Central tendencies, extreme points, and prototype ...catlab.psy.vanderbilt.edu/wp-content/uploads/File/Palmeri-QJEP2001.pdf · Indiana University, Bloomington, U.S.A. In three perceptual

THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY 2001 54A (1) 197ndash235

Central tendencies extreme points and prototypeenhancement effects in ill-de ned perceptual

categorization

Thomas J PalmeriVanderbilt University Nashville USA

Robert M NosofskyIndiana University Bloomington USA

In three perceptual classi cation experiments involving ill-de ned category structures extremeprototype enhancement effects were observed in which prototypes were classi ed moreaccurately than other category instances Such empirical ndings can prove theoreticallychallenging to exemplar-based models of categorization if prototypes are psychological centraltendencies of category instances We found instead that category prototypes were sometimesbetter characterized as psychological extreme points relative to contrast categories Extending aclassic and widely cited study (Posner amp Keele 1968) participants learned categories createdfrom distortions of dot patterns arranged in familiar shapes Participants then made pairwisesimilarity judgements of the patterns Multidimensional scaling (MDS) analyses of the similar-ity data revealed the prototypes to be psychological extreme points not central tendenciesEvidence for extreme point representations was also found for novel prototype patterns dis-playing a symmetry structure and for prototypes of grid patterns used in recent studies byMcLaren and colleagues (McLaren Bennet Guttman-Nahir Kim amp Mackintosh 1995)When used in combination with the derived MDS solutions an exemplar-based model of cate-gorization the Generalized Context Model (Nosofsky 1986) provided good ts to the observedcategorization data in all three experiments

One of the major research paradigms used for investigating perceptual categorization hasbeen the dot-pattern prototype-distortion task introduced by Posner and Keele (19681970) In this task prototype dot patterns are created and participants are trained to

Requests for reprints should be sent to Thomas J Palmeri Department of Psychology 301 Wilson HallVanderbilt University Nashville TN 37240 USA Email ThomasJPalmeriVanderbiltedu

This work was partially supported by Grant PHS R01 MH48494 from the National Institute of Mental Healthto Robert M Nosofsky and by Vanderbilt University Research Council Direct Research Support Grants toThomas J Palmeri We thank Carolyn Cave IPL McLaren Timothy McNamara and an anonymous reviewerfor their comments on earlier versions of this work Thanks to Jason Bechler Debra Coomes Andrew DomescikHeather Fahey Brenda Graves Kristen Hicks Stephanie Jurek Nicole Naylor Thomas Sharpe and StephanieTang for testing participants

copy 2001 The Experimental Psychology Societyhttpwwwtandfcoukjournalspp02724987html DOI10108002724980042000084

198 PALMERI AND NOSOFSKY

categorize statistical distortions of these prototypes During transfer old distortions newdistortions and the prototypes are presented to be classi ed without feedback As discussedfor example by Homa (1984) a major advantage of these experiments is that the dotpatterns are essentially in nitely variable and have a highly complex dimensional structureso that the properties of the arti cial categories that are created may mimic those of manynatural categories

The dot-pattern paradigm has been used to systematically investigate the effects ofnumerous fundamental variables on category learning and transfer including effects of cate-gory size (eg Breen amp Schvaneveldt 1986 Homa amp Vosburgh 1976 Posner amp Keele 1968Shin amp Nosofsky 1992) category variability (eg Barresi Robbins amp Shain 1975 Homa1978 Homa amp Vosburgh 1976 Posner amp Keele 1968) instance frequency (eg HomaDunbar amp Nohre 1991 Shin amp Nosofsky 1992) number of categories learned (egHoma amp Chambliss 1975) amount of training (eg Homa et al 1991 Homa GoldhardtBurruel-Homa amp Smith 1993) and delay between training and transfer (eg Homa CrossCornell Goldman amp Schwartz 1973 Posner amp Keele 1970 Strange Kenney Kessel ampJenkins 1970) In addition the paradigm has been used to investigate the relationshipbetween categorization and oldndashnew recognition memory (eg Homa et al 1993 Metcalfeamp Fisher 1986 Onohundro 1981 Shin amp Nosofsky 1992 Vandierendonck 1984) and hasalso served as a fundamental testing ground for investigating neuropsychological aspects ofcategorization (eg Knowlton amp Squire 1993 Kolodny 1994 Nosofsky amp Zaki 1998Palmeri amp Flanery 1999 Reber amp Squire 1997 Reber Stark amp Squire 1998 Squire ampKnowlton 1995) Indeed the dot-pattern paradigm has provided bedrock data for evaluat-ing numerous theories of categorization and memory including prototype models (egBusemeyer Dewey amp Medin 1984 Homa Sterling amp Trepel 1981) distributed memorymodels (eg Knapp amp Anderson 1984 Metcalfe 1982) connectionist models (egMcClelland amp Rumelhart 1985) and exemplar models (Hintzman 1986 Nosofsky 1988Shin amp Nosofsky 1992)

One of the most salient aspects of dot-pattern studies is a well-known effect called proto-type enhancement On average category prototypes that are not experienced during trainingare typically classi ed during transfer as well as and sometimes somewhat better than theold category instances and better than new category instances Although such prototypeenhancement effects were originally believed to provide solid evidence for the existence ofprototype abstraction processes theoretical work has shown that pure exemplar retrievalmodels can account for this phenomenon as well (eg Busemeyer et al 1984 Hintzman1986 Hintzman amp Ludlam 1980 Nosofsky 1988 1992 Shin amp Nosofsky 1992) It mayseem paradoxical that models that assume that categories are represented solely in terms ofstored exemplars can account for enhanced classi cation of unseen prototypes The keyintuition is that although any given old exemplar is highly similar to itself it may not bevery similar to any other old exemplars By contrast prototypes are typically similar to manyother exemplars stored in memory The similarity of prototypes to numerous stored exem-plars makes up for the lack of stored representations for the prototypes themselves

Shin and Nosofsky (1992) demonstrated an approach to modelling detailed aspects ofdot-pattern classi cation performance that combined an exemplar-based model of cate-gorization the Generalized Context Model (GCM Nosofsky 1986) with multidimensionalscaling (MDS) techniques (Shepard 1980) The stimuli they used were typical of most

EXTREME PROTOTYPE ENHANCEMENT 199

random-dot-pattern studies Random prototypes were generated for each category andstatistical distortions of these prototypes were created as category instances (PosnerGoldsmith amp Welton 1967) Some of these distortions were designated as training patternsand some were designated as transfer patterns In their studies all participants learned toclassify the same set of patterns Over numerous trials participants learned to classify thetraining patterns with corrective feedback During transfer they classi ed the trainingpatterns transfer patterns and category prototypes without feedback Across three experi-ments Shin and Nosofsky examined effects of several fundamental learning variables oncategorization including level of distortion of patterns category size delay of the transferphase and individual item frequency Their primary goal was to assess whether a pureexemplar-based model could account for the observed classi cation results of whetherprototype abstraction processes needed to be assumed as well

Several previous attempts to model dot-pattern classi cation have used randomly gener-ated multi-element stimulus vectors as inputs to simulation models (eg Hintzman 1986Knapp amp Anderson 1984 Metcalfe 1982 Nosofsky 1988) Although such representationsare intended to capture some elements of the physical instantiation of the dot-pattern stimulithey may fail to capture the true psychological relationships among these complex patternsMoreover these methods allow only gross-level predictions to be made such as predictionsof average classi cation performance for the prototypes and old and new distortionsInstead as a more detailed test of various competing models Shin and Nosofsky (1992)aimed to account for the classi cation performance of particular instances not just averageclassi cation of particular types of stimuli In their experiments participants provided pair-wise similarity ratings of the dot patterns These similarity rating data were then analysedusing standard MDS techniques to obtain a psychological scaling solution for the stimuliThe derived scaling solution was used in conjunction with the GCM a prototype modeland a mixed model to account for the observed classi cation data Theoretical analysesrevealed little evidence for the existence of a prototype abstraction process that operatedabove and beyond pure exemplar-based generalization Among other qualitative and quan-titative predictions the pure exemplar-based model could account for the prototypeenhancement effects that Shin and Nosofsky observed

It is important to note that many reported cases of prototype enhancement in experi-ments using the dot-pattern paradigm have compared classi cation accuracy for categoryprototypes relative to the average classi cation accuracy for other category instances inthose articles that report classi cation probabilities for individual stimuli the prototype isnot the best classi ed item overall For example the degree of prototype enhancementreported by Shin and Nosofsky (1992) was not very large and many old category instanceswere classi ed more accurately than the category prototypes A potential challenge for theGCM and other exemplar models is whether or not they could ever predict an extreme proto-type enhancement effect in this paradigm in which the prototypes are classi ed signi cantlymore accurately than other category instances

An observation of extreme prototype enhancement could provide a serious challenge tothe GCM and other exemplar models According to many theories of perceptual cate-gorization including the GCM objects are represented as points in a multidimensionalpsychological space (eg Ashby 1992 Homa 1984 Nosofsky 1986 Reed 1972 Shepard ampChang 1963) It is natural to assume that category prototypes in dot-pattern studies as well


as other experimental paradigms are psychological as well as physical central tendencies ofcategory instances (Homa 1984 Homa et al 1981 Nosofsky 1987 Posner 1969 Reed1972 Rosch 1975b 1978 Smith amp Medin 1981) For example in summarizing the resultsof Posner and Keelersquos (1968 1970) studies Anderson (1980 p 140 142) wrote ldquoOne of themost impressive demonstrations of subjectsrsquo ability to extract the central tendency of a setof instances is a series of experiments performed by Posner and Keele (1968 1970) Theprototype for Posnerrsquos dot patterns would have been the average of the studied dot patternsrdquoThis assumption about a central-tendency representation for the prototypes has beenlargely con rmed in multidimensional scaling studies involving randomly generated proto-types (Homa 1984 Homa Rhoads amp Chambliss 1979 Shin amp Nosofsky 1992)Simplifying the multidimensional representation of the patterns somewhat the left panel ofFigure 1 depicts category prototypes (cA cB and cC) as psychological central tendencies ofthe category instances A key point is that if the category prototypes are indeed centraltendencies in psychological space as well as being physical central tendencies of categoryinstances then the GCM cannot predict an extreme prototype enhancement effect Insteadcategory instances lying in extreme regions of the psychological space (relative to the con-trast categories) will tend to be classi ed more accurately than the central prototypes as isillustrated later

An alternative possibility however to be explored in the present research is that proto-types that are physical central tendencies of category instances may sometimes reside not aspsychological central tendencies but rather as psychological extreme points relative to thecategory instances The right panel of Figure 1 depicts a situation in which physical cate-gory central tendencies may be represented as extreme points (eA eB and eC) in thepsychological space relative to the category instances Under such conditions the GCMdoes predict extreme prototype enhancement as illustrated next1

To illustrate how predictions of the GCM can vary depending on whether the prototypesare central tendencies or extreme points we generated predictions of the model based onthe idealized con guration shown in the left panel of Figure 2 We compared predicted classi-fication accuracy for ldquocloserdquo exemplars which are close to members of contrast categoriesldquofarrdquo exemplars which are far from members of contrast categories central-tendencyprototypes and extreme-point prototypes The right panel of Figure 2 shows predictedclassi cation accuracy for each of these types of stimulus across a range of parameter values(this theoretical analysis allowed the sensitivity parameter c which scales distances betweeninstances in psychological space to vary across a range of values we also assumed equal

1 Extreme prototype enhancement has often been observed in experimental paradigms involving discrete-dimension stimuli However in experimental paradigms involving discrete dimensions such as those initially testedby Medin and Schaffer (1978) the interpretation of a prototype as a ldquocentral tendencyrdquo versus an ldquoextreme pointrdquois unclear Consider stimuli varying along four binary-valued dimensions Suppose the prototypes of Categories Aand B are 1111 and 2222 respectively and that category instances are generated by distorting these prototypes byvarying degrees (eg 1112 and 1121 might be examples of Category A and 2221 and 2212 might be examples ofCategory B) From one point of view these prototypes can be viewed as ldquocentral tendenciesrdquo in the sense that theyhave the modal values on each dimension However these prototypes act as extreme points in the multidimensionalcategory structure as well Therefore one needs to test experimental paradigms with complex continuous-dimension stimuli such as dot patterns to sharply distinguish between the roles of central tendencies and extremepoint representations in classi cation


response biases for the three categories and assumed equal attention to both psychologicaldimensions see the discussion of the GCM following Experiment 1 for details of themodel) As shown in the gure the close exemplars were predicted to be classi ed withrelatively low accuracy and the far exemplars were predicted to be classi ed with relativelyhigh accuracy The GCM could not predict prototypes to be the best classi ed items whencentral-tendency representations were assumedmdashprototypes were always classi ed with

Figure 1 A schematic illustration of central tendencies and extreme points in simpli ed two-dimensionalpsychological space In both panels three categories with six distortions each are depicted by the squares circlesand triangles In the left panel the prototypes (cA cB and cC) are central tendencies of the six distortions of theircategory In the right panel the prototypes (eA eB and eC) are extremes relative to the six distortions oftheir category and relative to the distortions of the other categories

Figure 2 The left panel displays a schematic illustration of the psychological space used in the simulationsreported in the text Grey symbols indicate close exemplars black symbols indicate far exemplars cA indicates acentral-tendency prototype eA indicates an extreme-point prototype The right panel displays the GCM predic-tions for each of these four types of stimulus as a function of the sensitivity parameter of the model c


intermediate accuracy In these simulations the failure to predict extreme prototypeenhancement with central tendency representations was observed regardless of whether theprototypes were allowed to be old training items (a point that will be important whenreviewing the results of the rst two experiments) However the GCM could predict theprototypes to be the best classi ed items when extreme-point representations were assumedIt is clear that the derived MDS solution is crucial to ascertain whether the GCM can orcannot predict an observed pattern of extreme prototype enhancement Certain ldquoobjectiverdquomeasures of pattern similarity de ned by such things as the average distances between dotsin pairs of patterns (eg Posner 1969) are probably insuf cientmdashusing such measures ofsimilarity the prototype being a physical central tendency of category instances wouldalways emerge as a psychological central tendency as well

The potential importance of psychological extremes in categorization has been noted insome other work For example Barsalou (1985 1991) demonstrated the importance of idealpoints in highly conceptual domains involving goal-derived categoriesmdashthe best example ofthe category ldquofoods to eat on a dietrdquo is one with zero calories not one with the averagecaloric content of typical diet foods In applications of their highly successful Fuzzy-LogicalModel of Perception (FLMP) Massaro and colleagues often tested paradigms in which thestimuli varied along two clear continuous dimensions and the prototypes to which peoplecompared objects were assumed to occupy extreme corners of the psychological space(Massaro 1987 Massaro amp Friedman 1990 Oden amp Massaro 1978)mdashnote that in theirparadigms the physically manipulated prototypes and their resulting psychological repre-sentations were both extreme points Although the potential importance of psychologicalextremes in categorization has been suggested by the work of Barsalou Massaro and others(eg Goldstone 1993 1996 Rosch 1975b) the idea that a prototype that is a central tendencyin the physically de ned space may emerge as an extreme point in the psychological spacehas not previously been suggested Although prototypes may indeed be physical centraltendencies of the distortions created from them it does not necessarily follow that they arepsychological central tendencies as well Rather various emergent dimensions based ondiagnostic con gurations among elements of a complex physical stimulus such as a dotpattern (eg Hock Tromley amp Polmann 1988) may be formed which cause the prototypesto be represented as psychological extremes within the context of learning particularcategories

One goal of the present research was to document that extreme prototype enhancementeffects could be observed empirically Whereas the prototypes that most recent experimentshave used were random dot patterns in one of the original Posner and Keele (1968) studieshighly recognizable dot patterns (eg a triangle an M and an F) were used as categoryprototypes instead We chose to use such recognizable patterns in the rst experimentreasoning that their use as prototypes might offer an excellent chance of empirically observ-ing an extreme prototype enhancement effect Another goal of the present research was toexamine the nature of the psychological representations of the physical category prototypesAre they best characterized as psychological central tendencies or as psychological extremepoints To assess this issue participants provided pairwise similarity ratings among allpatterns in the set and multidimensional scaling techniques were used to derive a psycho-logical space for those patterns If extreme prototype enhancement is observed and physicalprototypes are represented as psychological central tendencies then the GCM and many


other exemplar models would be falsi ed Finally even if the prototypes are represented asextreme points in the MDS solution the question still arises as to what type of decisionmodel will provide the best account of the detailed classi cation performance data Weassessed this question by comparing the quantitative ts of formal exemplar prototype andmixed exemplar-plus-prototype models

EXPERIMENT 1

This experiment was an extension of Posner and Keelersquos (1968) classic study Followingtheir design the categories were based on prototype patterns formed in the shape of a tri-angle a plus and an F2 Category instances were distortions of those prototype patternsDuring training participants learned to classify these instances with feedback To increasethe likelihood of observing an extreme prototype enhancement effect for one group ofparticipants the category prototypes were also presented during training (recall from thesimulations reported earlier that the inability of the GCM to account for extreme prototypeenhancement with central tendency representations was not modulated by the presence orabsence of prototypes during training) At transfer participants were tested on old distor-tions new distortions and prototypes Extending the Posner and Keele design participantsalso made pairwise similarity judgements multidimensional scaling techniques were used toanalyse the similarity data to derive the psychological coordinates for the patterns (Shin ampNosofsky 1992) The scaling solution should reveal whether the prototypes were centraltendencies or extreme points in the psychological space Exemplar prototype and exemplar-plus-prototype models were tted to the observed categorization data to test whether a pureexemplar-based model needed to be supplemented by special prototype abstractionmechanisms

Method

Participants

Participants were 280 undergraduates who received course credit All were tested individually

Stimuli

Patterns were composed of nine dots placed on a 50 3 50 grid As shown in Figure 3 the threeprototype patterns were in the shape of a triangle a plus and an F These prototypes tted within thecentre 30 3 30 of the grid From each prototype nine moderate-level distortions (6 bitsdot) werecreated by using a standard statistical distortion algorithm (Posner et al 1967) this algorithm moveseach dot of the prototype pattern some small amount in a random direction Six distortions wereselected as old training items and three were selected as new transfer items Stimuli were presentedon 14-in computer monitors

2 Unlike Posner and Keele (1968) we used a plus instead of an M so that the prototype would belong todifferent superordinate categories


Procedure

A standard category learningtransfer paradigm was used During training participants learned tocategorize the six training patterns from each category In the prototype condition participants also sawthe prototypes during training in the no prototype condition participants saw the prototypes only attransfer Patterns were presented once per block in random order for eight blocks On each trial apattern was presented and classi ed as an A B or C Corrective feedback was supplied for veseconds or until the space bar was pressed After an ITI of one second the next pattern was displayedThe assignment of category to response key was randomized for every participant Responses weremade by pressing labelled keys on a computer keyboard

During transfer all thirty patterns (one prototype six old distortions and three new distortionsfrom each category) were presented once per block in random order for three blocks No correctivefeedback was provided

Participants also rated the pairwise similarities among patterns Because of the large number ofpossible pairs each participant rated only half of them (randomly selected for each participant) Oneach trial two patterns were presented side by side Participants rated similarities by using a 10-pointscale (1 5 very dissimilar 10 5 very similar)

Results

Categorization data analyses

The observed category response probabilities for each individual stimulus in the noprototype and prototype conditions are reported in Table 1 A two-way repeated measuresanalysis of variance (ANOVA) was conducted on the categorization accuracy data (the prob-ability of classifying the item into the correct category) with no prototype vs prototype as abetween-subjects factor and type (prototype old or new) as a within-subjects factorAccuracies for prototypes old patterns and new patterns in the no prototype and prototypeconditions are summarized in the left panel of Figure 4 Accuracy was higher in the proto-type condition than the no prototype condition F(1 276) 5 9412 MSE 5 006 (alpha wasset at 05 for all statistical tests reported in this paper) The effect of type was signi cantF(2 552) 5 26116 MSE 5 002 planned comparisons revealed that prototypes were

Figure 3 The top part of the gure shows the three prototype patterns used in Experiment 1 Under each proto-type is an example of a moderate-level distortion created from that pattern


categorized more accurately than old instances which were categorized more accuratelythan new instances A signi cant two-way Prototype 3 Type interaction was observedF(2 552) 5 9686 MSE 5 002 in the prototype condition prototypes were categorizedsigni cantly more accurately than old instances however in the no prototype conditionprototypes were categorized roughly as accurately as old instances In both conditions new

TABLE 1Observed and predicted categorization response probabilities for the no prototype and

prototype conditions in Experiment 1

Observed Generalized context model

No prototype Prototype No prototype Prototype

Stimulus P(T) P(P) P(F) P(T) P(P) P(F) P(T) P(P) P(F) P(T) P(P) P(F)

TrianglesTP 798 088 114 926 045 029 745 122 134 903 047 049T1 774 095 131 805 093 102 774 102 124 841 074 085T2 829 071 100 912 031 057 792 095 113 860 064 076T3 429 283 288 491 264 245 447 224 329 449 229 323T4 738 119 143 810 107 083 736 136 128 800 110 090T5 862 071 067 940 048 012 813 088 099 883 057 061T6 614 136 250 690 107 202 642 149 209 691 129 180Ta 602 202 195 707 155 138 637 192 170 699 171 130Tb 521 219 260 502 252 245 551 192 257 607 163 231Tc 462 343 195 524 343 133 560 252 187 599 250 151

PlusesPP 260 555 186 038 914 048 211 522 268 025 934 041P1 217 498 286 133 588 279 224 497 279 143 639 218P2 138 607 255 088 714 198 142 568 290 091 674 235P3 162 598 241 100 779 121 179 589 232 114 703 183P4 143 610 248 074 762 164 120 670 209 071 788 141P5 131 600 269 064 652 283 151 538 312 087 639 274P6 098 710 193 062 810 129 108 704 187 062 814 124Pa 164 407 429 124 507 369 220 338 441 161 415 424Pb 219 390 391 143 502 355 219 427 355 167 504 329Pc 138 564 298 098 669 233 144 603 253 099 709 193

FsFP 243 333 424 012 052 936 177 280 543 025 053 922F1 231 305 464 162 312 526 191 280 529 125 299 576F2 074 181 745 076 183 740 108 274 618 072 243 685F3 188 174 638 129 148 724 185 196 618 113 124 763F4 271 233 495 183 181 636 213 214 573 160 201 639F5 174 274 552 152 243 605 189 214 597 132 194 674F6 105 174 721 057 095 848 127 185 688 063 096 842Fa 369 219 412 291 143 567 313 240 448 243 195 561Fb 150 291 560 124 255 621 168 331 501 127 312 562Fc 164 381 455 148 443 410 153 471 376 122 532 346

Note TP 5 prototype of triangle category T1 5 old instance of triangle category Ta 5 new instance oftriangle category Category T 5 triangles Category P 5 pluses Category F 5 Fs


instances were categorized with the lowest accuracy Note that the prototypes in the proto-type condition were the best classi ed of all stimuli (except for T5 in the triangle category)a nding that will prove important in the later theoretical analyses The following sectionsdetermine the psychological space of these stimuli and then t the categorization probabilitydata by using the GCM prototype and mixed models

Multidimensional scaling analyses

The average similarity matrices for the no prototype and prototype conditions were usedas input to the INDSCAL scaling model (Carroll amp Wish 1974 Shepard 1980) to derive asix-dimensional MDS solution (see Appendix A) The INDSCAL procedure gives an MDScon guration that is common to both groups together with individual dimension weightsunique to each group The overall t of the INDSCAL-derived distances to the observedsimilarity ratings was quite good (stress 5 069 r2 5 954) One potential concern was thatthe psychological space might vary depending on whether prototypes had been viewedduring training Because the INDSCAL solution is constrained to be structurally identicalfor both groups (subject only to dimensional stretching or shrinking) it is necessary toverify that the MDS solution is reasonable for both groups First the ts of the INDSCALsolution to each individual group were also quite good No prototype stress 5 069 r2 5 954prototype stress 5 068 r2 5 954 Second as indicated in Appendix A the INDSCALweights re ecting the importance of each dimension were comparable across both groupsTherefore presence or absence of prototypes during training did not seem to appreciablychange the locations of the dot patterns in the psychological space

Figure 5 plots the rst three dimensions of the MDS solution dimensions 1ndash3 accountfor 80 of the variance in the similarity ratings (percentage of variance accounted for by adimension equals the square of the INDSCAL dimension weight) Compare the left panelof Figure 5 with the right panel of Figure 1mdashthe scaling solution is consistent with the idea

Figure 4 The left panel displays observed and the right panel displays predicted categorization accuracycollapsed across individual stimuli for prototypes old items and new items in Experiment 1 The no prototypecondition is indicated by open triangles and the prototype condition is indicated by lled circles


that the prototypes were psychological extremes in relation to other exemplars of their cate-gories not central tendencies Although these patterns are indeed quite close to being phys-ical central tendencies of the old instances (as determined by physically averaging thedistortions) they are not psychological central tendencies That said we do believe that thelocation of objects in psychological space may be highly dependent on the context in whichsimilarity ratings are made (eg Medin Goldstone amp Gentner 1993) We observed extremepoint representations for prototypes in the context of contrast categories It seems quitepossible that if similarity ratings were collected for patterns from just a single category wecould observe central tendency representations for prototypes instead

To obtain converging evidence that the prototypes are well characterized as extremepoints in the psychological space we conducted the following analysis First we calculatedthe distance between each pair of stimuli in the MDS space as de ned in the INDSCALmodel For each individual stimulus we then computed its average distance to all membersof the contrast categories For example if a stimulus was from the triangle category wecomputed its average distance to all members of the plus and F categories We refer to thismeasure as the average between-category distance (DB) Likewise we calculated the averagedistance of each stimulus to all members within its own category We refer to this measureas the average within-category distance (DW) Finally we computed a composite measureDTOT de ned as the sum of DB and DW For each category we then rank-ordered thestimuli according to these measures To the extent that the prototypes occupy extremepoints in the psychological space their values of DB DW and DTOT should tend to belarge By contrast to the extent that the prototypes are central tendencies their values ofDW should tend to be small and their values of DB should tend to be intermediate In thepresent case the results were clear-cut In all three categories the prototypes were ranked rst on the between-category distance measure that is they had the largest values of DBLikewise for the plus and F categories the prototypes had the largest values of DW (In the

Figure 5 The left panel displays Dimensions 1 and 2 and the right panel displays Dimensions 2 and 3 of thesix-dimensional MDS solution given in Appendix A for Experiment 1 The triangles are indicated by triangles thepluses are indicated by the squares and the Fs are indicated by circles Numbers 1ndash6 indicate old exemplarsletters andashc indicate new exemplars and P indicates the prototype of each category


triangle category the prototype was ranked second on this measure) Finally in all threecategories the prototypes were ranked rst on the composite measure DTOT This analy-sis con rms the impression provided by visual inspection of Figure 4 namely that the pro-totypes occupied extreme points in the psychological space in which the category exemplarswere embedded

Finally although global measures of t such as stress and percentage of varianceaccounted for can be useful and informative sole reliance on them can sometimes leaveundetected systematic problems with the underlying scaling solution One possibility wethought important to investigate was whether the canonical prototype patterns were nearestneighbours of a relatively large number of category instances As discussed by Tversky andHutchinson (1986) spatial models of object similarity are bounded in the number of pointsany given point can be a nearest neighbour of Using measures of centrality and reciprocitythat Tversky and Hutchinson developed we were able to determine that the prototypeswere not nearest neighbours of a large number of points3 thereby strengthening the ideathat the derived MDS solution provides a reasonable description to the similarity structureunderlying these dot-pattern categories

Categorization theoretical analyses

Our next step was to t the GCM to the observed categorization data from this experi-ment We begin with a brief description of the details of the model According to the GCMevidence favouring a given category is found by summing the similarity of a presented objectand all category exemplars stored in memory Objects are represented as points in a multi-dimensional psychological space with similarity between objects i and j being a decreasingfunction of their distance in that space

sij 5 exp(2 c dij) 1

(Shepard 1987) where c is a sensitivity parameter that scales the psychological spaceDistance dij is computed using a simple (weighted) Euclidean metric

dij 5 Icirc S wm(xim 2 xjm)2 2

where xim is the psychological value of object i on dimension m (ie the derived coordinatesin the MDS solution) and wm is the attention weight given to dimension m The weightsldquostretchrdquo the psychological space along attended dimensions and ldquoshrinkrdquo it alongunattended ones (Kruschke 1992 Nosofsky 1984 1986) The probability of classifying i asa member of category J is given by

3 Measures of centrality (C) and reciprocity (R) as de ned by Tversky and Hutchinson (1986) were computedfor both the observed similarity matrices and the INDSCAL derived distances For the observed data in the noprototype condition C 5 1867 and R 5 2267 and in the prototype condition C 5 1733 and R 5 1967 For theINDSCAL derived distances in the no prototype condition C 5 1733 and R 5 2333 and in the prototype con-dition C 5 2000 and R 5 2133 Both the observed and the INDSCAL-derived centrality and reciprocitymeasures were small indicating that the prototypes were not nearest neighbours of a large number of pointsthereby strengthening our faith in the validity of the derived scaling solution Such small values of C and R aresimilar in scale to those obtained by Tversky and Hutchinson for perceptual stimuli such as colours soundsodours and simple shapes and for more complex categories such as birds fruits or weapons when superordinateterms were not included in the set


(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g

whereby bJ is the category J response bias R is the set of all categories and g controls thelevel of deterministic or probabilistic responding (Maddox amp Ashby 1993 McKinley ampNosofsky 1995 Nosofsky amp Palmeri 1997)

Our main theoretical goal was to determine whether a pure exemplar model the GCMcould account for the categorization probabilities reported in Table 1 Recall that extremeprototype enhancement was found in the prototype condition of the experiment Asexplained earlier the extreme point representations for prototypes which emerged from thesimilarity scaling solution allow the GCM to qualitatively account for extreme prototypeenhancement Another challenge is whether or not the GCM can make reasonable quanti-tative predictions as well the ts of the GCM were compared with those of a pure proto-type model and a mixed exemplar-plus-prototype model

In tting the GCM to the observed data for each condition (prototype and no proto-type) the full version of the model has eight free parameters an overall sensitivity para-meter (c) in Equation 1 ve free attention weights (wm) in Equation 2 (the six attentionweights sum to one) and two free category response biases (bJ) in Equation 3 (the threeresponse biases sum to one) for this particular dataset the g parameter could be set equalto 1 without much in uence on the t of the GCM to the observed data In the full versionof the model different parameters were assumed for the prototype and no prototype condi-tions yielding a total of 16 parameters Various restricted versions of the GCM were alsoinvestigated in which parameters were constrained to be the same across both conditions

The GCM was tted to the categorization probabilities given in Table 1 (120 free datapoints) using a maximum-likelihood measure of t (Wickens 1982) The predicted cate-gorization probabilities for each individual stimulus are reported in Table 1 and maximum-likelihood parameters and summary ts are reported in Table 2 (GCM-16) The averagedpredicted categorization accuracies for the three main types of stimulus (prototype olditems and new items) are shown in the right panel of Figure 4 The GCM tted the dataquite well accounting for 971 of the variance in the observed data Averaged acrossitems the GCM captured all important qualitative trends and showed very good quantita-tive predictions The GCM predicted higher categorization accuracy in the prototype con-dition than the no prototype condition Overall prototypes were predicted to be categorizedmore accurately than old instances which were categorized more accurately than newinstances Furthermore the model predicted correctly that prototypes would be classi edmore accurately than old instances in the prototype condition but that prototypes and oldinstances would be classi ed with roughly equal accuracy in the no prototype condition Inboth conditions new instances were categorized with the lowest accuracy

Several restricted versions of the GCM were also tested Although all yielded signi -cantly worse4 ts than the full version of the GCM some of these restricted versions t the

4 Likelihood-ratio tests were used to statistically compare models (see Wickens 1982) Let lnLF and lnLR denote thelog-likelihoods for a full and restricted model respectively Assuming the restricted model is correct the statistic2 (lnLF 2 lnLR) is distributed as a c2 with degrees of freedom equal to the number of constrained parameters If theobserved value of c2 exceeds a critical value then the restricted version ts signi cantly worse than the full version


data extremely well First as shown in Table 2 a nine-parameter version with attentionweights and biases constrained to be the same for both conditions t the data very wellaccounting for 967 of the variance this result provides evidence that selective attentionand response biases probably did not differ very much between the two conditions Second

TABLE 2Maximum-likelihood parameters and summary reg ts of the categorization data forthe full parameter GCM restricted versions of the GCM and prototype models in

Experiment 1

Parameter GCM-16 GCM-9 GCM-4 Prototype-A Prototype-B

No prototype c 1779 1723 1605 0798 2001w1 276 296 674 607 228w2 114 147 412 153 174w3 334 311 412 000 322w4 199 168 315 225 154w5 064 065 191 000 067w6 012 013 156 015 055bT 318 301 306 393 336bP 312 326 330 274 314bF 370 373 365 333 351

Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927

Prototype c 2168 2236 2051 1658 1941w1 320 296 714 357 360w2 188 147 474 173 274w3 260 311 262 000 000w4 149 168 281 122 195w5 070 065 206 227 092w6 013 013 170 121 079bT 279 301 306 306 313bP 344 326 330 287 328bF 377 373 365 408 360

Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946

Overall Fit 2 lnL 57173 59908 66306 99737 80247RMSE 0042 0045 0050 0075 0061Var 971 967 959 908 939

Note Var 5 variance accounted for (in percentages) c 5 general sensitivity parameterwm 5 attention weight given to dimension m bj 5 bias for making category j response2 lnL 5 negative value of log-likelihood RMSE 5 root mean squared error betweenobserved and predicted categorization probabilities GCM-16 5 full parameterized GCMGCM-9 5 GCM constrained with weights equal to INDSCAL weights and biasescommon between the no prototype and prototype conditions Prototype-A 5 prototypemodel using MDS-derived ldquoprototypesrdquo Prototype-B 5 prototype model using average ofold exemplars Underlined values are constrained parameters


a four-parameter version with attention weights set equal to the INDSCAL weights fromthe MDS solution and biases constrained to be the same for both conditions also t the dataquite well accounting for 959 of the variance Although the categories were presentedwith equal frequency for some reason restricted versions with equal biases across the threecategories tted the data signi cantly worse than the unrestricted versions

Prototype models which assume that categorization decisions are based on similarity to anabstracted prototype (eg Homa 1984 Reed 1972) were also formalized within the MDSframework The prototype models were identical to the GCM except that rather than com-puting the summed similarity of an item to all category exemplars one computes itssimilarity to the category prototype instead Two models were tested in Prototype-A thederived MDS coordinates of the prototypes were used in Prototype-B the prototype repre-sentations were generated by spatially averaging old category instance representations (Reed1972 Nosofsky 1988 Shin amp Nosofsky 1992) As shown in Table 2 both 16-parametermodels tted the data worse than the 4-parameter GCM Predicted categorization accura-cies for prototypes old items and new items are shown in Figure 6 Prototype-A did notcapture the qualitative trends in the data from the no prototype condition and under-estimated the difference in accuracy between old and new items in both conditionsPrototype-B captured most of the qualitative relations but systematically under- and over-estimated the accuracies for prototypes and new items respectively

A combined exemplar-plus-prototype model was also investigated (Shin amp Nosofsky1992) In this model evidence for a given category was equal to the summed similarity of anitem to all category exemplars plus a weighted similarity to the category prototype

[bJ(Sj e JSij 1 y SipJ)]g

P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g

where y is the weight for the prototype and sipJ is the similarity between item i and the proto-type for category J Note that y 5 0 yields the standard GCM In tting the combinedmodel to the data separate y terms were assumed for the no prototype and prototype con-ditions When the prototype was given by the MDS-derived coordinates the combinedmodel did not t signi cantly better than the standard GCM 2 lnL 5 57127 c2(2) 5 021When the prototype was given by the average of the old exemplars the combined model did t signi cantly better than the standard GCM 2 lnL 5 56819 c2(2) 5 708 withy(no prototype) 5 3408 and y(prototype) 5 1034 However note that the improvement in t was quite small Moreover if anything we expected to nd greater use of prototypeinformation in the prototype conditions rather the y term was greater in the no prototypecondition

Discussion

In Experiment 1 we were able to empirically document an extreme prototype enhancementeffect in which category prototypes were classi ed more accurately than other categoryinstances This nding contrasts with most other studies using the dot-pattern paradigm inwhich the category prototypes were typically classi ed with intermediate accuracy relativeto individual training instances of a category The ability of exemplar-based models such as


the GCM to account for extreme prototype enhancement hinges on the psychologicalrepresentation of the category prototypes depending on whether these physical centraltendencies are represented as psychological central tendencies or as psychological extremepoints

MDS analyses of participantsrsquo similarity ratings revealed that the prototypes wererepresented as extreme points in the psychological space relative to the categoryinstances This result is informative because the typical assumption expressed in thecategorization literature is that the physical manipulation of prototypes in generatingcategory instances has a fairly direct mapping onto the psychological representations ofthose prototypes and category instances (eg McLaren Bennet Guttman-Nahir Kimamp Mackintosh 1995)

The GCM can qualitatively predict extreme prototype enhancement when prototypesare represented as extreme points in the psychological space We were also able todemonstrate that the GCM could provide a good quantitative account of the observed cate-gorization data without needing to include adjunct prototype abstraction processes as wellEven in conditions in which the category prototypes were familiar shapes people stillseemed to rely on exemplar information for making categorization decisions

EXPERIMENT 2

The goal of Experiment 2 was to nd additional evidence for extreme prototype enhance-ment and for extreme point prototype representations using novel prototypes To ldquoinducerdquoextreme point representations for novel prototypes we chose to constrain the categoryprototypes to have bilateral vertical symmetry (see Figure 7) Symmetry in a variety offorms is pervasive in the natural world (Weyl 1952) and people are extremely sensitive to

Figure 6 The left panel displays Prototype-A and the right panel displays Prototype-B predicted categorizationaccuracies collapsed across individual stimuli for prototypes old items and new items in Experiment 1 The noprototype condition is indicated by open triangles and the prototype condition is indicated by lled circles


symmetry especially around the vertical axis (eg Baylis amp Driver 1995 Bornstein ampKrinsky 1985 Wagemans 1993 Wagemans Van Gool amp drsquoYdewalle 1992) Moreovercanonical forms of objects such as leaves crystals or sea shells are often pictorially repre-sented in nature eld guides with near perfect symmetry most people would agree that suchperfect natural forms are the prototypes of those categories

Method

Participants

Participants were 118 undergraduates who received course credit All participants were testedindividually

Stimuli

Three prototype patterns of 10 dots each were created each was constrained to be symmetricabout the vertical axis Five points were randomly located on the left side of the pattern and ve pointswere mirrored on the right side of the pattern One dot of a pair was rst randomly located in the30 3 30 grid the second dot was located having the same vertical coordinate and the negative hori-zontal coordinate For example if the rst point had a grid location of (1012) the second point had agrid location of (2 1012) A constraint was imposed that all dots be at least three units apart Ninemoderate-level distortions (6 bitsdot) of each of these three prototypes were created six were desig-nated as old training patterns Figure 7 displays the three prototype patterns along with an exampledistortion

Procedure

All procedural details were identical to those used in Experiment 1 except that ve training blockswere used

Figure 7 The top part of the gure shows the three prototype patterns for Categories A B and C respectivelyused in Experiment 2 Under each prototype is an example of a moderate-level distortion created from that pattern


Results


The observed categorization response probabilities for each pattern in the no prototypeand prototype conditions are shown in Table 3 A two-way repeated measures ANOVA wasconducted on the accuracy data with no prototype vs prototype as a between-subjectsfactor and type (prototype old or new) as a within-subjects factor A signi cant main effectof type was found F(2 232) 5 2712 MSE 5 001 planned comparisons revealed that oldpatterns were categorized more accurately than prototypes and prototypes were categorizedmore accurately than new patterns No main effect or interactions involving no prototypevs prototype were signi cant Inspection of Table 3 reveals that the effect of type dependsstrongly on the category The prototypes were classi ed better than the old items inCategories A and B but were classi ed worse than the old items in Category C Figure 8summarizes the observed categorization accuracy for prototypes old items and new itemsfor Categories A and B (as discussed later the MDS solution revealed a degenerate psycho-logical con guration for the patterns belonging to Category C so for illustrative purposesdata from Category C items were not included in the gure) The goal in the theoreticalanalyses will be to attempt quantitative as well as qualitative accounts of this pattern ofresults by the GCM prototype and mixed models


The average similarity matrices were used as input to the INDSCAL scaling model toderive a six-dimensional MDS solution (see Appendix B) The overall t of the INDSCAL-derived distances to the observed similarity ratings was quite good (stress 5 068r2 5 949) The ts of the INDSCAL solution to the no prototype and prototype groupsindividually were quite good as well (no prototype stress 5 073 r2 5 942 prototype stress5 063 r2 5 956)5

Figure 9 displays the rst three dimensions of the MDS solution The Category A proto-type clearly exhibits an extreme-point representation and the Category B prototype tendsmore towards an extreme point than a central tendency representation These impressionsare supported by calculation of the within- and between-category distance measures that weintroduced in Experiment 1 For both Categories A and B the prototypes were ranked rstin terms of the composite measure DTOT and were both highly ranked on the componentmeasures DB and DW However the Category C prototype exhibits neither an extremepoint nor a central tendency representation (The C prototype was ranked rst on the DWmeasure last on the DB measure and intermediate on the DTOT measure) Unfortunatelyby the chance nature of the prototype distortion procedure the instances of Category C

5 Nearest neighbour analyses (Tversky amp Hutchinson 1986) were conducted on the observed similarity matricesand the derived MDS distances For the observed similarity matrices in the no prototype condition C 5 2533 andR 5 2467 and in the prototype condition C 5 2133 and R 5 2333 For the derived MDS distances in the noprototype condition C 5 2467 and R 5 2700 and in the prototype condition C 5 2267 and R 5 2600 As inExperiment 1 it does not appear that the category prototypes were nearest neighbours of many category instances


apparently were extremely similar to one another as indicated by the compact clustering ofpoints along the three dimensions shown in Figure 9 Perhaps the lack of variance in thecategory instances limited peoplersquos tendency to abstract whatever emergent dimensionsmight cause the prototypes to be conceived as extreme points





Stimulus P(A) P(B) P(C) P(A) P(B) P(C) P(A) P(B) P(C) P(A) P(B) P(C)

Category AAP 915 045 040 912 037 051 857 074 069 881 062 058A1 921 028 051 910 025 065 812 093 095 821 085 094A2 751 090 158 771 085 144 763 135 103 756 142 102A3 870 045 085 884 028 088 841 067 091 850 061 089A4 825 051 124 845 065 090 848 076 076 857 070 073A5 819 102 079 831 071 099 830 078 092 839 071 091A6 881 051 068 890 042 068 857 069 073 869 061 070Aa 661 124 215 715 088 198 721 118 161 746 100 154Ab 576 158 266 551 178 271 671 158 170 669 154 177Ac 509 113 379 568 076 356 494 192 314 510 178 312

Category BBP 141 785 073 102 833 065 156 732 112 114 808 079B1 113 746 141 105 757 138 113 765 122 106 783 111B2 062 898 040 057 893 051 102 799 099 101 809 090B3 164 763 073 130 799 071 119 783 099 108 808 085B4 040 751 209 040 763 198 105 725 171 111 733 156B5 085 701 215 073 726 201 105 688 207 111 671 218B6 130 689 181 127 706 167 122 703 174 129 699 172Ba 062 751 186 071 768 161 119 724 157 127 722 152Bb 130 735 136 124 757 119 167 686 147 141 741 118Bc 085 729 186 082 737 181 138 702 160 138 718 144

Category CCP 305 215 480 274 172 554 142 222 636 122 167 712C1 141 147 712 150 122 729 088 112 800 092 095 814C2 141 107 751 130 099 771 091 099 811 099 088 814C3 096 181 723 110 147 743 098 132 770 098 109 793C4 136 102 763 138 096 766 096 121 783 110 111 779C5 085 113 802 073 105 822 090 102 808 101 098 800C6 073 141 785 096 107 797 081 116 803 087 100 812Ca 073 158 768 090 153 757 100 150 750 098 120 781Cb 136 147 718 170 153 678 119 152 729 117 124 759Cc 170 096 735 127 099 774 142 122 735 147 107 746

Note AP 5 prototype of Category A A1 5 old instance of Category A Aa 5 new instance of Category A



As in Experiment 1 the full version of the GCM had sixteen parameters eight for theprototype and eight for the no prototype condition ( ve free attention weights wm two freeresponse biases bJ and one free sensitivity parameter c in each condition g could be setequal to one without much in uencing the t of the model) The predicted response prob-abilities for each individual stimulus from all three categories for the full model are given inTable 3 the best- tting parameters and t values are given in Table 4 (GCMndash16) TheGCM t quite well accounting for 978 of the variance in the observed data in Table 3(predictions for just Categories A and B are summarized in Figure 8) For Categories A andB the model predicted correctly the prototypes to be classi ed more accurately than the olditems and for Category C the model predicted correctly the prototypes to be classi ed lessaccurately than the old items

Figure 8 The upper left panel displays observed categorization accuracy the upper right panel displays GCMpredicted categorization accuracy the lower left panel displays Prototype-A predicted categorization accuracy andthe lower right panel displays Prototype-B predicted categorization accuracy collapsed across individual stimuli forprototypes old items and new items in Experiment 2 (only categorization accuracy for Categories A and B areincluded) The no prototype condition is indicated by open triangles and the prototype condition is indicated by lled circles


Because no statistical difference was found between the prototype and no prototype con-ditions we expected that restricted versions that constrained parameters to be the same inboth conditions would also t the data quite well First an eight-parameter version was tted to the data in which all eight parameters were constrained to be the same in both con-ditions (GCMndash8) This model did not t the data signi cantly worse than the full versionof the GCM c2(8) 5 586 p 10 Second a one-parameter version with attentionweights set equal to the INDSCAL weights from the MDS solution equal biases for eachcategory and the same sensitivity parameter for both conditions tted the data quite wellaccounting for 969 of the variance however this model did t signi cantly worse thanthe full version c2(15) 5 4190 p 001

The two versions of the prototype model were also fitted to the observed data Bestfitting parameters and fit values for both models are given in Table 4 Prototype-A whichassumes the MDS coordinates of the prototypes fitted the data quite well accountingfor 955 of the variance in the observed data (predictions for just Categories A and Bare summarized in Figure 8) However note that this 16-parameter prototype modelfitted the data worse than the 1-parameter version of the GCM Furthermore thisversion of the prototype model is not the same version of the prototype model that faredwell in Experiment 1 (in that experiment it was Prototype-B that provided a reasonablygood fit) Thus considerations of parsimony favour the exemplar-based interpretation ofthe data

Prototype-B which assumes the prototypes to be central tendencies of the old exemplars tted the data quite poorly accounting for only 818 of the variance (predictions for justCategories A and B are summarized in Figure 8) This model performed most poorly in theprototype condition accounting for only 679 of the variance (note that the abstractedprototype in the prototype condition was assumed to be an average of the old distortions and

Figure 9 The left panel displays Dimensions 1 and 2 and the right panel displays Dimensions 2 and 3 of thesix-dimensional MDS solution given in Appendix B for Experiment 2 Category A exemplars are indicated bytriangles Category B exemplars are indicated by the squares and Category C exemplars are indicated by circlesNumbers 1ndash6 indicate old exemplars letters andashc indicate new exemplars and P indicates the prototype of eachcategory

218


Experiment 2


No prototype c 1856 1890 1875 1291 1329w1 354 344 707 521 516w2 320 337 600 479 441w3 000 000 158 000 000w4 007 006 154 000 027w5 156 171 133 000 002w6 163 142 127 000 015bA 315 324 333 402 345bB 315 317 333 344 314bC 371 359 333 255 342

Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965

Prototype c 1933 1890 1875 1365 2561w1 329 344 703 486 331w2 353 337 613 514 351w3 000 000 161 000 000w4 002 006 156 000 169w5 186 171 146 000 102w6 130 142 126 000 047bA 333 324 333 402 245bB 319 317 333 346 394bC 347 359 333 253 362

Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679


Note Var 5 variance accounted for (in percentages) c 5 general sensitivity parameterwm 5 attention weight given to dimension m bj 5 bias for making category j response2 lnL 5 negative value of log-likelihood RMSE 5 root mean squared error betweenobserved and predicted categorization probabilities GCM-16 5 full parameterized GCMGCM-8 5 GCM constrained with weights biases and sensitivities common between theno prototype and prototype conditions GCM-1 5 GCM constrained with weights equalto INDSCAL weights equal category biases and sensitivities common between the noprototype and prototype conditions Prototype-A 5 prototype model using MDS-derivedldquoprototypesrdquo Prototype-B 5 prototype model using average of old exemplars Underlinedvalues are constrained parameters


the presented prototype) The combined exemplar-plus-prototype model was also investi-gated As in Experiment 1 additional weighted prototypes were assumed to be representedin memory Neither a model that assumed the MDS coordinates of the prototype2 lnL 5 45039 c2(2) 5 00001 nor a model that assumed the average prototype2 lnL 5 44867 c2(2) 5 299 tted signi cantly better than the full version of the GCM

Discussion

In Experiment 2 we found that novel vertically symmetric dot-pattern prototypes gave riseto an extreme prototype enhancement effect (for two of the categories tested) and that theprototypes tended to be represented as extreme points rather than as central tendenciesAgain although the prototypes were indeed physical central tendencies of their distortionsthey did not emerge as psychological central tendencies Thus the nding that dot-patternprototypes may often have extreme-point representations appears to have some generality

Although the category prototypes tended to have a ldquospecialrdquo representational status asrelative extremes in the psychological space they did not have a special status with regardsto making categorization decisions Extending the theoretical results of Experiment 1 apure exemplar model (the GCM) provided a good account of the categorization datawhereas simple prototype models fared less well Furthermore the combined exemplar-plus-prototype models which supplement exemplar generalization with prototype abstrac-tion did not provide a signi cantly better account of the data than did the GCM

EXPERIMENT 3

The nal experiment in this study examined a nding of extreme prototype enhancementreported by McLaren et al (1995) These researchers had participants learn two categoriesof checkerboard patterns that were constructed so that the prototypes were physical centraltendencies of the category instances (see also McLaren 1997 McLaren Leevers ampMackintosh 1994 Wills amp McLaren 1998) The prototype of Category A was a randomcon guration of white and black squares as shown in Figure 10 The prototype of CategoryB was created by randomly switching a relatively large proportion of squares of theCategory A prototype from white to black or from black to white as shown in Figure 10From these category prototypes two types of instance were generated with differingrelationships to the prototype of the other category Close patterns were generated byswitching approximately 10 of the unique squares of one prototype to the colour of theother prototype This procedure created patterns that were physically more similar than wasthe prototype to members of the contrast category (an example from each prototype is shownin Figure 10) Far patterns were generated by switching approximately 10 of the squarescommon to both prototypes to the opposite colour This procedure created patterns thatwere physically more dissimilar than was the prototype to members of the contrast category(an example from each prototype is shown in Figure 10) Thus physically this procedureproduces patterns with roughly the following schematic arrangement

FAR-A PROTOTYPE-A CLOSE-A CLOSE-B PROTOTYPE-B FAR-B


McLaren et al (1995) found the category prototypes to be the best classi ed items a clearcase of extreme prototype enhancement Indeed McLaren et al conducted formal theo-retical analyses involving the GCM that assumed that the distance between checkerboardpatterns was directly related to the number of mismatching squares computed on a city-block metric In these analyses McLaren et al demonstrated that the GCM failed toaccount for the data and argued that the results posed a serious challenge to exemplarmodels in general Lamberts (1996) subsequently demonstrated that by making alternativemetric assumptions for calculating distances among the physically de ned patternsexemplar models such as the GCM could account for McLaren et alrsquos data

We hypothesized however that an even more fundamental issue might also be involvedIn particular the con guration of checkerboard-pattern stimuli in psychological space maybe quite different from the physical arrangement de ned by the number of overlappingwhite and black squares Once again although the prototypes are central tendencies in thephysically de ned space they may exist as extreme points in a psychologically de ned space

McLaren et al (1995) acknowledged the possibility that the individual black and whitesquares might give rise to higher order emergent features in which the prototypes were nolonger central tendencies To explore this possibility they tested a small subset of partici-pants in a separate identi cation training session In this session participants learned aunique label for two prototypes two close patterns and two far patterns the identi cationconfusion data were then used as input for a multidimensional scaling analysis AlthoughMcLaren et al found some evidence for a psychological arrangement of the stimuli com-parable to the physical arrangement they reported only a one-dimensional scaling solutionIt seems much more likely that these relatively complex checkerboard patterns should give

Figure 10 Stimulus used in Experiment 3 The top part of the gure shows the Category A prototype a closeexample and a far example The bottom part of the gure shows the Category B prototype a close example and afar example


rise to higher dimensional psychological scaling solutions Moreover the fact that only sixof the original patterns were examined in the McLaren et al study potentially limits thegeneralizability of their results

Our goal was to replicate the essential features of the McLaren et al experiments whilealso conducting a comprehensive similarity rating task in which similarities among allpatterns were measured Unlike in the McLaren et al study all participants in our studylearned the same set of stimuli By using this method the psychological scaling solutionre ects an arrangement of the complete set of actual stimuli that were learned rather thanan approximate arrangement of a subset of a general class of stimuli As was the case inExperiments 1 and 2 we predicted that extreme prototype enhancement would be observedand that the prototypes patterns would emerge in the scaling solution as extreme pointsrather than central tendencies

Method

Participants


Stimuli

Unlike the original McLaren et al (1995) study all participants in this experiment learned to cate-gorize the same set of patterns Therefore the stimulus generation procedure which follows wasperformed only once before the experiment was conducted

The stimuli were 16 3 16 checkerboard patterns of white and black squares 4 pixels on a side (seeFigure 10 for examples) The stimuli were presented at a video resolution of 640 3 480 on 14-in com-puter monitors The prototype for Category A was a completely random pattern of white and blacksquares The prototype for Category B was created by randomly changing 6 squares per row fromwhite to black or from black to white (96 squares changed) A xed set of 16 ldquocloserdquo patterns was gen-erated from each prototype by switching a randomly selected 25 of those 96 squares not in commonwith the other prototype from white to black or from black to white This procedure produces patternsthat have greater physical overlap with the other prototype than does their prototype (although theystill have physically more overlap with their own prototype) For each category 12 of the closepatterns were designated training patterns and the remaining 4 were designated transfer patterns A xed set of 4 ldquofarrdquo patterns was generated from each prototype by switching a randomly selected 25of the 160 squares in common with the other prototype from white to black or from black to whiteThis procedure produces patterns that have less physical overlap with the other prototype than doestheir prototype

Procedure

A standard category learningtransfer paradigm was used During training participants learned toclassify the 12 designated close patterns from each category Each of these 24 close patterns was pre-sented once per block for ve training blocks Corrective feedback was supplied for 3 s after everyresponse After an intertrial interval (ITI) of 500 ms the next pattern was displayed Participants wereurged to respond more quickly if their response times exceeded 4 s The assignment of category to


response key was randomized for every participant Responses were made by pressing labelled keys ona computer keyboard

During transfer all 42 patterns were tested (12 old close 4 new close 4 far and the prototype fromeach category) Each pattern was presented once per block for two transfer blocks No corrective feed-back was provided The computer merely reported ldquoOKrdquo for 2 s to indicate that a response had beenrecorded After an ITI of 500 ms the next pattern was displayed

Participants also rated the pairwise similarities among patterns Because of the large number ofpossible pairs each participant rated only one third of them (randomly selected for each participant)On each trial two patterns were presented side by side Participants rated similarities using a 10-pointscale (1 5 very dissimilar 10 5 very similar)

Results


Of the 133 participants 25 were removed for failing to reach a fairly lax criterion of atleast 60 correct on the last block of training The observed categorization probabilities foreach pattern in the transfer phase are shown in Table 5 As summarized in Figure 11 onaverage the prototypes were classi ed the best followed by the old close patterns the farpatterns and the new close patterns A one-way repeated measures ANOVA revealed a sig-ni cant main effect of stimulus type F(3 321) 5 6040 MSE 5 001 Planned comparisonsrevealed that prototype classi cation accuracy was signi cantly the best old close and farpattern classi cation accuracies were on average not signi cantly different from oneanother and new close pattern classi cation accuracy was on average the worst Extremeprototype enhancement was observed in which the prototypes were classi ed better thanany other pattern

Overall these results largely replicate those of McLaren et al (1995) The prototypeswere classi ed better than any other patterns even though the statistical distortion algo-rithm produced prototypes that were roughly physical central tendencies of the categoryinstances (Unlike McLaren et al however we did not nd that new far patterns wereclassi ed signi cantly better overall than old close patterns although this result appears todepend on the speci c category being consideredmdashsee Figure 11) The key question now isto examine the psychological scaling solution for these checkerboard patterns and to use itin conjunction with the GCM to test the exemplar-model predictions


The average similarity rating matrix was used as input to the KYST scaling model(Kruskal Young amp Seery 1973)6 The resulting scaling solution had a stress of 090 The rst two dimensions of the solution which are most informative are shown in Figure 12and the complete scaling solution is reported in Appendix C Along Dimension 1 the rstprincipal component of the KYST solution the category prototypes have extreme values

6 Because each subject only rated 13 of the possible pairs of checkerboard patterns an average similarityrating matrix had to be used for the scaling analyses Because only a single experimental condition was testedINDSCAL could not be used in the present study


and are not central tendencies of the old patterns However the prototypes had roughlyequal values along the other dimensions and calculations of the within- and between-category distance measures that we introduced in Experiment 1 produced intermediatevalues for the prototypes Although the prototypes for these grid patterns are not strictlyextreme points by our criteria proposed earlier the extreme values rather than centraltendencies along Dimension 1 may allow the exemplar model to adequately predict theextreme prototype enhancement effects and the other aspects of the categorization data


A version of the GCM with only two free parameters was tted to the observed dataParameters were a free sensitivity parameter c and a free response scaling term g UnlikeINDSCAL the orientation of the psychological dimensions is arbitrary in the KYSTmodel so incorporating the dimension-weighting parameters was not possible Thepredicted categorization response accuracies for the model are shown in Table 5 and are

TABLE 5Observed and predicted categorization response accuracy for

Experiment 3

Category A Category B

Stimulus Observed Predicted Stimulus Observed Predicted

PA 0944 0898 PB 0935 0882CA1 0796 0803 CB1 0620 0785CA2 0870 0745 CB2 0857 0836CA3 0833 0799 CB3 0838 0860CA4 0843 0863 CB4 0829 0842CA5 0796 0772 CB5 0685 0742CA6 0926 0888 CB6 0759 0750CA7 0875 0884 CB7 0847 0824CA8 0880 0855 CB8 0866 0874CA9 0815 0821 CB9 0810 0829CA10 0810 0838 CB10 0690 0758CA11 0866 0841 CB11 0759 0721CA12 0893 0845 CB12 0755 0816Ca1 0681 0727 Cb1 0722 0760Ca2 0815 0872 Cb2 0625 0776Ca3 0852 0802 Cb3 0745 0795Ca4 0875 0819 Cb4 0574 0576FA1 0662 0784 FB1 0884 0748FA2 0713 0726 FB2 0810 0751FA3 0759 0790 FB3 0894 0767FA4 0722 0767 FB4 0861 0806

Note CA1 5 old close pattern of Category A Ca1 5 new close patternof Category A FA1 5 far pattern of Category A PA 5 prototype ofCategory A


summarized in Figure 11 This two-parameter version of the GCM t fairly well account-ing for 959 of the variance in the observed data 2 lnL 5 22811 RMSE 5 0063 Thebest tting parameter values were c 5 1336 and g 5 4228

Most importantly the prototypes were predicted to be the best classi ed items as wasobserved Contrary to McLaren et al (1995) when combined with the multidimensionalscaling solution derived from the similarity ratings participants made a pure exemplarmodel was able to account for the qualitative nding of extreme prototype enhancementeffects observed in the present experiment

Inspection of Figure 11 does reveal that the predicted prototype classi cation prob-abilities are somewhat smaller than observed As in the previous experiments we examineda combined exemplar-plus-prototype model in which evidence for a given category wasequal to the summed similarity of an item to all category exemplars plus a weighted simi-larity to the category prototype The combined model did provide a signi cantly better tto the observed data than did the pure exemplar model c2(1) 5 2247 accounting for 966of the variance 2 lnL 5 20564 RMSE 5 0058 Although this mixed model somewhatmore accurately predicted the prototype classi cation probabilities it did not providenoticeably improved ts for the other stimulus types

Discussion

In this experiment we replicated the extreme prototype enhancement effects observed in aseries of studies by McLaren et al (1995)7 From these results McLaren et al argued thatldquoan exemplar theory of the type considered here is constrained to predict that far exemplarswill always be categorized at least as well as prototypesrdquo (p 671 but see Lamberts 1996)

Figure 11 Observed ( lled circles) and GCM predicted (open triangles) categorization accuracy collapsedacross individual stimuli for prototypes close (old) close (new) and far patterns in Experiment 3

7 We performed two additional replications and extensions of the McLaren et al (1995) study using differentstimulus sets These studies provided additional converging evidence for the results reported here In both studiesextreme prototype enhancement was observed and the prototypes were psychological extreme points Moreoverin one of the two studies the far stimuli were categorized signi cantly more accurately than the close stimuli repli-cating the original McLaren et al results


McLaren et alrsquos claim however assumes that the psychological space for the stimuli has adirect correspondence with the physical one In particular they assume that in addition tobeing physically de ned central tendencies the prototypes are psychological centraltendencies as well By contrast our MDS analyses revealed that the prototypes gave rise toextreme values along one of the emergent psychological dimensions Furthermore when aparticular exemplar model the GCM was used in combination with this derived MDSsolution it provided a reasonably good account of the extreme prototype enhancement thatwas observed (but see McLaren 1997 and Wills amp McLaren 1998 for other experimentalsituations involving these checkerboard stimuli that may involve prototype abstractionfollowing categorization or preexposure)

An assumption made by McLaren et al (1995) was that each 16 3 16 checkerboardpattern was represented within a 256 dimensional space In this space each psychologicaldimension corresponds directly to the primitive feature of some given square being white orblack We contend that this assumption is probably unjusti ed as our MDS analyses revealInspecting the patterns in Figure 10 it seems entirely reasonable to suppose that partici-pants might extract higher level con gurations of black and white squares Clearly nosimple feature-based account could capture properties such as the degree of symmetry in apattern or the ldquoclumpinessrdquo of a pattern Also depending on whether white is foregroundor background different kinds of con gurations can emerge For example informal dis-cussions with a subset of participants revealed that some noticed con gurations such as thewhite ldquostaircaserdquo in the upper left corner of the Category A prototype or the black ldquoislandrdquoin the upper right hand corner of the Category A prototype The extreme value of theprototypes along Dimension 1 may in part re ect a fairly sophisticated feature creationprocess which a simple elemental approach to understanding psychological dimensionsoverlooks (see Schyns Goldstone amp Thibaut 1998)

Figure 12 Dimensions 1 and 2 of MDS solutions for Experiment 3 Category A patterns are indicated by circlesand Category B patterns are indicated by squares White symbols indicate old close patterns grey symbols indicatenew close patterns black symbols indicate far patterns and Prsquos indicate prototypes


GENERAL DISCUSSION

The present article was initially motivated by a set of results from the classic dot-patterncategorization paradigm (Homa 1984 Posner amp Keele 1968 1970) In such experimentsprototypes are often classi ed as well as and often times better than old training patternsAlthough these results were originally taken as solid evidence for the existence of a proto-type abstraction process these effects have since been shown to be entirely consistent withexemplar models of categorization (eg Busemeyer et al 1984 Hintzman 1986 Hintzmanamp Ludlam 1980 Nosofsky 1988 Shin amp Nosofsky 1992) Although prototype enhance-ment effects are regularly observed in such studies they are typically not very large andalmost always at least some of the individual old exemplars are classi ed better than theprototypes Whereas innumerable dot-pattern studies have reported prototype enhance-ment effects in which the prototypes are classi ed as well as or better than the average of theold distortions few have reported extreme prototype enhancement effects in which theprototypes are classi ed better than all instances of a category8 As described earlier ndingextreme prototype enhancement for prototypes that are psychological central tendencies ofold distortions may prove extremely dif cult if not impossible for an exemplar model suchas the GCM to account formdashsuch a nding could potentially falsify the GCM

Therefore our goal in developing these studies was to attempt to empirically createextreme prototype enhancement effects that might prove exceptionally challenging toexemplar models In Experiment 1 we did so by using dot-pattern prototypes that weresimpli ed representations of highly familiar objects This method is precisely the one thatPosner and Keele used in their very rst studies It seemed reasonable to hypothesize that ifextreme prototype enhancement were ever to be found in a dot-pattern paradigm it would befound using such prototypes In Experiment 2 we used symmetric dot-pattern prototypesAgain it seemed reasonable that such prototype patterns might exhibit a special statusvis-agrave-vis their distortions which would cause them to be classi ed more accurately Finally inExperiment 3 we borrowed an experimental paradigm recently reported by McLaren et al(1995) using checkerboard patterns of white and black squares that was also shown to giverise to very high classi cation accuracy for the category prototypes In Experiments 1 and 3we found evidence for extreme prototype enhancementmdashcategory prototypes were classi edbetter than any of the old category examples on which participants were trained Tendenciesfor extreme prototype enhancement were also observed in Experiment 2

Recall these results pose a serious challenge to exemplar models only if the prototypeswhich are physical central tendencies of category instances are psychological centraltendencies as well For the dot-pattern stimuli used in Experiments 1 and 2 and for thecheckerboard patterns used in Experiment 3 the categories were constructed in such a waythat the prototypes were physical central tendencies of the category instances That is acomposite image made by averaging together all of the category instances looks almost iden-tical to the category prototype The fact that the prototypes are physical central tendencies

8 Unfortunately only quite recently have studies reported classi cation probabilities for all of the individualstimuli used in dot-pattern experiments Typically only average classi cation accuracies for various types ofstimulus (prototype old distortion new distortion) were reported making it impossible to assess whether extremeprototype enhancement was observed


does not necessarily imply that they are psychological central tendencies as well Forstimuli with clearly de ned psychological dimensions such as semicircles of varying sizescontaining radial lines of varying angles (eg Nosofsky 1986 Shepard 1964) tones vary-ing in loudness and pitch (Melara amp Marks 1990) or perhaps even schematic faces varyingthe location of facial features (eg Nosofsky 1991) a fairly direct mapping betweenphysical and psychological dimensions may exist However the mappings between physicalproperties and psychological dimensions of fairly complex stimuli such as arti cial dot-patterns or checkerboard patterns and perhaps more natural stimuli are not so clearlyde ned (Hock et al 1988 Shin amp Nosofsky 1992)

To determine the location of the category prototypes relative to the other categoryinstances participants provided similarity ratings between pairs of stimuli These ratingswere then subjected to a multidimensional scaling analysis to recover the underlying psycho-logical space of the stimuli used in each experiment Across all three experiments thescaling analyses revealed the category prototypes to be relative extremes in the psychologicalspace rather than central tendencies Across all three experiments when coupled with thisobtained psychological scaling solution one particular exemplar model the GCM couldboth qualitatively and quantitatively account for the observed extreme prototype enhance-ment effects as well as other aspects of the observed data By contrast various types ofprototype model either failed to account for the observed data or the best- tting version ofthe prototype model was not of the same type in different experiments In addition little orno evidence was found that pointed to the need to supplement an exemplar model with anadditional prototype abstraction process

It should be stressed that the MDS solutions incorporated in the GCM modelling werederived from similarity judgements following category learning Although research on thisissue is needed we think it is likely that the modelling approach would be considerably lesssuccessful if the similarity judgements were obtained in a completely independent contextThe present types of complex dot-pattern can probably be coded and represented in a varietyof highly exible ways Combined with the fact that similarity judgements are themselveshighly context dependent (Medin et al 1993 Tversky 1977) this exibility of codingsuggests a need to derive the MDS representation in the context of the category-learningsituation The most challenging future goal is to understand how the psychological repre-sentations are formedmdashhow and why did the prototypes come to be represented psycho-logically as extreme points Much of the work in categorization has assumedrepresentations to remain relatively unchanged with experience apart from changes in howdimensions are selectively attended (Kruschke 1992 Nosofsky 1986) Recent work how-ever has begun to suggest that an important component of categorization involves a morecomplex form of perceptual learningmdashnot only must a person learn what features are diag-nostic they must also create new diagnostic features (eg Lesgold et al 1988 Schyns et al1998 Schyns amp Murphy 1991 Schyns amp Rodet 1997) Rather than assume that theperceptual system provides a set of xed features to be used as inputs to higher level cate-gorization processes Schyns et al (1998) have suggested that exible functional featuresmay be created as part of the process of category learning A reasonable hypothesis is thatthe extreme-point prototype representations in the present studies arose from theemergence of these diagnostic functional features in the context of learning these particu-lar dot-pattern categories One possible starting point for incorporating perceptual learning


mechanisms into theories of categorization might be an associative model proposed byMcLaren and colleagues (McLaren Kaye amp Mackintosh 1989 McLaren 1997)

Given the nature of the dot-pattern stimuli and the exibility of coding that arises itseemed necessary to collect the similarity judgement data only following the completion ofcategory learning A potential concern that arises is that the prototypes emerged as extremepoints in the similarity representations because they were judged as the best examples oftheir categories According to this view the prototypes are indeed central tendencies in theldquotruerdquo psychological space in which the patterns are embedded Because observers mayallow judged category goodness to in uence their similarity ratings however the MDSanalyses of the similarity data revealed a distorted psychological space To the extent thatfunctional features are indeed created by the act of categorizing as hypothesized by Schynsand his colleagues (Schyns et al 1998 Schyns amp Rodet 1997) these alternatives becomeextraordinarily dif cult to disentangle Nevertheless we can point to some indirect evidencethat poses problems for the prototype-as-central-tendency view First consider category-learning situations in which stimuli vary along a few salient psychological dimensions andwhere creation of new functional features tends not to take place Examples include learn-ing to classify colours varying in their brightness and saturation or simple geometric formsvarying in their size and angle of orientation If the view is that the central tendency of thedistribution is privileged and similarity judgements among patterns simply follow judgedcategory goodness then when using such stimuli the central tendency should be the bestclassi ed item and an MDS analysis based on similarity data should place it at the extremesof the category distribution A vast literature however indicates that such is not the caseInstead when stimuli with these types of clear-cut psychological dimension are used thecentral tendency tends not to be among the most accurately or rapidly classi ed objects andscaling solutions based on similarity judgement data place it in the centre of the categorydistribution Thus although we cannot rule out the possibility that in the present experi-ments the prototype was a ldquotruerdquo psychological central tendency and that the similarityjudgements led to a distorted representation this view appears to be severely limited in itsgenerality

The bottom line is that we have provided evidence in favour of one particular theoreticalapproach to understanding extreme prototype enhancement effects that are observed forstimuli composed of highly exible and complex psychological dimensions In thisapproach the prototypes are to be interpreted as extreme points in the psychological simi-larity space that emerges from categorization experience The MDS-based exemplar modelhas been highly successful at accounting for details of classi cation performance in numer-ous domains involving stimuli varying along clear-cut and salient psychological dimensionsThe present research demonstrates generality for the approach by using the same types ofsimilarity scaling method as in this previous work but where the nature of the psychologicaldimensions that compose the objects is far more exible and complex It is an open questionwhether or not alternative models can be developed that match the generality and precisionof this MDS-based exemplar-modelling approach


REFERENCES

Anderson JR (1980) Cognitive psychology and its implications San Francisco WH Freeman and CompanyAshby FG amp Gott RE (1988) Decision rules in the perception and categorization of multidimensional

stimuli Journal of Experimental Psychology Learning Memory and Cognition 14 33ndash53Barresi J Robbins D amp Shain K (1975) Role of distinctive features in the abstraction of related con-

cepts Journal of Experimental Psychology Human Learning and Memory 104 360ndash368Barsalou LW (1985) Ideals central tendency and frequency of instantiation as determinants of graded

structure in categories Journal of Experimental Psychology Learning Memory and Cognition 11629ndash654

Barsalou LW (1991) Deriving categories to achieve goals The Psychology of Learning and Motivation 271ndash64

Baylis GC amp Driver J (1995) Obligatory edge assignment in vision The role of gure and partsegmentation in symmetry detection Journal of Experimental Psychology Human Perception andPerformance 21 1323ndash1342

Bornstein MH amp Krinsky SJ (1985) Perception of symmetry in infancy The salience of verticalsymmetry and the perception of pattern wholes Journal of Experimental Child Psychology 39 1ndash19

Breen TJ amp Schvaneveldt RW (1986) Classi cation of empirically derived prototypes as a function ofcategory experience Memory amp Cognition 14 313ndash320

Busemeyer JR Dewey GI amp Medin DL (1984) Evaluation of exemplar-based generalization and theabstraction of categorization information Journal of Experimental Psychology Learning Memory andCognition 10 638ndash648

Carroll JD amp Wish M (1974) Models and methods for three-way multidimensional scaling In DHKrantz RC Atkinson RD Luce amp P Suppes (Eds) Contemporary developments in mathematicalpsychology (Vol 2 pp 57ndash105) San Francisco Freeman

Goldstone RL (1993) Evidence for interrelated and isolated concepts from prototype and caricatureclassi cations Proceedings of the Fifteenth Annual Conference of the Cognitive Science Society(pp 498ndash503) Hillsdale NJ Lawrence Erlbaum Associates Inc

Goldstone RL (1996) Isolated and interrelated concepts Memory amp Cognition 24 608ndash628Hintzman DL (1986) ldquoSchema abstractionrdquo in a multiple-trace memory model Psychological Review 93

411ndash428Hintzman DL amp Ludlam G (1980) Differential forgetting of prototypes and old instances Simulations

by an exemplar-based classi cation model Memory amp Cognition 8 378ndash382Hock HS Tromley C amp Polmann L (1988) Perceptual units in the acquisition of visual categories

Journal of Experimental Psychology Learning Memory and Cognition 14 75ndash84Homa D (1978) Abstraction of ill-de ned form Journal of Experimental Psychology Human Learning and

Memory 4 407ndash416Homa D (1984) On the nature of categories Psychology of Learning and Motivation 18 49ndash94Homa D amp Chambliss D (1975) The relative contribution of common and distinctive information on the

abstraction of ill-de ned categories Journal of Experimental Psychology Human Learning and Memory104 351ndash359

Homa D Cross J Cornell D Goldman D amp Schwartz S (1973) Prototype abstraction and classi ca-tion of new instances as a function of number of instances de ning the prototype Journal ofExperimental Psychology 101 116ndash122

Homa D Dunbar S amp Nohre L (1991) Instance frequency categorization and the modulating effectsof experience Journal of Experimental Psychology Learning Memory and Cognition 17 444ndash458

Homa D Goldhardt B Burruel-Homa L amp Smith JC (1993) In uence of manipulated categoryknowledge on prototype classi cation and recognition Memory amp Cognition 21 529ndash538

Homa D Rhoads D amp Chambliss D (1979) Evolution of conceptual structure Journal of ExperimentalPsychology Human Learning and Memory 5 11ndash23

Homa D Sterling S amp Trepel L (1981) Limitations of exemplar-based generalization and the abstrac-tion of categorical information Journal of Experimental Psychology Learning Memory and Cognition 7418ndash439


Homa D amp Vosburgh R (1976) Category breadth and the abstraction of categorical information Journalof Experimental Psychology Human Learning and Memory 2 322ndash330

Knapp AG amp Anderson JA (1984) Theory of categorization based on distributed memory storageJournal of Experimental Psychology Learning Memory and Cognition 10 616ndash637

Knowlton BJ amp Squire LR (1993) The learning of categories Parallel brain systems for item memoryand category knowledge Science 262 1747ndash1749

Kolodny JA (1994) Memory processes in classi cation learning An investigation of amnesic performancein categorization of dot patterns Psychological Science 5 164ndash169

Kruschke J (1992) ALCOVE An exemplar-based connectionist model of category learning PsychologicalReview 99 22ndash44

Kruskal JB Young FW amp Seery JB (1973) How to use KYST A very exible program to do multi-dimensional scaling and unfolding Murray Hill NJ Bell Telephone Laboratories

Lamberts K (1996) Exemplar models and prototype effects in similarity-based categorization Journal ofExperimental Psychology Learning Memory and Cognition 22 1503ndash1507

Lesgold A Glaser R Rubinson H Klopfer D Feltovich P amp Wang Y (1988) Expertise in a complexskill Diagnosing X-ray pictures In MTH Chi R Glaser amp MJ Farr (Eds) The nature of expertise(pp 311ndash342) Hillsdale NJ Lawrence Erlbaum Associates Inc

Maddox WT amp Ashby FG (1993) Comparing decision bound and exemplar models of categorizationPerception amp Psychophysics 53 49ndash70

Massaro DW (1987) Speech perception by ear and eye A paradigm for psychological inquiry Hillsdale NJLawrence Erlbaum Associates Inc

Massaro DW amp Friedman D (1990) Models of integration given multiple sources of informationPsychological Review 97 225ndash252

McClelland JL amp Rumelhart DW (1985) Distributed memory and the representation of general andspeci c information Journal of Experimental Psychology General 114 159ndash188

McKinley SC amp Nosofsky RM (1995) Investigations of exemplar and decision bound models in largeill-de ned category structures Journal of Experimental Psychology Human Perception and Performance21 128ndash148

McLaren IPL (1997) Categorization and perceptual learning An analogue of the face inversion effectThe Quarterly Journal of Experimental Psychology 50A 257ndash273

McLaren IPL Bennet CH Guttman-Nahir T Kim K amp Mackintosh NJ (1995) Prototype effectsand peak shift in categorization Journal of Experimental Psychology Learning Memory and Cognition21 662ndash673

McLaren IPL Kaye H amp Mackintosh NJ (1989) An associative theory of representation of stimuliApplications to perceptual learning and latent inhibition In RGM Morris (Ed) Parallel distributedprocessingndashimplications for psychology and neurobiology Oxford Oxford University Press

McLaren IPL Leevers HL amp Mackintosh NJ (1994) Recognition categorization and perceptuallearning In C Umiltagrave amp M Moscovitch (Eds) Attention and performance XV Cambridge MA MITPress

Medin DL Goldstone RL amp Gentner D (1993) Respects for similarity Psychological Review 100254ndash278

Medin DL amp Schaffer MM (1978) Context theory of classi cation learning Psychological Review 85207ndash238

Melara RD amp Marks LR (1990) Perceptual primacy of dimensions Support for a model of dimensionalinteraction Journal of Experimental Psychology Human Perception and Performance 16 398ndash414

Metcalfe J (1982) A composite holographic associative recall model Psychological Review 89 627ndash661Metcalfe J amp Fisher RP (1986) The relation between recognition memory and classi cation learning

Memory amp Cognition 14 164ndash173Nosofsky RM (1984) Choice similarity and the context theory of classi cation Journal of Experimental

Psychology Learning Memory and Cognition 10 104ndash114Nosofsky RM (1986) Attention similarity and the identi cationndashcategorization relationship Journal of

Experimental Psychology General 115 39ndash57


Nosofsky RM (1987) Attention and learning processes in the identi cation and categorization of integralstimuli Journal of Experimental Psychology Learning Memory and Cognition 13 87ndash108

Nosofsky RM (1988) Exemplar-based accounts of relations between classi cation recognition and typi-cality Journal of Experimental Psychology Learning Memory and Cognition 14 700ndash708

Nosofsky RM (1991) Tests of an exemplar model for relating perceptual classi cation and recognitionmemory Journal of Experimental Psychology Human Perception and Performance 17 3ndash27

Nosofsky RM amp Palmeri TJ (1997) An exemplar-based random walk model of speeded classi cationPsychological Review 104 266ndash300

Nosofsky RM amp Zaki SR (1998) Dissociations between categorization and recognition in amnesic andnormal individuals An exemplar-based interpretation Psychological Science 9 247ndash255

Oden GC amp Massaro DW (1978) Integration of featural information in speech perception PsychologicalReview 85 172ndash191

Omohundro J (1981) Recognition vs classi cation of ill-de ned category exemplars Memory amp Cognition9 324ndash331

Palmeri TJ amp Flanery MA (1999) Learning about categories in the absence of training Profoundamnesia and the relationship between perceptual categorization and recognition memory PsychologicalScience 10 526ndash530

Posner MI (1969) Abstraction and the process of recognition In GH Bower amp JT Spence (Eds) Thepsychology of learning and motivation (pp 43ndash100) New York Academic Press

Posner MI Goldsmith R amp Welton KE Jr (1967) Perceived distance and the classi cation of dis-torted patterns Journal of Experimental Psychology 73 28ndash38

Posner MI amp Keele SW (1968) On the genesis of abstract ideas Journal of Experimental Psychology 77353ndash363

Posner MI amp Keele SW (1970) Retention of abstract ideas Journal of Experimental Psychology 83304ndash308

Reber PJ amp Squire LR (1997) Implicit learning in patients with Parkinsonrsquos disease Abstracts of thePsychonomic Society 2 9

Reber PJ Stark CEL amp Squire LR (1998) Cortical areas supporting category learning identi edusing functional MRI Proceedings of the National Academy of Sciences USA 95 747ndash750

Reed SK (1972) Pattern recognition and categorization Cognitive Psychology 3 382ndash407Rosch E (1975a) Cognitive reference points Cognitive Psychology 7 532ndash547Rosch E (1975b) Cognitive representations of semantic categories Journal of Experimental Psychology

General 104 192ndash233Rosch E (1978) Principles of categorization In E Rosch amp BB Lloyd (Eds) Cognition and categorization

Hillsdale NJ Lawrence Erlbaum Associates IncSchyns PG Goldstone RL amp Thibaut JP (1998) The development of features in object concepts

Behavioral and Brain Sciences 21 1ndash54Schyns PG amp Murphy GL (1991) The ontogeny of part representation in object concepts In DL

Medin (Ed) The psychology of learning and motivation (Vol 31 pp 305ndash354) San Diego CA AcademicPress

Schyns PG amp Rodet L (1997) Categorization creates functional features Journal of ExperimentalPsychology Learning Memory and Cognition 23 681ndash696

Shepard RN (1964) Attention and the metric structure of the stimulus space Journal of MathematicalPsychology 1 54ndash87

Shepard RN (1980) Multidimensional scaling tree- tting and clustering Science 210 390ndash398Shepard RN (1987) Toward a universal law of generalization for psychological science Science 237

1317ndash1323Shepard RN amp Chang JJ (1963) Stimulus generalization in the learning of classi cations Journal of

Experimental Psychology 65 94ndash102Shin HJ amp Nosofsky RM (1992) Similarity-scaling studies of dot-pattern classi cation and recognition

Journal of Experimental Psychology General 121 278ndash304Smith EE amp Medin DL (1981) Categories and concepts Cambridge MA Harvard University Press


Squire LR amp Knowlton BJ (1995) Learning about categories in the absence of memory Proceedings ofthe National Academy of Sciences USA 92 12470ndash12474

Strange W Kenney T Kessel F amp Jenkins J (1970) Abstraction over time of prototypes from dis-tortions of random dot patterns Journal of Experimental Psychology 83 508ndash510

Tversky A (1977) Features of similarity Psychological Review 84 327ndash352Tversky A amp Hutchinson JW (1986) Nearest neighbor analysis of psychological space Psychological

Review 93 3ndash22Vandierendonck A (1984) Concept learning memory and transfer of dot pattern categories Acta

Psychologica 55 71ndash88Wagemans J (1993) Skewed symmetry A nonaccidental property used to perceive visual forms Journal of

Experimental Psychology Human Perception and Performance 19 364ndash380Wagemans J Van Gool L amp drsquoYdewalle G (1992) Orientational effects and component processes in

symmetry detection Quarterly Journal of Experimental Psychology 44A 475ndash508Weyl H (1952) Symmetry Princeton NJ Princeton University PressWickens TD (1982) Models for behavior Stochastic processes in psychology New York FreemanWills AJ amp McLaren IPL (1998) Perceptual learning and free classi cation The Quarterly Journal of

Experimental Psychology 51B 235ndash270Original manuscript received 11 October 1999

Accepted revision received 22 March 2000


APPENDIX ASix-dimensional scaling solution for the dot patterns in Experiment 1

Dimension

Stimulus 1 2 3 4 5 6

TriangleTP 2106 2 0190 2 1490 1566 0151 2 0234T1 1602 2 0009 0174 0861 2 0110 2 0265T2 1687 2 0323 2 0219 0681 0051 2 1020T3 0003 2 0044 0717 0978 2 0123 2 2586T4 1428 0342 2 0076 0454 0496 0092T5 1850 2 0097 2 0401 1079 0247 2 0372T6 0964 2 0076 0316 0773 2 0978 2 0248Ta 1233 0392 0016 2 0004 0278 2 0800Tb 1095 2 0359 0067 2 0235 2 1552 2 1751Tc 1054 0586 2 0035 2 0122 0881 0402

PlusesPP 2 1082 1921 2 3199 2 0873 0432 0535P1 2 0528 1223 0041 0826 2 0745 1308P2 2 0614 0648 0726 2 0749 0815 1640P3 2 0532 0643 0061 2 0129 1941 2 0847P4 2 0733 1258 2 0101 2 1298 2 0676 0363P5 2 1107 0330 0437 0737 1919 2 0260P6 2 0791 1076 2 0436 2 1547 0465 0036Pa 2 0702 0782 0653 0701 2 1584 0385Pb 2 0713 0552 0441 0036 2 0299 2 2039Pc 2 0447 0991 0231 2 1754 1148 2 0468

FsFP 2 1200 2 2527 2 2646 2 0442 2 0022 0980F1 2 0784 2 0110 0759 1713 1706 2 0341F2 2 0457 2 0620 1490 2 1659 1459 1360F3 2 0499 2 1835 2 0044 0010 2 1127 2 0400F4 2 0292 2 0099 1129 0417 2 1428 1111F5 2 0724 2 0330 0959 0744 2 1513 0201F6 2 0756 2 2285 2 0857 2 0561 2 0848 0351Fa 2 0151 2 1489 2 0652 0736 2 0507 0890Fb 2 0687 2 0708 1130 2 0646 2 0595 1584Fc 2 0222 0357 0811 2 2293 0119 0395

Note INDSCAL weights for the no prototype group are 674 412 412 315 191 and 156and for the prototype group are 714 474 262 281 206 and 170 TP 5 prototype of trianglecategory T1 5 old instances of triangle category Ta 5 new instance of triangle categoryCategory T 5 triangles Category P 5 pluses Category F 5 Fs


APPENDIX BSix-dimensional scaling solution for the dot patterns in Experiment 2

Dimension


Category AAP 1834 2 0070 2 0913 0702 0051 0101A1 1374 0064 2 3266 2 0726 1154 2 0162A2 1240 2 0257 0738 2 0012 2 1406 0735A3 1578 0354 2 0434 2 0090 0943 0304A4 1628 0062 0870 0095 2 0396 0086A5 1391 0227 1597 0613 0166 2 0200A6 1611 0125 0385 2 0332 0283 2 0211Aa 1363 0441 2 0672 2 0462 0066 2 1461Ab 0941 0151 2445 2 0381 2 1078 0279Ac 0846 0605 0883 0068 2 1681 2 1958

Category BBP 2 0333 2 1627 2 0767 1078 2 1144 0211B1 2 0628 2 1226 2 1058 2 1136 2 1889 2 0647B2 2 0567 2 1661 2 0165 0428 0336 2 0730B3 2 0434 2 1630 2 0373 0280 2 0332 0498B4 2 0668 2 0636 1010 2 0202 2943 2 1491B5 2 0869 2 0672 2 0329 2 2380 2 0062 2 0154B6 2 0624 2 0917 0752 2 1497 0528 0412Ba 2 0752 2 1319 0704 2 1357 1090 2 0707Bb 2 0467 2 1455 2 0284 0244 2 1197 1103Bc 2 0590 2 1403 0094 1358 0894 0689

Category CCP 2 0742 0634 2 0498 2547 2 0317 0072C1 2 0863 1213 2 0577 1118 0429 2 0133C2 2 0746 1191 0085 0356 1041 1369C3 2 0866 1150 2 0171 2 0321 2 0801 2 0114C4 2 0675 1098 2 0452 1098 2 0127 2 2612C5 2 0763 1015 2 0751 2 1437 0171 2397C6 2 0957 1166 2 0225 0153 1264 0084Ca 2 0971 1101 0237 2 1208 2 0445 0427Cb 2 0733 0982 1205 0767 2 0006 0189Cc 2 0559 1292 2 0068 0634 2 0476 1625

Note INDSCAL weights for the no prototype group are 707 600 158 154 133 and 127and for the prototype group are 703 613 161 156 146 and 126 TP 5 prototype of CategoryA A1 5 old instances of Category A Aa 5 new instance of Category A


APPENDIX CSix-dimensional scaling solution for the checkerboard patterns in Experiment 3

Dimension


Category AC1 0429 2 0023 0214 2 0490 2 0175 0307C2 0462 2 0015 2 0440 0597 2 0082 0323C3 0489 0028 0481 2 0051 0755 0133C4 0602 0089 0335 2 0160 0173 0288C5 0520 2 0236 2 0122 2 0471 0359 0171C6 0836 0107 0050 0267 0144 2 0035C7 0802 0225 2 0122 0057 2 0146 0323C8 0624 0377 0361 0004 0164 2 0315C9 0471 0477 0179 0088 2 0668 0098C10 0515 0068 0353 2 0298 2 0199 2 0259C11 0526 2 0046 0168 0235 2 0412 0350C12 0705 0529 2 0128 0176 2 0256 2 0350C13 0388 0468 2 0140 2 0440 2 0334 0520C14 0839 0519 0137 0447 0045 0150C15 0682 0127 2 0236 2 0505 0017 2 0215C16 0719 0050 2 0186 0016 0250 2 0335F1 0677 2 0447 0078 0529 0008 2 0562F2 0847 2 0864 2 0899 2 0060 2 0121 0141F3 0611 2 0758 0471 2 0003 0028 2 0220F4 0662 2 0794 2 0085 0254 0211 0046P 0924 2 0114 0166 0041 2 0079 2 0086

Category BC1 2 0453 2 0203 2 0444 0530 0031 2 0196C2 2 0636 0262 2 0020 0165 0608 2 0022C3 2 0576 0187 2 0540 0170 0422 0108C4 2 0715 0167 2 0385 2 0028 2 0450 0062C5 2 0433 2 0534 0126 2 0594 0358 0088C6 2 0487 2 0368 0070 2 0542 2 0213 0073C7 2 0688 0446 0099 0363 0128 0243C8 2 0809 2 0367 2 0298 2 0388 0293 0078C9 2 0522 0619 2 0379 0273 0375 2 0177C10 2 0327 0515 2 0475 2 0303 0156 2 0426C11 2 0511 0563 0204 0059 2 0459 0138C12 2 0583 0034 2 0268 2 0086 2 0039 0484C13 2 0610 2 0070 2 0106 2 0269 2 0360 2 0631C14 2 0535 2 0361 2 0519 2 0087 2 0393 2 0322C15 2 0616 0520 0063 2 0217 0332 0143C16 2 0308 0246 0517 2 0597 0156 2 0335F1 2 0987 2 0320 0843 0261 2 0325 0043F2 2 0769 2 0590 0345 0457 0291 0564F3 2 0836 2 0415 0199 0114 2 0725 2 0058F4 2 0925 2 0075 0391 0481 0177 2 0459P 2 1006 2 0024 2 0054 0008 2 0048 0127

Note C1ndashC12 5 old close patterns C13ndashC16 5 new close patterns F1ndashF4 far patternsP 5 prototype


categorize statistical distortions of these prototypes During transfer old distortions newdistortions and the prototypes are presented to be classi ed without feedback As discussedfor example by Homa (1984) a major advantage of these experiments is that the dotpatterns are essentially in nitely variable and have a highly complex dimensional structureso that the properties of the arti cial categories that are created may mimic those of manynatural categories

The dot-pattern paradigm has been used to systematically investigate the effects ofnumerous fundamental variables on category learning and transfer including effects of cate-gory size (eg Breen amp Schvaneveldt 1986 Homa amp Vosburgh 1976 Posner amp Keele 1968Shin amp Nosofsky 1992) category variability (eg Barresi Robbins amp Shain 1975 Homa1978 Homa amp Vosburgh 1976 Posner amp Keele 1968) instance frequency (eg HomaDunbar amp Nohre 1991 Shin amp Nosofsky 1992) number of categories learned (egHoma amp Chambliss 1975) amount of training (eg Homa et al 1991 Homa GoldhardtBurruel-Homa amp Smith 1993) and delay between training and transfer (eg Homa CrossCornell Goldman amp Schwartz 1973 Posner amp Keele 1970 Strange Kenney Kessel ampJenkins 1970) In addition the paradigm has been used to investigate the relationshipbetween categorization and oldndashnew recognition memory (eg Homa et al 1993 Metcalfeamp Fisher 1986 Onohundro 1981 Shin amp Nosofsky 1992 Vandierendonck 1984) and hasalso served as a fundamental testing ground for investigating neuropsychological aspects ofcategorization (eg Knowlton amp Squire 1993 Kolodny 1994 Nosofsky amp Zaki 1998Palmeri amp Flanery 1999 Reber amp Squire 1997 Reber Stark amp Squire 1998 Squire ampKnowlton 1995) Indeed the dot-pattern paradigm has provided bedrock data for evaluat-ing numerous theories of categorization and memory including prototype models (egBusemeyer Dewey amp Medin 1984 Homa Sterling amp Trepel 1981) distributed memorymodels (eg Knapp amp Anderson 1984 Metcalfe 1982) connectionist models (egMcClelland amp Rumelhart 1985) and exemplar models (Hintzman 1986 Nosofsky 1988Shin amp Nosofsky 1992)

One of the most salient aspects of dot-pattern studies is a well-known effect called proto-type enhancement On average category prototypes that are not experienced during trainingare typically classi ed during transfer as well as and sometimes somewhat better than theold category instances and better than new category instances Although such prototypeenhancement effects were originally believed to provide solid evidence for the existence ofprototype abstraction processes theoretical work has shown that pure exemplar retrievalmodels can account for this phenomenon as well (eg Busemeyer et al 1984 Hintzman1986 Hintzman amp Ludlam 1980 Nosofsky 1988 1992 Shin amp Nosofsky 1992) It mayseem paradoxical that models that assume that categories are represented solely in terms ofstored exemplars can account for enhanced classi cation of unseen prototypes The keyintuition is that although any given old exemplar is highly similar to itself it may not bevery similar to any other old exemplars By contrast prototypes are typically similar to manyother exemplars stored in memory The similarity of prototypes to numerous stored exem-plars makes up for the lack of stored representations for the prototypes themselves

Shin and Nosofsky (1992) demonstrated an approach to modelling detailed aspects ofdot-pattern classi cation performance that combined an exemplar-based model of cate-gorization the Generalized Context Model (GCM Nosofsky 1986) with multidimensionalscaling (MDS) techniques (Shepard 1980) The stimuli they used were typical of most





















EXPERIMENT 1


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Participants


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(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





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Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


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Experiment 3









Discussion









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Participants


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(bJSje Jsij)g

P(J|i) 5 3S

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sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





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Participants


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Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



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Experiment 3









Discussion









GENERAL DISCUSSION














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Participants


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Results




































(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


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218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








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EXPERIMENT 1


Method

Participants


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Procedure




Results




































(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


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218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension











EXPERIMENT 1


Method

Participants


Stimuli




Procedure




Results




































(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


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Procedure




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218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension







EXPERIMENT 1


Method

Participants


Stimuli




Procedure




Results




































(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension






Procedure




Results




































(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension





































(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension


























(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension




















(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension
















(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension






(bJSje Jsij)g

P(J|i) 5 3S

K e R(bKSk e K

sik)g










Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension








Experiment 1



Fit 2 lnL 29144 30387 33601 58757 35072RMSE 0044 0046 0052 0085 0056Var 955 950 938 834 927


Fit 2 lnL 28029 29521 32705 40979 45175RMSE 0040 0043 0048 0063 0066Var 980 977 971 949 946








P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension










P(J|i) 5 4 S

K e R[bK(Sje K

Sij 1 y SipK)]g


Discussion






EXPERIMENT 2





Method

Participants


Stimuli


Procedure




Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension









EXPERIMENT 2





Method

Participants


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Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension







Method

Participants


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Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension






Results



























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension

























218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension














218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension










218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension





218


Experiment 2



Fit 2 lnL 22794 23036 24889 29056 25909RMSE 0046 0047 0054 0068 0057Var 977 977 969 950 965


Fit 2 lnL 22246 22500 24192 26790 92224RMSE 0045 0047 0050 0063 0179Var 979 978 975 960 679





Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension







Discussion



EXPERIMENT 3











Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension













Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension








Method

Participants


Stimuli



Procedure






Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension









Results












Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension










Experiment 3









Discussion









GENERAL DISCUSSION














REFERENCES






















































































Dimension








Dimension








Dimension









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Dimension





Documents

Central tendencies, extreme points, and prototype ...catlab.psy.vanderbilt.edu/wp-content/uploads/File/Palmeri-QJEP2001.pdf · Indiana University, Bloomington, U.S.A. In three perceptual