PSYCHOLOGICAL SCALING WITHOUT A UNITOF MEASUREMENT

BY CLYDE H. COOMBS*

University of Michigan

I. INTRODUCTIONThe concept of measurement has

generally meant the assignment ofnumbers to objects with the conditionthat these numbers obey the rules ofarithmetic (1). This concept of meas-urement requires a ratio scale—onewith a non-arbitrary origin of zero anda constant unit of measurement (3).The scales which are most widelymade use of in psychology are re-garded as interval scales in that theorigin is recognized to be arbitrary andthe unit of measurement is assumed tobe constant. But this type of scaleshould be used only if it can be experi-mentally demonstrated by manipula-tion of the objects that the numbersassigned to the objects obey the lawsof addition. The unit of measure-ment in psychology, however, is ob-tained by a combination of definitionsand assumptions, which, if regardedas a first approximation and associ-ated with a statistical theory of error,serves many practical purposes. But

1 This paper is a condensation of some of theideas contained in a chapter of a generaltheory of psychological scaling developed in1948-49 under the auspices of the Rand Cor-poration and while in residence in the Depart-ment and the Laboratory of Social Relations,Harvard University. While the author carriesthe responsibility for the ideas containedherein, their development would not havebeen possible without the criticism and stimu-lation of Samuel A. Stouffer, C. FrederickMosteller, Paul Lazarsfeld, and Benjamin W.White in a joint seminar during that year.Development of the theory before and afterthe sojourn at Harvard was made possible bythe support of the Bureau of PsychologicalServices, • Institute for Human Adjustment,Horace H, Rackham School of GraduateStudies, University of Michigan.

because we may sometimes questionthe meaning of the definitions and thevalidity of the assumptions which leadto a unit of measurement, it is ourintent in this paper to develop a newtype of scale not involving a unit ofmeasurement. This type of scale isan addition to the types set up by S.S. Stevens (3). Stevens recognizedratio, interval, ordinal, and nominalscales. The type which we shall de-velop falls logically between an inter-val scale and an ordinal scale. Weshall make no assumption of equalityof intervals, or any other assumptionwhich leads to a unit of measurement.We shall find, however, that on thebasis of tolerable assumptions andwith appropriate technique we areable to order the magnitude of theintervals between objects, We havecalled such a scale an "ordered met-ric." We shall develop the conceptsfirst in an abstract manner with ahypothetical experiment and then il-lustrate the ideas with an actual ex-periment. Under the limitations of asingle paper we shall not present thepsychological theory underlying someof the concepts and we shall placecertain very limiting conditions on ourhypothetical data in order to simplifythe presentation.

II. THE PROBLEM

When we set up an attitude scale byany of a variety of methods, for ex-ample the method of paired compari-sons and the law of comparativejudgment, we order statements ofopinion on the attitude continuumand assign a number to each state-ment. We recognize in this instance

145

146 CLYDE H. COOMBS

that the origin for the numbers isarbitrary. We then follow one ofseveral possible procedures (determin-ing which statements an individualwill indorse, for example) to locate thepositions of individuals on this samecontinuum. Because both individualsand stimuli have positions on thiscontinuum we shall call it a joint dis-tribution, joint continuum, or J scale.In general, with a psychological con-tinuum, we might expect that for oneindividual the statements of opinion,or stimuli, have different scale posi-tions than for another individual.Thurstone (4) has provided the con-cept of stimulus dispersion to describethis variability of the scale positionson a psychological continuum. Wehave recently (2) discussed an equiva-lent concept for the variability of scalepositions which an individual may as-sume in responding to a group ofstimuli. These two concepts havebeen basic to the development of ageneral theory of scaling to which thispaper is an introduction.

For didactic purposes we shallachieve brevity and simplicity for thepresentation of the basic ideas under-lying an ordered metric scale if weimpose certain extreme limiting condi-tions on the variability of the positionsof stimuli and individuals on the con-tinuum. These conditions are thatthe dispersions of both' stimuli and in-dividuals be zero. In other wordsthese conditions are that each stimu-lus has one and only one scale positionfor all individuals and that each indi-vidual has one and only one scaleposition for all stimuli. For purposesof future generalization we shall clas-sify these conditions as Class 1 condi-tions. Stimuli will be designated bythe subscript j and the position of astimulus will be designated the Qjvalue of the stimulus on the con-tinuum. The position of an indi-

vidual on the continuum will be desig-nated the Ci value of an individual *.

If we conceive of the attribute asbeing an attitude continuum, the dvalue of an individual is the Qj valueof that statement of opinion whichperfectly represents the attitude ofthat individual. In this case the dvalue of an individual is his ideal ornorm. We shall assume that the de-gree to which a stimulus represents anindividual's ideal value is dependentupon the nearness of the Q,- value ofthe stimulus to the d value of theindividual.

We shall then make the further as-sumption that if we ask an individualwhich of two statements he prefers toindorse he will indorse that statementthe position of which is nearer to hisown position on the continuum.

Thus if asked to choose between twostimuli j and k, the individual willmake the response

if(1)

\Qi-d\<\Q*-d\

where j > k signifies the judgment"stimulus j preferred to stimulus k."

Under the extreme limiting condi-tions we have imposed on the d andQj values the method of paired com-parisons would yield an internally con-sistent (transitive) set of judgmentsfrom each individual, though notnecessarily the same for each, andeach different set of 'such judgmentscould be represented by a unique rankorder for the stimuli for that indi-vidual. We shall call the rank orderof the stimuli for a particular indi-vidual a qualitative I scale, or, ingeneral, an I scale.

Thus if an individual placed fourstimuli in the rank order A B C D asrepresenting the descending order inwhich he would indorse them, then,

PSYCHOLOGICAL SCALING WITHOUT UNIT OF MEASUREMENT 147

A B C D

FIG. 1. A joint distribution of stimuli and individuals.

this would be equivalent to the con-sistent set of judgments A > B,A > C, A > D, B > C, B > D,C > D; where the symbol ">" signi-fies "prefer to indorse," as before.The order, A B C D, is the qualitativeI scale of this individual. Hence forClass 1 conditions it is sufficient tocollect the data by the method of rankorder; the greater power of the methodof paired comparisons would be un-necessary and wasted.

Let us assume now that we haveasked each of a group of individualsto place a set of stimuli in rank orderwith respect to the relative degree towhich he would prefer to indorsethem. Our understanding of the re-sults that would follow will be clearerif we build a mechanical model whichhas the appropriate properties. Thisis very simply done by imagining ahinge located on the J scale at the C,value of the individual and folding theleft side of the J scale over and merg-ing it with the right side. The stimulion the two sides of the individualwill mesh in such a way that the quan-tity | Ci — Qj | will be in progressivelyascending magnitude from left toright. The order of the stimuli on thefolded J scale is the I scale for theindividual whose d value coincideswith the hinge.

It is immediately apparent thatthere will be classes of individualswhose I scales will be qualitativelyidentical as to order of the stimuli andthat these classes will be bounded bythe midpoints between pairs of stimulion the J scale. For example, supposethat there are four stimuli, A B C D,whose Qj values or positions on a jointcontinuum are as shown in Fig. 1 andthat there is a distribution function ofthe positions of individuals on thissame continuum as indicated.

If we take the individual whoseposition is at X in Fig. 1, the I scalefor that individual is obtained by fold-ing the J scale at that point and wehave the scale shown in Fig. 2.

The qualitative I scale for the in-dividual at X is A B C D.

If we take the individual in positionY as shown in Fig. 1 and construct hisI scale, we have the scale shown inFig. 3.

The qualitative I scale for the indi-vidual at Y is C D B A.

Consider all individuals to the leftof position X on the J scale in Fig. 1.The I scales of all such individualswill be quantitatively different fordifferent positions to the left of X.For every one of them, however, theorder of the stimuli on the I scale willbe the same, A B C D. We shall re-

orA B C D

FIG. 2. The I scale of an individual located at X in the joint distribution.

148 CLYDE H, COOMBS

y

7*

orC D B A

FIG. 3. The I scale of an individual located at Y in the joint distribution.

gard these I scales as being qualita-tively the same. As a matter of fact,the I scales of individuals to the rightof X continue to be qualitatively thesame until we reach the midpoint be-tween stimuli A and B. For an indi-vidual immediately to the right of thismidpoint the qualitative I scale willbe B A C D. I scales immediately tothe right of the midpoint AB will con-tinue to be qualitatively the same,B A C D, until we reach the mid-point between stimuli A and C. Im-mediately past this midpoint thequalitative I scale is B C A D. Con-tinuing beyond this point a complicat-ing factor enters in which we shalldiscuss in a later section under metriceffects.

The distinction which has beenmade here between quantitative andqualitative I scales is of fundamentalimportance to the theory of psycho-logical scaling. In almost all existingexperimental methods in psychologicalscaling we do not measure the magni-tudes | C{ — QJ |, but only observetheir ordinal relations for a fixed d.2

8 The ordinal relations of | d — Cj \ mayalso be obtained experimentally for a fixed QJ,over a set of Q, We call such scales S scalesby analogy with I scales. For sake of sim-plicity they are not treated here but actuallythe treatment for I scales and S scales isidentical if the roles of stimuli and individualsare merely interchanged.

The kind of information which is ob-tained by the experimenter is essen-tially qualitative in nature.

As we shall see, data in the form ofI scales may tell us certain things:

1) whether there is a latent attributeunderlying the preferences or judg-ments,

2) the order of the stimuli on thejoint continuum,

3) something about the relative mag-nitudes of the. distances betweenstimuli,

4) the intervals which individuals areplaced in and the order of the inter-vals on the continuum i and

5) something about the relative mag-nitudes of these intervals.

III. A HYPOTHETICAL EXAMPLE

Let us now conduct a hypotheticalexperiment designed merely to illus-trate the technique. Of course thisexperiment, if actually conducted,would not turn out as we shall con-struct it, because we shall assume theextreme limiting conditions on Q andC values that were previously imposed.

Let us imagine that we have a num-ber of members of a political party andthat we have four individuals who arepotential presidential candidates. Letus ask each member of the party to

PSYCHOLOGICAL SCALING WITHOUT UNIT OF MEASUREMENT 149

place the four candidates, designatedA, B, C, and D, in the rank order inwhich he would prefer them as Presi-dent. With four stimuli the potentialnumber of qualitatively different rankorders is 24—the number of permuta-tions of four things taken four at atime. If there were no systematicforces at work among the partymembers we would get a distributionof occurrences of these 24 I scaleswhich could be fitted by a Poissonor Binomial distribution, everythingcould be attributed to chance, and theexperiment would stop there. In-stead, let us imagine for illustrativepurposes a different result equally ex-treme in the opposite direction. Letus imagine that from the N individualsdoing the judging only seven qualita-tively different I scales were obtainedand these were the following:

IiIsIthI,I,I?

A B C DB A C DB C A DC B A DC B DAC DB ADCB A

The significance of the deviation ofsuch results as these from pure chancewould be self-evident. Consequentlywe would look at these seven I scalesto see if there was some systematiclatent attribute represented by a jointcontinuum such that individual differ-ences and stimulus differences on sucha continuum could account for thesemanifest data. Studying the set ofseven I scales we observe that two ofthem are identical except in reverseorder, A B C D and D C B A. Fur-

thermore we see that in going from oneto the next, two adjacent stimuliin the one have changed positions inthe next. These are the characteris-tics of aset of I scales which have beengenerated from a single J scale. SevenI scales is the maximum number thatone can obtain from a single quanti-tative J scale of four stimuli under theconditions of Class 1. The systematiclatent attribute underlying this set ofI scales is represented by the J scalewhich generates them. Our objective,then, is to recover this J scale and dis-cover its properties or characteristics.

To recover the J scale we proceed asfollows. Every complete set of Iscales has two and only two scaleswhich are identical except in reverseorder. These are the I scales whicharise from the first and last intervalsof the J scale. Consequently thesetwo I scales immediately define theordinal relations of the stimuli on theJ scale, in this case A B C D (thereverse order, of course, is equally ac-ceptable). From the seven I scaleswe can order on the J scale the sixmidpoints between all possible pairsof stimuli. In going down the orderedlist of I scales as previously deter-mined, the pair of adjacent stimuli inone I scale which have changed placesin the next I scale specify the mid-point on the J scale which has beenpassed.

Thus in the first interval (Fig. 4),we have all the individuals to the leftof the midpoint between stimuli Aand B. The second I scale is B A C D,and as stimuli A and B have changedplaces in going from the 11 scale to the

FIG. 4. An example of how the midpoints"of four stimuli may section the jointdistribution into seven intervals, each characterized by an I scale.

150 CLYDE H. COOMBS

la scale we have passed the midpointAB. In going from la to Is stimuli Aand C exchange orders on the two Iscales and hence the midpoint be-tween A and C is the boundary be-tween 12 and Is. If we continue thisprocess we see that the order of the sixmidpoints is as follows: AB, AC, BC,AD, BD, CD. These six boundariessection the joint distribution intoseven intervals which are ordered asalso are the stimuli. From the orderof the six midpoints in the case of fourstimuli we have one and only one pieceof information about metric relationson the joint continuum. Becausemidpoint BC precedes AD we knowthat the distance between stimuli Cand D is greater than the distance be-tween stimuli A and B. We shall dis-cuss these points and characteristicsin more detail in the section on metriceffects. There are then an infinite va-riety of quantitatively different Jscales which would yield this same setof seven I scales—but there is only onequalitative J scale. The J scale inFig. 4 meets the conditions necessaryto yield the manifest data.

It must be emphasized that allmetric magnitudes in Fig. 4 are arbi-trary except that the distance fromstimulus A to B is less than the dis-tance from stimulus C to D.

With the qualitative informationobtained in this experiment about thelatent attribute underlying the prefer-ences for presidential candidates thenext task is the identification of thisattribute. Here all the experimentercan do is to ask himself what it is thatthese stimuli have and the individualshave to these different degrees as indi-cated by their ordinal and metric rela-tions. One might find in this hypo-thetical case that it appears to be acontinuum of liberalism, for example,or of isolationism. One would thenhave to conduct an independent ex-

periment with other criteria to vali-date one's interpretation*

• IV. METRIC EFFECTSWhile the data with which we deal

in the vast majority of scaling experi-ments are qualitative and non-numer-ical there are certain relations be-tween the manifest data-and the met-ric relations of the continuum. Theserelations have not all been worked outand general expressions have yet to bedeveloped. The complexity of the re-lations very rapidly increases with thenumber of stimuli; therefore to illus-trate the effect of metric we shall takethe simplest case in which its effect ismade apparent, the case of fourstimuli.

With four stimuli, A, B, C, and D,there are 24 permutations possible.Thus it is possible to find 24 qualita-tively different I scales. Also, obvi-ously, each of these 24: orders couldoccur as a J scale which could give riseto a set of I scales. Half of the Jscales may be regarded as merelymirror images of the other half. Thusif we have a J scale with the stimuliordered B A D C and ; identify thecontinuum as liberalism-conservativ-ism, then, in principle, we also havethe J scale with the stimuli orderedC D A B and would identify it as con-servativism-liberalism. Hence thereare only twelve J scales which may beregarded as qualitatively; distinguish-able on the basis of the, order of thefour stimuli. I scales which aremirror images of each other, though,are definitely not to be confused.They may well represent entirely dif-ferent psychological meanings. Thedirection of an I scale is defined experi-mentally—the direction of a J scale isa matter of choice.

Each J scale of four stimuli givesrise to a set of seven qualitative Iscales. We are interested in know-

PSYCHOLOGICAL SCALING WITHOUT UNIT OF MEASUREMENT 151

ing, of course, whether the J scalededuced from a set of I scales obtainedin an experiment is qualitativelyunique. The answer to this appearsto be yes and immediately obviouswhen it is recognized that a set of Iscales generated from a J scale hastwo and only two I scales which aremirror images of each other; thatthese two I scales must have beengenerated from the intervals on theopposite ends of the J scale, and thatthe order of the I scales within a set isunique. These statements are still tobe developed as formal mathematicalproofs and hence must be regarded astentative conclusions.

However, a given qualitative J scaledoes not give a unique set of I scales.For example, with four stimuli, wemay have the qualitative J scaleA B G D. This order of four stimulion a J scale can yield two different setsof I scales as follows:

set 1A B CDB A CDB C A DBe* n AC B DAC DB ADC B A

set 2A B C DB A CDB C A DC R A r>C B D AC D S ADC B A

It will be noticed that these two setsof seven I scales from the same quali-tative J scale are identical except forthe I scale from the middle interval.This arises from the following fact.There are six midpoints for the fourstimuli on the J scale. These are asfollows: AB, AC, AD, BC, BD, CD.The order and identity of the first twoand the last two are immutable; theymust be, in order: AB, AC, , , BD,CD. But the order of the remainingtwo midpoints is not defined by thequalitative J scale but by its quantita-tive characteristics. If the intervalbetween stimuli A and B is greaterthan the interval between C and D,

then the midpoint AD comes beforethe midpoint BC and the set of sevenI scales will be set 1 listed above. Ifthe quantitative relations on the Jscale are the reverse and the midpointBC comes before AD, then the set ofseven I scales which will result arethose listed in set 2 above.

Thus we see that in the case of fourstimuli, a set of I scales will uniquelydetermine a qualitative J scale andwill provide one piece of informationabout the metric relations. For fiveor more stimuli the number of piecesof information about metric relationsexceeds the minimum number thatare needed for ordering the successiveintervals. However, the particularpieces of information that are obtainedmight not be the appropriate ones fordoing this. It is interesting to notehere that this is a new type of scale notdiscussed by Stevens. This is a typeof scale that falls between what hecalls ordinal scales and interval scales.In ordinal scales nothing is knownabout the intervals. In interval scalesthe intervals are equal. In this scale,which we call an ordered metric, theintervals are not equal but they maybe ordered in magnitude.

As the number of stimuli increases,the variety of different sets of I scalesfrom a single qualitative J scale in-creases rapidly. This means that agreat deal of information is beinggiven about metric relations. For ex-ample a J scale of five stimuli yields a

set of eleven I scales f in general n

stimuli will provide ( * ) + 1 different

I scales from one J scale ). Depend-

ing on the relative magnitudes of thefour intervals between the five stimulion the J scale, the same qualitative Jscale may yield twelve different sets ofI scales. This means that for a given

152 CLYDE H. COOMBS

order of five stimuli on a J scale thereare twelve experimentally differenti-able quantitative J scales. Previ-ously, in the case of four stimuli, wefound only two differentiable quanti-tative J scales for a given qualitativeJ scale.

The particular set of I scales ob-tained from five stimuli may provideup to five of the independent relationsbetween pairs or intervals. For ex-ample, suppose we have the qualita-tive J scale A B C D E. Among thetwelve possible sets of I scales whichcould arise are the following two,chosen at random:

set 1A B C DEB A C DEB C A DEB C D A EC T> T» A T?

C T\ T> A T?

D P P A F

D C B E ADCE B ADE C B AE DC B A

set 2A B O D EB A C DEB C A DEB C D A EB r * ~n T? AC r> r\ T? Ap r> p p A

D C B E ADCE B ADE C B AE DC B A

Let us see what information is givenby each of these sets about the relativemagnitudes of the intervals betweenthe stimuli on the J scales. Considerset 1 first. The order of the ten mid-points of the five stimuli according toset 1 is as follows: AB, AC, AD, BC,BD, CD, AE, BE, CE, DE. Weknow immediately, from the fact thatthe midpoint BC comes after AD, thatthe interval between stimuli A and B(AB) is greater than the interval be-

SET 1

Order of midpoints

AD,CD,BD,CD,

BCBEAEAE

Relative magnitude ofintervals on J scale

CD < ABBC < DEAB <DEAC < DE

tween the stimuli C and D (CD). Wehave summarized this in the first rowof the table below. The other rowscontain the other metric relationswhich can be deduced from set 1.

Or, in brief form, the I scales con-tained in set 1 indicate that the follow-ing relations must hold between stim-uli on the J scale.

CD < AB < DEAB + BC < DE

In the same manner we may studythe implications of set 2 for the metricrelations between stimuli on the Jscale. The midpoints for this set arein the following order: AB AC AD,AE, BC, BD CD, BE, CE, DE.

SET 2

Order of midpoints

AD,CD,AE,AE,

BCBEBDBC

Relative magnitude ofintervals on J scale

CDBCDECE

< AB< DE< AB< AB

Or, in brief, the relative magnitudesof the intervals between stimuli on theJ scale are known to the followingextent.

BC < DE < ABDE + CD < AB

The different implications of thesetwo sets of I scales for the metric rela-tions on the J scale may be illustratedby sketching two quantitative J scaleswhich have the appropriate metricrelations (Fig. 5).

The two sets of I scales which areillustrated here were only two oftwelve possible different sets whichcould be generated from a single quali-tative J scale of five stimuli. Each ofthe twelve sets of I scales would implya different set of quantitative relations

PSYCHOLOGICAL SCALING WITHOUT UNIT OF MEASUREMENT 153

SetJ.

SftZ

H—h

FIG. 5. An example of two joint distributions with the same order of stimulibut different metric relations obtained from different sets of I scales.

among the distances between stimulion the J scale. The two sets of Iscales used here happened to differfrom each other in three of their par-ticular members. If we take thetwelve potential sets and make a fre-quency distribution of the number ofpairs of sets which have 1, 2, 3, 4, or 5particular I scales different or 10, 9,8, 7, or 6 I scales in common, we getthe following distribution.

Number of identicalordinal positions ina pair of sets withthe same qualita-

tive I scale

109876

Number of such pairsof sets

18241761

66

The surface has not even beenscratched on the generalizations whichcan be developed. Enough has beenpresented here to provide a generalidea of the type of information whichcan be derived.

V. AN EXPERIMENT

In order to study the feasibility ofthis unfolding technique and to com-

pare several different psychologicascaling techniques an experiment wasconducted in several classes in the De-partment of Social Relations at Har-vard University. A questionnairewas administered pertaining to gradeexpectations in a course. Data werecollected suitable for four differentkinds of analyses on the same contentarea from the same individuals. Thefour types of analyses for which datawere collected were (1) the generationof the joint continuum by the unfold-ing of I scales obtained by the methodof rank order, (2) the generation ofthe joint continuum by the unfoldingof I scales obtained by the method ofpaired comparisons, (3) the generationof the joint continuum by what weshall call the Guttman triangularanalysis, and (4) the generation of thejoint continuum by what we shall callthe parallelogram analysis.

We shall present here the analysisof only the rank order data. The ex-periment was first conducted in agraduate course in statistics and then,with a slight change in the wording ofsome questions, in an undergraduatecourse in sociology. Despite thesedifferences in subjects and questionsthe general resuts were practicallylidentical in the two groups and be-cause our primary interest is in thetechnique and not the content of this

154 CLYDE H. COOMBS

experiment we have lumped the dataof the two groups.

The questionnaire was arranged asa small booklet with each question ona different page. The questions ap-propriate to the different techniqueswere deliberately mixed. The natureof the instructions, the separate pag-ing for the questions, and the mixtureof the questions were part of a de-liberate effort to induce inconsistency,or at least to minimize a deliberateand artificial consistency. We feltthis was accomplished but the highdegree of consistency was surprising.

The content of the questionnairefollows. page 2.

page 1. Instructions: This is an experi-ment to test certain theoreticalaspects of psychological scal-ing techniques. It is entirely page 3.voluntary and you need notanswer the questionnaire ifyou so choose. However, you, page 4.as an individual, will not beidentified; complete anonym-ity is preserved. We are in-terested only in certain internal page 5.relations in the data. Thiswill become obvious to youbecause it will appear that weare getting the same informa- page 6.tion repeatedly in differentways.

You are free, of course, tomark these items entirely at page 7.random. It is our hope, how-ever, that enough of you willtake a serious attitude toward page 8.the experiment and make aneffort to respond to each itemon the basis of consideredjudgment. page 9.

The questions pertain toyour grade expectations in thiscourse. Of course, everyonewants an A or B, but we would page 10.like to ask you to give seriousconsideration to what you

really can expect to get. Wewant you to be neither modestnor self-protective. If youthink you will get an A orflunk the course, make yourjudgments accordingly. Re-member: there is completeanonymity.

There is one item or ques-tion on each of the followingpages. Consider each ques-tion or item independently ofthe others. Answer each onewithout looking back at previ-ous answers. Treat each itemas an item in its own right anddo not concern yourself withtrying to be logical or consist-ent. Work quickly.In the following list of gradescircle the two grades whichbest represent what you ex-pect to get in this course.A B C D EI expect to get a grade at leastas good as a B.yes noOf these two grades, which isnearer the grade you expectto get?A DOf these two grades, which isnearer the grade you expectto get?B COf these two grades, which isnearer the grade you expectto get?A BI expect to get a grade at leastas good as a D.yes noOf these two grades, which isnearer the grade you expectto get?D EOf these two grades, which isnearer the grade you expectto get?C AI expect to get a grade at leastas good as an A.yes no

PSYCHOLOGICAL SCALING WITHOUT UNIT OF MEASUREMENT 155

page 11.

page 12.

page 13.

page 14.

page 15.

page 16.

page 17.

page 18.

Of these two grades, which isnearer the grade you expectto get?B EOf these two grades, which isnearer the grade you expectto get?E CI expect to get a good grade,yes noOf these two grades, which isnearer the grade you expectto get?D BI expect to get a grade at leastas good as a C.yes noOf these two grades, which isnearer the grade you expectto get?E AOf these two grades, which isnearer the grade you expectto get?C DPlace the five grades in rankorder below such that the oneon the left is the grade youmost expect to get, then in thenext space is the grade younext most expect to get, andso on, until finally at the rightis the grade you least expecttovget.

1 2the gradeI mostexpect toget

4 5the gradeI leastexpect toget

Page 18 of the questionnaire con-tained the rank order data. Thetotal number of subjects for whomusable rank data were obtained was121 (statistics class 40, sociology class81). The individuals not included inthe data which follow were people whointroduced new grades (F), plus orminus grades, or left blanks. All in-dividuals who wrote down the fiveletters A B C D E in some order are

contained in the analysis. The Iscales obtained by the method of rankorder and the number of people ineach of the classes who so respondedare given in Table I.

TABLE I

NUMBER OF PEOPLE IN Two CLASSESGIVING EACH OF THE RANK ORDER

I SCALES

I scale

A B O D EB A C DEB C A DEC B A DEC B DAEC B DE AC DB E ADE C B AB C DA EB C DE AB A CE DC A B DE

Total

Statisticsclass14106112113001

40

Sociologyclass

622214

113007610

81

Let us first consider the two I scalesin the bottom two rows of Table I.These two scales, B A C E D andC A B D E, were each given by oneindividual. There is. no way thateither of these two scales could havearisen from a J scale on which thestimuli are in the order A B C D E re-gardless of the metric relations on theJ scale. All the evidence, both apriori and from the other responsepatterns, indicates that the orderof stimuli on the J continuum isA B C D E. So we must regard thesetwo I scales as errors on the part ofrespondents or assume some esotericpsychologies to explain them, thelatter completely unjustified. Hencewe shall drop these two individualsfrom further consideration.

Let us now look at the first eightscales listed in Table 1. From fivestimuli we can have eleven different Iscales to correspond to the eleven in-tervals into which the J scale is sec-

156 CLYDE H. COOMBS

tioned by the ten midpoints betweenstimuli. Consequently these eight Iscales constitute a partial set. Butbecause one of the missing intervals(interval 8) has two alternative Iscales, there are two possibilities forthe complete set of which these eightare a partial set, as follows:

I scale Total NA B C D E 20B A C D E 32B C A D E 27C B A DE 5C B D A E 12C B DE A 5C DB E A 1

D C B E A — C D E B A 0D C E B A 0DE C B A 1E D C B A 0

There were three intervals towardthe low end of the J scale which werenot occupied by any students. Ap-parently not very many students ex-pected to get a low grade. The factthat one of the intervals on the J scale(interval 8) is blank and could berepresented by two alternative I scalesmeans that one of the metric charac-teristics of the intervals between stim-uli on the J scale is not experimentallygiven. But from the remaining Iscales the order on the J scale and someof the metric effects are determined.

The indications are that the orderof stimuli on the joint continuum isA B C D E. We know this, of course,from a priori grounds, but the point isthat this fact need not be known be-forehand. Secondly, the individualsare placed in intervals and the inter-vals ordered on the continuum. Thenumber of ordered intervals is elevenbut individuals occupy only eight ofthe eleven intervals. Thirdly, weknow certain metric relations amongthe intervals between stimuli. Theseare obtained as follows. From theorder of the I scales within the set,

the successive midpoints betweenstimuli are: AB, AC, BC, AD, AE,

B D - (BE) ' (DC)- C E - D E - Be-

cause interval 8 was unoccupied andthere are two alternative I scaleswhich satisfy it, it is not knownwhether the midpoint DC or BEcomes first. Hence one piece of met-ric information is lacking. In thetable below are the metric relationswhich may be deduced and the basisfor the deduction.

Order of midpoints

BC, ADAE, BD

Metric relations

AB < CDDE < AB

Or, in brief, DE < AB < CD. Thepsychological distance between thegrades D and E is the least and thedistance between the grades C and Dthe largest, with the distance from Ato B in between. No information isgiven of the relative magnitude of thedistance between the grades B and C.

But now for another portion of thestudents there is a different interpre-tation. There are two I scales inTable I, B C D A E and B C D E A,which have not yet been considered.They are also members of a set thatis as valid as the first set. If we re-move I scales 4 and 5 from the partialset of 8 and substitute these two wehave the following:

I scaleA B C DEB A C DEB C A DEB C D A EB C DE AC B DE AC DB E A

Total N20322710651

D C B E A — C D E B A 0DC E B A 0DE C B A 1E DC B A 0

PSYCHOLOGICAL SCALING WITHOUT UNIT OF MEASUREMENT 157

This set differs from the precedingonly in I scales 4 and 5. If we analyzethe significance of this set to themetric relations we have the following:

Order of midpoints Metric relations

AE, BC AB >CE

It1 appears from this that for the 16individuals who gave the two I scalesB C A D E and B C D E A, if theyare treated as members of the sameset, the psychological distance be-tween the grades A and B is greaterthan either the distance from C to Dor from D to E or, in fact, is greaterthan the sum of these two distances.This is in contrast to the 17 indi-viduals in the first set who gave the Iscales C B A D E a n d C B D A E i npositions 4 and 5 for whom the rela-tive distances between_ grades was inthe order DE, AB, CD.

The reader must be aware of thefact that it is only these 33 individualsfor whom these metric relations arededuced. There is no way of know-ing, for example, that the 20 indi-viduals who gave the I scale A B C D Edid so from the first interval on oneof these two possible quantitative Jscales or from any one of the manymore differing in metric relations.An individual yielding the I scaleA B C D E is known to be to the leftof the midpoint AB and the relativedistances BC, CD, and DE do notaffect his I scale. It is just the criticalI scales which provide informationabout metric and this information isvalid only for the individuals whoyield these I scales. By putting themtogether we can construct a totalpicture, but our only evidence thatthey go together is one of internalconsistency.

The general picture of metric rela-tions among grades given by thesetwo sets of I scales is the following:

1) For 17 individuals the metricrelations a're DE < AB < CD

2) For 16 individuals the metricrelations are AB > CE

Thus we find from the data thatsome of these individuals are simplyon a different continuum from otherindividuals. To somehow computescale values for the stimuli which willbe assumed to hold for all these indi-viduals is to do violence to the experi-mental evidence. From the data wehave learned where an individual ison the continuum in relation to thestimuli, and in addition somethingabout how the whole continuum looksto him.

VI. SUMMARY

We have presented a new type ofscale called an ordered metric andhave presented the experimental pro-cedures required under certain limit-ing conditions to secure such a scale.

We have pointed out that the in-formation which could be obtainedunder these conditions is as follows:

1) the discovery of a latent at-tribute underlying preferences,

2) the order of the stimuli on theattribute continuum,

3) something about the relativemagnitudes of the distances be-tween pairs of stimuli,

4) the sectioning of the continuuminto intervals, the placing ofpeople in these intervals, andthe ordering of these intervalson this attribute continuum,

5) something about the relativemagnitudes of these intervals.

158 CLYDE H. COOMBS

These were illustrated with a hypo-thetical example and an experiment.

REFERENCES

1. CAMPBELL, N. R. Symposium: Measure-ment and its importance for philoso-phy, in Action, perception, and meas-urement, The Aristotelian Society,Harrison and Sons, Ltd., 1938.

2. COOMBS, C. H. Some hypotheses for theanalysis of qualitative variables. PSY-CHOL. REV., 1948, 55, 167-74.

3. STEVENS, S. S. On the theory of scalesof measurement. Science, 1946, 103,677-80.

4. THURSTONE, L. L. The law of compara-tive judgment. PSYCHOL. REV., 1927,34, 273-86.

[MS. received October 17, 1949]