Peter H. Schonemann
Some theory and results for metrics of bounded response scales
Journal of Mathematical Psychology, 1983, 27, 311-324
In an earlier note, a new metric for bounded response scales (MBR) was introduced which resembles the city-block metric but is bounded above. It was suggested the MBR may be more appropriate than minkowski metrics for data with bounded response scales.
In this article, some formal properties of the MBR are investigated and it is shown that it is indeed a metric.
Empirical predictions are then derived from the MBR and contrasted with those of a "monotonicity hypothesis", which holds that dissimilarity judgements tend to be biased towards overestimation of larger distances, and with predictions of the minkowski metrics, which imply additivity of collinear segments. Some empirical results are presented which contradict the mononicity hypothesis and the minkowski metrics, and favor the MBR.
Finally, the logic used to motivate the MBR is invoked to define a subadditive concatenation for bounded norms in the one-dimensional case, which may be useful in psychometric work where the upper bounds are often real, rather than due to the response scale. This concatenation predicts underestimation for doubling and overestimation for halving and middling tasks.
The "earlier note" referred to in the Abstract is an advance publication, A metric for bounded response scales. Bulletin of the Psychonomic Society, 1982, 19, 317-319. The intended proof that the MBR is a metric contains an error and is superseded by the JMP version. For help with the corrected proof I am indebted to Professor Herrman Rubin, Department of Statistics, Purdue University.
This paper was stimulated by a widely cited study by Krantz and Tversky (1975): Similarity of rectangles: An analysis of subjective dimensions. Journal of Mathematical Psychology, 12, 4-34. On first reading I was favorably impressed with this study. It appealed to me to first investigate the promise of extant scaling and measurement theories in a context where the results can be checked against a relatively solid empirical background, such as psychophsics, before applying them to areas where such background is still lackingt, such as consumer psychology. Since rectangles are simple stimuli that are easily parameterized, they can be systematically varied along well-defined physical dimensions such as height and width or area and shape. Yet, as we were soon to find out, even here appearances can be deceiving
Krantz and Tverski concluded that people use area and shape, not height and width when they judge similarity of rectangles, and that both these dimensions interact.
On reading the Krantz and Tversky paper more carefully, I noticed a contradiction: In the first part of their paper the authors go to considerable trouble to show by ordinal analysis that, even for these simple stimuli, the conventional metric assumptions were violated. In particular, all minkowski metrics were ruled out. However, in the second part, they analyzed the very same stimuli with a packaged program (INDSCAL, see Schonemann, 1972 ) that is predicated on a special case of minkowski metrics, namely the euclidean metric, this time without testing whether it fits their data.
Since the authors did not honor my requests for their data that would have enabled me to evaluate the fit of their model, I had to collect my own rectangle data. Assisted by some of my students, we eventually wound up with 5 or 6 data sets for different types of rectangular stimuli which we analyzed in various ways, On closer inspection we found they all had one striking feature in common: They were all subadditive (a+b > c for all 3 sides of a triangle). This meant that the raw data were already metric, prior to any monotone transfomation. This pervasive fact had been overlooked by most previous authors, including Krantz and Tversky, presumably because most scaling routines then in use automatically interpolated a monotone transformation between the raw data and the data fitted to the scaling model which destroyed the subadditivity.
This paper is the first in a series trying to account for this pervasive subadditivity in dissimilarity data for rectangles (Schonemann, Dorcey, and Kienapple, 1985; Schonemann and Lazarte, 1987; Lazarte and Schonemann, 1991), The basic idea was to postulate a very simple judgment process, that of the city-block metric, and amend it so as to introduce an upper bound to account for the upper bound of the rating scale. One way to do this is to invoke the hyperbolic tangent transformation, which also arises in physics to account for the boundedness of addition of high velocities. Its inverse is the well-known Fisher z transformation of statistics.
Of course, their are numerous other transformations that would equally have taken care of the upper bound. The appealing part about the hyperbolic tangent function is that it accounts for the pervasive finding that a few special cases of the minkowski family seem to do a fairly good job in modelling most dissimilarity data. The MBR accounts for this by predicting city-block distances for small distances (on a scale from 0 to1), the sup metric for large distances, and euclidean distances for intermediate distances. Borg and Groenen: Modern Multidimensional Scaling: theory and applications. New York: Springer, p. 295f present a readable account of the MBR.