject vectors in the common
R-dimensional space. The MDPREF model can be shown to be a limiting
special case of the Coombsian multidimensional unfolding model, in
fact, in which the ideal points are infinitely distant from the stimuli.
Another approach I'm associated with for analyzing preferential
choice data is called PREFMAP, a computer program (or set of two
programs-- PREFMAP1 and PREFMAP2) developed in collaboration with
Jacqueline Meulman and Willem Heiser for "mapping" preference data
into a predefined stimulus space via a hierarchy of preference models
ranging from the vector model, through the "simple" unfolding model,
in which the distances between stimuli and ideal points are defined
as ordinary Euclidean distances, the "weighted" unfolding model, in
which weighted Euclidean distances are assumed, to the "general"
unfolding model in which coordinate axes are differently rotated for
each subject's ideal point, with differential weights applied to
these idiosyncratically rotated axes. The R-dimensional stimulus
space into which the preference data are mapped is usually (but not
necessarily) defined via some form of MDS analysis of separately
collected proximity data (from the same or a completely different set
of subjects)-- so that the PREFMAP approach uses BOTH proximity AND
dominance data to produce a multidimensional representation of
preferences (or other dominance relationships) for a set of stimuli
by a number of human subjects or other data sources. Dominance data
are not limited to preferential choices, but can be data on dominance
relationships involving other aspects of the stimuli or other objects
or entities; e.g., height, length, weight, overall "size", lightness,
warmth, speed, or any other measurable attribute on which judgments
can be made or measures taken indicating that "A dominates B" vis a
vis that attribute. As such, dominance data can be data defining
order or rating scale values on essentially any variable whatever,
with data sources other than judgments by human subjects-- e.g.,
purchase patterns or other behavior of subgroups of consumers or
other people, physical measurements implemented by instruments such
as scales, rulers, light meters, thermometers, or stopwatches-- or
any other source of data defining ordinal, interval, ratio or
absolute scale values for a given set of entities.
I might be wrong in my initial assumption that "PREFSCAL" is the name
of the Heiser and (present or former student) procedure for
multidimensional unfolding analysis-- it may be the name of some
other approach for analysis of preferential choice, or may even be a
generic term referring to any form of analysis of preference or other
dominance data. I'm certain, however, that this is NOT the name of
any specific model or methodology for analysis of preferences with
which I am personally associated!
Best regards.
Doug Carroll
At 01:26 PM 3/11/2007, Art Kendall wrote:
I haven't used PREFSCAL in many years.
The SPSS documentation, says that PREFSCAL uses proximity data, but
I have always thought of preference as dominance data.
I believe the Leiden group, created this section of SPSS software.
Is it the same/similar to Doug Carroll's PREFSCAL?
Does it make a difference in PREFSCAL if the data is proximity or
dominance data?
Back in the 70's, if I had a set of stimuli, e.g., potential
Presidential candidates, for each pair I asked for a zero/one
variable from each subject.
However, I have the impression that these days the degree of
preference, e.g, on a zero to seven response scale, can be used to
analyze not only which member of the pair dominates, but also by how much.
Will the SPSS PREFSCAL handle a matrix of pairwise preference extent
ratings per respondent?
Art Kendall
Social Research Con
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