Similarity of rectangles

JOURNAL

OF MATHEMATICAL

16, 161-165

PSYCHOLOGY

(1977)

Note Similarity PETER

Department

of Psychological

Sciences,

of Rectangles H.

SCH~NEMANN

Purdue

University,

West Lafayette,

Indiana

47906

Krantz and Tversky found that neither (log-) height (y) and width (x), nor area (x + y) and shape (x - y) qualify as “subjective dimensions of rectangles” because both pairs violate the decomposability condition for their dissimilarity data. However, the data suggest a nonlinear transformation of x, y into a pair of subjective dimensions u(x, y), v(x, y) for which decomposability should be approximately satisfied. An explicit statement of this mapping is given.

Krantz and shape and area the techniques to test the data

Tversky (1975) analyzed dissimilarity data for 17 rectangles of differing for two groups of eight and nine subjects, respectively. They employed for ordinal analysis described in Beals, Krantz, and Tversky (1968) for “decomposability” for two pairs of potential “subjective dimensions,” u*(x, y) = x

and

zJ*(% y) = y,

(1)

where y = (log-)

x = (log-) width, are the physical

dimensions u**(x,

They

found

of the rectangles,

in both instances

systematic

= ~Wi

(2)

7J**(x, y) = x - y.

(3)

and also

and

39 = x + Y

height

violations

of the decomposability

condition

, uA; w% > %h

(4)

where 6(x, , yi ; xj , yj) denotes the observed dissimilarity value for the pair of rectangles respectively. The function F(g, h) strictly (i,i) with width xt , xi and height yi , yi , increases in each argument g, h, and g, h are two positive symmetric functions on (u, r~), i.e., satisfy g(u, , uj) = g(uj , IQ) > g(ui , ui) = g(z+ , ZQ) = 0, with an analogous definition for h(v, , Vj). This decomposability definition can be viewed as a generalization of the well-known Minkowsky metric in two dimensions h@* , fJi ;

Uj

9 Wj) = (I Ui -

Uj

1’ + 1Vi - zlj /r}l”,

(5)

161 Copyright All rights

Q 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSN

0022-2496

162

PETER

H.

SCHiiNEMANN

where g, h capture the “intradimensional subtractivity” and F the “interdimensional additivity” of this class of distance functions. Since (5) is a special case of (4), all Minkowsky metrics and, in particular, the Euclidean metric (r = 2), satisfy the decomposibility requirement (4). Since Krantz and Tversky found it violated for both (u*, n*) and (u**, v **) upon ordinal analysis of the dissimilarity data, it follows that these data cannot be monotonically regressed (via F) on any of these distances defined on that the data cannot be (a*, v*) and (u**, v**). This d oes not mean, however, monotonically regressed on Euclidean distances for some other choice of u and v. This point is sometimes misunderstood. It is easy to show that any set of ps real numbers which satisfy the first two distance axioms can always be carried into Euclidean distances by a simple parabolic transformation. A very simple proof of this result was given in Schonemann (1971). A somewhat deeper argument is necessary to establish the stronger claim that this can always be done in no more than p - 2 dimensions (Lingoes, 1971; Guttman, 1968). In the second part of their study Krantz and Tversky show that the dissimilarity data can indeed be monotonically regressed on Euclidean distances which, on the assumption of a fit of Horan’s (1969) subjective metrics model, permit an embedding of all 17 stimuli in a two-dimensional common space. They achieved this by first regressing the dissimilarity judgments for each subject separately by M-D-SCAL IV (Kruskal, 1964) on Euclidean distances. They report a satisfactory fit for each subject in two dimensions. They then employed Carroll and Chang’s (1970) INDSCAL program for fitting Horan’s subjective metrics model, but apparently did not test the fit at this stage. Since the main theme of the study revolves around the falsification of models and decomposition rules it should be noted that Horan’s model is a relatively strong model with two clear-cut falsifiability constraints. As was shown in Schijnemann (1972) the two basic assumptions which characterize the model underlying INDSCAL are: (i) All scalar product matrices which can he derived from the presumed Euclidean distance matrices upon fixing an origin must be in the same m-dimensional column space, and (ii) all subject-specific metrics (C,) must have the same eigenvectors. These are evidently two fairly strong conditions which cannot be taken for granted to hold for all conceivable data. A program for testing both these constraints explicitly, subject by subject, has recently been completed by Schiinemann, Carter, and James (1976). Although such detailed checks on the fit on Horan’s model were apparently not made in this study a reasonable fit is likely, nevertheless, because both the common space plot and the individual weights checked well with the results of the previous ordinal analysis. The most striking result of this combined scaling and measurement analysis was that perceived shape differences increase with perceived area differences. This interaction between the two subjective dimensions was apparently the main reason that decomposability was violated for both (u*, v*) and (u**, v**). The main point of this note is to show that the results of this analysis also provide the key for defining two subjective dimensions of rectangles, (a, v) in terms of (x, y), for which decomposability should be approximately satisfied, provided the fit of both scaling models was satisfactory, as assumed. To express these two dimensions in terms of x and y it is only necessary to compare the configuration in the two-dimensional

SIMILARITY

OF RECTANGLES

163

physical space, a square, with the two-dimensional configuration in the subjective space, a symmetric trapezoid (Krantz and Tversky, 1975, Figs. 2, 4). If we can find a mapping U(X, y), V(X, y) which carries the square into the trapezoid, then U, z, are two subjective dimensions for which we must have decomposability, because Krantz and Tversky found that the stimuli can be embedded in a Euclidean space of two dimensions, u, o, and Euclidean distances satisfy decomposability. The function F in (4) is provided by M-D-SCAL which relates the subjective dissimilarities to the subjective Euclidean distancesin the (u, V) spaceby a monotonic transformation. The construction of such a mapping of the physical into the psychological coordinate system is more readily explained with reference to Fig. 1. Figure la is a somewhat simplified reproduction of Fig. 2 in Krantz and Tversky (1975). Since the original physical measures,height and width, are on a common ratio scale,the derived measures x and y are a common difference scale. To remove this additive indeterminacy the derived measuresare expressedrelative to those of the central stimulus 17 in Fig. lb. Moreover, since the designwas chosenso that the physical measuresof adjacent stimuli stood in constant ratios (I = 1.3), such adjacent stimuli are separatedby equal intervals in Fig. lb. On taking logarithms relative to baseY, the coordinates range from -k to k if there are 2k + 1 levels on each dimension. The central stimulus 17 thus has coordinates (0,O). Stimulus 7, which is k levels above 17 on the height dimension, hascoordinates (0, k). These logarithmized scalescan be used to define two new physical scales a* = y* + x* (log) area,

s* = y* - x* (log) shape.

(6)

Note that “shape,” as defined here, is the reciprocal of “shape” as defined by Krantz and Tversky. In the (a*, s*) coordinate system the stimuli plot is as shown in Fig. lc. To obtain the mapping of this derived physical coordinate systeminto the psychological coordinate system which resulted from the Euclidean embedding of the common space analysis (Fig. 4 in Krantz and Tversky; here reproduced as Fig. Id) it is necessary to account for the differential distortion of the ordinate with increasing abscissavalues. Since the trapezoid in Fig. Id is symmetric this is accomplishedby adding a correction term ca*s* to s*, where c is a suitably chosenconstant. A rough estimate for ck, based on Fig. Id, is ck = 0.3. We thus have u = a*

and

v = s*(l + ca*>

(7)

for the mapping which carries the derived physical coordinates a*, s* into the psychological dimensionsu, 0. The definition of z, in (7) precisely reflects the main empirical finding reported by Krantz and Tversky: perceived shape differences increase with perceived area. More importantly, u and v should be endowed with the “defining properties for subjective dimensions” as Krantz and Tversky (p. 5) demand. In particular, they should satisfy decomposability to a much higher degreethan either (u*, v*) or (u**, v**), especially for those subjectswhosedissimilarity data gave a good fit for the commonspaceanalysis. These predictions could be tested empirically, e.g., with the program mentioned above.

164

PETER

Y=1m 7 6 00 1

17

2

H.

SCHijNEMANN

5

4

-k

3 x = 1nw at

b

-k/2

-k

0

k/2

k

' k(l+ckJ k

‘7

1~

k (l+ck/ZJ/Z .17

0

a*=x*+y* -k Cl-ck/ZJ/Z .k(l+ck/ZJ/Z

-k

-5

3'

; -k(ltckJ

c

-k

b

k

d

FIG. 1. (a) Physical space (see Krantz & Tversky, 1975, (c) Translated and rotated physical space. (d) Subjective Fig. 4).

-k

k

Fig. 2). (b) Translated physical space (see Krantz & Tversky,

space. 1975,

It has been asked how the present approach relates to the alternative models which Krantz and Tversky discussat the end of their paper (p. 32). As I seeit the basicdifference is that Krantz and Tversky first define the subjective dimensionsad hoc and then check on decomposability. If they find it violated, they go on to investigate more complicated dissimilarity functions, e.g., Eq. (6) in their paper. In contrast, in the approachsuggested here, the subjective dimensionsare somewhatmore complicated, but empirically derived functions of the physical dimensions. The dissimilarity function is simpler, namely a Euclidean distance, and decomposability is guaranteedasa consequenceof the successful Euclidean embedding. Finally it should be noted that the basic idea, to infer the relation between physical and subjective dimensionsfrom a comparison of the configuration in both coordinate systems, could be useful more generally, provided, of course, an explicit mapping

SIMILARITY

OF RECTANGLES

165

between both coordinate systemscan be found. This may not always be as easy as it was here, largely as a result of the careful planning and execution of the experiment and the insights provided by the thorough theoretical analysis of the investigators. REFERENCES BEALS, R., KFUNTZ, D. H., & TVERSKY, A. Foundations of multidimensional scaling. Psychological Review, 1968, 75, 127-142. CARROLL, J. D., & CHANG, J. J. Analysis of individual differences in multidimensional scaling via an N-way generalizations of “Eckart-Young” decomposition. Psychometrika, 1970, 35, 283-319. HORAN, C. B. Multidimensional scaling: Combining observations when individuals have different perceptual structures. Psychometrika, 1969, 34, 139-165. GUTTMAN, L. A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 1968, 33, 465-506. KRANTZ, D. H., & TVERSKY, A. Similarity of rectangles: An analysis of subjective dimensions. Journal of Mathematical Psychology, 1975, 12, 4-34. KRUSKAL, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 251-263. LINGOES, J. C. Some boundary conditions for monotone analysis of symmetric matrices. Psychometrika, 1971, 36, 195-203. SCHBNEMANN, P. H. On two points of view. Revision. Unpublished mimeo, Purdue University, 1971. SCHBNEMANN, P. H. An algebraic solution for a class of subjective metrics models. Psychometrika, 1972, 37, 441-451. SCHBNEMANN, P. H., CARTER, F. S., & JAMES, W. L. Contributions to subjective metrics scaling. I. COSPA, a fast method for fitting and testing Horan’s model, and an emperical comparison with INDSCAL and ALSCAL. Institute Paper No. 587. Purdue University Krannert Graduate School, 1976.

RECEIVED: July 13, 1976