Applied Psychological Measurement

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Applications of Multidimensional Scaling in Cognitive Psychology Edward J. Shoben Applied Psychological Measurement 1983 7: 473 DOI: 10.1177/014662168300700406

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Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 Applications of Multidimensional Scaling in Cognitive Psychology Edward J. Shoben

, University of Illinois

Cognitive psychology has used multidimensional view is intended to be illustrative rather than ex- scaling (and related procedures) in a wide variety of haustive. Moreover, it should be admitted at the ways. This paper examines some straightforward ap- outset that the proportion of applications in com- plications, and also some applications where the ex- and exceeds the planation of the cognitive process is derived rather di- prehension memory considerably rectly from the solution obtained through multidimen- proportion of such examples from perception and sional scaling. Other applications examined include psychophysics. Although a conscious effort has been and the use of MDS to assess cognitive development, made to draw on more recent work, older studies as a function of context. Also examined is change have been included when deemed appropriate. how an ideal representation is selected, whether, for This will examine of both tra- example, a space or a tree is more appropriate. Fi- paper applications nally, some inherent limitations of the method for ditional two-way MDS (e.g., Kruskal, 1964a, 1964b; cognitive psychologists are outlined, and some pitfalls Shepard, 1962a, 1962b) and three-way MDS (e.g., and potential misapplications are identified. Carroll & Chang, 1970). Three-way analyses which can extract on the same data Although there are many examples of multidi- higher dimensionality set, are but not mensional scaling (MDS) and related methods ap- generally, exclusively, performed when individual differences are considered. When plied to problems in cognitive psychology, ~lI7S these individual variations are not at issue, two- has not been used to its fullest potential in this way MDS is branch of psychology. Not only are there many usually employed. The most and the most common circumstances where a straightforward application straightforward use of MDS has been of these of MDS might be profitable, but there are also descriptive. Many dark, 1968; & instances where some more subtle application of early applications (e.g., Rapaport Fillenbaum, 1972) have used MDS to characterize these procedures might be useful in providing an- the structure a set of stimuli. swers to cognitive questions. The present paper spatial underlying This use makes no however, to will attempt to illustrate some of these less straight- attempt, specify how the stimuli are processed. Although research- forward applications by example, as well as noting ers can to the dimension some of the more routine applications. No effort attempt identify along which the stimuli in a multidimensional has been made to cover every application of MDS vary space, little or no evidence has been provided in the cognitive literature, however, since this re- frequently that subjects are actually using these dimensions. More recent studies have progressed beyond this descriptive use: 1. Studies suggesting that subjects are actually using the spatial representation recovered by 473

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MDS. For example, Rumelhart and Abraham- sals (ma and na) from the other consonant pho- sen (1973) have used Henley’s animal names nemes. Within the remaining consonants, there was and found that distance in the ~1~~ solution also a separation between those that are formed at would predict solutions to analogies. Simi- the front of the mouth (fa and ba) and those formed larly, Rips, Shoben, and Smith (1973) found at the back of the mouth (ga and zha). These results that distance in a multidimensional space would thus placed some order on this domain. predict reaction times (l~’~’s) in a categorization The color domain has long been of interest to task. users ofMDS (~l~~a~9 1954). In his original paper, 2. Studies comparing the structure of the stimulus Shepard (1962a) provided an analysis of the judged space over time. For example, Arable, Kos- similarity of colors. In the resulting t~l®-di~e~- slyn, and Nelson (1975) and Howard and sional solution, a color circle was recovered in Howard (1977) have looked at changes in the which the various colors formed a circle with a gap stimulus space from a developmental perspec- between the color with the shortest wavelength (vi- tive. Voss and his colleagues (~ah®r~~ ~ ~~®ss9 olet) and the one with the longest wavelength (red). 1979; Bisanz, l~~P®~°&dquo;~~9 Vesonder, & Voss, If the other points are connected by drawing the 1978) have used MDS procedures to examine circle, the resulting ordering is monotonic with changing representations of prose with changes wavelength. Moreover, the fact that red and violent in context. are quite close to each other in the two-dimensional 3. Studies using MDS analysis to evaluate var- space (while they are maximally distant in wave- ious theories (e.g., Friendly, 1977; Soli length) accords with the knowledge that red and Arable, 1979). violet are quite similar to each other. This paper will examine some of the descriptive For semantic domains, the results are generally uses of MDS before moving on to some of the less less clear-cut. There are several possible reasons straightforward applications. Alternatives to MDS for this difference. First, semantic domains are as a representation of data will be examined; and generally of functionally infinite size (although there finally, there will be an attempt to specify the role are exceptions, such as kin terms). Thus, it is not of MDS in cognitive psychology. practicable to scale all a~Air~als9 some selection must be made. Surprisingly, this selection is often made haphazardly. Perhaps not surprisingly, altering the Uses sample of exemplars can affect the resulting solution. Second, semantic domains are ~~y~~~l~~1~~31 Domains Concrete Objects potentially more heterogeneous than the other two domains Many stimulus sets have been subjected to MDS that have been discussed. Whereas any group of analysis. These range from consonant phonemes colors will have some hue and some saturation, it to color names to semantic domains. A brief and is not easy to come up with shared dimensions for useful review has been provided by Shepard (1980). the ~®~ns jus~~ace9 c®~se~°v~~~®~a9 and laughter. Third, The data set that has probably been subjected to some domains that are commonly subjected to MDS analysis by the largest number of multidimensional analysis (categories) may pose technical problems scaling programs is the confusion data collected by for computer programs that are commonly used. Miller and Nicely (1955) on consonant phonemes. This last point will be discussed in greater detail Shepard (1972) incorporated the fitting of an ex- in a subsequent section. ponential decay function on the original confusion One of the most frequently scaled semantic do- proportions in performing the MDS analysis. The mains is animal names. Henley (1969) scaled 30 resulting two-dimensional solution was readily in- animal names based on the dissimilarity judgments terpretable. The first dimension distinguished be- of undergraduates. In the three-dimensional solu- tween voiced (za and da) and unvoiced (fa and ka) tion that Henley obtained, the first dimension was phonemes. The second dimension separated the na- clearly identifiable as size, with elephant, giraffe,

Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 475 and camel at one extreme and rat, mouse, and melhart, 1975), the dual code theory (Paivio, 1971), chipmunk at the other. The second dimension, la- the SAM model (Raaijmakers & Shiffrin, 1980), beledferocity by Henley, had the pr~d~t~rs-li®~, the random walk model (Ratcliff, 1978), the schema tiger, and Ieopard-ai one extreme and the do- model (Rumelhart & Ortony, 1977), the MOPS mesticated animals―cow, sheep, and donkey-at model (Schank, 1980), and Tulving’s (1975) mem- the other extreme. Finally, the third dimension, ory model. labeled humanness by Henley, placed the apes at Using expert subjects, Ross (1983) employed one extreme and pig, dog, and wolf at the other. SINDSCAL (Carroll & Chang, 1970; Pruzansky, Henley’s last dimension is particuarly difficult 1975) to obtain a two-dimensional solution. The to interpret. It is not at all clear why giraffe, chip- resulting solution ordered the theories in terms of munk, and camel are all &dquo;more human&dquo; than horse, the unit of analysis on one dimension and the de- dog, and leopard. Although the third dimension gree of rigor on the other dimension. In terms of does separate the anthropoid apes from the rest of the unit of analysis, MOPS and the schema model the animals, it is difficult to maintain that this third were at one extreme, and J. A. ~~derst~n9s dis- dimension is interpretable. Similarly, the label for tributed memory model and l~~tcliff’s random walk the second dimension has been disputed even though model were at the other. Theories such as MOPS the ordering of objects is quite consistent across are often concerned with paragraphs or stories as studies (see Storm, 1980, for an exception). For units, whereas Ratcliff model and J. A. Ander- example, rather than ordering the animals on ftt- son’s model are usually applied to single words. rocity, Rips, Shoben, and Smith (1973) labeled a On the degree of rigor dimension, theories that similar dimension in their MDS solution predacity, have strong mathematical or computer underpin- arguing that this dimension separated the wild an- nings were at one extreme, and thus MOPS, the imals from the domesticated farm animals. LNR model, HAM, SAM, lZ~tcliff’s model, and In addition to this dispute over labeling dimen- J. A. Anderson’s model all scored very high on sions, there has also been an apparent controversy this dimension. At the other extreme, theories that over how many dimensions are required. Rips et are not mathematical, such as Tulving’s model, al. (1973), for example, used only two dimensions. levels of processing, and the dual code theory were Shoben (1976) also used two dimensions (in ad- low on this dimension. dition to one that separated the birds from the mam- Although Ross did not attempt to apply his scal- mals in his solution). However, King, Gruenewald, ing results to any kind of behavioral data, it is and Lockhead (1978) have argued for a three- interesting to note that memory theories are orga- dimensional result, and Rumelhart and Abraham- nized in this way, and it may perhaps belie a gen- sen (1973) were unable to obtain satisfactory re- eral trend in that two of the theories examined by sults in their analogies task with any dimensionality Ross (levels of processing and the dual code theory) lower than three. have recently been receiving considerable criti- cism ; interestingly, both are theories without strong or and both Theory mathematical computer underpinnings are theories whose prominent unit of analysis is MDS analysis has also been profitably applied pairs of words. to theories of memory (e.g., Ross 1983). Ross examined 12 well-known cognitive psychology theories: the distributed model (J. A. An- memory of MDS Analysis derson, 1977), ~1~~ (J. R. Anderson & Bower, 1974), the short-term memory model (Atkinson & Some early studies (Clark, 1968; Fillenbaum & Shiffrin, 1968), levels of processing theory (Craik Rapaport, 1971) have allowed the MDS analysis ~ Lockhart, 1972), the prepositional theory to stand by itself. Although the authors typically (Kintsch, 1974), the LNR model (Norman & Ru- made some comment about the dimensions re-

Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 476 covered and their significance, no attempt was made Task Processing Account to provide evidence that the derived space is actually used in processing information. In contrast, most recent studies have attempted The I~1~S solution could be used here to provide to obtain evidence for the general validity of the a processing account of the task. If subjects com- derived multidimensional space. Rips et al. (1973), pared exemplars to each other, then it might have for example, showed that distances in a multidi- been expected that the distance between the ex- mensional space would predict categorization times. emplars would be the best predictors of RT. Dis- Specifically, they obtained the mean reaction time tances in the multidimensional space and regression (RT) to verify statements such as &dquo;A robin is a analyses were able to place severe constraints on bird.&dquo; For the 12 birds and 12 mammals they em- any viable processing account of this Same-Dif- ployed, Rips et al. found that semantic distance ferent task. correlated with categorization time quite highly. Rumelhart and Abrahamsen (1973) also used MDS Similarly, Shoben (1976) used semantic distance to provide a processing account of an analogies to predict RTs in a Same-Different task. The stim- task. Although more sophisticated accounts of an- ulus domain in this study was a set of six mammals alogical reasoning are now available (Stemberg, and six birds. Scaled together (in contrast to most 1977), Rumelhart and Abrahamsen’s work is in- earlier studies), these 12 exemplars and the two structive in that it shows another way in which superordinate terms bird and mammal yieldcd a MDS analysis can be applied to provide some the- three-dimensional solution that was readily inter- oretical constraint on a cognitive task. pretable. One dimension, as usual, was size with Rumelhart and Abrahamsen were concerned with bear at one extreme and robin, sparrow, and mouse four-term analogies such as &dquo;f®x:h~rsee:chipm~~ko at the other. A second dimension was labeled pre- .&dquo; Subjects were asked to select the best dacity with bear and lion at one extreme and goose completion (in one experiment) from among alter- and chicken at the other. A third dimension simply r~ativcs-a~tel®pe, donkey, elephant, and wolf- clustered the birds at one extreme and the mammals where elephant is the best answer. Spatially, Ru- at the other. For Same category judgments, such melhart and Abrahamsen noted that the ideal point as hawk-eagle, Shoben was able to show first that or solution to the analogy in the multidimensional the distance between hawk and bird and the dis- space can be located by first determining the re- tance between eagle and bird predicted Same RT lationship (in all three dimensions in this case) be- quite well. Moreover, the distance between hawk tween the first two terms and then by applying those and eagle was not predictive of Same ~’1’, contrary differences to the third term. Thus, people have to some semantic memory theories (such as Schaef- solved the analogies by constructing a parallelo- fer & Wallace, 1969, 1970). In addition, Shoben gram whose vertices are the first three terms of the was also able to use the distances obtained from analogy. In the present example, fox is smaller than MDS to predict the ease with which people can horse (Dimension l ), somewhat more ferocious than determine that the exemplars are from different horse (Dimension 2), and slightly less human than categories. For the first item, the distance between horse (Dimension 3). The ideal point is thus an the exemplar and its category was predictive of animal that is larger than chipmunk, less ferocious RT. For the pair bear-goose, for example, the bear- than chipmunk, and slightly more &dquo;human&dquo; than mammal distance was predictive ofRT. Moreover, chipmunk. Elephant comes closest to this ideal point. the distance between the second exemplar and the Interestingly, the alternatives can also be rank- category of the first was also predictive, in this ordered (as done by Rumelhart and Abrahamsen), example, the goose-mammal distance. Here, small and subjects will judge that antelope is the second- distances led to long RTs. Once again, the distance best completion to the problem, donkey is third, between exemplars was a relatively poor predictor. and wolf is poorest.

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In addition to predicting subjects’ first choices mation to the recall data. A different and promising by distances from an ideal point, Rumelhart and approach is one developed by Friendly (1977). He Abrahamsen also developed a theory about the dis- derived the average distance between any pair of tribution of subjects’ responses. Specifically, they objects !’ and j by the following equation: assumed that subjects’ choices would be in proportion to their distances from the ideal point. More formally, they proposed that the distribution of re- lit is the position of item i on trial t and ~,J, sponses would follow Luce’s (1959) choice rule, is an indicator variable which is 1 if items i and j are both recalled on trial t and zero otherwise. If giving greater weight to items recalled in close Here, di = Xi - 1: the distance between alternative contiguity to each other is desired, then this expres- Xi and the ideal point, and v( ) is a monotonically sion can be modified by extracting the square root decreasing function, and P(Xi X,, .... X,,) is the of the squared difference in recall position, i.e, probability of selecting the ith item from the n alternatives. Following Shepard’s (1972) original work, Rumelhart and Abrahamsen (1973) assumed = where <2 is constrained to be v(x) exp ( - c~x)9 With this proximity matrix, Friendly was able to because obtained fits to re- positive, Shepard good obtain a highly interpretable two-dimensional so- call data using this kind of function. In addition, lution that had a very low stress value. this function has the virtue of requiring only one Whereas Friendly derived his similarity matrix parameter. from free recall pr®t®c®ls, a related question is The results corresponded quite well to the the- whether distances obtained from an MDS solution oretical predictions, both qualitatively and quan- can predict recall data. For example, Caramazza, titatively. Examination of even the data for third Hersh, and Torgerson (1976) found little correla- and fourth choices showed a close correspondence tion between distance and recall proximity on early between theory and data. Moreover, this corre- trials, but positive and usually significant correla- even when the distance spondence held up among tions on later trials. For categories whose exem- alternatives was varied. the possible systematically plars varied the most in the multidimensional space, case in which MDS This exemplifies a analysis such as mammals, these correlations were quite appears to have actually aided in the formulation high; for categories whose exemplars showed little of a psychological processing model. Because it variation, such as fish, the correlations were much was relatively easy to determine the distance of lower. Obviously, there must be variation in the each of the alternatives from the ideal point, given space in order to predict patterns of free recall. the availability of the multidimensional space, Ru- Generally, similarity as measured by direct ratings were led melhart and Abrahamsen (1973) to for- or by distance in a multidimensional space is a mulate the expanded theory based on Luce’s ( 1959) reasonably good prediction of results in free recall choice rule. experiments (Friendly, 1979; Schwarz & Hum- phreys, 1973). Free Recall Analysis 1~~~~~ Perceptioxx 1~~5 analysis can also play a useful role in the analysis of free recall. The distances derived from Perhaps the area in which MDS has contributed the multidimensional space can be used as predic- most to cognitive psychology in the past 5 years tors of output order in free recall. Shepard ( 1972) is in the area of music perception. Early psycho- did this analysis by fitting an exponential transfor- logical investigations focused solely on the dimen-

Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 478 sion of pitch height ~Ste~ ~~s ~ Volkmann, 1940),9 Thus, it appears that more than a chroma circle and subsequent work has provided evidence of more is present when the tones are presented in a scale complicated structure largely in its emphasis on the context. Regardless of musical training, people make octave (Shepard, 1964). According to this for- the classification of tones as suggested by music mulation, the notes of the scale could be thought theory; this representation can be recovered through of in terms of a chroma circle, analogous to the MDS. circle observed for colors. Krumhansl and her colleagues (Krumhansl, Recently, the investigation has proceeded in a ~h~~h~9 ~ Kessler, 1982) have generalized this musical context. That is, many studies now ask result from tones to chords. In a scale context, subjects, not for their judgment of similarity of two people judged the major chords (for C major: CEG, tones in the absence of explicit context, but for the FAC, and GBD) as central unless the scale context similarity of two tones in the context of a diatonic was a minor key, in which case the corresponding scale, or the similarity of two passages in the con- chords occupied the center of the MDS represen- text of a melody. This practice has led to the re- tation. More specifically, for the A minor scale covery (through MDS) of much more complicated Krumhansi et al. employed, the corresponding chords representations. were A minor, D minor, and E major, respectively. Perhaps the seminal work along these lines was Chords that were not part of the scale sequence done by Kmmhansl (1979). She presented subjects (such as the E major for the C major scale) were with a simple musical context: a major triad chord, at the periphery of the space. This pattern held an ascending major scale, or a descending major across scale types. scale. Differences among these contexts were slight. Recently, Pollard-Gott (1983) has extended this Subjects were then presented with a pair of tones line of work to passages from classical music. Morse whose similarity they were to judge in the context. specifically, her subjects listened to a Liszt sonata The raw similarity results alone are interesting. over three sessions. In each, the listeners were en- For stimuli in the major triad, other major triad couraged to take notes and to think about the re- stimuli were judged to be most similar, followed lations among the passages. After each session, by diatonic tones. Nondiatonics were judged as subjects judged the similarity of the 28 pairs of least similar. This same pattern was observed for stimuli that were generated from the eight passages diatonic tones. For nondiatonic tones, there was (4 to 16 measures in length) Pollard-Gott studied. little effect of this categorical variable. Instead, the These similarity data were analyzed using similarity in pitch height was the primary deter- SINDSCAL (Carroll & Chang, 1970; Prozansky, minant of the similarity between a nondiatonic tone 1975). The data are particularly interesting when and another one. examined across sessions. After the first session, The MDS representation that Kmmhansl ob- the dimensions recovered by the program represent tained is a more complicated version of the chroma fairly gross physical features of the passage; the circle. Of particular interest is the three-dimen- dimensions are most adequately labeled, for ex- sional solution that resembles an inverted cone. At ample, as happy-sad, high-low, simple-complex, the base of the cone are the components of the and loud-soft. However, by the second session, a major triad. For the C major scale she employed, different dimension emerges: a thematic dimen- these items are middle C, 9 ~ ~ G, and high C, reading sion. By the second session, this dimension sep- clockwise around the circle. The diatonic tones arated without overlap passages that deal with Theme cluster at the next level. These, too, can be viewed A from those that deal with Theme B. This sepa- as a circle. Reading clockwise, the tones here are ration is even more pronounced after the third lis- D, F, A, and B. Finally, at the base of the inverted tening session. Thus, it seems that a thematic di- cone are the nondiatonic tones, which, like the mension emerges as the subject becomes more other clusters, are arranged in ascending order when knowledgeable about the composition. the circular base is read in a clockwise direction. Strong support for this interpretation is provided

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by the results obtained in an expert condition con- mension now reflected the temporal ordering of ducted by Pollard-Gott (1983). In this condition, events in the story such that nouns that occurred a one-dimensional MDS solution was obtained; that early in the story appeared at one extreme and single dimension accounted for 84% of the vari- nouns that appeared at the end of the story appeared ance. As with the fourth dimension obtained with at the other end of the dimension. Thus, it seems the naive subjects, this single thematic dimension that a story context can alter the relationship among clustered all the passages that dealt with Theme A terms. The results for a second story used by LaPorte at one extreme and all the passages that dealt with and Voss showed similar, although more compli- Theme B at the other. cated, results. A more extensive framework for studying prose representations has been developed by Bisanz, Measurement off Change in Structure LaPorte, Vesonder, and Voss (1978). They pre- Some of the most impressive applications of MDS sented subjects with stories involving animal names have been in demonstrating a change in the rep- that had a clear thematic structure. In one condi- resentation of stimuli. For example, by examining tion, for example, the animals varied along a lead- a set of stimuli in several conditions, it may be ership dimension and along a ~el~f a~l~~ss~ dimen- possible either to show an effect of context and/or sion. Bisanz et al. compared the MDS solution to extract a more complicated multidimensional so- obtained prior to reading the story with one derived lution. Less dramatically, it may be demonstrated after reading the story. In the former case, they that context does have an effect. obtained a typical animals solution, similar to the One straightforward application of this strategy one obtained by Rips et al. (1973) in which the was performed by LaPorte and Voss (1979), who two dimensions could be characterized as size and presented subjects with 20 nouns taken from one ferocity. After reading the story, and being in- of two simple stories. For one story, these nouns structed to rate the animals in terms of their sim- included fields, clouds, vegetation, train, ap- ilarity to each other in the story, the themes (such proach, decade, troops, plague, eggs, and food. as helpfulness and leadership) were recovered. In- Initially, subjects all performed a pairwise rating t~r~stir~~ly, these two dimensions were not re- of similarity for all pairs of nouns. The resulting covered equally well. matrices were subjected to a three-way 1~~5 anal- In addition to recovering the story structure, ysis, and a two-dimensional solution was obtained Bisanz et al. were also able to demonstrate that the where one dimension separated man-made objects recovered structure could predict performance in a (train, approach, and troops) from natural ones memory task. In particular, they asked subjects to (fields, vegetation, and food). The second dimen- determine if pairs of animals were both helpful or sion was interpreted by the authors as separating both not helpful. At least for affirmative responses, animals from nonanimals, with eggs and food at it was clear that distance in the multidimensional one extreme and fields and clouds at the other. space was predictive of RT, and the authors were Subsequent to this initial rating, subjects either able to formulate a regression model (where dis- repeated the task (a control condition) or read a tance in the multidimensional space was one of the story about how grasshoppers become a pest every independent variables) that predicted the latencies 10 years. These experimental subjects then per- of all the trials quite well. formed the rating task on the same words once Actually, it is possible that Bisanz et al. might again. Interestingly, the resulting space had both have been able to obtain greater predictability by similarities and dissimilarities to the original so- looking at another distance in the space they ob- lution. tained. The only distance that Bisanz et al. ex- The first dimension was virtually the same as amined is the distance between the two stimuli. For before and could be characterized as a man-made example, in the pair lion-tiger, they looked only vs. natural dimension. However, the second di- at the distance between the two animals. Assume

Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 480 for the moment that lion is helpful and that tiger one extreme; mouse, rabbit, and deer all cluster is not, and further assume that the subject processes near the other pole of this dimension. lion first. In this case it might be profitable to ex- Howard and Howard assessed age differences amine the distance between the point for lion and by looking at the weight assigned to each dimen- the point for helpful and between tiger and helpful, sion in the subject space. Averaging over subjects if the decision required is whether both are helpful. within age groups, they found that younger subjects Logically, these are the distances that should be tended to emphasize the size dimension, while older involved in the model that Bisanz et al. proposed, subjects emphasized the more abstract domesticity and it is surprising that they appear not to have and predativity dimensions. More specifically, first- looked at these distances. In this case, a small lion- and third-grade subjects placed much greater weight helpful distance should be facilitory and a small on the size dimension than on either of the other tiger-helpful distance should be inhibitory. This two. Sixth-graders placed equal weight on the size kind of analysis (used earlier by Shoben, 1976, in and predativity dimensions and less weight on the a Same-Different task with semantic categories) domesticity dimensions. College students were es- would provide additional support for the model sentially the opposite of the young children; they espoused by Bisanz et al. placed greatest weight on the predativity dimension and least weight on the size dimension. Thus, it does indeed seem that increasing age is accom- Cognitive Development panied by an increasing reliance on abstract di- One obvious place to look for examples of MDS mensions, at least for this stimulus domain. as an index of change is in work with children. A more complex analysis along similar lines has Particularly in cognitive development, it should be been performed by Miller and Gelman (1983). They possible to use MDS to assess changes in people’s used techniques developed by Arable, l~~ssly~, and subjective representations over time. It would be Nelson (1975) to investigate the development of particularly desirable, for example, if it were pos- the concept of number. As in Howard and Howard sible to show that tended to were young subjects organize (1977), four age groups used: kindergartners,9 concepts along perceptual dimensions, whereas third-graders, sixth-graders, and graduate students. adults tended to employ more abstract dimensions. Using a modification of the method of triads, Miller One fairly straightforward illustration of this ap- and Gelman (1983) had subjects judge which of proach is a study by Howard and Howard (1977). three digits was most similar and which was least They obtained similarity judgments for a set of 10 similar. To cut down on the number of triads to be animals that were chosen from Henley’s (1969) judged by very young children, Miller and Gelman original set. The subjects were first-graders, third- used a balanced incomplete sampling procedure graders, sixth-graders, and college students. The developed by Arable et al. (1975). The resulting resulting matrices obtained from all subjects were similarity matrices were analyzed separately for analyzed with Carroll and Chang’s (1970) IND- each age group using KYST2A (Kruskal, Young, SCAL program. A three-dimensional space was & Seery, 1977). Additional analyses were also per- obtained in which the three dimensions extracted formed; these will be discussed briefly later. were size, clojnesticaty, and predativity. These last For the 10 digits, two-dimensional solutions were two dimensions are usually thought of as equivalent obtained for all four age groups. For the young (see Rips et al., 1973), but Howard and Howard children, the solution can be described as a semi- (1977) make a good case that these dimensions are circle in which the digits are ordered by magnitude. distinguishable although not uncorrelated. For the The kindergartners appear to have some difficulty domesticity dimension, mouse, rabbit, lion, deer, with zero, but they and the third-graders otherwise and bear are on one side of the midpoint, and horse, provide a perfect ordering in terms of magnitude cow, sheep, dog, and pig are on the other side. along the semicircle. Older children and adults, in For the predativity dimension, lion and bear are at contrast, reveal more than magnitude information

Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 481 in their solutions. For sixth-graders, the odd-even age. Methodologically, it is interesting to note that dimension is clearly present in addition to a mag- the clustering analysis performed by Miller and nitude dimension. Particularly for adults, other re- Gelman (1983) parallels the scaling work done by lationships are also present; and their solution, in Howard and Howard (1977). In both cases, one some cases, does not order the digits perfectly along representation was obtained: a three-dimensional the magnitude dimension. For example, the powers solution in the former and a set of seven clusters of two (2, 4, 8) are closer together in the two- in the latter. In both cases, the argument for de- dimensional solution than they should be according velopmental change relies on an orderly change in to the magnitude dimension. the weights assigned to each dimension (in the for- Thus, Miller and Gelman (1983) were able to mer case) or each cluster (in the latter case). The demonstrate rather elegantly that there is increasing Miller and Gelman (1983) paper is particularly con- complexity in children’s conception of number as vincing because, in addition to this logic, these age increases. For young children, the concept of authors also examined another representational number is strongly associated with counting. With measure (in this case MDS solutions) indepen- age, the concept becomes richer, as other relations dently, in which the representation was obtained (oddness, or powers) become part of the concept. independently for each age group. Supporting these conclusions, Miller and Gel- man (1983) employed the ll~I~CLIJS’ program (Arabie & Carroll, 1983) as a further means of ~~~~°~a~e~ Dimensionality from the change with age. In this analysis, investigating Examination of Different Contexts all age groups must have the same clusters; what differed (in addition to the additive constant) were Previous examples have described a few of the the weights assigned to these clusters by each group studies that have used MDS to index change. Phrased of subjects. Five of the seven clusters pertained to more generally, these studies have looked at the counting, and the other two were the odd numbers representation of a stimulus domain in several dif- excluding 1 (3, 5, 7, 9) and the powers of 2 (2, ferent contexts. One of the interesting byproducts 4, 8). For the kindergartners and the third-graders, of this approach is that it may be possible to extract the five counting clusters were all assigned higher more dimensions in this circumstance than if the weights than these last two clusters. For the adults, same stimulus domain is examined in only one on the other hand, the powers-of-2 cluster had the context. It is suggestive, for example, that Howard highest weight, and the weight for the odd-num- and Howard (1977) were able to extract three di- bers-excluding-I cluster was fourth, only slightly mensions from a set of 10 animal names, where behind the large-numbers (6, 7, 8, 9) cluster. Sixth- Rips et al. (1973) were able to extract only two graders were between these two extremes. The from a set of 14 animal names, with both using weight for the powers-of-2 cluster ranked third, the INDSCAL model. and the weight for the ®dd-r~~~nbers-ea~cludin~-1 The most striking example of this higher di- cluster ranked fifth. mensionality is an analysis of consonant phonemes Thus, this clustering analysis very nicely com- performed by Soli and Arabie (1979). In this study, plements the MDS analysis in that it shows an they used the data from Miller and Nicely (1955), increasing complexity in the concept of number which they fir st transformed to conform more closely with increasing age. Young children’s conception to the INDSCAL model (see Arabie & Soli, 1982, of number is almost entirely dependent on count- for the details and justification of this procedure). ing ; greater complexity occurs with increasing In contrast to earlier studies, Soli and Arabie (1979) also used the full set of confusion matrices (in- those with severe in 1INDCLUS (for INdividual Differences CLUStering) is the three- cluding relatively distortion) way version of MAPCLUS, the Arabie and Carroll (1980) al- their analysis. They obtained a four-dimensional gorithm for fitting the Shepard-Arabie (1979) ADCLUS model. solution that accounted for 69% of the variance.

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Their first dimension ordered the phonemes in mensional representations are structures where each terms of periodicity/burst order with /n/ and /m/ at stimulus is represented by a continuous value on a one end of the dimension and /p/, /t/, /k/, /f/, and number of dimensions. Tree structures are hierar- /s/ at the other extreme. The second dimension chical representations, and networks are undirected ordered the phonemes in terms of their first for- graphs. Friendly has argued that there is a uniquely mants, with the voiced consonants at one extreme appropriate method for analyzing data for each of and the nasals and voiceless stops at the other. The these three types of memory models. Specifically, third dimension was ordered in terms of the second MDS is the proper procedure for dimensional rep- fornants, with phonemes with falling second for- resentations, and hierarchical clustering is the ap- mants at one end and phonemes with rising second propriate representation for tree structures. For pur- formants at the other extreme. Finally, the fourth poses of this discussion, Friendly has restricted dimension was ordered in terms of spectral dis- himself to minimum-spanning trees (thereby im- persion. This dimension separates two groups of posing a hierarchical requirement) as the appro- fricatives from the other phonemes and is largely priate tool of analysis for network representations. relevant to a particular listening condition. Unfortunately, the most straightforward conclu- Soli and Arabie (1979) have also been able to sion does not appear to be warranted. That is, it is show that the salience of these dimensions varied not the case that data are analyzed with all three markedly with the kind of listening condition. For procedures and then the correct representation de- example, both the periodicity/burst dimension and termined on the basis of which procedure best fits the first formant dimension were salient under ex- the data. In fact, Friendly was careful to make this treme degradation. For the second fornant and point. However, it does appear that such a pro- spectral dispersion dimensions, increasing degra- cedure can provide some evidence in favor of one dation decreased the weight given to these dimen- method or another and that the methods are rea- sions. sonably sensitive to variations in underlying struc- Thus, by their ability to make the Miller-Nicely ture. data conform to the INDSCAL model, Soli and Friendly’ approach can be characterized as the- Arabie have been able to extract higher dimen- ory driven. Instead of examining the data in an sionality from this data set than had been previously effort to infer the appropriate underlying represen- done. l~~re®ver9 this higher dimensionality has en- tation, he starts with a theoretical structure and then abled them to make some arguments concerning looks to see if that hypothesized theoretical orga- the relative importance of acoustic as opposed to nization was evident in the data. In his examples, phonemic properties in the underlying representa- he assumes a particular structure and then analyzes tion. By examining a wide variety of listening con- the data by an appropriate method to ascertain if texts, Soli and Arable seem to have obtained ad- the results are consistent with the hypothesized ditional information out of an old, and often structure. In his examples, the data are always free analyzed, set of data. recall, where the proximity matrix is derived by the method summarized earlier in this paper. How- ever, this methodology is suitable for any prox- matrix. MDS the Underlying Representation imity This general approach has been carried a step There have been some efforts to use MDS and further by Reitman and Rueter (1980). They held related techniques to determine the appropriate un- a particular theory of free recall and wished to derlying representation. Friendly (1977) for ex- obtain some confirming evidence. In their theory, ample, examined three possible representations of chunks are an important component, and the order information in memory: dimensional representa- in which the stimuli should be recalled is also pre- tions, tree-structure representations, and network scribed. Their technique, only peripherally related representations. According to his definitions, di- to MDS, 9 first identifies the chunks, and then par-

Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 483 titions these chunks into a lattice. Lastly, this lattice elongation, is more complicated. This statistic is is converted into an ordered tree. Although the motivated theoretically by the nature of binary rooted details of this procedure are not important here, trees. For any triple of terminal nodes, it is usually Reitman and Reuter have been able to show that the case that two of the terms form a subcluster their method can recover the structure of a known that does not include the third. Thus, ~l; = ~~ stimulus set. In a subsequent example, they were <{)~ if i and j are the two terms that form the sub- able to illustrate nicely the differences in the or- cluster. It would thus also be expected that the ganization of ALGOL terms for novice and expert differences would have the relationship t)~ - ~;,< users. Hirtle (1982) has recently extended this work. -- (~ - Looking at the triangle of distances, Thus, it appears that one useful application of the middle side is closer in length to the long side ~1~5 and related techniques is in evaluating the- than it is to the short side. The elongation measure ories. If the underlying representation is suffi- used by Pruzansky et al. is simply the proportion ciently well specified, then it should be possible to of triangles in which this difference relation holds. obtain data from which the underlying represen- Pruzansky et al. (1982) examined the utility of tation could be recovered. At a minimum, a suc- these measures by calculating them for 20 data sets. cessful outcome argues for the plausibility of the Generally, when the proportion of elongated tri- theory. angles exceeded .65, ADDTREE provided a better .fit to the data; otherwise, KYST did. Similarly, when the skewness was less than -.4, ADDTREE The Nature off the Underlying Representations provided a better fit. Generally, these two measures Although the examples cited in the preceding tended to be negatively correlated, and there where section suggest that these techniques can profitably no cases where the two measures truly conflicted. be used to assess the viability of a particular theory, The result of greatest interest is that the data sets they explicitly do not attempt to compare compet- for which ADDTREE provided the better fit could ing theories directly. However, a recent paper by generally be described as conceptual stimuli, whereas ~ruz~i~sl~y9 Tversky, and Carroll (1982) has sug- the data sets that could be classified as perceptual gested that such comparisons may be both possible were almost always better fit by KYST. The data and profitable. Pruzansky et al. compared MDS (as sets that involved colors, forms, or letters were exemplified by KYST2A) with tree structures (as usually fit better by KYST, but data sets that con- exemplified by Sattath & Tversky’s, 1979, ADD- tained names of category exemplars were usually TREE). Pruzansky et al. first demonstrated that better fit by ADDTREE. Although Pruzansky et each provided a better fit to artificial data generated al. noted that factorial designs (as commonly used from its appropriate representation than the other in perceptual data sets) tended to favor KYST and method did. Specifically, KYST provided a better that natural selection (as commonly used in con- fit when the data set was generated from a plane, ceptual data sets) tended to favor ADDTREE, they and ADDTREE provided a better fit when the data offered no other explanation for this dichotomy. set was generated by a tree. Moreover, this rela- One problem in fitting the conceptual data sets tionship held over varying numbers of stimuli and could be in the nature of the stimuli. Although most levels of noise. of the conceptual data sets examined by Pruzansky Subsequently, Pruzansky et al. found two em- et al. are unpublished, there is sufficient familiarity pirical measures that predicted which procedure with the procedural details of the first eight they would provide the better fit (as measured by either studied to offer some speculation. The first seven the product-moment correlation or by Stress For- data sets (referenced by Pruzansky et al. as the mula 2). The first measure is the skewness of the Mervis et al. data sets) involved 19 exemplar names data, as determined by the standard measure of and one category label. The eighth data set (Hen- skewness: the third central moment divided by the ley, 1969) consisted of 30 animal names with no cubed standard deviation. The second measure, category label. For the first seven data sets, the

Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 484 superiority of ADDTREE, as measured by the dif- cle of tones as the horizontal structure. For other ference in both of Pruzansky et al. 9 indices of domains, there may be no comparable underlying goodness of fit averaged .15 for r,2 and .07 for structure available; the applicability of this ap- r~. The superiority of ADDTREE for the eighth proach is thus not immediately obvious. data set was about half these means, .08 and .03, Another approach is to ask the purpose for which respectively. Only one of the seven data sets ex- the MDS solution is obtained. If, for example, one amined by Mervis et al. (1980) had smaller dif- does not wish to claim that the underlying repre- ferences between the two procedures than Henley’ss sentation is spatial but rather to use the distances (1969) data set. in the multidimensional space to predict some dependent variable (such as RT) in another experiment, then other options may be available. For Problems with Categories example, Shoben (1976) was particularly interested Because of the way in which people judge cat- in the distances obtained by MDS between category egories, it is difficult for MDS programs to accom- exemplars and their superordinate. His first scaling modate categorical data sets. In particular, people solution used three dimensions to depict six mam- seem to regard all exemplars as similar to their mal exemplars, six bird exemplars, and two su- category name. For example, Shoben (1976) found perordinate terms. An examination of the Shepard that the similarity between goose and bird was vir- diagram, which plots fitted distances against the tually equivalent to the similarity between hawk original data, indicated that the greatest disparity and eagle, which in turn was less than the similarity between the actual data and the fitted distances between hawk and bird. More abstractly, even occurred for pairs that involved superordinate terms. atypical exemplars of a category are regarded as To use precisely these distances in subsequent anal- highly similar to their category name. This general yses, Shoben rescaled the data and used the weights finding causes problems for scaling algorithms. For option to ensure that the disparity between actual example, the similarity between robin and goose and fitted distances would be minimal for pairs is quite low, but each is highly similar to the con- involving a superordinate term. In this case, this cept of bird. In terms of distances in a multidi- manipulation was successful in that the exemplar- mensional space, robin and goose should be quite superordinate distances obtained from the weighted far from each other, but both must be quite close MDS solution predicted RT in a Same-Different to bird. Obviously, it is impossible to satisfy both task quite well. conditions. This kind of conflict is not present if superordinate terms are not among the test stimuli. It is not clear how this problem can be solved. Choosing an Appropriate Representation One possibility is to adapt a proposal of Krum- hansl’s (1983) and try to measure typicality of terms There are no simple rules for choosing the ap- separately. Krumhansl has argued that similarity is propriate representation. Instead, there are a num- a function, not only of the distance between the ber of criteria whose utility will depend on the two objects, but also of the distance (typicality) of particular circumstance. One obvious benchmark each of the objects to the superordinate. In the is the interpretability of the solution. Clearly, an context of the musical scale, mhansl was able uninterpretable solution is not convincing evidence to vary the context (which scale is used) and thereby for the plausibility of the underlying representation. to show that the vertical structure (typicality) has More often, there are competing arguments con- an effect on the similarity judgment independent cerning which of two representations is more in- of the effect of the distance between the two objects terpretable. For example, Sattath and Tversky (1979) (horizontal structure). In fitting the function that have argued that the ADDTREE representation of described the contribution of both horizontal and Henley’ animal data is more understandable than vertical structure, Krumhansl used the chroma cir- the one provided by MDS.

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Sattath and Tversky have also used another cri- to account for her data at more microcosmic levels terion for deciding the appropriate representation: as well. For example, she could account for the goodness of fit. Pruzansky et al. (1982) have also similarity of i to j being greater than the similarity employed this criterion in their comparison of of j to a when j was closer to the superordinate ADDTREE and KYST. From the two measures term. they used, it is clear that different measures of Thus, it seems that there is no easy answer to goodness of fit may give different results, and, the question of the single most appropriate repre- moreover, that a particular index of goodness of sentation. All of the methods individually have fit may favor one method over another. For ex- problems, and there is no magic rule for combining ample, one of the indices used by Pruzansky et al. approaches. Moreover, slight variations in tech- is directly related to stress, which is minimized by nique can have fairly substantial effects on the out- KYST. Not surprisingly, KYST fares better when come. Perhaps the most dramatic example of the this particular index is used. large improvement in the scaling solution as a func- Pruzansky et al. have also suggested that there tion of a relatively small (to a cognitive psychol- are statistical features of the data that may also help ogist) change in procedure is the work of Soli and in making a decision. As noted earlier, they sug- Arabie (1979) on the Miller-Nicely data. By using gested that the skewness and the elongation of the INDSCAL rather than MDSCAL and employing a data will enable determining in advance whether transformation of the original data to make them ADDTREE or KYST will yield a representation better conform to the assumptions of the INDSCAL that is more faithful to the original data. These two model, they were able to recover many more di- values can therefore be measured with the expec- mensions than Shepard (1972) did in his seminal tation of having some idea of the procedure that work on this data set. Conscquently, it is premature will give a better fit. to specify any rule or rules for determining the All of these procedures are useful, but it can be preferred underlying representation for any partic- argued that they do not provide a definitive answer, ular data set. either individually or collectively. As Krumhansi (1983) has n®ted9 thcrc are often theoretical reasons for adopting a particular approach. She has given ~~®b~e and Pitfalls in 1~~~ Analysis an excellent example of this theoretical motivation It might be speculated that one of the reasons in her work with music. As noted earlier, in the why MDS analysis is applied so infrequently to absence of specific context, the notes of a scale cognitive problems is that it is a complicated set may be conceived as a chroma much like of techniques with which few cognitive psychol- the results for colors. With context, however, she ogists are familiar. Because of this complexity, was able to show that the similarity of a pair of there are a number of pitfalls into which cognitive on distance tones depended not only the in the psychologists are prone to fall and there are a num- chroma circle, but also on the distance to the su- ber of interesting cognitive problems to which the perordinate, which in this case was the scale. In a power and potential applicability of MDS analysis C major scale, for example, G is closer to C than have not been applied. might be expected from the chroma circle alone. in this C# is further from C than Similarly, scale, Number of ~~~~uh would be expected in the absence of explicit context. The simplest problem to avoid is using too few This approach has proven very fruitful. Not only stimuli for the number of dimensions. In two di- was Krumhansl able to provide a satisfactory spa- mensions, for example, it is trivially easy to ac- tial representation of her own data, but with some commodate the similarities among six stimuli. It additional assumptions related to those in her spa- is very likely that a seemingly low value of stress tial density model (Krunhansl, 1979), she was able will be obtained even if the similarities are gen-

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erated randomly. Klahr (1969), for example, has Gamer (1972), many researchers have gone on to performed monte carlo simulations that provide users assume that their stimuli were wholistic or integral, with stress values beyond which there can be re- whereas others never considered any other metric. jection of the hypothesis that the data are random. There are at least two other metrics that are of These values are provided, of course, for a given inherent interest to cognitive psychologists. In most number of stimuli and a given number of dimen- general terms, the Minkowski family of metrics is sions. described by Equation 4 where r -- 1:e As a general rule of thumb, it would appear that it is very unwise to use fewer than 9 or 10 stimuli in a two-dimensional solution and fewer than 14 If the exponent r is equal to 2, Equation 4 gives or 15 when three dimensions are employed. More- the familiar Euclidean case. Two other cases are over, it should be noted here that the stress values of particular interest: when r equals 1 and when r given in Klahr are artificially high (Arabie & Boor- approaches infinity. In the latter case, the distance man, 19 73) in that they used a now obsolete and between two objects reduces to the maximum dif- clearly inferior initial configuration. Nevertheless, ference between the objects on any dimension. Thus, these tables (or their equivalent) are the best avail- two objects that differ from each other by a mod- able and they should be consulted. The major im- erate amount on each of three dimensions are ac- port of this discussion is that there should be war- tually closer to each other than two objects that are iness of MDS solutions performed with too few identical on two dimensions but very different on stimuli. Thus, Shafto’s (1973) argument for case a third. Although there are few applications of this grammar, which depends heavily on a two-dimen- dominance metric, Arnold ( 19~ 1 ~ has for sional representation of eight stimuli, should be this particular metric in the semantic te he ex- viewed with a high degree of skepticism. amined. Although some general guidelines for nonmetric When r equals 1, the distance between two points ~1~5 can be formulated, there are no such easy is determined by adding the differences on all di- rules for other commonly used algorithms, such as mensions. In two dimensions, this procedure re- INDSCAL or SINDSCAL. Here, the degree of sembles distances in a city where a person must constraint provided by the data depends, not only travel three blocks north and two blocks east, for on the number of stimuli, but also on the number example, to get from 43rd Street and 7th Avenue of subjects. For example, it would be incorrect to to 46th Street and 5th Avenue in New York City. fault Soli and Arable (1979) for extracting four Because of this analogy, this value of ~ is often dimensions from the Miller-Nicely data when there referred to as the city-block metric. This metric is were 16 stimuli and 16 &dquo;subjects&dquo; (actually lis- particularly interesting to cognitive psychologists tening conditions). Clearly, some kind of simula- because it implies that the dimensions recovered tion work must be done before treating this issue by MDS are separable, rather than integral. definitively. At a minimum, however, when the Despite the inherent appeal of the city-block number of subjects is very small, it would seem metric, there are understandable, though not ex- prudent to follow the strictures outlined above for cusable, reasons for its lack of prominence. The nonmetric MDS analysis. Moreover, when the first reason is that numerical problems are more number of subjects is small, the orientation of the likely to be encountered in non-Euclidean metrics. axes may not be well defined. For the Euclidean distance model, either a small number of random initial configurations or a ra- tional can be selected and r Metrics starting configuration there will be reasonable confidence that a near op- Nearly all of the applications of MI7S in cog- timum solution will be obtained. This procedure nitive psychology have assumed Euclidean dis- will rarely yield satisfactory results for the city- tance. Consistent with Shepard (1964) and with block metric. Local minimum problems are far more

Downloaded from apm.sagepub.com at UNIV OF ILLINOIS URBANA on September 3, 2012 487 common; it appears that many more random start- There are a number of more technical options ing configurations must be used (Arabie, 1973) within the often used programs that are seldom when r equals 1 than when the Euclidean metric used but are potentially useful. For example, KYST is used.0 permits the similarities to be weighted so that, for Fortunately, Arnold ( 1971 ) has developed a pro- example, similarities could be weighted according cedure that seems to surmount most of these prob- to their reliability or consistency across subjects. lems. In a little noticed paper, Arnold approached Friendly (1977) has shown that this kind of weight- both the city-block metric and the dominance met- ing procedure can be used in obtaining a satisfac- ric (he set r = 32 to approximate the dominance tory MDS representation of free recall data. If tied metric) by a successive approximation procedure. data are viewed as particularly meaningful, then it He first obtained a solution in Euclidean space; this might be desirable to elect the secondary approach solution was then the initial configuration for r to ties instead of the primary approach. There are = 1. ~ and r = 2.5. Approaching = 1, he used other examples, as a careful reading of the program the resulting configuration when r = 1.5 as the documentation will reveal; the important point is starting configuration for a solution with r set at that there are options available that can be usefully 1.2~, and so on, until he obtained a solution with applied to problems in cognitive psychology only ~° = 1. In a similar way, Arnold obtained a rep- if users are aware of them. resentation of his stimuli using the dominance metric. this Arnold found that the Using procedure, Conclusions Euclidean solution gave the poorest fit (as measured by stress) to the data. Stress declined mon- MDS has been usefully and frequently applied ®t®nically as r decreased from 2 to 1, and it also to problems in cognitive psychology. In many areas, declined monotonically as r increased from 2 to such as the work in music perception and the de- 32. Interestingly, the best fit was obtained with the velopment of models of analogical reasoning, the dominance metric. applications have been particularly inventive. Some Although this procedure appears very successful of the examples described above illustrate that there and well worth using, it does appear that it has one are many more problems in cognitive psychology important limitation: It does not seem to work well to which MDS and related procedures might prof- with two-dimensional solutions (Carroll & Arabie, itably be applied. Particularly with the advent of 1980). The reasons for this failure are not presently new methods for determining the underlying struc- known. tural representation, such as the work of Pruzansky et al. (f9~2) and Krumhansl (1983), it appears that MDS may become less a method of exploratory analysis and more a method of theoretical analysis. Using the Options In addition, developments in individual difference Finally, it appears that many users of MDS are models, such as the introduction of INDCLUS, not aware of all the options that most computer suggest that it may be possible to examine the struc- programs permit. For example, although INDS- ture of a set of stimuli as a function of condition CAL is characterized as three-way (stimuli by stim- or experimental context by using these algorithms. uli by subjects), the last way need not be actual From the perspective of cognitive psychology, individual subjects. Soli and Arabic (1979), for it appears that the limitation on MDS is that it tells example, have used conditions in lieu of subjects little about processing. Cognitive psychology has very successfully. Miller and German (1983) used many examples of problems where two very dif- age as their individual difference variable in their ferent structures coupled with two different sets of INDCLUS analysis. It would seem that INDSCAL processing assumptions can predict the same re- might be particularly helpful in examining the ef- sults. Two prominent examples are the proposi- fects of context in cognition. tional imagery debate and semantic memory models.

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Anderson (1978) has even argued that some of Arnold, J. B. A multidimensional scaling study of semantic distance. Journal these fundamental structural questions can never of Experimental Psychology Monograph,1971,90, 349-372. be resolved. At a minimum, it appears that choices Atkinson, R. C., & Shiffrin, R. M. Human memory: A can not be made without a consideration certainly proposed system and its control processes. In K. W. of the accompanying processing assumptions. Spence & J. T. Spence (Eds.), The psychology of Thus, if the goal of the cognitive psychologist learning and motivation: Advances in research and is to construct sensible theories of cognition, then theory (Vol. 2). New York: Academic Press, 1968. Bisanz, G. L., LaPorte, R. E., Vesonder, G. T., & Voss, it is clear that MDS analysis can be quite helpful J. F. On the representation of prose: New dimensions. in half of the battle. Although it may not help Journal of Verbal Learning and Verbal Behavior, 1978, directly in determining the underlying process, it 17, 357-358. does provide evidence about the structural prop- Caramazza, A., Hersh, H., & Torgerson, W. Subjective erties of the data and how that structure might change structures and operations in semantic memory. Jour- nal and Verbal under various conditions. Moreover, MDS of Verbcal Learning Behavior, 1976, analysis I5,103-117. might actually lead to the development of a pro- Carroll, J. D., & Arabie, P. Multidimensional scaling. cessing model, as it did for Rumelhart and Abra- Annual review of psychology, 1980, 31, 607-649. hamsen (1973). With the current interest of cog- Carroll, J. D., & Chang, J. J. Analysis of individual nitive psychology in the nature of the underlying differences in multidimensional scaling via an N-way of and the current interest in the effects generalization ’Eckart-Young’ decomposition. Psy- representation chometrika, 1970, 35, 283-319. of the of 1~175 and context, potential applicability Clark, H. H. On the meaning and use of prepositions. related techniques in this field appears to be large. Journal of Verbal Learning and Verbal Behavior, 1968, 7, 421-431. Craik, F. I. M., & Lockhart, R. S. Levels of processing: A framework for memory research. Journal of Verbal Refeirences Learning and Verbal Behavior, 1972,11,671-684. Ekman, G. Dimensions of color vision. Journal of Psy- Anderson, J. A. Neural models with cognitive appli- chology, 1954, 38, 467-474. cations. In D. LaBerge & S. J. Samuels (Eds.), Basic Fillenbaum, S., & Rapaport, A. Structures in the sub- processes in reading : Perception and comprehension. jective lexicon. New York: Academic Press, 1971. Hillsdale NJ: Erlbaum, 1977. Friendly, M. L. In search of the m-gram: The structure Anderson, J. R., & Bower, G. H. Human associative of organization in free recall. Cognitive Psychology, memory. Washington DC: Winston, 1974. 1977,9, 188-249. Arabie, P., & Boorman, S. A. Multidimensional scaling Friendly, M. Methods for finding graphic representations of measures of distance between partitions. Journal of associative memory structures. In C. R. Puff (Ed.), of Mathematical Psychology, 1973, 10, 148-203. Memory organization and structure. New York: Ac- Arabie, P., & Carroll, J. D. MAPCLUS: A mathematical ademic Press, 1970. programming approach to fitting the ADCLUS model. Garner, W. R. Information integration and form of en- Psychometrika, 1980, 45, 211-235. coding. In A. Melton & E. Martin (Eds.), Coding Arabie, P., & Carroll, J. D. INDCLUS: An individual processes in human memory. New York: Wiley, 1972. differences generalization of the ADCLUS model and Henley, N. M. A psychological study of the semantics the MAPCLUS algorithm. Psychometrika, 1983,48, of animal terms. Journal of Verbal Learning and Ver- 157-169. bal Behavior, 1969,8, 176-184. Arabie, P., Kosslyn, S. M., & Nelson, K. E. A mul- Hirtle, S. C. Lattice-based similarity measures between tidimensional scaling study of visual memory of 5- ordered trees. Journal of Mathematical Psychology, year-olds and adults. Journal of Experimental Child 1982,25, 206-225. Psychology, 1975, 19, 327-345. Howard, D. V., & Howard, J. H. A multidimensional Arable, P., & Soli, S. D. The interface between the scaling analysis of the development of animal names. types of regression and methods of collecting prox- Developmental Psychology, 1977, 13, 108-113. imity data. In R. G. Golledge & J. N. Rayner (Eds.), King, M. C., Gruenewald, P., & Lockhead, G. R. Clas- Proximity and preference: Problems in the multidi- sifying related stimuli. Journal of Experimental Psy- mensional analysis of large data sets. Minneapolis: chology: Human Learning and Memory, 1978,4, 417- University of Minnesota Press, 1982. 427.

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Shepard, R. N. Multidimensional scaling, tree-fitting, Tulving, E. Ecphoric processes in recall and recognition. and clustering. Science, 1980, 210, 390-398. In J. Brown (Ed.), Recall and recognition. London: Shepard, R. N., & Arabie, P. Additive clustering: Rep- Wiley, 1975. resentation of similarities as combinations of discrete overlapping properties. Psychological Review, 1979, 86, 87-123. Acknowledgments Shoben, E. J. The verification of semantic relations in a same-different paradigm: An asymmetry in semantic Preparation of this paper was se~pportecl in part by Grant memory. Journal of Verbal Learning and Verbal Be- No. BA~~ 82-17674 from the National Science Founda- havior, 1976, 15, 365-379. tion. The authors thanks Lawrence E. Jonesfor his careful Soli, S. D., & Arabie, P. Auditory versus phonetic ac- reading

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