Multidimensional similarity: an experimental and theoretical comparison of vector, distance, and set theoretical models: I. Models and internal consistency of data

Acta Psychologica 41 (1977) l-46 0 North-Holland Publishing Company

MULTIDIMENSIONAL SIMILARITY: AN EXPERIMENTAL AND THEORETICAL COMPARISON OF VECTOR, DISTANCE, AND SET THEORETICAL MODELS* I. MODELS AND INTERNAL CONSISTENCY OF DATA

I-Iannes EISLER** and Edward E. ROSKAM

University of Stockholm, Sweden and University of Nijmegen, Holland

In an experiment (conducted by the senior author) subjects were instructed to give

three kinds of quantitative judgment on pairs of simple geometrical stimuli: similarity,

commonality ratio and magnitude ratio. It was assumed that the different kinds of

judgments could be interpreted in terms of one cognitive structure, and that formal

models for each of these judgments should be validated by predicting one kind of

judgment from knowledge of another kind of judgment. The present report proposes

two different, though related, systems of formal models for similarity, commonality and

magnitude ratio judgments. One, called E-model, interprets the judgments in terms of

set- and vector-representations, and connections between them; the other, called R-

model, interprets the judgments in terms of set- and distance-re,presentations and their

connections. The results are slightly in favor of the E-model. In a subsequent paper, the

authors will report the results of multidimensional analyses for the same data, which

slightly favor the R-model.

0. Introduction’

This paper presents models which provide an interpretation of three judgmental measures used to express relations between stimuli, or

* This work has been supported by the Swedish Council for Social Science Research. We are

indebted to Ulf Forsberg, Sten Nilsson and Cees Elzinga for experimental and computational

assistance.

** Requests for reprints should be sent to Hannes Eisler, Department of Psychology, University

of Stockholm, Box 6706, S-113 85 Stockholm, Sweden.

’ This report presents the main results of the two authors’ efforts to achieve a theoretical

interpretation of data from an experiment done by the senior author. His original report stimulated the second author to continue the analysis along somewhat different lines. Rather

than publish two papers, the authors have tried to unite their hypotheses and findings as coherently as possible, notwithstanding the fact that they proceeded differently in many

details, and have different opinions on certain aspects of the material.

2 H. Eisler, E. E. RoskamfMultidimensional Similarity, I

rather percepts: similarity, multidimensional magnitude ratio, and commonality ratio. The first one is well known, the second was devised by Ekman (1963), and the third was developed for the present study in order to operationalize and test a theoretical model implied by an argument by Ekman (196 1). These three measures will be identified in the first place by the instruction to the experimental subjects used to elicit the data. Section 2 presents these instructions and the experimental procedure, together with some preliminary data on consistency and interindividual correlations.

Two forms of models have been developed to interpret the data. In Section 3, a model is described which was developed by Eisler (un- published report 1967) following Ekman’s theories, but differing from them in some respects. This model is referred to as E-model.

In Section 4 a modified model developed by the second author is presented. This is refirred to as R-model.

Each model permits identical tests for internal consistency of the commonality ratio and the magnitude ratio ratings. These tests are discussed in Section 5.

Section 6 presents the specific predictions between the three data sets for both models. Finally, both models permit multidimensional analyses of the data, presented in Part II of this paper.

1. General considerations

The concept of similarity has been widely used and has been investigated by many authors (e.g. Landahl 1945; Attneave 1950; Torgerson 1952; Ekman 1954; Shepard 1962; Gregson 1975). It has been impor- tant in many areas of experimental psychology, and has often been used as a convenient tool to explain, for instance, generalization effects.

The present investigation aims at understanding the concept of similarity by analyzing data from subjects who are instructed to express judgments of similarity. Such data can, of course, be analyzed by invoking a theory about similarity judgments. The present study is concerned with the explanatory and predictive power of different models regarding similarity judgments and the subsumed subjective (perceptual) space.

In general, a model may or may not be externally testable. By the latter, we mean that a theory connects several sets of independent data,

H. Eisler, E. E. Roskam/Multidimensional Similarity, I 3

so that one set can be predicted from another. Some measure of the precision of these predictions will testify to the appropriateness or validity of the theoretical interpretation of each of the sets of data. For the case at hand, where we use mathematical models to interpret the data, we may assume that perceptions can be represented in a ‘subjective’ space, which may give rise, for different instructions, to different judgments having specific quantitative relations.

If a model is not externally testable, it can at best be validated by the internal consistency of the data, or otherwise merely serve as a tool for data reduction. The distance model for similarity data, for instance, is becoming a widely used tool for analyzing data (cf. Shepard et al. 1972: Green and Rao 1972) whereby a (large) empirical data matrix is replaced by a comparatively small amount of, hopefully, more essential information. The adequacy of such a model can be assessed by (a) the internal structure of the data, if the model implies certain testable conditions, (b) the precision with which the original data are repro- duced from the fitted quantities, and (c) by the reasonableness of the outcome of the analysis, with the experimenter as the judge.

Ekman’s (1963) vector model for multidimensional magnitude ratios and Hays’ (1958) model for trait implication are other examples of models which are often not externally tested.

The aim of the present study is to apply such external tests. Two classes of models’ for similarity can be distinguished. The first con- siders similarity as a monotonically decreasing function of differences in perceived attributes, represented as distances in subjective space; this class is probably used most frequently. Its theoretical and computational characteristics have been investigated by, e.g., Attneave 1950; Torgerson 1958; Shepard 1962; Kruskal 1964; Beals et al. 1968; Tversky and Krantz, 1970. These models may be called difference modek3 The perceptual space is characterized by the absence of a fixed origin; the percepts are represented on a multidimensional interval

2 Usage of the terms ‘model’, ‘theory’, ‘representation’, and ‘interpretation’ is not entirely

satisfactory in most of the recent literature. The mathematical representation of an empirical

system is sometimes called a ‘model’, sometimes a ‘theory’. Also it is sometimes said that

mathematics (theory or model) provide an interpretation of the data or that a particular mathematical system represents a theory. We have reframed from trying to be very rigorous in

our usage of terms and appeal to the goodwill of the reader to understand our objectives. 3 Ekman and Sjoberg’s (i965) denotation ‘distance model’ seems to refer to the (geometrical)

model’s representation rather than to the model itself.

4 H. Eisler, E. E. RoskamlMultidimensional Similarity, I

scale. The dimensions have the property of intra-dimensional subtracti- vity and inter-dimensional additivity.

The second class of models, christened content models by Ekman and Sjijberg (1965), is based on magnitude measures of single percepts and of the common elements in pairs of percepts. Thus, similarity is considered as the ratio between some measure of commonality and some measure of totality. The perceptual space has a fixed origin, and the percepts are represented on a multidimensional ratio scale. Related models are Hays’ (1958) and Restle’s (196 1). Content models seem to invite first and foremost a set-tlzeoreticd representation. But, as will be shown, a representation in terms of vector space is also possible.

If experimental data appear to fit both a content model and a difference model - as is the case with the data of the present experiment - the question arises whether a theory can combine the two models. Such a theory would, in any event, have to reinterpret each of the models; an attempt to develop such a theory is made in Section 4.

2. The experimental design Fifteen simple geometrical stimuli were used. Three kinds of numerical judg-

ments, that is ratings, were collected on each pair of stimuli: (i) S: similarity ratings; the subject rated the similarity of a pair of stimuli on a

O-l 0 scale (0 meaning no similarity at all; 10 meaning identity). Ratings were then converted to a O-l interval through division by ten. These data are denoted oSii,4 where i and j refer to the two stimuli constituting the pair.

(ii) C: commonality ratio ratings; the subject gave a ratio, ocii, expressing how much of percept J is common to 1 and J.

(iii) Q: multidimensional magnitude rutio ratings: the subject gave a ratio, oqii, stating how much of or how many times percept J makes percept I.

2. I. Stimuli

The stimulus material consisted of points in a plane, that is, each stimulus consisted of two luminous points. An example of a pair of stimuli is shown in fig. 2.

The choice of stimulus material was determined by the requirement of having as few dimensions as possible, but more than one, which would be easily identified a

’ We use the subscript o preceding a symbol to indicate observed quantities as distinguished from theoretical quantities, and from computed estimates (the latter will be indicated by ^ , e.g.

2~). The reader is asked to excuse a slight lack of elegance since later WC will use only mean values (averaged over subjects) as data, and use the subscript o not to indicate data from individual subjects, but these mean values.

H. Eisler, E. E. Roskam/Multidimensional Similarity, I

1 0 0 0 0 0 0 0 0 5

cm T 0 0 0 0 0 0 0 0

Fig. 1. The panel for the stimulus display

Fig. 2. Left: Arrangement and names of the stimuli. Right: The stimulus display as seen by a subject for the stimulus pair FM.

priori. The purest two-dimensional space, at least physically, are points in a plane;

even with such stimuli, however, the subjective, perceptual space need not be two-dimensional. The results in Part II indicate indeed that the perceptual space definitely is not two-dimensional.

The experiment was carried out in a light-proof room. On a table there was a black box, the front side of which was a metal sheet where two sets of 16 small holes were drilled, each set making up a square. With the exception of the lower left hole in each, and two variable holes, one in each square, all the holes were closed with tightly fitting removable nails (see fig. 1). Inside the box was a lamp, and from behind the E had access to the removable nails. The S was seated in front of the box on a chair adjusted in such a way that his line of sight was midway between and halfway up the two squares. The S’s distance from the display was 1 m.

Each stimulus consisted of a pair of luminous points in one of the squares, one fixed (the lower left) and one variable. Stimuli were changed by moving a nail from one hole to another while the light was off.

We shall refer to the 15 stimuli by letters indicating the variable point, as indicated in fig. 2.

It should perhaps be made quite explicit that one stimulus consisted of a pair of luminous spots, and that each presentation consisted of a pair of stimuli. Also, each stimulus (two luminous spots) shall be represented by one point in a subjective space.

The 15 stimuli provided 120 stimulus combinations (pairs), including identical stimuli. The total number of presentations was 210 (omitting pairs of identical

6 H. Eisler, E. E. Roskam/Multidimensional Similarity, I

Table 1. Experimental tasks and groups of subjects

Group I (12 = 13) Group II (II = 12) Group III (n = IS)

S-data C-data Q-data M-data x X X X X X X

stimuli), with the left and right positions of a given stimulus pair randomized. The

sequence of presentation of the stimulus pairs was likewise randomized, with different orders for different 5’s.

2.2. Subjects und experimental groups

There were three groups of Ss. They produced the data as indicated in table 1. Half of the Ss in each group made the similarity ratings before the other task, the other half afterwards.

Each S judged the same stimulus pair two times (switching ‘standards’ [see below] among the two stimuli for the two ratings of each pair) for the commonality ratio rating (Groups 1 and II) or the magnitude ratio rating (Group III), and two times for the similarity rating (all groups). Pairs of equal stimuli were estimated in all three tasks but are not included in the analysis.

Thus, each S made 480 estimations altogether for the S- and C- or Q-data. They were divided between four sessions. Each session lasted about one hour or less.

The commonality data from Group I had to be discarded for reasons given below. In addition to the magnitude ratio ratings of pairs of stimuli, the subjects in Group III also made direct magnitude estimation of the length and of the angle of each stimulus. These data are referred to as M-data in table 1, and will be dealt with in Part 11.

2.3. The sim ilurity ru ting data

For the similarity ratings, the Ss were instructed to ‘Estimate the similarity between the two patterns (in front of you, each pattern consisting of two bright points) on a scale between zero and ten, where zero stands for no similarity at all and ten stands for identity. Make your estimates as precise as you can. Use decimals if you like. Here are a few examples for practice’. The words in parentheses in this quotation of the literal instruction were actually given as a part of a prior general instruction about the experimental situation.

The data for each stimulus pair were pooled over all Ss and replications by taking the arithmetic mean. For each S, the correlation of his data (105 values) with the pooled values was computed, and two Ss, who correlated only 0.646 and 0.424, were removed (the next lowest correlation was 0.782). The arithmetic averages, after removing these two Ss and after dividing by ten to obtain values between 0 and 1, constitute the S-data, which are in the sequel referred to as oSii (each based on 76 values from 38 Ss).

H. Eisler, E. E. RoskamfMultidimensional Similarity, I I

We also calculated the correlation coefficients between the group means, taken two at a time; these were all 0.98.

The mean os taken over all pairs (i, i) was 0.499 and the standard deviation of oSii was 0.20 (the same values for each of the groups separately were within 0.02 of the pooled values). The oSii data are given in Appendix A.

2.4. The commonality ratio rating data

Two groups (Group 1 and Group 11) made commonality ratio ratings. We will not include the instruction for Group 1, since it apparently failed. The commonality data of this group showed certain peculiarities which raised the suspicion that the Ss had misunderstood the instruction. In general, we expect that the commonality estimate, ‘how much of percept J is common to both I and J’, (Cii), is different from the estimate ‘how much of percept I is common to both I and J’, (Cii), and that neither is equal to the similarity rating. Plots of c,. o 4, oCii and oSii as given by the SS

in Group I showed no great differences between these values. Their data were therefore discarded, and the commonality experiment was repeated with Group 11, using a new instruction. The Ss in Group II were trained to use and interpret the concepts of set theory, by means of ellipses with different degrees of overlap, before carrying out the experiment proper.

Although one might object that the Ss in this experiment were certainly not naive, one should consider that our interest is not in the ‘average’, nor in the ‘typical’ subject, but in a particular form of cognitive judgment which requires fairly sophisticated subjects who are capable of carrying out the task required.

The instruction for the commonality rating experiment in Group II was given before the training just mentioned, and reads as follows.

‘In front of you, you see two point patterns, each consisting of two bright points. One pattern lies to the left, the other to the right. Each pattern gives rise to an undefined sensation, and the two sensations have something in common. One of the patterns will become your reference pattern. Your task is to estimate how large a part of the reference pattern constitutes the common sensation. Make your estimate as precise as you can. Give your estimates in any way you like, fractions, decimal fractions, percentages or otherwise. Before each presentation, I shall tell you which pattern you are to regard as the reference pattern by saying ‘left’ or ‘right’, which means that the left or right point pattern is your reference pattern. To check that we agree about which of the patterns is your reference, I want you to say clearly in every estimation which one you regard as the reference pattern. An estimation may for instance be like this: “2/9ths of the right”.’

After this, the Ss were given a sheet with ten pairs of ellipses with varying degrees of overlap and the overlapping parts shaded. The sheet was headed with an instruction:

‘Here are a few figures designed to emphasize the line of thought in the instruction. Areas are used instead of point patterns. How large a part of the reference pattern is made up by the common sensation? Write your answer in the columns headed “reference pattern LEFT”, “reference pattern RIGHT” to the side of the figures’.

8 II Eisler, E. E. Roskam/Multidimetzsional Similarity, I

Clearly incorrect responses (for nested ellipses, or ellipses overlapping complete- ly or not at all) were corrected by the E. Then the following instruction was read:

‘You are now to transfer the line of thought from the areas to the point patterns. The instruction is thus (repetition of the foregoing). Here are a few practice runs’. A few point patterns were shown and the instruction continued: ‘You have now decided how you will make your estimations. Is there anything you want to ask about? Remember to state clearly which pattern you regard as the reference pattern. Your task is thus to estimate how large a part of the reference pattern is made up by the common sensation’.

Of the 13 Ss, only eight could be retained. The judgments of one of those excluded correlated about zero or negatively with the rest, and one showed a plot of ocii and ocji vs. osii which was symmetric about the 45O line, as if he had carried out similarity ratings (like the unsuccessful Group I). The three other excluded Ss appeared to base their judgments solely on the size (length) of the point patterns, since they almost always gave a commonality ratio equal to unity when the pattern with the smaller distance between the points was the reference.

The relation between “ii and Sij of a ‘good’ S is shown in fig. 3.’ The means of the ratings of the eight Ss are given in Appendix A. The averages are referred to as

o 11 c.. when i is the reference, and o”ij, when j is the reference.

Similarity 0Sij

Fig. 3. Commonality ratio ratings vs similarity ratings, illustrating the assymmetry around the line y = x.

’ The figure actually shows averages.


2.5. The multidimensional ratio rating data

The Ss in Group III made multidimensional ratio estimations. They were instructed as follows:

‘(You are now familiar with the experimental situation, but this time we want another kind of judgment.) Look at one of the point patterns, say the left one. It is called the standard and is thought to contain a certain magnitude of an undefined sensation. Your task is to state how much of this sensation or how many times this sensation can be found in the other pattern. You may make your estimation in any way you like: fractions, decimal fractions, percentages, etc. Before each presentation I shall tell you which one is to be considered the standard by saying ‘left’ or ‘right’. To check that we agree on which is the standard, I want you to state clearly which one you regard as the standard in every estimation. As examples I can mention a few estimations in which the left is regarded as the standard: ‘l/3 of the sensation to the left is found in the right’, or ‘exactly the same sensation as the left is found to the right’, or ‘right contains 7.3 times the left sensation’, or ‘87% of the left sensation is found in the right one’.’

Fifteen Ss took part in this experiment. Two were excluded since their estimates correlated only 0.217 and 0.351 with the group mean, the next lowest correlation being 0.607.

Occasionally, one S’s estimate for a pair of stimuli was removed, when it was clearly recognizable as an ‘outlier’, probably due to confusion of the standard with the variable stimulus (a typical series of responses to a particular stimulus pair was: 0.10, 0.25, 0.30, 0.30, 0.33, 0.40, 0.40, 0.40, 0.60, 0.60, 3.00 with 3.00 the outlier). The magnitude ratio judgments, given in Appendix A, are the averages of 13, sometimes 12, and occasionally 11 Ss. These data are referred to by the symbol

o4ij, when j is the standard.

3. Set (content) and vector models (E-model)

Content models (Ekman and SjBberg 1965) consider the similarity of a pair of stimuli as the ratio between the percept of what is common to both stimuli (‘commonality’) and the percept of the two stimuli taken together (‘totality’). This concept of similarity is reminiscent of Restle’s (196 1). Restle, however, defines similarity as the measure of the intersection between two sets of stimuli, their commonality, without norma- lizing through division by the totality. Since the measure of the intersection is independent of the measure of either of the sets alone, two ‘small’ stimuli could be identical and yet have a smaller similarity than two ‘big’ stimuli which have only a small fraction in common. This seems unreasonable. Besides, when normalized as in the content model, similarity varies between 0 (no similarity at all) and 1 (identity), which

10 H. Eider, E. E. Roskam/Multidimensional Similarity, I

seems plausible and is expedient for ratings and subsequent data treatment.

Whereas the interpretation of commonality as the measure of the intersection of both sets is rather obvious, totality may be defined in at least two ways, namely, as the measure of the union of two sets, or as the sum of the measures of each set. In Section 6.4 the two definitions of totality will be compared empirically. Since the latter one agrees well with the data, we write similarity. in set theoretical terms, as (cf. Eisler 1964):

2m(l n J) Sii = ~.____.__

/n(I) + m(J) ’ (3.1)

where Sii denotes the similarity between percepts I and J, and TM(I), m(J) are measures associated with each percept conceived of as sets of perceptual elements. The ?,z(I n J) is the commonality measure and the commonality ratio6 with j as the reference stimulus is:

It is easily seen that sii is equal to the harmonic mean of cii and cii:

1 /Sij = + (1 /Cij + 1 /Cji), i.e.

2CijCji

sij= --->-* Cij + Lji

(3.2)

(3.3)

Both Cij and Cji lie between 0 and 1, and in the C;IS~ of nested sets, e.g. I c J , we have Cji = 1.

We may conceive of the commonality measure ~H(I n J), the common part of the two percepts, as a fraction of the smaller percept. Denoting this fraction by +ij, 0 G $ij < 1, we see that it equals the larger commonality ratio, since, if rn(InJ) = $ij - min[l?z(l), m(J)], then

+ii = -: m(In J)

p= ITlLlX(Cij,Cji). mm [ nz( I),m( J)] (3.4)

6 Note the formal identity of the commonality ratio and Bush and Mosteller’s (1951) similarity

index.


Written in these terms for later comparison with the vector model similarity becomes

2 min[MI),dJ)l tiij V = ’ m(I) + m(J) (3.5)

(For future reference we identify the larger commonality ratio max (CibCji) as +ii).7

The content model is compatible not only with a set theoretical, but also with a vector model. In this model, as developed by Ekman (1963), percept I is represented as a vextor xj, whose length is hi; the angle between a pair of such vectors, Xi and Xi, say, is @ii. The quantity qij specified in the instruction, ‘how much of J is found in I?’ (see Section 2.5), is interpreted as the proportion of hj constituted by the projection of Xi on Xi. This projection being hiCOS @ii, we obtain the multidimeri- sional magnitude ratio

hi Yij =- COS @ij

hj (3.6)

where qij refers to the magnitude of I relative to J as the standard. An alternative way of writing (3.6) is

bij qij = 5

where bii ’ Zp XipXjp, p=l,...,r (3.7)

which expresses qij in terms of the scalar products of vectors in r-dimensional space. By appropriate methods (see Section 5.2), one can find best fitting values for the scalar products and then carry out a principal component analysis of the scalar product matrix, giving the coordinates of the vectors x relative to the principal axes of the configuration. Part II of this paper deals with these calculations and the subjective spaces obtained.

Assuming that similarity is a function of the percept vectors, that is,

’ To be more precise: both the E- and the R-model require measures - which may be denoted by r and [ respectively - to represent a certain relation between two percepts which, although

they have different substantive interpretations, can both be equated with the max(CipCii). We

decided to use J, as a formal symbol defined by ~ij = max(ci~c~i). Rigorously, eq. 3.4 should use the symbol r and eq. 4.12 the symbol c, to keep the two interpretations of max(cipcii)

apart.

12 H. Eider, E. E. RoskamlMultidimensional Similarity, I

sii = f(xi,xj), a functional relation can be established between magnitude ratios and similarity ratings. Ekman et al. (1963) presented such a model, defining totulity as the vector sum of the two pertinent vectors, and commonality as the sum of the two vectors projections on each other. Though this model fits a number of studies nicely (Ekman et al. 1964; Ekman 1965), it clearly does not hold for a number of others (Kiinnapas 1966, 1967, 1968; Kiinnapas et al. 1964), nor for the data of the present experiment. According to Ekman et al. (1964), the model may hold for homogeneous stimulus sets, but not for stimuli with more clearly distinguishable features. Shepard (1964) and Torger- son (1965) make a similar distinction.

As already mentioned, totality may be more suitably conceived of as the sum of two vector lengths, and not as the length of the vector sum.

Commonality may be represented by

“ij m(1 n J) = ____ ,

max(hj,hi) (3.8)

that is the projection of the smaller vector on the larger one, and totality by hi + hi. Equating the vector length with the percept measure, Iii = m(I), and keeping in mind that Dii = llillj cos @ij, we

obtain from (3.8)

(3.9)

In this way the fraction $q in the set theoretical model is equated with cos @ii in the vector model. Consequently, we can express similarity as a function of vector lengths and angle:

2 niin@;,&)

sii = __ _ ,

lli + llj COS $bij,

cf. eq. 3.5. Also,

Rlill(Cij,Cji) = COS $ij ___-~ . ITlZtX(lZi,Ilj)

(3.10)

(3.11)

Eq. (3.10) agrees with the similarity function originally suggested by Eisler and Ekman (1959) for unidimensional percepts, and is supported by several other studies (Eisler 1960; Ekman, Goude and Waern 1961),


although a recent study by Sjeberg (197 1) questioned its validity. For unidimensional variation we have

Sij =

2 min(hi,hi)

hi + hj

and for purely qualitative variation, where hi = hi, we have

(3.12)

Sij = COS $ij = IllaX(Cij,Cji> 3 $ ij. (3.13)

From the commonality ratio ratings and the multidimensional magnitude ratio ratings, independent estimates of the cosines and the vector lengths (or rather, the ratios of the vector lengths) can be obtained. From the first data set we have cos @ii = max(cijcji), eq. (3.9), and from the second cos @ii = d/(qijqji), eq. 3.6. The ratio hi/hi is obtained as cij/cji, eq. 3.2, or d(qii/qji), eq. 3.6. AS section 6.3.1 will show, these estimates do not agree as they are, and furthermore, for a number of ratings, qvqji, i.e. cos2$ii, exceeds unity. Therefore, three correction constants are introduced. -The first one is well known from unidimensional ratio estimation: the scale composed of the standards differs in its unit from the scale composed of variables (see, e.g. Eisler 1960; Goude 1962; Sjijberg 1971). This entails a correction of the ratio estimates by a factor k, which may be seen as an under- or overestimation factor. The other two corrections consist in relating the variables by power functions. The rationale for these latter corrections is the following. All three judgmental tasks require a weighting of ‘quantitative variation’ as expressed by the vector length, and ‘qualitative variation’ as expressed by the cosine (see Ekman 1970).

Let us assume that the weight is in some sense correct in the similarity ratings. In the commonality ratio ratings attention is directed to the qualitative variation, which may entail too low a weight for the quantitative factor. For the magnitude ratio estimation the reversed weighting may take place: correct estimates of the vector lengths, and an underestimation of the effect of the cosines. Since here the factors of a product are differently weighted, the weights are expressed as exponents.

The introduction of correction parameters might seem unpalatable. However, the equations given above may be seen as valid for an ideal case, in analogy with, say, the gas laws for ideal gases in physics. A distinction between real and ideal conditions has to be made in psycho-


logy as well as in physics (cf. Eisler and Montgomery 1974). With these corrections, the multidimensional magnitude ratio model

is amended and modified into

1 /li qij = I_ c (cos @ii)”

‘ I

(3.14)

and so

and the commonality ratio model into

l/r CG =

i i

min(hi,/zj)

hi

COSQij (3.16)

and so

hi Cij y __= -

/lj (1

(3.17) icji

Likewise, the relation between similarity and commonality ratios changes from eq. (3.3) to

2c max cmax = IllaX(Cii,Cji)

sij= pp 7 c min = IlliIl(Cij, Cji> (3.18)

4. A modified model, relating content and distance (R-model)

A major indication for a modified model was found in the earlier analyses of Eisler’s ( 1967 [ 19701) data: nonmetric multidimensional scaling of the similarity data showed a fairly well fitting and well interpretable two-dimensional result when a city-block metric was em- ployed (see part 11 of this paper). Once this was recognized, it appeared natural to try and find a formal interpretation which would establish relationships among all three variables (similarity, commonality and magnitude ratio) and a dimensional representation based on the city- block metric. To this end, we introduce first a modified relation between similarity and set measures which links both to a multidimensional interpretation by a city-block metric. Then, a reformulation of the commonality and magnitude ratio models suggests itself almost naturally.

H. Eisler, E. E. Roskam/Multidimensiona~ Similarity, I 15

Although the set measure, m(I), is usually thought of as a positive numerical value, there does not seem to be any compelling reason why one should not allow negative values for a set measure. Let us consider mi as a measure of how much a set (or a percept in this context) ‘possesses’ in excess of some reference set, or null set, or is short of with respect to this reference. This implies that mi is an interval

measure. For clarity we shall reserve m(I) to refer to the standard usage, as in the preceding section, and use mi with the implication that an interval measure is meant.

By similar reasoning we can define /.Ji as a ratio of excess or shortage with respect to the reference set. If the interval and the ratio measurement are uniquely related, and independent of the unit of measurement, it follows (cf. Lute 19.59) that

Pi = eXP(mi) and mi = ln(pi). (4.1)

A vector-representation of sets can be conceived of as a representation with respect to the different kinds of elements in a set. For instance, let I be a set consisting of two apples, three pears, one orange and four peaches. The vector Xi representing this set of fruits would be an ordered 4-tuple (2, 3, 1, 4). The set I is thus a linear combination of constituent sets, each consisting of equivalent elements.

Accordingly, we assume that I = u IP, IP n Iq = @ if p # q and that

Ip C_ Jp or Ip >_ Jp for all percepts I, J, . . . and all ‘aspects’ p. q, . . . Let m be a measure on sets, and let, in particular, xip = m(Ip), mi = m(I), and m be additive. So we have:

mi = =p xip, p=l,...,r (4.2a)

WZinj = Cp Illill(Xip,Xjp) (4.2b)

(4.2~)

where Xip, p = 1, . . ., Y are the components of the vector xi. The vector representing the intersection I n J consists of the min(xip,xip), p = 1, . . ., Y and the vector representing the union consists of the

max(x+jp). We take it that the vectors x are defined with respect to an arbitrary

origin. A geometric example is shown in fig. 4. It is easily verified that

16 H. Eisler, E. E. Roskam/Multidimensiorlal Similarity, I

Fig. 4. Vector (x) representation of set I and J, their union (IuJ), and their intersection (InJ)

in two dimensions.

the symmetric set difference dii equals

dij = lniuj - Wlinj = Zp IX@-Xjp], (4.3)

and satisfies the axioms of a metric distance (cf. Lute and Galanter 1963: 297). The distance function is a ‘city-block’ metric.

It should be noted that the usual additive relations among set measures are satisfied by m:

Wliuj = Wli + mj - minj (4.4)

but not by ,u, since

PiPj PiCJj= --= eXp(miuj) (4.5a)

Pinj

and

Pir*j Pinj = ___ = eX&$WZinj)

PiUj

and so

Pinj IJfnj eXp(lT?inj - WZiuj) = -= __ = exp( ~ dij).

P*iUj P#j

(4.5b)

(4.6)

The ideas put forward here have appeared off and on in the psycho- metric literature in one form or another, e.g. Restle (1959, 196 1: 43 sqq.), Lute (1963: 114) and Lute and Galanter (1963: 296-303), who treat the notion of the set-symmetric difference as a metric


distance. Lute and Galanter (1963) appear to think of embedding these distances in an r-dimensional Euclidean space. Gregson (1970, 1975: 170) refers to Halmos’ (1950) textbook on measure theory to introduce equations very similar to our (4.2) but does not appear to think of a space with a Minkowski l-metric (city-block metric). The essential idea is that a stimulus is represented as a set which is the union of non-overlapping subsets, i.e. I = U I,, (p = 1, . . ., Y), and a measure m($,) is introduced. Explicit employment of a metric multidimensional scaling by a city-block metric, in connection with the data of the present paper, was published by Roskam (1972), and a different theory for embedding similarity measures in a metric space, via a set-theoretical interpretation, was subsequently presented by Rosenberg (1974) using a model of Gregson (1970). Recently, Becker and Pipahl (1974) also noted that a measure function m, defined on stimuli (sets) with aspects (components) satisfying certain disjunctiveness and subset conditions, satisfies the property of a Minkowski l-metric.

A continuous function which maps dii in (4.3) into the (0,l ] interval and is antimonotone, is the exponential function of -dii, given in (4.6). We propose to express similarity in terms of this exponential function, but we will make allowance for an arbitrary scale constant; this in fact means that we allow a free choice of the basis of the exponential function. Using exp to denote the exponential function to the basis e, we define

sij = exp (- p”ii) = [ exp(-dij)]@/2, P>O (4.7)

or

dij = - $ In(Sij>.

The way of writing the scale parameter, p/2, is merely a convenience for later expressions.

Using (4.6), we have

(4.8)

Next, in a manner analogous to Eisler’s definition of commonality ratios. we define the model


(4.9)

where 6 > 0 is another scale parameter, not necessarily equal to p. Since we can set one such parameter at an arbitrary value, we will in the sequel set 6 = 1.

It follows that sij is a power function of the geometric mean of cij and Cji,

(4.10)

which equals the geometric mean if p = 6.

A further justification for having powers as free parameters can be found in the circumstance that only the extreme values of the (O,l] interval of Cij and sij have a definite meaning. Different subjects and different instructions may give rise to similarity a.rd commonality functions with different steepness in respect to the 7ulderlying continuum, even if a subject is instructed to express the ratio as if it were a proportion of some absolute nature. There is no absolute standard to tell how much should be common between two things, in order to have, say, 0.5 similarity. In the same way, no such standards exist to define a 0.5 commonality ratio. As is generally the case in psychophysics, the exponent may be different for different stimulus modalities. Both sij and Cij, as implied by their definitions, are independent of the origin of Xi and Xi. A shift of origin will add a constant to m, and it will multiply y, but not affect Sij or CQ.

The magnitude ratio of two percepts can now ~zot be expressed in terms Of ?YZi, since the latter depends on the arbitrary origin of a coordinate space. It seems natural to define

(4.11)

where $ ij is 8 some measure of the ‘overlap’ between two percepts. The exponents, y and CY, are justified by the same considerations as before, and k is an over- or underestimation parameter, cf. Section 3. We choose to define + ii in a way corresponding to (3.9), namely:

’ See footnote 7. Rigorously, eqs. 4.11 and 4.12 should use the symbol r; rather than $J.


$ij E ITlaX(Cij,Cji) = Fiflj

fin(Pi,Pj) ’ (4.12)

This definition now appears to have attractive properties. First it equals 1 when two sets (percepts) are nested, that is, whenever I C J then minj = mi. Secondly, it bears a certain analogy to a geodesic angle, like the cosine, which equals 1 when two lines have the same direction. Specifi- cally, consistent with the city-block metric in (4.3), we have

I C J C K * dij + djk = dik and $'ij=$jk=$'ik = I (4.13)

(like COSij = COSjk = COSik whenever dij + djk = dik in the case of Euclidean metric). Eq. 4.13 is true even if Xi, Xj and xk are not multiples of each other. For nested sets we have

sij = [k min(qikqji)] (kP)/r = K2 , Say (t’i < rj>, r’ J (4.14)

that is, similarity is a power function of the smallest magnitude ratio. The model in (4.14) was supported by Sjoberg (1971), as against sij = 2ri/(ri + rj), which follows from Ekman’s and Eisler’s model (cf. eq. 3.12). Some authors (Hiiijer 1969; Waern 1970) have reported high correlations between similarity ratings and magnitude ratios for unidimensional stimuli. Waern found that for most of her subjects, the similarity rating was related to a power transformation of magnitude ratios.

By eliminating pini from eq. 4.8 and 4.12 we obtain

Sij = (iijJ'lm~lPmax)'t I*min - - Inin(Pi,Pj)

Pmas = max(Pi,Pj) ’ (4.15)

5. Internal consistency of the data

We investigate the internal consistency of the Q-data and the C-data independently. No internal consistency test of the S-data by themselves is possible.

For both Eisler’s (Section 3) and Roskam’s (Section 4) model, the C quantities and the Q quantities can be expressed in the same generic


form. The criteria for internal consistency are therefore the same in both models. We decided to estimate tile paramaters of the model using logarithmic least squares procedures.

Writing

wij cij = Y- , with VVjj = yj and

\vij = m(InJ) or

"ij = (PinjP (5.1)

we estimate aij = In Wij and aj = ajj by minimizing

Cij (In OCij - uîj + lij)‘, i,j = 1 , . . _ ,?I, (5.2)

where o denotes data (observed values) and ^ denotes estimates. By setting ocii = 1 we will force ij = djj. However, no restriction is imposed which will make “ii < min (at&j), although that should actually be a requirement. Differentiating with respect to aîj = &ji and putting the derivatives to zero leads to

a*.. = $ 11

[ ln(0Cij ’ OCji) + aî + ijl * (5.3)

Since, obviously, the minimum log least squares is invariant under addition of an arbitrary constant to all aii and ai, we may set ZZia*i = 0.

Next, differentiating eq. 5.2 with respect to &j, and using 5.3, we have, when Ci Bi = 0

1 &ji bj = -Ci 111

( > -. (5.4)

71 ocijf

Table 2 gives the values ci. From (5.4) and (5.3) we obtain Bii.

The fitted C values are computed from a*ij and sj by

A exp iij

cij = - eXQ&?j (5.5)

A plot of ocij VS. Eij is shown in fig. 5. The proportion of explained

H. Eisler, E. E. Roshzm/Multidimensional Similarity, I 21

variance’ is 0.962, corresponding to a correlation of 0.98. These values indicate good internal consistency. The bisector of the two regression lines passes virtually through the origin with slope equal to 1 .O.

0 0.2 0.‘4 0.6 0.8 1.0

Fitted commonality ratio zij

Fig. 5. Observed commonality ratio ocii vs fitted commonality ratio ~ij.

5.2. Intemul consistency of the Q-data

Proceeding in a similar way, we write

lnqij=gij +.f -43

where”

(5.6)

9 In the sequel, goodness of fit will be indicated by the proportion of explained variance, U,

since in most cases the predicted variables do not allow a linear transformation and accordingly

the correlation may overcstirnate the goodness of fit. U= l-C@ - z')' / X:(z -2)', where z are given values and z’ estimated values.

” We choose not to aim at separate estimates of cos @ii (or $Q), k, o and y at this point, preferring to do so later from the relations between (e.g.) commonality and magnitude ratio

data (see Sections 6.3 and 6.4).

22 H. Eisler, E. E. Roshzm~Multidimensional Similarity, I

Table 2 Estimates of percept measures from C-data and from Q-data (see also fig. 8).

C-data Q-data

Stimulus expm exPti

-0.4396 1.2590 -0.6419 1.2700

-0.0026 1.9490 -0.0440 2.3093

0.1687 2.3129 0.3490 3.4210

-0.6699 1 .oooo -0.8809 1 .oooo

-0.3060 1.4389 -0.4726 1.5042

-0.0310 1.8943 0.0382 2.5070

0.1914 2.3661 0.3523 3.4323

0.0943 1.7782 -0.2755 1.8320

-0.0720 1.8183 -0.1688 2.0383 0.1180 2.1986 0.1481 2.7982

0.3177 2.6848 0.4604 3.8241 0.1381 2.2433 0.0479 2.5316

0.1419 2.2519 0.1580 2.8261

0.1599 2.2927 0.3417 3.3960

0.3795 2.8559 0.5881 4.3449

E-model interpr.

R-model interpr. log k(I) S ini

W) fip

log ii y ??zi

Note: Columns 3 and f are scaled so that Xj = Xf = 0. The columns exp(j) and exp(n are resealed to Set exP@~) = exPCfD) = 1. (Percept D has the smallest magnitude.)

gii = (II ln(cos @ii) - In k, and fi = In Iii, (5.7)

or

gij=ln$i-Ink and fi = In /.Lr= ymi,

and minimize

x ij (ln04ij - 2ij -fi+&)‘, i, j= 1,. . . ,n. (5.8)

Differentiating with respect to 8~ = iii and putting the derivatives to

zero gives

!jij = f lll(&ij ’ 04ji). (5.9)

H. Eisler, E. E. RoskamjMultidimensional Similarity, I 23

Differentiating with respect to fi, putting to zero, and setting Zp.=O, we have i’

fj=I lZlIn o’lij

2n j ( ) - . oqji

(5.10)

This solution imposes no restrictions on gii vis-a-vis fi,jj, and in particular the scalar product matrix associated with Ekman’s model for the Q’s may not be positive semi-definite. Table 2 gives thepvalues.

We reproduce fitted Q values by:

$ii =

exP fi

exp fj exp gii. (5.11)

The goodness of fit is shown in fig. 6. The correlation between oqij and $ij is extremely high, 0.99 (proportion of explained variance O-979), showing excellent internal consistency, in this sense. Some of the estimates gq are, however, larger than zero, implying estimates of $ij or cos @ii which are larger than 1, indicating the need for the overestima-

3.5

3.0 := (T

0” 2.5 .- ;;i L

s 2-O 3 .-

67 ; 1.5

P 2 1.0 z

:: 0.5

Fitted magnitude ratio 4ij

Fig. 6. Observed multidimensional magnitude ratio oqij vs fitted mdltidimensional

ratio $ij. magnitude

24 fi Eider, E. E. Roskam/Multidimensional Similarity, I

tion parameter k-, introduced in eqs. (3.14) and (4.1 1). In the sequel the percept measures (vector lengths) will be denoted

by II if they are computed from Q-data and by’p if they are computed from C-data, regardless of the interpretation.

6. Relations between the three data sets and model comparisons

6.1. Introduction

In this section we present the results of a number of analyses where both models entail predictions.

The following points should be noted in order to understand the computations presented below:

(1) A predicted variable is computed directly as a function of the observed or fitted variables, or is fitted in the least squares sense, when one or more of the four parameters, h-, (Y, p, y are required.

(2) Since both the predicted and the observed or fitted variables are fallible, the fitting is often carried out in both directions.

(3) An indication will be given as to whether the variables themselves or their logarithmic transform are used. In the first case the program STEPIT (Chandler 1969) is used, in the latter the common fitting of an overdetermined system of linear equations. Though strictly speaking these two methods reflect different models ~ minimizing the error of the dependent variable or its logarithmic transform - the fit usually is so good that this difference can be neglected.

(4) The goodness of fit is indicated by the proportion of explained variance, because this measure is not misleading (see fn. 9). If the log-transformed variables are used, this percentage is calculated for the antilogs, i.e. the variables themselves.

6.2. A brief comparison between the E- and R-models

We have seen that, in spite of partly different interpretations, the two models are formally very similar. Three of the parameters, k, 01, and y, are the same. In the E-model the magnitudes (hi) are considered ratio scaled, and the dimensional compositions (cos @ii) as well as the similarities (sii) absolute scales. The magnitudes and the dimensional

H. Eisler, E. E. Roskum/Multidimensional Similarity, I 25

compositions are furthermore endowed with weights. In the R-model the measures pi are on a log interval scale, and the overlap measures ($ ii) as well as the similarities (Sii) may be denoted bounded power scales. Whether these two points’ of view constitute a difference in substance may be a moot question.

Regarding the predictions that can be made from the two models, they are identical with two exceptions.

(i) The relation between similarity and commonality ratio differs (eqs. 3.3, 3.18 and 4.1 O), and

(ii) the relation between similarity and the subjective space is different (eqs. 3.10 and 4.3,4.7).

We will see that the data do not give a clear-cut answer as to the correctness of either model.

6.3. Relations between commonality ratios and multidimensional magnitude ratios

63.1. Percept measures (vector lengths) The ratios of the percept measures for the C-data and the Q-data,

CiiIcji and qijIqji> are related by a power function, yielding, after taking logarithms,

(6.1)

The regression was carried out in both directions” for both the observed (ocii,oqii) and the log-least-squares fitted (cî~~~) values. The parameter y was computed from the bisector of the two regression lines and is given in Appendix B. The correlation between the logarithms of the variables amounted to 0.90 for the observed and 0.98 for the log-least-squares fitted values, an excellent fit, especially for the fitted values, see fig. 7.

Essentially the same test is, of course, performed by regressing the estimates fif (&(I) in the Elmodel), as obtained in the analysis of the C matrix, on the estimates fiy ($i is the E-model), as obtained from the

I’ Rather than using (y) = 105 values, requiring an arbitrary selection between 4ii/4ii and

qji/qii, we used all n*-n pairs, that is both qii/qii and qji/qv. The same goes for the cs. AS a

result, the regression lines intersect at (0, 0).

26 H. Eisler, E. E. RoskamlMultidimensional Similarity, I

Fig. 7. Ratio of multidimensional magnitude ratios vs ratio of commonality ratios (fitted

values) in log-log coordinates.

analysis of the Q matrix, since

(6.2)

corresponds to eq. 6.1. The result of the regression analysis (after taking the log of eq. 6.2) of these values is given in Appendix B and in fig. 8. The correlation is 0.98.

The regression coefficients from eqs. 6.1 and 6.2 are in almost perfect agreement, as would be expected. Inspection of the plot in fig. 8. shows, however, that the power parameter may not be the same for all stimuli. The value of y for the ‘larger’ stimuli appears to be larger than for the ‘smaller’ stimuli.

As a conclusion from the foregoing analysis, we find that the percept measures, as estimated from the C-data, are not simply proportional to those obtained from the Q-data. A power function appears to be more appropriate.

H. Eisler, E. E. Roshxm/Multidimensional Similarity, 1 27

In al O.G- 0

2 E 0

.-

5 O.U- 80

0 .-

2 0.2- L 00

ca 0 0 5 o.o- 0 .- In C iif -0.2 - 0

._ D 0

._

; -OeU

5 -O.G- : 0

tr _ -o.a-

F 0 / / / I J -I -0.G -0.11 -0.2 0.0 0.2 0.4

Log(Fi) from commonalily ratio estimates

Fig. 8. Vector lengths from multidimensional ratios vs set measures from commonality ratios in

log-log coordinates.

6.3.2. Max Ccipcji) from qibqii Both models predict that the maximum commonality ratio is a

power function of the product of the two magnitude ratios, viz. l2

c,, E $ij = (k2qij *qji)112”s (6.3)

We calculated the regression coefficients of the log-transformed variables. The results are given in Appendix B and the fit is illustrated in fig. 9.

The values of k agree very well with those found from fitting the Q-data to the S-data, as presented later. We estimate (Y from the two regression lines, yielding the two estimates 0.51 and 0.39 respectively; averaging, we may set 01 = 0.45; this result differs but little from the estimate presented later in this section, using a different method.

The conclusion to be drawn at this point is that the angle function in

I2 we set 6 = 1 throughout this and later sections.

28 H. Eisler, E. E. Roshzm/Multidimensional Similarity, I

Fig. 9. Larger commonality ratio as a function of the corresponding multidimensional magnitu-

de ratios (fitted values).

the model for the magnitude ratios is not simply equal to the larger commonality ratio, but that it is a power function of it.

4.3.3. Mill (cij,cji) jhom qij<lii Both models predict the relation

qij < Sji “Cij < Cji (6.4)

It is mildly violated by 16 out of the 105 stimulus pairs: we conclude that it is fairly well supported by the data. More specifically, the models specify the following:

R: C,,in &!!!%. c !J m in = min(pi, yj) 111 ‘LX ’ (6.5)

p 111 ;,x 1-1,~~~~ = max(pi, ~j)

l” COS 4ij,

II, in = mill(Ili, /Ii)

/I,,,~~ = max(lzt /li j (6.6)

H. Eisler, E. E. RoskamjMultidimensional Similarity, I

Accordingly for both models

4min ‘/zy C min = C-1 (k2 4jj. 4ji)

1Y2a Y max

+l/ol _qmin Sl/~+l/Y) 4 Ydlb max

After taking logarithms we can write (6.7) as

ln(c,,,i,)=$lnk+%(!&+~)lnqmi~ +%(A

We fit the linear equation

In (Cmi*) = Ul + b In qmin + C In qrnax.

29

(6.7)

(6.8)

(6.9)

From this equation we obtain (Y = l/(b + c), y = l/(&c), k = exp [a/(b + c)] . The results are given in Appendix B and fig. 10. The fit appears to be good. Comparison of this result with the earlier estimates of k, CY and y shows good agreement.

1 .

0

Fig. 10. Smaller commonality ratio as a function of the corresponding multidimensional magnitude ratios (fitted values).

30 H. Eider, E. E. RoskamlMuItirlimerlsional Similarity, I

6.3.4. q from ci? cji In a manner corresponding to the one in the previous section we

obtain (see eqs. 3.14-3.17 and 4.11, respectively)

1 c,,,y+a Yrnax = - ___-

h- CminY (6.10)

1 CminY (4 =-. mln k_ cm,,Y-~ . (6.1 1)

The parameters were evaluated using STEPIT to obtain a least squares fit for both qm,, and qmin at the same time, minimizing the sum of the sums of least squares deviations from eqs. 6.10 and 6.11. The result is found in Appendix B. Whereas the values for & and + for the fitted variables agree well with those obtained from other relations, use of the observed variables yields a strong underestimation, probably due to statistical regression.

6.3.5. q from p, $ The E-model yields

qij = k F (COS @ij)& I

which after replacing hi and cos @ii by pi and $ii leads to the R-model formulation:

1 ‘l-ii ’ qij = k ,~j

(-) $”

(6.12)

Taking the logarithms of (6.12) and applying multiple regression, the parameters shown in Appendix B are obtained. The multiple correlation is 0.96. The fit is illustrated in fig. 1 1.

6.4. Relation between similarity and commonality ratio

Here the R- and E-models differ, though the difference is not very decisive numerically.

R-model: sij = (cij. cii)Pl’ (6.13)

H. Eisler, E. E. RoskamjMultidimensional Simibity, I 31

2.5 .-

<[T

.o 20 ;;; * L

% 3 1.5

._ &

r” 1.0

0 0.5 1.0 1.5 2.0 2.5 3.0

Fig. 11. Multidimensional magnitude ratio as a function of set measures fi and the larger commonality ratio cmax (fitted values).

(6.14)

cf. section 3. Eq. 6.14 assumes totality to be the sum of the two measures m(I) and m(J). If on the other hand totality’is conceived of as the measure of the union, m(IuJ), we obtain

m(lnJ) m(In J) m(I)9 ij

sii = m(Iz) = m(I)+m(J)-m(InJ) = m(I)+m(J)-m(I)&j=

Qij =- 1 -*ii + m(J)/m(I)

m(1) G m(J). (6.15)

Taking the correction y for the quantitative variation into account, we obtain

c max Sij =

(6.16) P.

32 H. Eisler, E. E. RoskamlMultidimelzsional Similarity, I

0.

0.

Fig. 12. Similarity as a function of commonality ratios according to the R-model.

Eqs. 6.13, 6.14 and 6.16 were fitted, using STEPIT. Eq. 6.16 gave T = 0.44 and a proportion of explained variance of 0.747 (for the fitted values). Both the disagreement of the y-value with what was obtained in previous sections and the comparatively bad fit leads to a rejection of eq. 6.16, i.e. of the interpretation of totality as the measure of the union of the two sets.

The result for eq. 6.13 is fi = 1.15 with a proportion of explained variance of 0.9 14. The corresponding plot is found in fig. 12. The result for eq. 6.14 is given in Appendix B, and fig. 13. The difference in fit between R- and E-models (proportion of explained variance 0.9 14 and 0.938 respectively) is too small to allow any conclusions as to which may be the correct one. Both models make use of one fitted parameter.

6.5. Specl:fic predictions 0.f the E-model

From eqs. 3.10 and 3.15 we derive the relation

2 min(hi,hj)

‘ii = ‘YXY

hi + hi ’ Ck2qij L, qji) . (6.17)


0

0

Fig. 13. Similarity as a function of commonality ratios according to the E-model.

The vector lengths, IZi, were estimated by the method in Section 5 (log-least-squares fitted values). To study eq. 6.17 for observed values, too, the vector lengths hi were replaced by e The product qij’qji is estimated by oqij*oqji. The parameters k and Q were estimated using STEPIT to minimize the discrepancies between the left and right members of eq. 6.17, using estimates in the right member and the observed dij in the left member. The results are given in Appendix B.

Inverting the relation in (6.17), we have

(6.18)

Inserting h and gii in the right member of (6.18), we now estimate a and k to have the right member match the $ij. The results are given in Appendix B.

Since both oSij and oqij are fallible data, we also minimized the following sum of squared deviations, with respect to CY and k, where s’ and q’ are predicted values according to the right members of eqs. 6.17 and 6.18:

34 H. Eider, E. E. Roshzm/Multidimensional Similarity, I

/ / I I

0 0.2 0.4 0.F 0.8 1.0

93 .(\TclijqjiPL6 . MIN(~i,F;l)

^hi*^hl

Fig. 14. Similarity as a function of vector lengths and multidimensional magnitude ratios

according to the E-model.

Eq. 6.19 attempts their difference in dix B and fig. 14 and 15.

to correct for the unequal number of qs and ss and absolute magnitude. The results are given in Appen-

(6.19)

The same equation was also used to reanalyse a number of previous studies comprising similarity and magnitude ratio estimation. These results are given in table 3. It is seen that the multiplicative correction constant k is near, but for most cases somewhat lower than, unity. This is in agreement with the usual findings in unidimensional ratio estimation. Of more interest is the exponent Q, the psychological meaning of which would need further clarification. A low exponent implies an overestimation of the qualitative variation in the multidimensional ratio estimation task relative to similaritv estimation. No clear-cut pattern

H. Eisler, E. E. RoskamlMultidimensional Similarity, I 35

3.0

2.5

:: <CT

.o 2.0

3

4 1.5 2 ._ & : 1.0

0.5

0 0.5 1.0 1.5 2.0 2.5 3.

1 i;i i;i*i;’

>

.46

-‘T. .93 hi 2’MIN(C;i ,Zj)

‘Sij

Fig. 15. Multidimensional magnitude ratio as a function of vector lengths and similarity according to the E-model.

emerges from the 14 studies; the variation in OL may depend on the stimulus material (all experiments with low exponents are concerned with visual patterns), or on experimental conditions (compare study XII and XIII).

Next, Eisler’s model estimates, using (3.10), (3.1 l), and (3.17):

C ii+ij irnin llr

min = 0% . ___ - Fil min ( ) il max

(6.20)

where ~min = min(Li,ij) and i,,, = max(fii,ij), and the is are estimates from the analysis in Section 5. The constant y was estimated using STEPIT and the results are given in Appendix B.

Finally we have the relation

Sij = 2 min(hi,hj)

hi + hj G ij (6.2 1)

Tab

le

3

The

re

latio

n be

twee

n si

mila

ritie

s an

d m

ultid

imen

sion

al

ratio

s:

01, k

and

pr

opor

tion

of

expl

aine

d

vari

ance

fo

r 13

exp

erim

ents

.

No.

Ref

eren

ce

Stim

uli

No.

of

stim

uli

I II

III

IV

V

VI

VII

VII

I

IX

X

XI

XII

XII

I

Kiin

napa

s et

al.

1964

Kiin

napa

s et

al.

1964

Kiin

napa

s 19

66

Kiin

napa

s 19

67

Kiin

napa

s 19

68

Kiin

napa

s 19

68

Ekm

an

& L

indm

an

1961

Ekm

an

et a

l. 19

64

Ekm

an

1965

Ekm

an

& E

ngen

, 19

62

Ekm

an

et a

l. 19

64

Ekm

an

1965

Eng

en

1962

Ekm

an

et a

l. 19

64

Ekm

an

1965

Ekm

an

et a

l. 19

64

Ekm

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1 .o

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31

7

V I I I I 0 0.2 0.4 0.G 0.8 1.0

2’MIN(~i,~j)

i;i+Cj Tnax

Fig. 16. Similarity as a function of vector lengths and the larger commonality ratio (fitted

values) according to the E-model. This relation does not require any parameters.

Table 4 Proportion of explained variance from eq. 6.21.

Predicted variable Observed values Log-least-square-fitted

values

‘ij 0.940 0.954

$ij 0.879 0.907

which allows parameter-free mutual prediction of s and J/ = cm,,. The results are given in table 4, where for the column ‘observed values’ hi/hi is replaced by Jm in the computation. The fit is shown in fig. 16.

6.6. Specific predictions of the R-model

From eq. 4.1 1 we obtain, by dividing qii by qji, pi/pi = (qij/qii)“‘Y which, together with $ii from eq. 6.3, allows sij in eq. 4.15 to be


expressed as a function of yij and qji, corresponding to eq. 6.17 for the E-model:

(6.22)

The further treatment parallels that in Section 6.3.3: simplifying eq. 6.22 and taking logarithms yields

Eq. 6.23 is fitted by linear regression. Table 5 gives the result. The values of 01 and y can be estimated making use of the p value obtained from eq. 6.13. Though the fit of sij as a function of qij and qji according to the R-model is good (see fig. 17) with multiple correlations of 0.97 for observed and 0.96 for fitted values, the parameters & and particularly + deviate strongly from the corresponding estimates as

Fig. 17. Similarity as a function of the multidimensional magnitude ratios according to the

R-model.

H. Eisler, E. E. Roshzm/Multidimensional Similarity, I 39

Table 5 Parameter values and goodness of fit from eq. 6.23.

Least squares

solution using

j = 1.15 from eq. 6.13 d/d T/j 5/d h

Observed values 0.53 0.68 1.26 0.62

Fitted values 0.50 0.68 1.38 0.57

Values &, j, and + taken from the results of eqs. 6.13 and 6.24

Proportion of

explained

9 variance

0.78 0.921

0.79 0.904

Observed values 0.46 1.20 2.61 0.53 1.38 0.884

Fitted values 0.44 1.23 2.76 0.51 1.41 0.876

given in Appendix B. This result cannot be due to an incorrect estimate of p, since the ratios y/cu may be crudely assessed to about 3 from the cr and y values in Appendix B, more than twice what is obtained from eq. 6.23, see table 5. Keeping in mind the good fit, a possible explanation of this discrepancy might be that the subjects experience both the quantitative and the qualitative part of the complex sensation differently in all three judgment tasks.

It is instructive to study the corresponding relation with cii*cji as the dependent variable, because, according to eq. 6.13, this expression equals Sii 2/fl. Eqs. 6.3 and 6.7 yield

Cii . Cji = (k2 4ij4ji)’ 1~ (6.24)

Using the same procedure as for eq. 6.22 we obtain a good tit, and also good agreement with the 6 and T values obtained from other relations, see Appendix B. Comparing the outcome for eqs. 6.22 and 6.24 we find that though the parameter values obtained from eq. 6.22 are not in good agreement with those obtained from other relations, the fit of this formula is quite insensitive to the particular values. If we insert the values of & and T computed from eq. 6.24, and b from eq. 6.13, into eq. 6.22, the fit deteriorates very little, see table 5.

Finally, at first sight it is surprising that both the E- and the R-model as expressed in eqs. 6.17 and 6.22 yield practically the same fit, though they are different. A closer scrutiny of the two formulas shows that the component concerned with the qualitative variation, containing the

40 Ii. Eisler, E. E. Roskam/Multidimensional Similarity, I

product CJijCJji, is almost the same (6 is close to unity, see eq. 6.13). It is easy to show that the remaining components approximate each other. Let min(hi,hj)/max(hi,hi) = (~~ti/~~,,,,)t = t. By using t as the independent variable we obtain, after rearran ‘ng

B for the different

components of eqs. 6.17 and 6.22,2t/( 1 +t) and to 27, respectively. These two expressions can be compared by plotting u = In [t/(l+t>l as a function of v = In t. This plot, u = .f(v>, shows only a weak curvilinearity, indicating that a straight line with the slope fl/2y might be a fair approxima- tion. With the values of b/2? from table 5, the slope can be computedas 0.735. Tllis slope can be compared to the derivative ./“(v) = I/( 1 + t).

For the permitted range 0 < t < 1. we obtain 0.5 < .f”(~) < 1. The slope value calculated above is in the midst of this region.13

7. Summary and conclusion

Perhaps the most interesting result from the present investigation is that it proved possible to obtain systematically different judgments for the three instructions, similarity estimation, commonality ratio estimation, and multidimensional magnitude ratio estimation, for the same stimulus set. This is by no means a trivial finding, as illustrated by the failure of the first group of subjects to make commonality ratio judgments. Furthermore, the commonality ratio estimation data, as well as the multidimensional ratio estimation data, showed good internal consistency. (No corresponding test for the similarity estimates exists.) In so far as the E- and the R-models make the same predictions regarding the interrelations among the three data sets, the data confirm the models, in terms of both goodness of fit and agreement between the same parameters derived in various ways.

It did not prove possible to establish the superiority of either model. They differ in the similarity relation Sii, both for the similarity as a function of the commonality ratios (eqs. 6.13 and 6.14), and as a function of the multidimensional magnitude ratios (eqs. 6.17 and 6.22).

However, the fit for these two relations is the same for either model,

” For unidimensional similarity, Kiinnapas and Kiinnapas (1973, 1974) propose sii = t12 and

obtain a good tit to data from four experiments. As just shown, however, a good fit to their similarity equation implies a good tit to Sii = 2 t/(l+t) (in logarithmizcd form), and vice vcrs;1.

Their exponents vary between 0.7 and 0.9, agam within the range off“(Ll).


though the E-model uses two and the R-model three parameters. However, as presented until now, the data seem to tip the scale

slightly in favor of the E-model for the following reasons. (1) The number of parameters of the models: The R-model as a

whole makes use of four parameters, viz. h-, cy, y, and /3. The E-model uses only the first three of these.

(2) The agreement of the same parameter when computed from different relations: In the relations that distinguish the two models, the E-model estimates Q from eq. 6.17 and y from eq. 6.14, whereas the R-model estimates both (Y and y from eq. 6.23. The E-model gives fair agreement for (Y and good agreement for y; the R-model acceptable agreement for CY and not so good for y. This last finding is mitigated by eq. 6.23 not being sensitive to the particular y value.

The comparison becomes somewhat less unfavorable for the R-model if the parameter 3 is omitted, i.e. ,? = 1.15 is replaced by p = 1. This change of the model does not change the fit of eq. 6.13 much (the proportion of explained variance drops from 0.914 to 0.868); it would make the agreement between all 01 values good (see table 5, first column, with p = l), but still leave the optimal y value derived from eq. 6.23 half as great as when derived from all other relations. We might be left with a different appreciation of the quantitative variation for all three tasks and thus still with four parameters (two different y values) for the R-model. As will be shown in Part II, the multidimensional analysis, there the scale tips slightly the other way, in favor of the R-model.


Appendix A

Raw data from three judgment tasks: similarity estimation, os, commonality

ratio estimation, nc, and multidimensional ratio estimation, ,_q.

No. Stimulus pair ij o'ij o'ij o'ji oqij oqji

1

s 4 5 6 7 8 9

1C 1: 1: 13 14 15 16 17. 18 19 20 21 2: 2: 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4c 41 42 43 44 45 46 47 48 49 50 51 52 53 54

A0 0.677 AC 0.504 A0 9.443 AE 0.633 AF 0.479 AG 0.315 AH 0.319 AI 0,334 AJ 0.3qs AK 0.252 AL n-258 AM 0.252 AN 0.245 A0 O.lQ! DC 0.773 30 0,255 BE 0.474 BF C.Q)3 3G 0.645 BH 0.497 n1 0,5r)9 QJ 0.599 3K 0.5'9 3L 0.335 RM 0.397 BN 0,436 BO 0.411 co 0.171 CE 0.304 CF 0.662 CG 3,853 CH 0.297

CI 0.369 CJ 0,593 CK cl.'?3 CL 0,4',1 CM q.3')4 CY 0.544 CCI 0.5w DE 0.455 OF 0.232 OG O.!.-i6 OH 0.708 01 0,333 OJ Q.209 OK 0.116 OL @.5W OM 0.273 ON 0,173 00 C.l?' IIF 0,666

fG 0.492 EH 0.454 EI 007q1

3.599 0.975 3.446 2.956 0?397 0.468 0.577 0.654 I.420 0.731 5.373 3.769 cl*?97 0.3Ql 3.44', Oc466 0.31' C-524 0.299 ,3.627 0,262 0.399 0.224 3.542 00305 0=374 0.211 0.46G 0.749 0.939 0.297 0,195 O.h69 0.495 Cl.796 0,822 0.714 0.833 0.391 0.381 0,515 0.455 cl;595 0.706 O.fi?S 0.706 0.361 0.312 0.385 0.414 9,454 0.462 0.3?6 0.581 3.319 0.127 0.499 0.337 0.719 0.574 0.967 0.934 0.322 G.243 (J-455 0.356 cl.599 0.566 0.714 0.774 c),33' 0. ‘b.?

0.335 0.357

0.594 0.543 3,519 0.562 0.495 0.495 3,?65 0.299 0.147 0.393 J-524 0.927 0,398 0.563 0.194 0.47c 3.107 5.362 0.325 0.987 0.253 a.573 O,~b't 0.546 0.113 0.418 0.6Cl.1 0.97c 0.470 0.739 0.449 0.372 0,49? 0,774

0.58 1.72 0.39 2.56 0.95 0350 G.69 0.97 c.47 1.50 C.36 2.18 c.49 1.07 0,50 1$26 c.37 2.00 0.23 2.47 0.34 1.13 c.29 1.58 0.29 2306 G.2c) 2.81 C.75 1.41 1.49 0.35 1.30 0.56 0.85 c.98 0.72 1.34 0.95 0.40 1901 0.71 0.76 0.99 a.55 1.26 c.55 0.87 0.52 1.05 0.61 1111 0.46 1.65 2.66 0118 1.94 0.30 1.25 0.67 0,9? 1.00 1.19 0.45 1.50 c-49 0.96 Cl.64 0.83 1.09 1,04 cl,59 c.97 0.49 O.Ql 0.94 0.69 I.25 0.61 1.09 0~27 1.90 0.21 2.46 0.65 1.72 0.39 1.50 0.24 1.62 0214 2709 0.41 2.41 0.3? 2.37 0,15 2357 0.15 3.33 0363 le.52 c.39 2.03 0.62 1.02 0.82 1-16

H. Eisler, E. E. Roshzm~Multidimensional Similarity, I

Appendix A (cont’d)

43

No. Stimulus pair ij o'ij o'ij o'ji oqij oqji

55 EJ 0,439 0.519 56 EK 00458 0,349 57 EL 0.290 0.310 58 EN 0.515 0.483 59 EN 0.542 5,597 ho EO 0.474 O-42? 61 FG ‘30 749 0.7?R 62 FH 0.418 O-37? 63 FI 0.h70 0.707 64 FJ 00 961 0,931 65 FK 0.75’7 0.65' 66 FL 0.37’) 00333 67 FM 0.576 0.499 68 FN 0.69: 0.643 69 FCI 0.634 0.570 70 GH 0.394 0.333 7: GI cl,337 0.46 72 GJ 0.731 0.767 73 GK 0.992 0.931 74 GC 0,409 0,349 75 GM ct.477 0.493 76 GN 0.4’;7 0.67') 77 GO 0.793 0.834 78 HI 0.602 0.593 79 HJ 0.397 0.445 80 HK 0.353 0.271 81 HL cl.9?1 0.744 82 HM 0.493 0.499 83 HN 0.345 0.194 84 HO 30 z 40 SC94 85 IJ (3.4597 0.759 86 XK 0.483 0.495 07 IL oe5’)r 0.499 88 IM 3.935 0.993 89 IN ‘).a 49? 3.593 QC IO cl.493 0.453 9: JK 0.745 0.727

9: JL 0049') 0,456 93 JM 0.703 0.727

94 JN 0.9'5 0,912 95 JO 0.7'5 0.707 96 KL o-391 0.449 97 KM 0,5?J 0,593 98 KN cl.'qs 0.937 99 KG 0.9'.6 0.937

100 LM 0.427 0.433 101 LN 0.439 0.535 102 LO 0,341 0,349 103 MN 9.76,0 5.794 104 MO 0.490 0.435 105 NO 0.789 0.756

0.974 0.756 0.440 0.906 0.881 0.806 30824 0,412 0.712 J-R87 0.908 0.456 0.506 0.685 0,843 0.288 0.47C 0.696 0.903 0.381 0.419 0.431 0.854 0.604 00393 0.409 3.931 0.714 0.402 3.&01 0.834 0.740 5.537 0.912 00931 O.Rl8 0.832 0.381 0.745 5.923 O.A99 0.424 0.460 0.906 00 8713 0.718 0.493 0.490 0.843 5.610 0.834

0.55 0040 0.32 0.49 0.50 0.37 0.49 1.02 1.06 0.87 0.70 O-67 O-RI) c-74 0.61 1.31 1.50 1.12 0.92 1.08 0.95 d.95 C.83 0.75 0.53 cl.32 C-8') 0.61 0.39 0,29 c-71 0.54 0.69 0.81 0065 0.46 0.77 S.89 1.31 0390 0.69 l-19 1.44 1.09 0.92 G.72 0.57 0.54 0.79 C059 c-79

:-E 1127 1.48 2.45 2.75 1.41 0.46 c-75 1.18 1.55 0,58 1.20 1.22 1.62 0.41 0.49 0.77 l-C7 0.53 0.76 0.90 1.17 0.91 1.05 1.40 1.37 1.36 1.60 2021 1.33 1.42 1.02 1.52 1040 1.91 1.33 0.73 0.98 1023 1.47 0.51 0.66 0.97 10 17 0.98 1.13 1.25 1.20 I.40 1.33

44 H. Eisler, E. E. RoskamfMultidimensional Similarity, I

Appendix B

Values of fitted parameters o, 0, y, k, and goodness of tit measured as proportion of explained variance, U.

Equation _-

6.1

6.2

6.3”

6.3b

6.8

6.10,6.11

6.12

6.14

6.17

6.18

6.19

6.20 6.24 ..___

i

1

Observed values ______ 01 Y k (I

1.37

0.52 0.90 0.768

0.38 0.97 0.673

0.53 1.36 0.91 0.887 0.18 1.05 0.99 0.802

1.35 0.938

0.50 0.92 0.922

1.45 0.933 0.53 1.38 0.91 0.859

Log-least-squares fitted values

Q Y k u -

1.48

1.48

0.51 0.91

0.39 0.96

0.51 1.43 0.92

0.37 1.35 0.94

0.39 1.45 0.96

1.43 0.48 0.93

0.39 0.96

0.46 0.93

1.44

0.5 1 1.41 0.91

0.812

0.759

0.880

0.929

0.924

0.949

0.911

0.987

{ i;{‘::$!

0.848

a) $ .. dependent variable.

b, 4;. qii dependent variable.

References

Attneave, F., 1950. Dimensions of similarity. American Journal of Psychology 63, 516-556.

Beals, R., D. II. Krantz and A. Tversky, 1968. Foundations of multidimensional scaling.

Psychological Review 75, 127-142.

Becker, J. and W. Pipahl, 1974. The additive difference model and a metric and an ordering on

sets. Psychometrika 39, 361-362.

Bush, R. and F. Mosteller, 1951. A model for stimulus generalization and discrimination.

Psychological Review 58,413-423.

Chandler, J. P., 1969. STEPIT - Finds local minima of a smooth function of several para-

meters. Behavioral Science 14, 81-82.

Eisler, H., 1960. Similarity in the continuum of heaviness with some methodological and

theoretical considerations. Scandinavian Journal of Psychology 1, 69981.

Eisler, H., 1964. A choice model for paired comparison data based on imperfectly nested sets. Psychometrika 29, 363-370.

Eisler, H., 1967 [ 19701. Multidimensional similarity, an experimental comparison between

vector, distance, and set theoretical models. First version, 1967 (mimeo); second version, 1970 (unpubl. manuscript).

Eisler, H. and G. Ekman, 1959. A mechanism of subjective similarity. Acta Psychologica 16, l-10.

Eisler, II. and If. Montgomery, 1974. On theoretical and realizable ideal conditions in psychophysics: Magnitude and category scales and their relation. Perception and Psychophysics 16,

1577168.

H. Eisler, E. E. RoskamfMultidimensional Similarity, I 45

Ekman, G., 1954. Dimensions of color vision. Journal of Psychology 38,467-474.

Ekman, G., 1961. Some aspects of psychophysical research. In: W. A. Rosenblith (ed.), Sensory

communication. New York: Wiley.

Ekman, G., 1963. A direct method for multidimensional ratio scaling. Psychometrika 28,

33-41.

Ekman, G., 1965. Two methods for the analysis of perceptual dimensionality. Perceptual and

Motor Skills 20,557-572.

Ekman, G., 1970. Comparative studies on multidimensional scaling and related techniques.

Reports from the Psychological Laboratories, University of Stockholm. Supplement 3.

Ekman, G. and T. Engen, 1962. Multidimensional ratio scaling and multidimensional similarity

in olfactory perception. Reports from the Psychological Laboratories, University of Stock-

holm. No. 126.

Ekman, G. and R. Lindman, 1961. Multidimensional ratio SCahg and multidimensional simi-

larity, Reports from the Psychological Laboratories, University of Stockholm. No. 103.

Ekman, G. and L. Sjoberg, 1965. Scaling. Annual Review of Psychology 16,45 L-474.

Ekman, G., T. Engen, T. Ktinnapas and R. Lindman, 1963. A quantitative principle of qualita-

tive similarity. Reports from the Psychological Laboratory, University of Stockholm.

No. 152. Ekman, G., T. Engen, T. Ktinnapas and R. Lindman, 1964. A quantitative principle of qualita-

tive similarity. Journal of Experimental Psychology 68,530-536. Ekman, G., G. Goude and Y. Waern, 1961. Subjective similarity in two perceptual continua.

Journal of Experimental Psychology 61,222-227. Engen, T., 1962. The psychophysical similarity of the odors of aliphatic alcohols. Reports from

the Psychological Laboratories, University of Stockholm. No. 127.

Goude, G., 1962. On fundamental measurement in psychology. Stockholm: Almqvist &

Wiksell.

Goude, G., 1972. A multidimensional scaling approach to the perception of art. I. Scandinavian Journal of Psychology 13, 258-27 1.

Green, P. E. and V. R. Rao, 1972. Configuration synthesis in multidimensional scaling. Journal

of Marketing Research 9, 65-68.

Gregson, R. A. M., 1970. Quadratic similarities. British Journal of Mathematical and Statistical

Psychology 23,53-68.

Grcgson, R. A. M., 1975. Psychometrics of similarity. New York: Academic Press.

Halmos, P. R., 1950. Measure theory. New York: Van Nostrand.

Hays, W. L., 1958. An approach to the study of trait implication and trait similarity. In:

R. Tagiuri and L. Petrullo (eds.), Person perception and interpersonal behavior. Stanford:

Stanford Univ. Press.

Hoijer, B., 1969. Mechanisms of cognitive judgments. Report from the Department of Psycho-

logy, University of Uppsala. No. 67.

Kruskal, J. B., 1964. Multidimensional scaling by optimizing goodness of tit to a nonmetric hypothesis. I, II. Psychometrika 29, l-27, 115-129.

Kiinnapas, T., 1966. Visual perception of capital letters: multidimensional ratio scaling and

multidimensional similarity. Scandinavian Journal of Psychology 7, 189-196.

Ktinnapas, T., 1967. Visual memory of capital letters: multidimensional ratio scaling and multidimensional similarity. Perceptual and Motor Skills 25, 345-350.

Kiinnapas, T., 1968. Acoustic perception and acoustic memory of letters: multidimensional ratio scaling and multidimensional similarity. Acta Psychologica 28, 161-170.

Kiinnapas, T. and U. Kiinnapas, 1973. On the relation between similarity and ratio estimates. Psychologische Forschung 36, 257-265.

46 H. Eider, E. E. Roskam/Multidimensional Similarity, I

Kilnnapas, T. and U. Kiinnapas, 1974. On the mechanism of subjective similarity for unidimen-

sional continua. American Journal of Psychology 87,215-222.

Kiinnapas, T., G. Malhammar and 0. Svenson, 1964. Multidimensional ratio scaling and multi-

dimensional similarity of simple geometric figures. Scandinavian Journal of Psychology 5,

249-256.

Landahl, H. D., 194.5. Neural mechanisms for the concepts of difference and similarity. Bulletin

of Mathematical Biophysics 7, 83-88.

Lute, R. D., 1959. On the possible psychophysical laws. Psychological Review 66, 81-95.

Lucc, R. D., 1963. Detection and recognition. In: R. D. Lute, R. R. Bush and E. Galanter

(eds.), Handbook of mathematical psychology. Vol. 1. New York: Wiley.

Lute, R. D. and E. Galanter, 1963. Psychophysical scaling. In: R. D. Lute, R. R. Bush and

E. Galanter (eds.), Handbook of mathematical psychology. Vol. 1. New York: Wiley.

Restle, F., 1959. A metric and an ordering on sets. Psychometrika 24, 207-220.

Restle, F., 1961. Psychology of judgment and choice. New York: Wiley.

Rosenberg, W., 1974. On the validity of Minkowski metrics. Nederlands Tijdschrift voor de

Psychologie 29,229-233.

Roskam, E. E., 1972. Multidimensional scaling by metric transformation of data. Nederlands

Tijdschrift voor de Psychologie 27,486-508.

Shepard, R. N., 1962. The analysis of proximities: multidimensional scaling with an unknown

distance function. I, II. Psychometrika 27, 125-140, 219-246.

Shepard, R. N., 1964. Attention and the metric structure of the stimulus space. Journal of

Mathematical Psychology 1, 54-87.

Shepard, R. N., A. K. Romney and S. B. Nerlove, 1972. Multidimensional scaling. 2 ~01s. New

York: Seminar Press.

Sjoberg, L., 1971. Three models for the analysis of subjective ratios. Scandinavian Journal of

Psychology 12,217-240. Torgerson, W. S., 1952. Multidimensional scaling. I. Theory and method. Psychometrika 17,

401-419.

Torgerson, W. S., 1958. Theory and methods of scaling. New York: Wiley. Torgerson, W. S., 1965. Multidimensional scaling of similarity. Psychometrika 30, 379-393.

Tversky, A. and D. H. Krantz, 1970. The dimensional representation and the metric structure

of similarity data. Journal of Mathematical Psychology 7, 572-596. Waern, Y., 1970. Preference and perception. A study on individual subjects of the unfolding

model. Reports from the Psychological Laboratories, University of Stockholm. No. 309.

Documents

Multidimensional similarity: an experimental and theoretical comparison of vector, distance, and set theoretical models: I. Models and internal consistency of data