Doug,
Thanks for such a swift, informative and thorough response!
Now that you say it, I recall that the programs I tried to convert to
DEC-10 in the mid-70s *were* MDPREF, PREFMAP, and PREFMAP. (and SINDSCAL).
I have data for the 1976 and a few later elections. Unfortunately,
they are are on old magtapes that I am trying to restore. *If *I can
be sure I can recover my data before the deadline for the CSNA
submissions, I am going to try to analyze these data to present at CSNA.
If I can recover it, for 1976 I'll have data from 210 politically
elite respondents who provided pairwise comparisons for similarity
on a 1 to 6 response scale and pairwise comparisons for preference on a
1 vs 0 response scale. The stimuli were 10 people who were potentially
US Presidential candidates. In my dissertation I used SINDSCAL to look
at the similarities. I have not yet related the similarities and
preferences.
Art
J. Douglas Carroll wrote:
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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