multi dimension scaling

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PSYCHOMETRIKA--VOL,~0, NO. 4

D~CEMBEa, 1965

MULTIDIMENSIONAL SCALING OF SIMILARITY*

WARREN S. TORGERSON

T H E J O H N S H O P K I N S UNIVERSITY

A colleague of mine once summarized the use of statistics in psychology

in the following way. "The trouble is that too many people know more sta-

tistics than they unders tand." I am no t at all sure but what this is beginningto be an appropriate sta tement for those of us who are actively working in the

field of multidimensional scaling.

It was not too long ago that the problems, approaches and solutions

in multidimensional sealing seemed rather straightforward. There was first

of all --i f you will pardon me-- the tradi tional approach: by Torgerson [14, 15]

out of Richardson [10] with help by such midwives as Gulliksen [4], Green, and

Messiek and Abelson [9]. This procedure or set of procedures knew what it was

doing and also knew what it required. Briefly, its formal requirements were

too severe. It asked not only that the perceptual or cognitive structure of

the set of stimuli be Euclidean in nature, but also, tha t observations on simi-

larity of pairs of stimuli be l inearly related to distances between points in

the space. But when the given requirements were not seriously violated, one

could in fact use similarity to determine the underlying perceptual or cogni-

tive structure of a set of stimuli. The model was used, and is still being used,

in quite a number of different areas with success.

There was also the approach of Attneave [1], who attacked the problem

from the other direction: Assuming tha t one knows the dimensionality and di-

mensions of a set of stimuli, can one determine how the differences on each of

the dimensions are combined to give an over-all impression of similarity.Attneave's results suggested tha t an additive space would be more appropriate

than the Euclidean space just mentioned. But more about that later.

Then there was the unfolding approach, begun by Coombs for the unidi-

mensional case, generalized by Hays to more than one dimension, and later

improved upon by Coombs and other workers in his laboratory [2]. In this

approach, one begins with nonmetric data and ends with nonmetrie results.

In all of the approaches mentioned thus far, s imi lar i ty was considered

to be the complement of distance in a space of one kind or another. GSsta

Ekman [3] provided an alternative. In his model, similarity, perhaps aftersuitable transformations, was interpre ted directly as a scalar product or angle

*Presidential address delivered at the annual meeting of the Psychometric Society,Chicago, Illinois, September 7, 1965.

379



380 PSYCHOMETRIK&

between a pair of stimuli rather than as a distance. Ekman's model seems to

me to be both theoretically sound and likely to be useful. However, it is also

my feeling that the data to which it has been applied thus far are betterdescribed by the distance models.

This is where things stood for a number of years: A distance model

with severe formal requirements, which nevertheless seemed to work quite

well; a nonmetric approach which, though it appeared to require little from

the data, also yielded little in the way of information about the spatial con-

figuration; and an alternative vector model that seemed to give complex

solutions for rather simple data.

But then came Shepard's papers on the analysis of proximities [11, 12].

In a sense he was anticipated by Helm [5], who, in tracking down the cause of

some distressing results obtained by Mellinger [8], discovered some of the

effects of monotonic bias in the original measures of similarity on the final

multidimensional solution, and indeed, transformed these measures to obtain

results more in line with expectations. Bu t while Helm showed us some of the

effects of monotonic bias in the distance measures, and how they could be

eliminated when the appropriate transformation was applied to the data, it

was Shepard who provided us with the first computerized, iterative program

for finding tha t appropriate transformation.

Since then, Torgerson and Mueser, Kruskal [6] and Guttman and Lingoes

[7] have all developed what each considers to be an improvement over theoriginal Shepard program. We all like our own best, of course, but, I really

suspect that goodness of approach is not strictly ordered. Which program is

in fact superior will depend upon the particular circumstances involved. As

time goes on, there surely will be---and indeed ought to be---other alternative

solutions offered. And each probably will have its own advantages and

disadvantages.

The aim of all of these procedures is to require no more information

than is needed by the nonmetric approach of Hays, and end with results as

definitive and informative as those obtained by the tradit ional methods whentheir requirements are met. Indeed, they go a little further. Iterative pro-

cedures can also be used for a space requiring the additive distance metric

favored by Attneave as well as the Euclidean metric ordinarily used, or for

that matter, for compromise metrics living anywhere in between the two.

The over-all aim of the new procedures is ambitious: to provide methods

that give us metrically invariant solutions when given only ordinally invariant

observations. Now, why should such an objective be reasonable? Two con-

siderations are relevant. First, considerably more than ordinal information

on the location of points in a space is required before the rank order of dis-

tances between these points is determined. Hence, ordinal information on

distances does imply a considerable amount of interval information on the

location of the points. Second, except in degenerate cases, any serious mono-



WARREN S. TORGERSON 381

tonic distortion of distances results in an increase in dimensionality, ordinarily

all the way to N -- 1 dimensions.

Hence, if one uses the device of finding the monotonic transformationof the original distances that allows the points to be fitted in a space of

minimum dimensionality, then, except for those degenerate cases, he will end

up with the solution desired--providing such a solution exists. And this

is the device the new iterative programs have used. In actual practice, of

course, empirical data is not likely to be strict ly monotonic with distance, but

rather to contain some amount of extraneous variation. The practical problem

has thus been to develop procedures that find the monotonic transformation

that gives a best fit in a specified number of dimensions. Then the smallest

number of dimensions that yields a fit considered to be adequate can be

taken as the solution.

Tile new procedures would thus seem to offer nothing but advantages

over the old; to require very little and to yield very much. Yet there are many

problems connected with their use which, at least to me, have not been at all

obvious. Nowadays, it is not difficult to "know more than one understands."

The methods can easily be applied to any symmetric table of data--it is as

easy as doing a factor analysis. And, like factor analysis, the methods always

yield an answer. But it can be even more difficult to comprehend fully the

meaning of that answer.

The problems have to do with the minimum dimensionality criterion, therequirements of the model, the nature of similarity itself, and finally, the

tendency of experimenters to interpret results in the manner to which they

have become accustomed. These factors can interact with one another to

form a rather complicated picture.

First, consider the minimum dimensionality criterion. It ought not

to be accepted blindly. It is, after all, only a device. When we are interested

in perceptual structures we really want solutions that stay put when they

ought to, that is, structures that remain invariant regardless of monotonic

distortion in the original data caused by variations in such things as experi-mental procedure, stimulus context, subject sample, or occasion. The mini-

mum dimensionality criterion is only a means to this end. Given enough

points and not too many dimensions, it does work--except for those degen-

erate cases mentioned earlier. But later on I will present some experimental

results that suggest that the world of similarity may be a good bit more

degenerate than one might have supposed.

There is also the problem of multiple solutions. The i terative procedures

seek a configuration that maximizes the goodness of fit of the distances

between points in the configuration to the best fitt ing monotonic transforma-

tion of the original observed distances. The surface over which the proceduresiterate is a complex one, where, I suspect, multiple peaks or potholes are more

common than not. It need not always be easy to find the very best solution,



382 PS Y C HOMETR I KA

and the second best might well yield a configuration vast ly different from the

first. Actually, given empirical data that contains both systematic and un-

systematic error, it is not even certain tha t the configuration that best meetsthe statistical criterion will necessarily always be the one desired.

The requirement that the observations on similarity be monotonically

related to distances cannot be taken lightly, either. Several kinds of data th at

we tend to associate with the concepts of similarity or distance present diffi-

culties. The stimulus-response confusion matrices, for example, typically are

not symmetric, and therefore cannot be handled directly by the approach. For

the subject-by-stimulus preference matrices, the assumption of monotonicity

across subjects ordinarily makes no sense. Also, situations surely must exist

where a vector model like Ekman's would be more reasonable. One of the

indices used to measure pattern similarity, for example, is the product-

moment correlation coefficient, and we all know what any red-blooded

American psyehometrican does when he gets his hands on a table of those!

There are several other indices for measuring pattern similarity also. I will

only note here th at since they are not monotonically related to each other,

they can not be expected to yield the same structure. The monotonic re-

quirement, there, means considerably more than just having a set of numbers

that one feels must have something to do with similarity.

But it seems to me th at the most interesting question having to do with

multidimensional scaling concerns the over-all reasonableness of the ap-proach itself. The question cannot be considered without at the same time

inquiring into the natu re of similarity. This, you will recall, was the problem

directly attacked by Attneave. My early attempts to reconcile Attneave's

results, which pointed to an additive spatial structure, with those obtained

by Richardson, and me, which pointed to an Euclidean structure, led to the

distinction in Theory and Methods of Scal ing between a multidimensional

att ribute on the one hand, and a set of stimuli varying with respect to several

different attributes on the other. Richardson, and I, and a host of others

since, carried out our experiments on the multid imensional att ribute of color,and here the Euclidean model appears to fit quite well. A tight coupling seems

to exist between the requirements of the formal model and the behavior of

the observable relation. Similarity appears to be a unique relation for stimulus

pairs and it does appear to act like a distance.

Attneave, on the other hand, carried out his experiments using sets of

stimuli that varied with respect to several different quantitative attributes.

His stimulus sets consisted of such things as parallelograms varying in size and

tilt, and triangles varying in size and angle. Shepard (13) carried out a similar

experiment using as stimuli, circles varying in size and angle of a radial line.

For stimulus sets such as these, the separate attributes are the primitive, or

obvious, or basic features. I t is easier to judge each attr ibute separately than

to form the over-all judgment of similarity. Attneave and Shepard both found



WARREN S. TOR~ERSON 3 8 3

tha t the additive spatial model provided a closer fit to the data than the Eu-

clidean model.

One is tempted to make a generalization: the Euclidean model goes withmultidimensional attributes; the additive model with sets of stimuli varying

on several different attribu tes. But, a sequence of experiments carried out by

Gayle Mueser Schulman and me indicates tha t this would be an oversimplifi-

cation. A more suitable generalization would now appear to be that as one

adds more and more obvious perceptual structure to a set of stimuli, the

process underlying the judgments of similarity changes from what appears

to be a rather basic perceptual one, to one which contains more and more

cognitive features. And as the contribution of cognition goes up, the ap-

propriateness of the multidimensional representa tion goes down.

For example, one of the nice things about distance is that it stays put.

Baltimore is closer to Washington than to Philadelphia and it stays closer

whether or not we decide to include Chicago in our table of inter-point dis-

tances. Physical distance is as much a property of a pair of points as, say,

physical length is a property of a single line. But similarity does not behave

that way for sets of stimuli varying on different attributes. Under these

circumstances, similarity does not exist as a unique, invariant relation between

a pair of stimuli, but rather depends upon such things as stimulus context

and the cognitive strategy taken by the subject. Schulman and I showed that,

for stimuli varying with respect to two simple quantitative attributes, theorder of judged differences depends upon the total set of stimuli used. Whether

a stimulus A is judged more like stimulus B than like C, depends upon the

remaining stimuli used in the experiment. We used two sets of bottom-

heavy, kite-shaped stimuli varying in size and in bottom heaviness. An

example is shown in Fig. 1. In one set, the range of variation on physical size

relative to bottom-heaviness was twice that of the other set. Yet multidi-

FIGURE 1

E x a m p l e o f s t i m u l i v a r y i n g o n tw o s i m p l e p h y s i c a l a t tr i b u t e s . S u b j e c t s r a t e d t h e s i m i -

l a r i t y o f a l l p a i r s o n a s c a l e f r o m z e ro ( i d e n t i c a l ) t h r o u g h t e n ( m o s t d i f fe r e n t) .



384 PSYCHOMETRIKA

mensional analysis of each set gave configurations th at were almost identical.

The results, given in Fig. 2, show th at the subjective range of size to shape was

essentially the same for both sets in spite of the two-to-one ratio in thephysical variables. The reason seems clear: in order to render a judgment of

over-all similarity, the subject must, in a sense, decide how to weight a given

difference in size relative to a given difference in shape. Other things being

equal, this weighting factor appears to be based on the over-all variation in

size relative to shape over the entire stimulus set used. We have here some-

thing very much like the typical context or anchoring effect of psych•physics,

except that in this case, the observable results are a little more striking:

whereas context and anchoring effects in unidimensional psych•physics just

change the shape of the psychological scale monotonically, in the multi-

attribute case, not even the rank order of the judgments remains invariant

over changes in stimulus set.

In spite of this, however, each set of stimuli, when considered separately,

appeared to meet all of the requirements imposed by the multidimensional

scaling program. In each case the configuration was two-dimensional, with

one dimension corresponding to size and the other to shape. Both fit the data

very well, with the additive metric fitting a little better than the Euclidean.

Hence, for structures such as these, the spatial representation of stimuli is

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O b t a i n e d c o n f i g u r a t i o n s f o r s t i m u l u s s e ts A a n d B . T h e o r d i n a t e s h o w s :A - - t h e o b t a i n e d a v e r a g e ra n g e o f s e t A ,

B - - t h e o b t a i n e d a v e r ag e r a n g e o f s e t B ,

C - - t h e p r e d i c te d a v e r a g e r a n g e o f s e t B .



WARREN S. TORGERSON 3 8 5

a p p r o p r i a t e a n d n o t m i s le a d i n g o n l y i f t h e e x p e r i m e n t e r re a li ze s t h a t t h e

o b t a i n e d c o n f i g u r a ti o n s a r e n o t i n v a r i a n t o v e r d i f fe r e n ti a l s t r e t c h i n g o f a x e s .

S p a t i a l m o d e l s a l s o t e n d t o i m p l y c o n t in u i t y . W e t e n d t o in t e r p r e t ad i m e n s i o n o r d i r e c t i o n i n th e s p a c e a s a c o n t i n u o u s v a r i a b l e . S i n c e s p a c e

i ts e lf is n o t h i n g b u t a h o le , i t s e e m s t o m e t h a t t h is a s s u m p t i o n i m p l ie s t h a t

t h e h o l e c a n b e f il le d ; t h e h o l e s h o u l d n o t h a v e u n f i l la b l e h o l e s i n i t. B u t w h e n

w e a d d a l it tl e m o r e o b v i o u s s t r u c t u r e t o a s e t o f s t im u l i , w e c a n e n d u p w i t h

a s p a c e t h a t c a n n o t b e f il le d . F i g . 3 s h o w s s u c h a s e t o f s t i m u l i . H e r e , w e h a v e

t w o o b v i o u s p h y s i c a l v a ri a b l e s, b u t n o w o n e o f t h e m a l s o in c lu d e s a n o b v i o u s

n a t u r a l o r i g in w i t h i n t h e p h y s i c a l c o n fi ~,m r a tio n . S i x t e e n s u b j e c t s r a t e d t h e

s i m i l a r i t y o f al l p a i r s o f th e s e s t im u l i . T w o o f t h e s u b j e c t s b e h a v e d d i f f e re n t l y

f r o m t h e r e s t so th e y w e r e a n a l y z e d s e p a r a te l y . T h e s e t w o s u b j ec t s g a v e u s

a t w o - d i m e n s i o n a l s o l u t io n e s s e n ti a l l y l ik e t h a t o f F i g. 3 , so m y r e m a r k s d o

n o t a p p l y t o th e m .

T h e d a t a f r o m t h e o t h e r f o u r t e e n s u b j e c ts , h o w e v e r , r e q u i r e d t h r e e s u b -

j e c t i v e d i m e n s i o n s r a t h e r t h a n t w o . T h e f i r s t d i m e n s i o n c o r r e s p o n d e d t o t h e

p h y s i c a l w i d t h o f t h e s t i m u l i a n d , h e n c e , b e h a v e d a s o n e w o u l d e x p e c t . T h e

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F m U R E 3Stimuli v~rying on one simple ~nd one bipolar physical at t r ibu te. Th e st imuli were drawn

by connecting tw o circles with arcs.



386 PSYCHOMETRIKA.

next two subjective dimensions, which are shown in Fig. 4, are more interes t-

ing. Here the bipolar physical variable appears to be separated into two

psychological dimensions: a quantitative dimension corresponding to degreeof asymmetry and a trichotomous, qualitative dimension corresponding to

sign of asymmetry. Assuming this interpretation is correct, then all of the

points have to lie on a U-shaped surface within the three-dimensional space,

and the rest of the space simply cannot be filled.

Here, the nature of the similarity structure only becomes clear after an

appropriate rotation of axes. For most orientations, the configuration appears

properly dimensional, with points projecting all along the three axes.

A thi rd experiment was carried out in order to verify this split of bipolar,

physical variables into qualitative and quantitative parts. Judgments of

degree of similarity were obtained for all pairs of the 25 stimuli shown in

Fig. 5. As you can see, these stimuli vary with respect to two bipolar aN

tributes. The two maverick subjects of the earlier experiment had warned us

tha t subjects can use different strategies and that these strategies can result

in very different configurations. Hence, before averaging, we needed to make

sure that the subjects averaged over were doing the same thing. Use of an

inverse factor analysis procedure enabled us to divide the subjects in two

fairly homogenous groups. Again, one group ignored mirror-imageness and

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Fmu~ 4Factors II and III. The dotted l i n e s a r e d r a w n a t p l u s a n d minus the mean of the absolute

values on the sign dimensionof the twelve asymmetricstimuli of Fig. 3.



WARREN S. TORGERSON 387

gave us a two-dimensional solution corresponding closely to the physical

configuration. Their results are shown in Fig. 6.

The other group was the interesting one. We expected, of course, tofind four dimensions: a qual itative sign dimension and a quan titative degree-

FIGURE 5Twenty-five stimuli varying on two bipolar physical attributes.

Factor ] I

~0-0 . . . . . . . . . . . . . . . . . . . . . O - - - . 0

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Factors I and II for the group that ignored mirror-imageness.

Factor I



3 8 8 P S Y C H O M E T R I K A

of-asymmet ry dimension for each of the bipolar physical variables. Our itera-

tire program was applied to the data as usual, and the best four-dimensional

solution obtained. Three of the dimensions were as expected, but the fourthwas hash. We attempted to help the program in various ways, but each time

it gave us a fourth dimension that was hash. Pride of ownership eventually

gave way to preservation of a good idea, so we next tried Kruskal's pro gra m--

and obtained essentially the same results: three good dimensions and one bad.

Bu t we do not give up easily. For a final try, we returned to the old-fashioned,

non-iterative, fully-metric, traditional approach. We extracted all of the di-

mensions and t hen rotat ed th em to the four-dimensional sub-space tha t best

fit our notions. The results are shown in Figs. 7 and 8. The configuration is

weak and it is errolfful, but it is there.

We are left with a question of why the iterative procedures failed us.

I suspect that this might well be one of those degenerate cases mentioned

earlier. The iterative procedures work best when the space is relatively well-

filled, but a space with quali tat ive dimensions cannot be well-filled. In the

present case, since, mirror-imageness on one of the axes did not contribute

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l u t e v a l u e s o n t h e s i g n d i m e n s i o n f o r t h e t w e n t y a s y m m e t r i c s t i m u l i , a n d a t t h e a v e r a g e

v a l u e o n t h e d e g r e e d i m e n s i o n f o r e a c h o f th e t w o s u b s e t s o f t e n a s y m m e t r i c s t im u l i .



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Fmcm~ 8Factors II and IV . T he theoret ical values are computed as in Fig. 7.

v e r y much d i s si m i la r i t y t o t h e j u d g m e n t s , t h e i t e r a t i v e p r o g r a m s c o u l d a t t a i n

a b e t t e r s o l u t io n b y e l i m i n a t i n g t h e c o n t r i b u t i o n e n t i r e l y a n d t h e n c a p i t a li z i n g

o n e r r o r .

T h e m o r a l i s o b v i o u s b u t w o r t h m e n t i o n i n g a n y w a y . T h e i t e r a t i v e p r o -

g r a m s do d i sc a r d i n f o r m a t i o n t h a t is i n t h e d a t a . S o m e o f t h e m - - l i k e o u r s - -

d o s o o n l y r e lu c t a n t l y , b u t o t h e r s r o u t i n e l y e li m i n a t e a ll b u t r a n k - o r d e r

i n f o r m a t i o n a t t h e o n s e t. T h e p o i n t t o r e m e m b e r i s s i m p l y t hi s: a l t h o u g h o n e

c a n o f t e n th r o w a w a y s o m e t h in g a n d t h e n r e c o v e r it la t er , s o m e t i m e s i t j u s t

g e t s l o s t .L e t m e s u m m a r i z e b r i e f l y . F i r s t , w i t h r e s p e c t t o m e t h o d o l o g y : t h e n e w

p r o c e d u r e s a re c l e a r l y a g r e a t a d v a n c e o v e r t h e o ld , b u t t h e y o u g h t n o t t o b e

u s e d e x c l u si v el y . W h e n t h e r e q u i r e m e n t s o f t h e t r a d i ti o n a l m o d e l a re n o t

v i o l a t e d o u t ra g e o u s l y , i t c a n a t t h e v e r y l e a s t p r o v i d e a v a l u a b l e c h e c k o n

t h e o b t a k l e d s o l u t i o n .

S e c o n d , w i t h r e s p e c t t o t h e n a t u r e o f s i m i l a r it y : I h a v e a r g u e d t h a t

s i m i l a r i t y is n o t a u n i t a r y c o n c e p t . T h e m a j o r d i s t i n c t i o n is b e t w e e n s i m i -

l a r i t y a s a b as ic , p e r h a p s p e r c e p t u a l , r e l a t i o n b e t w e e n i n s t a n c e s o f a m u l t i d i-

m e n s i o n a l a t t r i b u t e a n d s i m i l a r i ty a s a d e r i v a t i v e , c o g n i t i v e r e l a t i o n b e t w e e n

s t i m u l i v a r y i n g o n s e v e r a l a t t r i b u t e s . S i m i l a r i ty in t h e f o r m e r c a s e a p p e a r s

t o h a v e t h e p r o p e r t i e s o f d i s t a n c e i n E u c l i d e a n s p a c e . S i m i l a r i t y in t h e l a t t e r

c a s e is c o m p l e x , a n d i s s e n s i ti v e t o a l l o f t h e d e l i c a t e p r o b l e m s o f a t t i t u d e a n d



390 PSYCHOMETRIKA

strategy involved in decision-making tasks in general. Here, degree of simi-

larity is not an invariant relation between a pair of stimuli, but rather depends

upon such things as stimulus context. The shape of the configuration and evenits dimensionality varies with the set or strategy taken by the subject. And

under some circumstances, the dimensions obtained turn out to be qualita-

tive, class variables, rather than quan titative measures of degree.

Since many, if not most of the stimulus sets of interest to psychologists

seem to be multi-attribute rather than multidimensional in nature, the over-

all usefulness of the multidimensional scaling approach might appear to be

limited. However, I am beginning to suspect that the reverse will turn out

to be true. For although the similarity structures in the cognitive case do

not appear to be inherently spatial or dimensional in nature, yet it does

seem that such structures can always be imbedded in an appropriate space.

This appears to be so even though the similarity relation itself does not possess

all of the properties of a distance, and even though a dimensional interpre ta-

tion of the resulting configuration might often make li ttle sense. The danger

of course is that we will fail to discover the special natu re of these configura-

tions and will instead invent a dimensional interpretation, and then expect

the results to have all of the invariance properties associated with distances

in space. But that would be a fault of ours, and not a fault of the methods.

The methods themselves say nothing about kinds of invariance across

subjects, stimulus samples, or occasions. For any given set of data, theysimply show us the configuration of stimuli implied by the observations. They

do describe the configuration by projections of the stimuli on axes, but we

need not accept that particular form of description. Instead, we can--and

probably must --t ake two additional steps.

First, we can see how the configuration changes from one experiment

to another. We can determine whether stimulus context or instructions matter .

This is what Schulman and I did for the first experiment described today.

There, context did matte r, and the analyses showed us just how it matte red.

The configuration changed in a very simple and straightforward manner--merely a differential stretching of axes. And this in spite of the fact t ha t ob-

servations on stimulus pairs common to the two sets would not even have been

monotonically related.

As a second step, we can think a b o u t - - a n d l o o I ~ at--the shape of the

configuration itself. Wit h the proper rotation, the mixed qualitat ive-quantita-

tive structure in the last two experiments became clear. Other kinds of under-

lying structures ought also to yield characteristic configurations of the stimuli.

If so, then th~ techniques might serve as very useful tools for enabling the

experimenter to determine just what kind of structure does underlie the data.

Some preliminary work carried out this past year indicates th at a number ofother reasonable structures do indeed generate recognizable configurations.

Let me describe two or three of the more obvious ones.



WARREN So TORGERSON 391

Considcr, for example, a pure ly nominal, exclusive, class structure . Here ,

each stimulus is a member of one and only one class: stimuli in the same class

are similar and stimuli in different classes are dissimilar. If there were Nclasses, the multidimensional scaling procedures would give a solution in

N -- 1 dimensions, with stimuli of each class at a corner of an N - 1 dimen-

sional simplex. Three classes would therefore form clusters at the corners of

an equilateral triangle in two-space.

I suppose it could be argued that even so, a dimensional interpretation

would be more parsimonious than a class interpretation, since we need only

two dimensions as against three classes. But the argument would be weak.

A dimensional interpretation would not tell one that stimuli can only occur

at just three locations, and an adequate verbal description of the dimensions

would be likely to tax even a confirmed factor analyst. Suppose the classes

were animal, vegetable and mineral, wi th viruses and slime molds not allowed.

An interpretation that says simply "it's an animal" is somehow more satis-

fying than one that says that it has a high loading on animalness and a low

one on the bipolar vegetableness-mineralness dimension.

Consider another related structure. I mention this one only because I

have not y et used the words "common elements," and of course no discussion

of similarity can be complete without them. The structure in mind is sort of

a class structure, but one where the individual stimuli can be members in

different degrees of several classes. As a specific example, consider the alloysof copper, tin, and zinc. Here we have some hypothetical common elements

that have a little substance to them. Th e stimuli --alloys --differ from one

another in the proportions of each of the elements they contain. Let me make

the probably unwarranted assumption tha t alloys with nearly the same com-

position are similar, and that they get more different in a reasonable way as

their compositions get more different.

Once again multidimensional analysis would yield a two-dimensional so-

lution, but now all of the alloys would lie within a triangle with curved sides,

such as is illustrated in Fig. 9. We would not expect clusters of course. But

the configuration is recognizable. In fact, it is not unlike that of a positive

manifold simple structure, except that it occurs in one less dimension. Once

again, a dimensional interpretation would not be very informative.

Finally, consider what ought to happen to either of the class structures

just described if the stimuli also differed with respect to some independent

quantitative variables. If things behaved reasonably, the configuration ob-

tained would differ only by the addition of an extra quantitative dimension

for each independent quant itat ive source of difference; the projections of the

stimuli on to the subspace determined by the class structure would not change.

Fig. 10 differs from Fig. 9 only by the addition of one such quantitativevariable.

The examples just given are only a small sample of the simplest of the



392 PSYCHOMETRIK.4.

I I

!

C u

I

ls n

m I

FIGUI~E 9

Alloys of copper, tin, and zinc. All possible alloys would lie within the curved triangle.

l I I

FmUR~ 10Dimensions I and II represent the three elements. Dimension II I represents an independ-

ent, quant itat ive variable. All stimuli lie within the curved prism.



W A R R E N S . T O R G E R SO N 3 9 3

p o s s i b l e c o g n i t i v e s t r u c t u r e s t h a t m i g h t o c c u r. I f o t h e r b a s i c a l l y n o n - s p a t i a l

s t r u c t u r e s y i e l d s p a t i a l c o n f i g u r a t i o n s a s c h a r a c t e r i s t i c a s t h e s e , t h e n i t s e e m s

t o m e t h a t t h e f u t u r e o f t h e m u l t i d i m e n s i o n a l a p p r o a c h i s s e cu r e . A m e t h o dt h a t c a n b e u s e d to d e t e r m i n e t h e n a t u r e o f a n u n d e r l y i n g s t r u c t u r e s e e m s

v e r y v a l u a b l e , in d e e d . I t m a y e v e n t u r n o u t t h a t t h e t e c h n i q u e s w i l l b e c o m e

m u s t u s e fu l in t h o s e v e r y a r e a s w h e r e a p u r i s t c o u l d a r g u e t h a t t h e y s h o u l d

n o t h a v e b e e n u s e d a t a l l .

R E F E R E N C E S

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[3 ] E k m a n , G . D ime n s io n s o f c o lo r v i s io n . J. Psychol., I954 , 38 , 467-474 .

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[5 ] H e lm , C . E . A s u c c e s siv e i n t e rv a l s a n a ly s i s o f c o lo r d i ff e re n c e s . E d u c a t io n a l T e s t i n gS e rv i c e T e c h n ic a l R e p o r t , P r in c e to n , N o v e m b e r 1 9 6 0.

[6 ] K ru s k a l , J . B . M u l t i d im e n s io n a l s c a l i n g b y o p t imiz in g g o o d n e s s o f fi t t o a n o n m e t r i c

h y p o th e s i s . Psychometrika, 1964, 29, 1-28 .

[7 ] L i n g o e s, J . C . A n I B M 7 09 0 p r o g r a m f o r G u t t m a n - L i n g o e s s m a l l e s t s p a c e a n a l y s i s - - I .Behav. Sci. , 1965 , 10, 183-184.

[8 ] M e l l i n g e r , J . S o me a t t r i b u t e s o f c o lo r p e r c e p t io n . P h D th e s i s , U n iv . C h ic a g o .

[9 ] M e s s i c k , S. J . , a n d A b e l s o n , R . P . T h e a d d i t i v e c o n s t a n t p ro b l e m in m u l t i d ime n s io n a lsca l ing . Psychometrika, 1956, 21, 1-17.

[10] : [~ ichardson , M . W . Mu l t id im ens io na l psych oph ys ics . Psyehol. Bull., 1938, 35, 659-660.[11 ] S h e p a rd , R . N . T h e a n a ly s i s o f p ro x im i t i e s : M u l t i d ime n s io n a l s c a l i n g w i th a n u n -

k n o w n d i s t a n c e fu n c t io n . I . Psychometrika, 1962, 27, 125-140.

[ 12 ] S h e p a r d , R . N . T h e a n a l y s i s o f p r o x im i t i e s : M u l t i d i m e n s i o n a l s c a l in g w i t h a n u n k n o w nd i s t a n c e fu n c t io n . I I . Psychometrika, 1962, 27, 219-246.

[13 ] S h e p a rd , R . N . A t t e n t io n a n d th e m e t r i c s t r u c tu r e o f t h e s t imu lu s s p a c e . J. math.Psychol., 1964, 1, 54-87.

[14 ] T o rg e r s o n , W . S . M u l t i d im e n s io n a l s c a l i n g : I . T h e o ry a n d me th o d . Psychometrika,1952, 17, 401-419.

[ t5 ] T o rg e r s o n , W . S . Theory and m ethods of scaling. N e w Y o rk : W i l e y , 1 9 5 8 .

Ma nuscr ip t rece ived 9 /2 2 /6 5

multi dimension scaling

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