Educational Testing Service

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1 RB-63-l3 An Individual Diffeences Model fo Multidimensional Scaling Ledyad R Tucke and Samuel Messick National Institute of Mental Health, United States Public Health Sevice Reseach Gants M-2878 and M-4l86 to Educational Testing Sevice and Office of Naval Reseach, Contact Non-18;4(39) Univesity of Illinois Educational Testing Sevice

2 AN INDIVIDUAL DIFFERENCES MODEL FOR IvIULTIDlMENSIONAL SCALING Abstact A quantitative system is pesented to pemit the detemination of sepaate multidimensional peceptual spaces fo individuals having diffeent viewpoints about stimulus inteelationships. The stuctue of individual diffeences in the peception of stimulus elationships is also detemined to povide a famewok fo ascetaining the vaieties of consistent individual viewpoints and thei elationships with othe vaiables.

3 AN INDIVIDUAL DIFFERENCES MODEL FOR ~ultidimensional SCALING The pesent pape attempts to develop a quantitative system to povide fo diffeential epesentations of peceptual stuctues fo individuals having diffeent viewpoints about stimulus inteelationships. In past attempts at stimulus scaling, two majo appoaches have been employed in dealing with data obtained fom goups of individuals: One appoach has been to ascetain goup aveages and then to genealize findings to the "aveage peson" in each goupj the othe pocedue has been to wok with each peson sepaately and to enumeate the esults individual by individual. The fist method, which is the moe usual in pactice, may lead to a staightfowad but possibly false intepetation, in that the esults fo the aveage peson may not descibe vey accuately the consistent esponses of each individual in the sample. The second method of woking with each individual sepaately also possesses seveal dawbacks, among which ae the extensiveness of expeimental obsevations equied to obtain stable esults fo each individual and the difficulties involved both in descibing the esults fo goups of individuals and in compaing the esults fo seveal individuals and goups. This eseach was suppoted in pat by the National Institute of Mental Health, United States Public Health SeVice, unde Reseach Gants M-2878 and M-4186 to Educational Testing Sevice, in pat by Educational Testing Sevice, and in pat by the Office of Naval Reseach unde Contact Non-1834(39) and the Univesity of Illinois. The authos wish to thank Ds. Haold Gulliksen and Douglas N. Jackson fo thei helpful comments and Miss Henietta Gallaghe fo supevising the computations. Potions of this pape wee pesented at the Ameican Psychological Association meetings in Chicago, Septembe This pape was witten while D. Messick was a Fellow at the Cente fo Advanced Study in the Behavioal Sciences.

4 -2- The intention in the pesent pape is to develop a system which will povide not only multidimensional desciptions of individual peceptual stuctues and a basis fo compaisons between individuals and goups, but also a supestuctue to epesent the vaieties o types of consistent individual peceptions. The pesent model is thus concened both with the multidimensional scaling of stimuli and the stuctue of consistent individual diffeences in peception and judgment. Many of the multidimensional scaling methods based upon an aveaging of esponses ove all individuals in a sample, eithe in tems of simple aveages as in the method of equal-appeaing intevals o in tems of moe complicated scaling functions as in the methods of successive intevals and of complete tiads, have been eviewed by Messick (1956b) and Togeson (1958). The application of this multidimensional scaling of the 'aveage individual" in a goup pesents cetain difficulties, howeve, when compaisons ae attempted between peceptual stuctues obtained fom divese goups that pesumably have diffeent oientations to the stimuli. A common finding has been that only subtle diffeences appea in these stuctues and that the main attibutes of the peceived spaces ae essentially identical (Abelson, 1955; Messick, 1956a; Messick, 1961). It may be that in these studies all individuals peceived the stimulus inteelations in moe o less the same manne, thus yielding the obtained obsevations of only mino diffeences between goups. On the othe hand, it may be that extensive diffeences existed in individual peceptual spaces, but the scaling method blended them togethe in deiving the aveage stuctue fo each goup. We

5 -3- might not have discoveed yet how to sot individuals into contasting goups that would have diffeent peceptual stuctues fo thei aveage pesons. The vaiables employed so fa fo establishing such compaison goups may be only slightly elated to individual diffeences in peceptual stuctues. It would be desiable, then, to develop a pocedue fo uncoveing diffeential peceptual spaces that does not equie pio soting of individuals into subgoups on the basis of vaiables pesumed to diffeentiate between peceptual stuctues, but one that would instead indicate the vaiety of individual peceptual stuctues epesented in the total goup. A technique is theeby equied that would fist isolate empiically any consistent individual viewpoints about stimulus diffeences and would then povide fo the deivation of sepaate multidimensional spaces fo each viewpoint. In the pesent discussion, the multidimensional scaling model developed by Richadson (1938), using the Young and Householde (1938) theoems, and extended by Togeson (1952), Messick and Abelson (1956), and Shepad C1962a; 1962b) will fom the basis fo desci.ption of the peceptual stuctue fo each individual. In this model, each stimulus is epesented by a point in a Euclidean space, with the peceived diffeence o dissimilaity between two stimuli epesented by the distance between the two stimulus points. Measues of peceived dissimilaity among stimuli may be obtained by seveal expeimental pocedues (c. Shepad, 1960), such as judging which of two stimuli is moe simila to a thid stimulus (Richadson, 19}8j Togeson, 1951), ating the dissimilaity between membes of pais of stimuli on some ating scale

6 -4- (Attneave, 1950; Abelson, 1955; Ekman, 1954; Messick, 1956c; Jackson, Messick & Solley; 1957; Coombs, 1958; Reeb, 1959; Abelson & Semat, 1962), anking the similaity of the emaining stimuli to each stimulus in tun (Klingbeg, 1941; Moton, 1959), using intusion eos in identification leaning as measues of poximity (Shepad, 1958), and estimating intestimulus distance diectly on a atio scale (Helm, 1959; Indow & Kanazawa, 1960; Indow & Uchizono, 1960). Vaious efinements in analysis ae possible, including the application of scaling techniques to ecove inteval popeties fo the dissimilaity o distance Bcale (c. Adams & Messick, 1958; Togeson, 1958) and the solution fo an additive constant to establish an advantageous scale oigin to yield the simplest multidimensional epesentation (Messick & Abelson, 1956; Togeson, 1958). The multidimensional peceptual space can then be deived by facto analyzing scala poducts between stimulus vectos computed fom the distance estimates (Young & Householde, 1938; Togeson, 1958), o by applying Shepad's (1962a; 1962b) compute model fo constucting a Euclidean metic configuation diectly fom nonmetic poximity infomation. The poblem of uncoveing and epesenting consistent individual viewpoints about stimulus popeties has been addessed fo the case of unidimensional scales by Tucke (1955; 1960). Tucke (1955) developed a vecto model fo paied compaisons that pemits Judges, when evaluating stimuli with espect to a unidimensional attibute such as pefeence, to diffe among themselves in thei peceived odeing and spacing of the stimuli. Each individual viewpoint is epesented as a vecto in a multidimensional space of stimulus objects, with stimulus

7 -5- pojections on each vecto epesenting scale values fo that viewpoint. The numbe of dimensions equied to span the space of individual viewpoint vectos is detemined by facto analysis, and the esulting otated facto loadings epesent stimulus scale values fo the vaious viewpoints. A simila ationale undelies the facto analysis ofcategoy atings (Mois & Jones, 1955; Messick, 1960). The dimensions isolated in these vecto models summaize consistencies in atings of sepaate stimuli with espect to some specified attibute, and they epesent consistent individual viewpoints about that stimulus popety. The dimensions in the distance model of multidimensional scaling, on the othe hand, ae deived fom judgments about pais of stimuli with espect to similaity, and they epesent diffeential attibutes of peceived stimulus vaiation. In the vecto model, then, the multidimensional space epesents the diffeent viewpoints of the Judges, each viewpoint being a one-dimensional scale of the specified stimulus attibute. In the distance model, the multidimensional space epesents the diffeent ways in which the stimuli ae peceived to vay, each judge peceiving the space in essentially the same manne. The pesent pape attempts to combine these two appoaches by applying the vecto model of stimulus scaling to measues of similaity between pais of stimuli, theeby isolating dimensions of individual viewpoints about stimulus similaity o poximity. Stimulus pojections (in this case fo pais of stimuli) on each otated dimension of viewpoint will then povide measues of similaity to be analyzed accoding to the distance model of multidimensional scaling (Messick & Abelson, 1956; Togeson, 1958; Shepad, 1962a; 1962b). A sepaate multidimensional epesentation of the

8 -6- peceived stimulus space is thus povided fo each consistent viewpoint about stimulus similaity. A nontechnical discussion of these methods and thei development was pesented in the context of social peception by Jackson and Messick (1963), and some ecent applications wee descibed by Gulliksen (1961; 1963) and Tucke (1963). Helm and Tucke (1962) applied these method.s to colo peception. Analysis of Consistent Individual Viewpoints in Multidimensional Scaling The pesent model assumes that estimates of intestimulus distances ae available fo each individual. As indicated peviously, these estimates may be obtained expeimentally by seveal pocedues, such as ating the dissimilaity between stimuli on a ating scale o constucting the intestimulus distances diectly by atio judgments. Let x(jk)i = an estimate of dissimilaity o intepoint distance between stimuli j and k by individual i. i, h = individuals 1:2,3,..,N j, k = stimuli 1,2,3,...,n. (jk) = stimulus-pais 12, 1), 23, etc.; k > J Numbe of stimulus-pais = n(n-l)/2. Thee i6 one such distance measue fo each pai of stimuli and each individual, so that these measues may be aayed in a ectangula table with a ow fo each pai of stimuli and a column fo each individual. All cells in this table (designated matix X) should 'be filled; i.e., thee should be no missing data.

9 -7- x = matix of x(jk)i' having n(n-l)/2 ows fo the stimulus-pais and N columns fo the individuals. The typical multidimensional scaling analysis (cf. Togeson, 1958) involves an aveaging, fequently weighted in tems of a scaling function, of the x(jk)i values ove the individuals to obtain a single numbe o scale value to epesent the dissimilaity o distance between each pai of stimuli j and k. These aveaged o scaled distance values ae then usually analyzed accoding to the Young-Householde theoems to obtain a multidimensional epesentation (Messick & Abelson, 1956; Togeson, 1958). This pocedue assumes that these aveage distance measues adequately summaize the infomation in the distibution of values ove the i individuals and that the vaiation in these values is due to andom dispesion o eo of measuement. The pesent analysis, on the othe hand, fist asks whethe thee is consistent covaiation among individuals in these x(jk)i estimates by factoing X into its pincipal components. fo the consistent vaiance in If only one facto is found to account X, then the appopiate facto loadings, o othe types of aveage distance values, may be analyzed as usual to obtain a single epesentative multidimensional space. If, on the othe hand, moe than one facto is necessay to account fo the vaiance in X, then moe than one set of distance values will be obtained fom the facto loadings to be subsequently analyzed by multidimensional scaling pocedues. Seveal multidimensional spaces would theeby be deived epesenting diffeent points of view about the peceived stimulus aangements.

10 -8- In this pocedue, dimensions of viewpoint ae obtained fo consistent individual diffeences in the dissimilaity o distance estimates. Since thee will pesumably be fewe consistent viewpoints than thee ae individuals, the technique appeas moe efficient than analyzing each individual's distance estimates sepaately. Also, as will be seen below, the pesent method povides a famewok fo compaing the vaious viewpoints and fo elating them to outside vaiables. The cental point in the above discussion was the statement that consistent co-vaiation in the X(jk)i estimates is evaluated by factoing X into its pincipal components. Since X is an asymmetic, ectangula matix, howeve, the usual diect factoing equations ae not appopiate (e.g., Hamon, 1960). This poblem has been solved by factoing X accoding to a theoem of Eckat and Young (1936). Detemining Dimensions of Individual Diffeences Since the numbe of stimulus-pais is elated to the squae of the numbe of stimuli, the numbe of ows in matix X is likely to be elatively lage. If 20 stimuli wee used, fo example, the numbe of stimulus-pais would be 190; fo 25 stimuli thee would be 300 pais. Consequently, it is advantageous to use a modeately small sample of individuals and to pefom a type of obvese analysis in which elationships ae computed between individuals athe than vaiables. The basic matix in this analysis, designated matix P, is composed of sums of squaes of measues fo the individuals in the main diagonal and sums of coss-poducts of measues between pais of individuals off d.iagonal, all sums being taken ove the pais of stimuli. Thus, in tems of matix algeba,

11 -9- p = X'X, (1) an N x N matix of sums of coss-poducts between columns of X. The analysis next follows the pocedue developed by Eckat and Young (1936) to obtain a matix X of lowe ank than matix X that appoximates X in a least squaes sense. This analysis paallels Host's development (1963, pp ). Essentially, the matix A X is constucted to the desied degee of appoximation fom the lagest chaacteistic oots and vectos of matix X. A X = U w, a matix of ank that appoximates matix X in a least squaes sense (Eckat & Young, 1936), whee U ::; n(n-l)/2 x section of an othogonal matix (UI U = I), ::; x diagonal matix of latent oots VI = x N section of an othogonal matix (w \~ I ::; I) This analysis is simila to the pincipal components method developed by Rotelling (193)), but diffes in that the components ae deived fom the matix of sums of squaes and coss-poducts of aw measues inetead of fom a matix of intecovaiances as in the Hotelling pocedue.(cf. Nunnally, 1962). 'fhe components U, and W ' in the basic Eckat-Young theoem [Equation (2)) ae detemined fom the chaacteistic oots and vectos of the coss-poducts matix P. Since P unlike X, is a squae, symmetic matix, it may be analyzed diectly into pincipal components by standad pocedues (Hamon, 1960).

12 -10- AP A A 2 XI X = WI W, whee 2 is a diagonal matix composed of the lagest chaacteistic oots of P, and W contains, as ow vectos, the coesponding chaacteistic vectos of P. Note that the chaacteistic oots of Pae the squaes of the diagonal enties in matix, so that the diagonal matix in Equation (2) must be constucted fom the squae oots of the values in (p2) fom Equation (3). The matix U may now be computed by u = XW' p-l, (4) since W WI = UI U = I If in some expeiments the numbe of individuals is geate than the numbe of stimulus-pais, the coss-poducts matix of Equation (1) should instead be computed between stimulus-pais, summing ove individuals. The coss-poducts matix in the pesent analysis should always be computed between the vaiables on the shote side of X, summing ove the vaiables on the longe side. Thus, if N > n(n-l )/2, P = X Xl J (la) an n(0-1)/2 by n(n-l)/2 matix of sums of coss-poducts between ows of X The emainde of the analysis follows by symmety: A X = U w (2a) A A A. P = X XI = U p2 UI (3a)

13 -11- ( 4a) The elements in W' epesent pojections of points coesponding to individuals on unit-length pincipal vectos of X (and p). The elements in U epesent pojections of points coesponding to stimulus-pais on unit-length pincipal vectos of X. These stimuluspai pojections, when appopiately weighted, scaled, and otated to oientations possibly moe appopiate psychologically than the pincipal axes position, will constitute measues of distance between pais o~ stimuli. Thee 'Will be at least as many sets of distance measues as thee ae columns in the U matix, each set being subsequently analyzed by multidimensional scaling pocedues. Scaling fo Diffeences in Sample Size The above analysis poduces coefficients fo stimulus-pais and fo individuals that ae scaled so that W WI = I. Since W is a matix of ode by N, the esulting coefficients ae a function of the numbe N of individuals in the sample. ThUS, even if two multidimensional scaling studies diffeed only in sample size, i.e., if the same stimuli wee involved and the judges consisted of two diffeent-sized andom samples fom the sam.e population of individuals, the esulting numbes would not be compaable. into a matix V : It is desiable, then, to escale W V=KW, (5 ) so that the coefficients in V ae independent of sample size; i.e.,.! V V' = I N (6)

14 -12- whee K and (~) ae scala matices with diagonal elements K and ~,espectively. SUbstituting (5) into (6) and solving, and 1 V = N2 w (8 ) To maintain the basic elationship of Equation (2), be escaled to U must then Thus, fom (2) "X = y V 1 1 = U N-2' N2 W = U w (10) since N2 and N-2 ae scala matices. The matix Y now contains scaled stimulus-pai pojections on the pincipal vectos, and the matix V contains, as ow vectos, scaled individual pojections on the pincipal vectos. Fom Equation (4), (11) The V matix of scaled pojections of individuals on pincipal vectos may be conveted into a facto matix A of scaled pojections of individuals on pincipal factos by weighting each vecto by the squae oot of the coesponding chaacteistic oot (Hamon, 1960): 1 A = V = N2 W (12)

15 -13- Then, fom Equation (10), "X = Y A Rotation to Stuctue in the Space of Individuals Since the pincipal axes location may not be the most appopiate oientation f'o dimensions of' viewpoint about stimulus similaity, a otation of the obtained A and Y matices might be consideed. This possibility is analogous to the otation of axes in facto analysis. One citeion fo such a otation would be a seach fo simple stuctue, by eithe gaphical o analytical pocedues (Hamon, 1960), in the facto space of the individuals. An x nonsingula tansfomation matix T is sought to otate these pincipal factos to simple stuctue o some othe citeion: B=TA (14) A matix Z of scaled stimulus-pai pojections on these otated axes is obtained by (15 ) It should be noted that the basic condition of the Eckat-young theoem in Equation (2) is still satisfied unde these tansfomations: (16) The matix Z contains scaled stimulus-pai pojections on the otated axes. Each column of Z thus povides a set of measues epesenting distances between pais of stimuli in tems of a otated dimensian of viewpoint about stimulus similaity. The n(n-l)/2 coefficients

16 -14- in each of the columns of Z constitute measues of distance between the n(n-l)/2 possible pais of n stimuli, which may then be aayed in sepaate n x n distance matices. Each distance matix is then analyzed by the methods of multidimensional scaling to obtain sepaate multidimensional spaces (Messick & Abelson, 1956; Togeson, 1958; Shepad, 1962a; 1962b). The matix B contains, as ow vectos, scaled individual pojections on the otated axes. The coefficients in the ows of B may be consideed scoes fo the individuals on viewpoint vaiables. The size of each coefficient in a ow indicates the extent to which that individual's point of view about stimulus similaity coesponds to the paticula otated dimension of viewpoint epesented by the ow. Since each individual eceives a scoe on all viewpoint dimensions, coelations may be computed between these viewpoint vaiables and scoes on othe outside measues--pehaps of pesonality, cognitive, o social vaiables--to ascetain popeties and coelates of the viewpoint dimensions. Scoes fo the individuals on outside measues may also be used in a kind of multiple-coelation pocedue to oient viewpoint dimensions in the facto space of individuals (matix B ) so that they coelate as highly as possible with paticula outside measues (Mosie, 1939; Cliff, 1962). In multiple-coelation tems, if the pojection of an outside measue into the individual facto space of B is found to account fo most of the measue's vaiance (high multiple-coelation), then a viewpoint can be located (using B -weights as diection numbes) and the attendant multidimensional space deived to epesent high scoes on the outside measue, whethe they wee actually pesent in the sample o not.

17 -15- Idealized Individuals Since the enties in matix B epesent coodinates of points fo individuals on otated axes, this space may be eadily plotted gaphically. The facto space of individuals would also usually be plotted pio to otation fom the enties in matix A. If cetain individuals ae of paticula inteest, pehaps because of thei scoes on othe vaiables o because of thei deviant o cental location in the facto space, it may be desiable to deive sepaate multidimensional spaces fo each of these pesons. This may be accomplished by estimating distance measues fo each of these i individuals by post-multiplying matix Z by those column vectos of B coesponding to the selected individuals. If the selected column vectos of B ae efeed to as B i, then fom Equation (16), (17) whee A X. is an n(n-l)/2 by i matix of estimated distance measues ~ fo i selected individuals and B. is an x i matix of selected ~ individual coefficients on otated viewpoint dimensions. The i sets of distance measues in A X. ae estimated only fom the facto vaiance ~ in the -dimensional Viewpoint space. Since much of the eo vaiance in the oiginal x(jk)i measues has theeby been eliminated, the epoduced distance measues should be moe stable than the aw A atings fo subsequent analysis. Each of the i columns of Xi' then, contains n(n-l)/2 measues of distance between pais of stimuli. These can then be analyzed sepaately by multidimensional scaling methods to poduce i sepaate spaces, one fo each of the selected individuals.

18 -16- It is also possible to inset into the plots of the facto space of individuals additional points at any desied location. These points may be intepeted as "idealized individuals. II Thei location may be detemined fom any desied citeion, such as placing an idealized individual nea o Within clusteings of points fo eal individuals, o at the extemities of the aay of eal points, o at positions detemined by outside measues. Any desied numbe of idealized individuals may be inseted into the facto space. Sepaate multidimensional spaces may be deived fo each idealized individual as follows: Fist, ead the coodinates of each idealized point fom the facto plots of matix B, and ecod the coodinates of each point in a column vecto. Assemble these column vectos fo g idealized individuals into a matix G. Analogously to EqUa.- tions (16) and (17), co~pute,,'v X g = Z G (18) whee A Xg is an n(n-l)/2 by g matix of estimated distance measues fo g idealized individuals and G is an by g matix of idealized individual coodinates on the otated axes. The elements in each "'- column of X epesent estimates of distance among the possible pais g of n stimuli fo each idealized individual These distances can then be analyzed sepaately by multidimensional scaling methods to poduce g sepaate spaces, one fo each idealized individual. If the idealized individual points ae inseted into the facto plots pio to otation, then the coodinates would be ead ;fom the efeence fame of matix A, and

19 -17-, whee G A is an by g matix of idealized individual c60dinates on the unotated factos of matix A The extent to which each eal individual's point of view about stimulus similaity is elated to each of selected idealized viewpoints may be detemined by otating the dimensions of the facto space of individuals to positions defined by idealized individuals. That is, a dimension is located fo each selected idealized individual on which that idealized individual has a loading of unity and the othe idealized individuals have loadings of zeo. Then the pojections of the eal individuals on each dimension will indicate the extent of elation between the eal individuals and the selected idealized viewpoint. Fo this pupose, G can be consideed to be an extension of the B matix. G can be defined as an x 1.' squae section of G. ThUS, an 1.' x 1.' nonsingula tansfomation matix A is sought that will otate the idealized individual vectos of G into new positions such that each tansfomed vecto has one unit loading with the emaining enties zeo loadings: A G :: I (20 ) Although any numbe of idealized individuals may be defined and thei coesponding distance measues deived by Equation (18), Equation (20) can be solved only if G is a squae matix, possessing an invese. The coefficients elating eal individual viewpoints to idealized viewpoints can be computed in stages fo vaious squae sections of G.

20 -18- -~.i\.=g (21) H =.i\. B = G- 1 B (22 ) whee H is an by N matix of pojections of eal individuals on selected idealized individual dimensions. The size of these coefficients indicates the extent of elationship between each eal individual viewpoint and the idealized individual Viewpoints. ThUS, the computation of distance measues fo idealized individuals is seen to be the esult of anothe otation on the facto space, since A X = Z B = Z G G- 1 B H Each column of a Z matix of otated stimulus-pai coefficients, then, may be intepeted as measues of dissimilaity o distance between pais of stimuli fo an idealized individual. Each ow of the coesponding B matix of otated individual coefficients elates each eal individual to a paticula idealized viewpoint. The esultant peceptual spaces fo the idealized individuals ae indicative of the vaiety of spaces existent fo the eal individuals in the sample. If all subjects in the sample happen to have simila peceptual spaces fo the selected set of stimuli, thee will be only one column in matix Z, and the peceptual space f'o the single idealized individual will epesent the space fo all eal individuals in the sample. At the opposite exteme, the peceptual space f'o each subject might be unelated to the spaces of' evey othe SUbject. In this case, thee would be as many idealized individuals as eal individuals, and each subject could be consideed his own idealized individual. Between these two extemes thee

21 -19- ae many possible degees of complexity in the stuctue of individual diffeences in peceptual spaces which may be investigated expeimentally. The Peceptual Space fo the Goup Aveage Since coss-poducts ae analyzed in the pesent pocedue athe than intecovaiances, the infomation contained in the means of the dissimilaity atings o distance scoes x(jk)i is etained in the analysis. Consequently, the fist chaacteistic oot of the cosspoducts matix P is vey lage elative to the subsequent oots, since the coesponding fist pincipal vecto 1n U essentially ecoves these mean scoes. Indeed, the fist chaacteistic oot 1s often so lage elative to the emainde that some ules-of-thumb caied ove J fom facto analyses of covaiance and coelation matices 'Would usually indicate the pesence of only one consistent facto in P. Theefoe, to avoid giving undue weight to the consistently lage fist oot of a coss-poducts matix, citeia fo deciding the numbe of factos should include, in addition to elative vaiance accounted fo, the seach fo pattens in the distibution of oots and fo sudden beaks in the distibution of successive diffeences in oots. Although the coefficients in the fist unotated pincipal vecto in U ae not pecisely popotional to the aveage values (since the fist pincipal component accounts fo somewhat moe vaiance than would the unweighted mean dissimilaity atings), the loadings on this fist vecto will be vey highly coelated with the mean x(jk)i values and may be intepeted as distance measues fo the liaveage peson" in the goup. An aveage peceptual space may then be deived by teating the n(n-l)/2 loadings on the fist unotated pincipal

22 -20- vecto in U as measues of distance among the n stimuli and apply ing the standad pocedues of multidimensional scaling analysis (Messick & Abelson, 1956; Togeson, 1958; Shepad, 1962a; 1962b). In this analysis of the aveage peceptual space, the fist unotated U vecto may be scaled, if desied, by N -~ ~ as in Equation (9) o weighted by the coesponding latent oot, since the distances and the associated peceptual space ae detemined only to within multiplication by positive constants (Togeson, 1958). Thus, a peceptual space obtained by teating the coefficients on the fist pincipal vecto in U as distance measues would be oughly equivalent to a multidimensional scaling of the aveage X(jk)i values. Howeve, only in the case whee a single viewpoint dimension is found to be necessay in the pincipal components analysis of P would these aveage distance values adequately epesent the individual spaces. An Illustative Analysis of Political JUdgment Data An analysis of judgments of dissimilaity among cetain political leades with espect to thei political thinking was pefomed accoding to the pesent model to illustate the pocedue. Data wee selected fom a lage set peviously analyzed by Messick (1961) by taditional multidimensional scaling methods. Messick (1961) asked 57!1 male and 262 female undegaduates to ate on a nine-point scale the similaity of all possible pais of 20 political leades with espect to thei political thinking. The multidimensional method of successive intevals (Messick, 1956c; Diedeich, Messick)& Tucke, 1957) was then applied to two sepaate 8ubsamples of 267 students who endosed the Democatic

23 -21- Paty and 464 subjects who aligned themselves with the Republicans. The two esulting peceptual spaces each consisted of seven dimensions with essentially identical aangements of stimulus points. Fo the pesent pupose, a smalle sample of 39 students was selected in tems of thei answes to the fou questionnaie items listed in Table 1. To illustate the advantages of the pesent method, an Inset Table 1 about hee attempt was made to insue consistent individual vaiation in judgments of similaity by including in the analysis fou goups of individuals epesenting fou diffeent pattens of esponse to those items: libeal Democats in favo of labo, consevative Democats in favo of management, libeal Republicans in favo of labo, and consevative Republicans in favo of management. Ten subjects wee selected fo each of these goups except the consevative Democats in favo of management; it was not possible to find 10 students with this latte combination of esponses. Even when the selection citeion was elaxed to include consevative Democats who did not indicate a sympathy towad labo, only nine such subjects could be found fom the lage sample of ove 800. Dimensions of Viewpoint The 39 individual atings of dissimilaity fo the 190 possible pais of 20 political leades wee aayed in the matix X, which in this case had 190 ows fo the stimulus-pais and 39 columns fo the individual ates. Each cell enty x(jk)i was an intege fom 1 to 9, epesenting the categoy of dissimilaity into which individual i placed stimuluspai (jk). The 20 political leades used as stimuli ae listed in Table 8.

24 -22- The matix P of sums of coss-poducts between columns of X was computed by Equation (1) and is pesented in Table 2. P was next analyzed by the method of pincipal components as in Equation (3). The Inset Table 2 about hee diagonal matix of chaacteistic oots (f!) contained one vey lage oot as expected (258,784.74), then two smalle oots ( and ) that appeaed in tems of the total patten to be somewhat lage than the SUbsequent oots, which tailed off in faily egula steps to nea zeo ( , , , , , , , , , , , , , , , , , , , , , , , , , , , , , \~ , , , , , ). Consequently, it was decided to chaacteize the stuctue of individual diffeences in tems of thee dimensions. The squae oots of the thee lagest chaacteistic oots of P wee usen to constuct the diagonal matix (having diagonal elements , 58.15, and 50.24), and the thee coesponding chaacteistic vectos of P compised the matix W (Table 3). At this point, W would odinaily be escaled to fom the matix V by multiplying each element by.f39 as in Equation (8). Since only one sample was involved in the pesent example, this step was left out of the computations fo the sake of simplicity. Each ow of W was weighted by the coesponding value to poduce the coesponding ow of matix A, as in Equation (12)(see Table 3). The enties in the fist

25 -23- Inset Table 3 about hee ow 6f the A matix (o the fist column of Al as in Table 3) wee faily unifom lage positive values fo all the individuals. As expected, howeve, these fist facto loadings wee vey highly coelated (Speaman ank of.97) with the aveage atings made by these individuals to all 190 stimulus-pais. The matix U of stimulus-pai pojections on the unotated pin cipal vectos was computed by Equation (4) (Table 4). Since the scaling Inset Table 4 about hee facto fo sample size was not included in the pesent example, this step also coesponds to the computation of the matix Y by Equation (11). Vaiation in Individual Additive Constants The pesent model assumes that the input data o x(jk)i values epesent estimates of distance between stimuli j and k fo each individual i. Thus, in the case of one aveage dimension of viewpoint the elements of the matices in Equation (16) could be epesented by, (2~. ) whee is a loading fo the (jk) -th stimulus-pai on the single dimension and b i is a weight fo the i -th individual. Fo dimensions of viewpoint, A E b X(jk)i C mel z(jk)m mi, (25 )

26 -24- whee z(jk)m is a loading fo the stimulus-pai on the m -th dimension of viewpoint and b mi is a weight fo the individual on the m -th dimension. In tems of the model, these distances should be measued on a atio scale and if they ae not, cetain vaiations in individual scale popeties might be mistaken fo individual diffeences in viewpoint. Fo example, even though inteval popeties maybe eflected in a distance scale based upon categoy atings of stimulus dissimilaity, such as the pocedue used in the pesent study, the zeo points of each individual's scale might not be compaable. Thus, fo the case of one undelying viewpoint dimension, Equation (24) would become, (26) whee is an additive constant to tanslate each individual's scale to a atio scale with a fixed zeo-point (cf. Messick & Abelson, 1956). Fo dimensions of viewpoint, The ight side of Equation (26) can be epesented in matix tems as a Z matix (compised of a column vecto of z(jk) loadings on the single viewpoint dimension and a column vecto of unities) times a B matix (composed of a ow vecto of b. weights and a ow vecto of :1. individual additive constants c i ). The X matix of Equation (26) is thus seen to be of ank 2 even though only one undelying viewpoint dimension was postulated. Similaly, the ight side of Equation (27)

27 -25- can be epesented as a Z matix with columns of z(jk) values and a column of unities times a B matix with ows of b. weights J. and a ow of c. values. The coesponding X matix is thus seen to J. be of ank + 1 even though only viewpoint dimensions wee postulated. Thus, the possibility that one of dimensions obtained with the pesent pocedue epesents vaiations in individual scale constants and not a dimension of viewpoint should be caefully evaluated, paticulaly if inteval scaling o ating pocedues had been used to estimate the distances oiginally. This evaluation can be achieved by detemining an by tansfomation matix L that will otate Z (o u ) as closely as possible in a least squaes sense to a matix Q that contains a column of unities (o, because of fee tansfomation within a scale facto, a column of constants): U L = '" Q, (28) '" whee Q is a least squaes estimate of the citeion matix Q. If '" Q is found to contain a column of unities o constants, within some acceptable ange of vaiation, then one of the dimensions of X is intepetable in tems of individual vaiations in scale constants, with the emaining ( -l ) dimensions epesenting diffeent viewpoints about stimulus similaity. The invese tansfomation -1 L applied to the appopiate matix of individual weights will then povide the coesponding C. values. J. A least-squaes solution to Equation (28) has been outlined by Cliff (1962), in which

28 -26- and ( Q' U (u' U )-1 U1 Q ~) 1m ~ 1-)1) = m ~~ (30) whee ~ is the m -th column of' Q. In the pesent case, since U' U = I, these fomulas simplify to (Q{ U) (Q{ U) I (Q{ Ql) (31) whee Q 1 is the column of Q containing the unities, (Q' U) is a 1 ow vecto of column sums of U ' and (Q~ Q1) is the numbe of stimuluspais = n(n-l)/2 whee Q 1 is a least-squaes estimate of the column of unities, L 1 is the column of L that tl~nsfoms U '" into Ql. ' and is a column vecto of column sums of U If it is desied to maintain "- Q 1 as a unit-length vecto, then the tansfomation simply. entails postmultiplying U by a column vecto containing the column sums of U this column vecto being nomalized to unit length (i.e., scaled so that the sums of squaes of the column sums become unity). vecto was computed fom the U matix in the pesent example, and instead of containing elatively constant values, its enties vaied Widely between.0389 and (If all of the enties wee equal, each of the 190 ~ values would have been appoximately.0726 in the unit-length vecto fom.) Since it was thus not possible to detemine a

29 -27- diection in the obtained thee-dimensional facto space that would clealy coespond to vaiation in individual scale constants, it was concluded that all thee dimensions should be intepeted in tems of diffeential viewpoints about stimulus similaity. Idealized Individual Dimensions Gaphs wee plotted fo the thee dimensions of the facto space of individuals by taking the enties in each column of matix A (each ow of AI in Table 3) to epesent the coodinates of a point in thee-dimensional space, thee being one point fo each of the 39 individuals. The fist facto in matix A geneated plots in which all individuals bad lage positive coodinates. The plot fo the second and thid factos is pesented in Figue 1. Inset Fig. 1 about hee At this point, matix A could be otated to simple stuctue as in Equation 14 to poduce matix B of individual coefficients on otated axes. The invese tansfomation applied to matix Y as in Equation 15 would then poduce matix Z of stimulus-pai pojections on the otated axes. Each column of Z would povide a set of distance measues to be analyzed by standad multidimensional scaling methods to obtain thee sepaate peceptual spaces. Points epesenting idealized individuals could also be inseted into the facto space of matix B, and thei pojections used to constuct matix G. Intestimulus distance estimates fo each idealized individual would then be computed by Equation (18), each set of distances being subsequently analyzed by

30 -28- multidimensional scaling methods to poduce a peceptual space o each idealized individual. In the pesent example, howeve, the acto space of matix A (see Figue 1) was simple enough to pemit location of some idealized individual points diectly, without equiing a pio otation. An examination o Figue 1 eveals a concentation of individuals with positive coeficients on dimension II and small o nea-zeo coefficients on dimension III. Many of these individuals wee Republican students (as indicated by the and R notation on the figue). The points towad the left in Figue 1, coesponding to negative coefficients on dimension II, wee moe spead out on dimension III than the points on the ight, giving the appeaance of a tiangle to the ent{e goup of points. Two lines wee dawn nea the boundaies o this tiangle, intesecting at the point A. Points Band C wee chosen nea the exteme eal individual points at the left on the two lines. These thee points wee taken to epesent thee idealized individuals, chosen to span the boundaies of the eal points. Thei pojections on dimensions II and III wee obtained fom Figue 1 and used to constuct the matix G (Table 5); the pojections of these idealized individuals on dimension I wee taken as the Inset Table 5 about hee mean of the coefficients fo neaby eal individuals on dimension I. It 1s of inteest to note the pogession in Figue 1 fom libeal Republicans below the AC line though a goup of consevatives above this line to a concentation of libeal Democats towad the point B. Such an

31 -29- aangement suggests that this diection may be elated to individual diffeences on some measue of political ideology. Since the coodinates in the matix G wee based on the efeence axes of matix A athe than the otated axes of matix B, the matix "X of intestimulus distance estimates fo the thee idealized individg uals was computed by Equation (19) (Table 6). The coefficients in each Inset Table 6 about hee " of the thee columns' of X epesent measues fo each of the thee g idealized individuals of the 190 intepoint distances spanning the 20 stimulus points. The elation of each eal individual to these thee idealized viewpoints can now be computed by Equation (22). Multidimensional Peceptual Spaces The distance measues in each column of wee soted into the appopiate ode and aayed in a sepaate distance matix D. These thee distance matices wee then sepaately analyzed by the oultidimensional scaling pocedues outlined by Messick and Abelson (1956) and by Togeson (1958). Even though insufficient vaiation was found above in individual additive constants to geneate a dimension in the facto space of individuals, each of the pesent distance matices might still equie the detemination of an additive constant, pesumably of oughly compaable size fo the thee matices, fo an optimal dimensional esolution. In the pesent example, howeve, afte one cycle of tlle iteative solution with an initial constant of zeo (Messick & Abelson, 1956), the additive constants wee judged to be negligible fo these thee distance

32 -30- matices. (Incidentally, the size of the distance estimates and the pesence o absence of negative distances can be manipulated by moving the location of the idealized individual point in the facto space, paticulaly in elation to the lage "aveage" fist facto. Thus, it is possible to ecast the additive constant poblem as a poblem of otational placement of the idealized individual dimension with espect to the lage aveage dimension.) Thee scala poducts matices, computed fom the distances in each of the thee D matices, wee then analyzed sepaately by the method of chaacteistic oots and vectos. The thee sets of chaacteistic oots ae given in Table 7. An examination of these oots suggests that the peceptual space fo idealized individual A is stongly Inset Table 7 about hee unidimensional, that the space fo idealized individual B has t1v'o lage dimensions followed by a possible thid small dimension, and that the space fo idealized individual C is somewhat moe complex, involving possibly five o six dimensions. Stimulus pojections on these lage dimensions fo each of the thee idealized spaces ae pesented in Table 8. These stimulus values Inset Table 8 about hee ae detemined within a otation, tanslation, and multiplication by positive constants. The lage single dimension in peceptual space A

33 -31- appeas to eflect an evaluative distinction among the stimuli. One of the two lage dimensions in space B contasts Republicans and Democats, and the othe dimension appeas to be evaluative in natue. The five o six dimensions of space C pesent no immediate clea distinctions of the elatively simple type that emeged in the othe two spaces. This suggests that a moe ela,boate otation of space C is equied befoe intepeting the dimensions. An analysis of the peceptual space fo the goup aveage, as detemined fom the total sample, poduced seven dimensions, which ae descibed in detail by Messick (1961). In addition to the possibility mentioned above of elating vaiation in the facto space of individuals to pesonality and cognitive vaiables, it is also of inteest to inquie about possible coelates of the shift in complexity of peceptual stuctues" fom simple spaces fo individuals on the AB line to the complex space fo idealized individual C. Pehaps this dimension of individual diffeences contasts pesons that might be temed "abstact" with othes that equie consideable concete and specific detail fo thei decisions (cf. Havey, Hunt, & Schode, 1961). Pehaps it is elated to individual diffeences in "cognitive complexity" (Biei &Blacke, 1956; Scott, 1962) o is a consequence of consistencies in pefeed categoywidths o equivalence anges (Messick & Kogan, 1963; Sloane, Golow, &Jackson, 1963). In conclusion, the pesent analysis illustates the powe of the poposed method to yield a. multidimensional desciption of the peception of elations between stimuli by vaious individuals, in a famewok that pemits the vaieties of consistent individual peceptions

34 -32- to be ascetained ~~d elated to othe pesonality and cognitive vaiables. Summay of the Pocedue fo DeteminiLg Dimensions of Individual Diffeences in Multidimensional Scaling 1. Obtain estimates of distance o dissimilaity between all possible pais of n stimuli fo each of N individuals. Aay these distance estimates in a matix X, having n(n-l)/2 ows fo the stimulus-pais and N columns fo the individuals. 2. Compute an N by N matix of coss-poducts P = XIX. [If N > n(n-l)/2, compute the coss-poducts matix sunnning ove the vaiables on the longe side of X. In this case, P = XX', and a symmetic analysis is used in place of the following steps See Equations (la) - (4a).J 3. Facto P by the method of pincipal components and constuct the diagonal matix (2) fam the lagest chaacteistic oots of P and the matix W fom the coesponding chaacteistic vectos. P = WI 2 W 4. Scale W fo diffeences in sample size to poduce the 1 matix V::;; N 2 W 1 5. Compute the matix Y::;; U N- 2 eithe by fist obtaining U = X WI - 1 o diectly fom Y = X VI - 1 N Compute the facto matix of individuals 1 A ::;; V = N2 W 7. Plot the factos of A gaphically to detemine (a) otation to stuctue in the facto space of individuals and (b) locations fo idealized individuals.

35 Detennine, eithe gaphically o analytically (Hannon, 1960), an x nonsingula tansfomation matix T to otate the pincipal factos of A to a desied stuctue, denoted matix B = TA. 9 Compute matix -l Z ::: YT 10. Each column of Z contains scaled stimulus-pai pojections on otated axes. These enties epesent distances between pais of stimuli accoding to otated dimensions of viewpoint. Next, distance matices ae constucted, one fom each column of Z. These distance matices ae analyzed sepaately by standad multidimensional scaling pocedues to obtain peceptual spaces (Messick & Abelson, 1956; Togeson, 1958j Shepad, 1962a; 1962b). 11. If desied, the enties in the fist unotated pincipal facto of Y (o of U ) may be similaly used to constuct a distance matix. Multidimensional scaling of this distance matix poduces a peceptual space fo the goup aveage. 12. If desied, locate points to epesent g idealized individuals in the facto space of matix B. Read the coodinates of each idealized point diectly fom the facto plots and ecod the coodinates of each point in a column vecto. Assemble these column vectos fo g idealized individuals into an by g matix G. Compute A X = ZG, an n(n-l)/2 by g matix of estimated g distance measues fo g idealized individuals. (If the coodinates of the idealized points ae detennined fom the unotated axes of matix A to fom G A, compute X g = YG A.) 14. Constuct a distance matix fom each of the g columns of X g and analyze them sepaately by multidimensional scaling methods to obtain g peceptual spaces, one fo each idealized individual.

36 If desied, fo by squae sections of G J compute H = G- 1 B to obtain a matix H of pojections of eal individuals on selected idealized individual dimensions.

37 Refeences Abelson, R. P. A technique and a model fo multidimensional attitude scaling. Publ. Opine Quat., , 18, Abelson, R. P., & Semat, V. Multidimensional scaling of facial expessions. J. expo P lchol., 1962, 63, Adams, E., & Messick, S. An axiomatic fomulation and genealization of successive intevals scaling. Psychometika, 1958, 23, Attneave, F. Dimensions of similaity. Ame. J. Psychol., 1950, 63, Biei, J., & Blacke, E. The geneality of cognitive complexity in the peception of people and inkblots. J. abnom. soc. P~chol., 1956, 53, Cliff, N. Analytic otation to a functional elationship. Psychometika,. 1962, 2'7, Coombs, C. H. An application of a nonmetic model fo multidimensional analysis of similaities. Psychol. Reps, 1958, ~, Diedeich, G. W., Messick, S., & Tucke, L. R. A geneal least squaes solution fo successive intevals. Psychometika, 1951, 22, Eckat, C., & Young, G. The appoximation of one matix by anothe of lowe ank. Psychometika, 1936, 1, Ekman, G. Dimensions of colo vision. J. Psychol., 1954, 38, Gulliksen, H. Linea and multi.dimensional scaling. Psychometika, 1961, 26, Gulliksen, H. judgments. The stuctue of individual diffeences in optimality In G. Byan and M. Shelly (Eds.), Human judgments and optimality. New Yok: Wiley, 1963, in pess.