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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 Wisan’s 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 Chauvenet’s Criterion for the elimination of outliers, an option now called “MAXVAL.”

It is also clear from Wisan’s dissertation that the Galileo group at Illinois at that time were very uncertain about the dimensionality of the Galileo space, and nearly clueless about it’s non-euclidean character. At the time of Wisan’s 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.

It’s 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 it’s 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.

It’s 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 everyone’s 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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