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The Galileo Group at the University
of Illinois
Curiously, the psychometricians who developed multidimensional scaling
seldom if ever measured anything. For the most part, these workers
were mathematicians, statisticians, computer programmers and theorists
who tested their algorithms on small, well known data sets which
were passed from team to team. Meanwhile, the Galileo group at the
University of Illinois in Urbana-Champaign conducted extensive measurements.
The first Galileo measurement was a hand drawn picture of Alice
in Wonderland, and undergraduate students were asked to estimate
how far apart objects in the picture were from each other. The mean
distance estimates provided by the students were submitted to an
unstandardized factor analysis implemented in SOUPAC (Statistically
Oriented Users Package), a proprietary University of Illinois
statistical package, and coordinates were plotted on the inside
of a large cardboard box. A peephole was cut into the side of the
box that provided a perspective view that corresponded to the drawing.
Encouraged by the ability to reproduce the picture, several more
measurements were initiated. Gail Wisans dissertation (1972)
showed the state of the art at the time. She tried several mathematical
procedures to produce the Galileo space, including factor analysis
of cross products, factor analysis of scalar products, and non-metric
multidimensional scaling (TORSCA). Of these, unstandardized factor
analysis of. the scalar products was most promising, particularly
when scaling known configurations, such as the distances among the
corners of a brick, which both the unstandardized factor analysis
and TORSCA could scale perfectly. Whens small amounts of random
error were added to the data, however, the factor analysis solution
stayed roughly the same, but the non-metric solution deteriorated
badly. Similar results were found with the distances among the features
of the human face. Due to the many symmetries, TORSCA solutions
were very distorted, even though stress measures remained low, giving
a misleading representation of the quality of the solution. She
also showed that the ratio scaled paired comparisons were sensitive
to extreme values with small sample sizes, which subsequently led
later users to adopt modifications of Chauvenets
Criterion for the elimination of outliers, an option
now called MAXVAL.
It is also clear from Wisans dissertation that the Galileo
group at Illinois at that time were very uncertain about the dimensionality
of the Galileo space, and nearly clueless about its non-euclidean
character. At the time of Wisans dissertation, the group clearly
knew that the orientation of spaces was arbitrary, and had written
special software to select a common origin and rotate to a common
orientation, but this software only worked up to three dimensions.
Clearly, even Woelfel expected (or hoped) that the space would be
euclidean and of low dimensionality, but none of the known methods
for determining the dimensionality of the space appeared very useful.
As always, the group turned to careful empirical measurement to
resolve the issue, using procedures for evaluating reliability and
validity learned while developing the Wisconsin Significant Other
Battery.
Barnett showed that the ratio scaled paired comparisons were precise
and reliable at very small sample sizes, including the imaginary
eigenvectors, which he speculated might be due to the concepts scaled
having volume instead of being dimensionless points. If the objects
scaled were hyperspheres of non zero radius, all the inter point
distances would be attenuated as an additive function of the radii.
Its difficult to convey the extent to which the preconception
that the space of cognition ought to be low dimensional and euclidean
dominated workers during this period. Psychometricians abandoned
the use of classical Young Householder Torgerson multidimensional
scaling virtually completely in favor of the new non metric programs
such as TORSCA (TORgerson SCAlar products) and KYST (Kruskal, Young,
Shephard, Torgerson), and later ALSCAL, even though the coordinate
systems made by these programs were frequently terrible. Even the
Galileo researchers, whose measurements led them inexorably to accept
and embrace the Riemanian character of the space of cognition, initially
spent copious amounts of energy and worry over the non-Euclidean
character of the Galileo spaces.
The contrast between the response of the psychometric community
and the Galileo group to the crisis of the non-euclidean
structure of classical MDS is interesting and instructive. The psychometric
community, with its platonic orientation and considerable
mathematical skill, spent its research efforts developing sophisticated
mathematical and computational algorithms to transform away any
non-euclidean components, while the Galileo group, first at the
University of Illinois and later at
Michigan State University the East
West Center in Honolulu, and the University
at Albany, implemented a series of studies to measure
the reliability, stability and extent of the non euclidean components
under a variety of conditions. As already noted, Wisan showed that
the ratio scaled pair comparisons were precise and reliable, and
produced meaningful spaces which corresponded to external events.
Gillham
and Woelfel showed considerable stability in a three
point in time study.
Meanwhile, returning to the original question of how to average
discrete expectations, John Saltiel, originally a graduate student
at the University of Illinois and at that time on the faculty at
Montana State, arrayed the most chosen occupations for a group of
Montana high school students in a Galileo space and showed that
the occupation most likely to be chosen by a given student lay at
the mean of the coordinates of the discrete occupations his/her
significant others expected, confirming athe model hypothesized
by Woelfel earlier.He also showed that important attributes known
to be related to occupations, such as socioeconomic status, percent
female incumbancy, and so on, could be projected into the space
with considerable precision.
Its instructive to note that Saltiel used an unstandardized
factor analysis of scalar products, the method that worked best
in Wisan, but which also reveals substantial negative eigenvalues,
which indicates a non-euclidean space. Saltiel removed these negative
eigenvalues by subtracting the largest negative root (lambda min)
from all roots and renormalizing, a method suggested to the Illinois
group by Jim Lingoes of the University of Michigan. The only reason
he did so was everyones discomfort with a non-euclidean model,
and the suspicion that the non euclidean character of the space
was the result of measurment error.
Woelfel,
Kincaid, Newton and Holmes later showed that the
space of occupations was stable over repeated measures and in places
as different as New YHork and Hawaii, and Woelfel
and Barnett showed that the imaginary eigenvectors
could not be attributed to error of measurement, but were precisely
measured characteristics of the space of cognition. In powerful
testimony to the hold of Plato over the American psychometric community,
this important article was published in Europe. Tom
Gordon showed that the ratio scaled paired comparisons
method scaled consistently under changes of scale when analyzed
with the Galileo program.
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