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Ræeived February 23, 1998Revision received February 11,2000

Accepted April 6, 2000 r

Joumnl ol BFñmcntd Psycholoty: Gcæhl2ml - Vo l , 130. No.2 .273-298

Grounding Spatial Language in Perception:An Empirical and Computational Investigation

Terry Regier Laura A. Ca¡lsonUniversiiy of Chicago Univercity of Not¡e Dame

The prcsent paper grounds the linguistic calegoriation of space in aspecG of visual Perception:specifically, the structure of projætive spâtial tems such as ¿óove ae grounded in (he Process of

attention md in vector-sùm coding ofoveEll direction. This is fomaliæd in lhe atrentional vector-sum

(AVS) model. This computational ñodel accuntely predicts linguistic acceptabilityjudgments for spatial

tems, under a veiety of spatial configurations, In 7 experiments, the predictions of the AVS model {c

t6ted against those of 3 compering models. The results sùppofr the AVS model and disconfim its

competitoß. The authore conclude that the shcture of linguistic spatiål catego¡ies can be palially

explained in tcms of independently motivated perceptual Processs.

The mapping of language onto space is being actively studied inmany of the disciplines of cognitive science. Exmples may bedrawn from neuroscience (e.9., Fãah, Brunn, Wong, Wallace, &Ca¡penter, 1990; Shallice, 1996; Stein, 1992), from psycholinguis-tics (e.g. Cillson-Radvansky & Iruin, 1993, 1994; Culson-Radvansky & LogN, 199'l; Gmham, 1989; Landau & Jacken-doff, 1993; lævelt, 1984; Talmy, 1983; H. A. Taylor & TYersky,1996\ vaî der zre, 1996), including crosslinguistic reseuch (e.g.,

Cora: Casad, 1988; Dutch: Helmantel, 1998; American Sign Lan-guage: Emmorey & Casey, 1995; Russian: Janda, 1988; Tzettal:Brown & Levinson, 1993; Levinson, 1996; Italian: Taylor, 1988),anC from ccmputalional modeling (e.g., Gapp, 1994, 1995; Haris,1990; Herskovits, 1986, 1998; Munro, Cosic, & Tabasko, 1991;Nenov & Dyer, 1994; Regier, 1996; Schina, 1993; Siskind, 1994;Suppes, Liang, & Boettner, 1992). Across these disciplines, re-

CoDyíphr 2m¡ by úc Ànêd@ Psrcholo8ic¡l As6ì¡üon, hctu&3øtoUSs.m ml: 10.¡03?ÆG85.130.2.273

This micle reflects e4ual contibutions ofbolh authors; otder ofauthor-

ship was rbitary. This work was supponed in pd by National Science

FoùndÂtion Cmnt SBR97-2?638. Podions of this work were Presented at

the långuâge üd Space Workshop at tìe l4th National Confcrence on

Anificial Intelligence in PFvidence, Rhode Islud, July 1997; ùe work-

shop on Interfacing Models of Language at the Neural Infomation Pro-

cssing Systems conference in Brækenridge, Colondo, Decembe¡ 1997; at

the Grounding Word Meaning Workshop at the lsth Nationâl Conferencc

on Aftificial Intelligence in Mâdison, Wisco¡sin, Jùly, 1998: at the 71st

Annual Meeting of the Midwcstcm Psychological Associãtion in Chicago,

Illinois, May 1999: ând at the 1998 mæting of the Chicago Linguistic

Society in Chicago, Illinois. April 1998.We thank paticipants at the confercnces, as well as three anonymoùs

reviewers, and BebM Landau, Gordon logan, and Annette Heßkovib fo¡

helpfirl commenrs. We âlso tìank Derick HiSSins and Bryce CoriSân for

their help in prog¡amming and running the modcls.

Coftespondence concem¡ng this anicle should be addressed to Laura A

Carlson, Depdment of Psychology, Unilersity of Nore Dame, I l8-D

Haggar Hall, Norc Dâmq Indima 46556, or Terry Regier, Depdment of

Psychology, Univeßity of Chicago, Græn 414, 5848 south University

Avenue, Chiøgo, lllinois 60637. Elecùonic mail may be senl to

Lcdlson @ nd.edu.

seachers ue asking how we tink up spatial exPressions with our

representations of objects and their relations.Why all this interest? One reason is that the linguistic catego-

dzation of space sefles as an interface between language and theperceptual world- This makes it likely that the structure of linguis-

tic spatiâl categories may be explicable in tems of percePtual

processes that re not themselves linguistic in chilacter (Haywild

& Tæ. 1995: Landau & Jackendofi 1993; see also Kay &

McDaniel. 1978, for related work in the semmtic domain of color).

Thus, one reason to exmine spatial lmguage is that it offers thepossibility of grounding some aspects of lmguage in nonlæguage.

wc britrg togetheÍ twc of the above strems of reseæch in

spatial language, the psycholinguistic and the computational, to

explore this issue ofsemantic grounding. Specifically, we ask what

nonlinguistic percepn¡a.l processes may underlie the semmtics of

English prcjective spatial tems such as ¿àove A number of

reseuchem have examined how the acceptability of spatial tems

vuie across different spatial conFrgurations (Cælson-Radvmsky

& Logm, 1997; Frmklin, Henkel, &Zengas, 1995; Gapp, 1994'

1995; Haywild & Tm, 1995; Logan & Sadler, 1996). For exm-ple, Logm md Sadter presented participants with a small O in the

middle of an invisible 7 x 7 grid (in cell 4, 4). Across trials' a

small X was placed in each of the remaining cells in tho grid, ild

participæts were æked to mte the acceplability of the sentence

"The X is above the O," as a description of the spatial scene.

In general, we use the tem landuark to refer to the central

object relative to which another object is located; that other objcct

is refened to as the trujector (Langacker, 1987; these õe also

known as the reJ&r¿nce ald located obiects, resPectively; C{lson-

Radvansky & I¡win, 1993, 1994; Levelt, 1984). Thus' in logæ

and Sadler's (1996) study, the landmuk was the O, the Ùajector

was theX and acceptability ratings concemed the relation betwæn

the two.These acceptability mtings were presented in a three-

dimensional plot. The x-õis md y-dis reprcented the rows 8d

columns, respectively, ofthe 7 x 7 grid, and the z-ilis rePresented

the mem acceptability rating, given a placement of the X at each

274

location within the grid. Such plots, rêfered to as spa¡¡al tem_plates, provide a yisualization of the shape of the spatial categoryabove rclati\e to a particultr objæt,

Figure I shows a spatial template for ¿åoy¿, constructed fromLogm md Sadler's (1996) data (Experiment 2). Cell (4, 4) isempty because that is where the lmdmuk was placed. Logil andSadle¡ identified thræ regions along the surface of the spatialtemplate: The good region consisted of the highst ratings andcoresponded to the peâk in the template. T\e øcceptable rcgionconsisted of intemediate ntings md conesponded to the regionsflmking the good region. The åød region consisted of the remain_ing unifomly low mtings. \{e refer to these regions as direcr,oblique, and.other, ræpectively, bæause tbis classification fæuseson the locations of the regions rathe¡ thil on the acceptabilityjudgments comonly encountered in those regions. The ovejtshape of the plor is nor spæific to these particulr lmdmuks mdtrajætoß (cf. Ctrlson-Radvmsky & Logan, 1997; Haywild &Tu, 1995; Logm & Sadler, 1996). Nor is it specific to a paIticulilrelation (lrgm & Sadler, 1996): The spatial templates for aàove,below, under, over, lef, and rìght aI had similü shaps mddiffered only in orientarion.

What underlis such ratings, relative to this lüdmilk ild oth_eß? More pointedly, what pereptual or cognitive structurs dercflected in these linguistic judgments? Does spatiat perceptionshape spatial lmguage in this instance, md if so, how? Our soal inthis anicle is to ilswer these qustions,

We procæd as follows. We focus on tbe spatial relation ¿åovein the expectation that our rsults will generalize to other prcjec-tive relations that shile the sme spatial template shape. Weconsider fou computational models, each one spæifying a possi-ble mæhmism underlying linguistic spatial rem ratings. Signifi-cantly, one of thse four models is in part independently motivatedby perceptual considerations. We present a set of seven experi-ments dsigned to discriminate mong the models, using a vuietyof Imdmuk shap*. r¡y'e exmine both the humm behavioral datamd the fits of the models ro these dâta. To preview the findings,only the itrdependently motivated model passes all our empiricalmd computational tests. It also p¡ovides a úghter overall fit to ourdata. This model thus prcvides a preliminary account of the rela-tion between spatial perception md spatial lmguage.

The Models

Each of the four models views a point-trajector object loqtedrelative to a two-dimensional lildmilk object ild retums ilacceptability judgment indicating how well the spatial tem øåovcdescribes the relation between the two obiects. These Dredictedjüdgmenls re then compued with empirically obtained àata. Wedescribe the models md then present a set of contasting p¡edic-tions that the models nake. These predictions motivate our em-pirical and computational work.

The Bounding Box Model

The bounding box (BB) model is bæed on cleu intuition.rAccording to this model, a trajætor object is above a lmdnrkobject if it is higher than the highest point of rhe lædmuk ædbetwæn its rightmost and leftmost points. Thus, this model selæts

RECIER AND CARLSON

the region ofspace prcjecting upwud from the rectmgulil bound-ing box of the landmdk object. This is illusrrated in Figure 2.

The model hð two conceptually distinct elements. The fNt ele_ment detemins whether the hajætor is verticalty higher thm thelmdmtrk æd the sond detemins whethe¡ it is horizontally cen_teEd with Hpæt to the lildmilk. Both model elemenrs mav beimplemented uing the sigmoidal squæhing ñrnction, æ in Equatiãn l:

s iÊ lx ,Êa in )=- I

F expþE xl:! l ' ( l)

This fr¡nction retums values ner 0 if x is negative and valuesneil I if r is positive. More fomally, lim.-_- sfg(a gain) = 0;lim,* sig(r, gair) : l; and there is a smooth interpolarion in theintemediate region, with stg(0, gaø) = .5. The value gai¿ adjuststhe abruptness of the transition from 0 to 1. In two-dimensionalspace, this function implements a soJ? half plme: x : 0 defines aline, md the function retums values neõ I for points on one sideof that line and values neil 0 for points otr the orher side. The gainpümeter adjusts the ¿ard¿eJs ot soflness of the half plane.

Given this building block, we firet detemine whether the tra-jector is higher than the ludmuk. The specific fomulation wechose for this purpose accomodates a vtriety of landmilkshapes. We staft by defining rhe top of the lildm{k. Intuitively,the top is the set of lmdmæk points that ile exposed ftom above:the onæ that would get wet in the min. More fomally, it is the setof lmdmilk points (x, y) so rhar there exists no landmuk poinr (x,y') with y' at a greater elevation than y. The height of rhe trajectorwith respæt to the lmdmilk is given by Equation 2:

sig(y - hightop, highgaín) + sig(y - towtop, l\n ¿ ¡ g n ¡ = - . \ 2 )

Here, hìghtop is the elevation of rhe highest point on the latrdmilktop. Thus, siggt - híghtop, highgain) defines a horizontal linetouching this highest point md selæts the half plme above it.Similæly, /owtop is the eleyarion of the lowesr poinr on rhelandmilk top. Thus, sigþ - lowrop, l) dehnes a horizontal linetoüching this point md selects the half plane above it. The value,¡eig¡¡t is the average of these two horizontal sigmoids. The rsult-ing value is I if the trajecror is well above all landmuk top poinrsild 0 if it is well below all landmrk top points. Intemediatepositions ræeive intemediate values, The gain of the upper sig-moid is detemined by a fræ puamete¡ highgain, aid, the gao ofthe lower sigmoid is fixed at l. This gives the model some limitedflexibility in fitting the data.

In this aficle, the horizontal line græing the very rop of thelædmrk is refened to as the graziag line. We explicitly rst for itsinfluence.

The other element of rhe BB model detemines whether thet¡ajætor is horizontally centered. To do this, it selæts the centeredregion: a venical bil, bounded by the leftmost and righrmost pointsof the lmdmilk, as in Equation 3:

center = sig(x - lef, lrgain)h*F x sig(right - x, lrgd¡n)t- p.

(3 )

rè....iiÈIf.¿;

"#tT¡;'1ì,:i; .

::!'

,.Ñ',,Èl,ù:=ii\it\\\iÈi:,È¡Ér

Ratine I- 8

I

ACCEPTABLE 6(Oblique) 5

_ > 4

3210

"-"','--:ïiä;;iJ

' Vy'e thank d eonymous reviewer for suggsring the basic featuG ofthis model.

Here, srg(r - lelt, lrgein) draws a vertical line touching thelefünost lmdmdk point and selecs the half plane to its right.Analogousty, sfg(r€ht - x, lr¿ain) dmws a venical line touchingthe rightrnost lúdmdk point and selæts the half plile to its left.The product of these two sigmoids selæts the intersection of thesetwo half plme: the centered venical bil. There ile two freepffieteß in this element of the model: lrgain, the gain on theseleft-right sigmoids, md l¡¿rp, the power to which they ue raised.

The BB model as a whole combinæ these two comDonents. asin Equation 4:

BB model: above = height x center- (4)

Thus, the model picks out that portion of the centered verticalbu that is higher than the landmuk top. The model has th¡æ ftee

Fig!¡€ -¡. Spatiâl teñplate for øbove.'lhc data in the figure æ from "A Compuhtional Analysis of

Apprchension of Spatial Relations," by G. D. t gan and D. D. Sødle4 in Inngudge and Space (p. 512) by P

Blæm, M. A. Petenon, L. NÂdel, & M. F. Gæn (Eds.), 1996, Cambridge, MA: Mff Prcss. Copyright 1996

by MIT Press, Adapted with pemission.

ACCEPTABLE(Oblique)

4-

275

2 3 4 sColumn

ln l l|

(other)

ptrmeters: the gain and exponent of the left- and ilght-flmkingsigmoids md the gain of the uppemost sigmoid in the heightft¡nction.2

The Proximl and Center-of-Mass Model

The proximal and center-of-mass (PC) model assumes thatprcjective spatial tems are defined in tems of ¿ngular fqturs,that is, in polil coordinates rather than the Ca¡tesian coordinates ofthe BB model. There is empirical support fo¡ such a notion. Givena point lildmùk md a point trajecto¡, one may define a rayprcjecting from the lædriilk to the hajector and obtain the mgu-lu deviation of that ray f¡om the reference äis (upright verticalfor above\. Capp (1995) hö noted that acceptability for Prcjætivespatial tems drops offroughly tineilly with this angulü deviation.In his data, this lineility holds for deviations from 0 thoughapproximately 68'. Logm md Sadler's (1996) data also supportthe lineility of this initial drop-off, for deyiations of 0 throughapproximately 72'. At angulil deviations of 90'and beyond,acceptability mtings te essentially 0. AccePtability ratings in theintemediate range, between 72" nd 9O", ile not detemined bythis eulier work. Thus, we propose a function that smoothlyinterpolates in this intemediate nnge-the product of a line ild asigmoid, illustated in Figure 3 md fomalized in Equation 5:

2 Further free pmmeteß for the sigmoids of the height ñ¡nction werenot found to be neessary.Fi8!¡e 2. lllustration of the botrnding box model. LM = ludmrk.

¿ I O

0.8

E 0.8

: 0.5

7 o.o

0,3

o.2

0.1

0

"MYo"€! â tonsoon otodonbüon:Lhã¡ Slgmld

{- ñ. 6lopo lnd y{nbMpt of 60 ¡tnâ ¡Þ tu FnñeÞE,

0 30 60 & 120 fs l8o

tugshrdovt¡üor In øgm!

Figzre 3. Alignment fi¡nction of the proximal ud center-of-mæs model.

fla): l(slope x a) + y-interceptl x sþ(90 - a, gain\. (5)

Here, a is the mgulu deviation. fa) presents a lineu drop-offfor a < 90 md then a smooth træsition to 0, a rqsonablefomaliation of these eulier empirical findings. This function hæthree free pmetere: the slope and y-interæpt of the litre md thegain of the sigmoid. A cetrEal fæture of this chuacterization ofspatial tem acceptability is that it is dependent only on thedirection, not the length, of the vector connæting the lildmilk tothe t¡ajætor.

This fomalization assumes a point landmùk. But what if thelandmark objæt is not a point; whar if it has (ar least) rwo-dimensional extension? Regier (1996, 199'1) has suggested that¿åove judgments may in fact rely on two orientational features,features that are conflated in the case of a point landmark. Thefirst feature is tíe center-of-mass orientation: the orientation ofthe ray connecting the cetrter of the landmilk to the center ofthe trajector. The second feature is the proximal orientation; theorientation of the ray connecting the edge of the landmark to theedge of the trajector, where they ue closest. Regier's (1996,1997) suggestion was that above judgments depend on theextent to which each of these two orietrtations âligns withupright vertical. Figure 4 illust¡ates this idea. Figure 4(a) con-tains a scene in which both the center of mass and Droximalorientations are perfectly aligned with upright vertical; theresult is a milimally acceptable example of above. Fígure 4(b)contains a scene itr which the proximal orientation is stillperfectly aligned with upright veftical, but rhe cente¡-of-massorientation is not; the result is a good but less-than-petfectexample of above. The comparison of Figures 4a and 4b thussingles out the possible role of the center-of-mass orientation inaáave judgments. Figure 4c contains a scene in which thecenter-of-mass orientatioo is the sme as it was in Figure 4b butthe proximal orientation is more deviant from upright vertical;the result is an even less good example of aàoye. Thus, thecompilison of Figures 4b and 4c sùggests a role for the prox-imal orientation.

The PC model extends the work of Gapp (1995). As notedeillier, capp (1995) has found a roughly liner drop-off inacceptability with angular deviation. His data are based on the

REGIER AND CARIION

center-of-mass orientation, and the lineility is not perfect. Thenature of this imperfection led Gapp to conjecture that peoplebase theh judgments not on the center-of-mass orientation, butrather on the proximal orientâtion. tn the PC model, the idea ofa linear drop-off is generalized and applied to both the proximaland center-of-mass orientations, in the expectation that bothorientations are relevant. The PC model has not yet beenthoroughly tested against challenging empirical data-rhar isone goal of this anicle.

Fomally, the PC model chtracterizes aåoy¿ as a lineil combi-nation of the degrees of alignment of the center-of-mass andproximal orientations with upright vertical (Regier, 1996, 1997), asin Equation 6:

PC rudel: above = al(con\ + (l - a)J(ptox). (6)

Here, com is the deviation of the center-of-mass orientation fromupright vertical, pro¡ is the deviation of the proximal orientationfrom upright vertical, mdl ) is the alignmenr function discussed

(a) @

erlier; a is a ftæ p{meter conholling the relative contribution ofthe center-of-mass and proximal orientations. ln addition, there uethræ ftee pmmeteß in the line X sigmoid alignment function{ ):the slope md y-intercept of the line ild the gain of the sigmoid.These pa¡meteß ile shued byflcom) ædffprox). The model asa whole thus has a total of four free pumetere.3

The Hybrid Model

The hybrid (PC-BB) model is a væiant of the PC model that alsoincorporates the height element of the BB model. Given a tmjectorÌocated relative to a landmæk, the model detemines an initialacceptability judgment much as the PC model would. This initialjudgment is then adjusted depending on the height of the ûajectorrelative to the lmdmilk. These two elements may be seen in thefomal definition of the model shown in Equation 7:

PC-BB model: above : fag(com) + (l - dlg(prox)f x height.

(7 )

The fißt factor, [a8(com) + (l - d)g(pro¡)], is boûowed ftomthe PC model. It is a lineu combination of the degræs of align-ment of the center-of-mass md proximal orientations with uprightvedical. Here, pror ud com ile the proximal and center-of-massorientations, g(¿) measures the degræ ofalignment of angle ¿ withupright venical, md d is a free püameter cont¡olling the relativecontributions of ænter-of-mass æd proximal orientations. Theorientational alignment function g(a) is purely lineu, as expressedin Equation 8:

e@) = l(slope x a) + y-interceptl. (8)

Here, a is mgulil deviation from upright vertical, md rrope mdy-intercept re free ptmetere. A diffe¡ence from the PC moCel isthat there is no sigmoid gating this orientational alignment func-tion. Instead, the model as a whole is gated by trajector height,relatiYe to the top of the lmdmilk.

The second model factor, height, is borowed from the BBmodel md is defined in Equation 2. As before, this qumtitycaptures the extent to which the trajætor is higher thm the top ofthe lmdmuk. Because the PC-BB model is gated by this qumtity,it is sensitive to height, as well as the proximal and center-of-massorientations. The PC-BB model has four f¡ee p{meters: a, therelatiye süength of the center-of-mass and proximal orientations,the slope andy-intercept of the aligment functions, and lligàgøin,the gain of the uppemost sigmoid in the height function.

The Attentional Vector-Sum Model

The attentional vector-sum (AVS) model is similu ín design tothe PC-BB model, in that it combines an orientational componentwith a height ñrnction. However, its dchitecture is also infomedby two independent obsenations.

The first obseryatiotr is that the humil apprehension of spatialrelations involves attention Q-ogÃ, 1994, 1995). For example,Logm (1994) has found that visual seuch for a tdget in a fìeld ofdistractoß is slow when t{gets differ from distmctors only in thespatial relation (either above, below, left, or right) mong theirelements. This suggests that spatial relations ue not preattentivelyperceptually available, that they do not "pop out" ofthe visual Freld

(b) @

(c)

GROUNDING SPATIAL LANGUAGE

@

Figure 4. lî pmcl a, both the prcxiñal md the centcr-of-mass orienb-tions de perfætly aligned with upright venical; in pa¡el b, the prcximalorienbtion is still aligned with upright vefical bur ûrc cemer-of-massorientation deviates somewhat; in puel c, the center-of-mass orientâtionrchins the value it had in b, but the prcximal orientation is more devimlLM = Iildmdk; TR = mjæbr.

(freisman & Gomican, 1988), Rather, their perception seems torequi¡e the focus of atfention (see also Logm, 1995).

The second observation is that in several neuml subsystems,overall di¡ection is represented as the vector Jrm of a set ofconstituent directions. For example, in their studies of rhesusmonkey motor cortex, Georgopoulos, Schwartz, and Kettner(1986) examined a population oforientation-tuned cells in the teaof motor cortex that represents the monkey's m. Each cell wasbroadly tuned, with a prefened direction, so that the cell wouldrespond muimally when the monkey's intended am moyementwas in the prefened direction. They found that the direction ofmotion of the m was accurately predicted by a vector sum overthe population of cells, as in Equation 9:

V "''' (e)

Here, ¿i is the activation of cell i, ¿i is the prefered direction ofcell i, md the sum is taken over the population as a whole. Similürepresentations govem saccadic eye movements (Lee, Roker, &Spuks, 1988). In perception, Wilson md Kim (1994) have foundthat when palicipilts view moving pattems, they perceive motionin the direction of the vector sum of two constituent modoncomponents. These results suggest that the representation is awidely used one.

The AVS model brings together these two appãetrdy utelatedobseryations, conceming attention ild vector-sum representation.In the model, il anentional beam is focused on the lmdmdk. Inparticulæ, the bem is focused on that point on the lildmilk topthat is vertically atigned with the trajector or closest to being soaligned, PaÍs of the lmdmtrk nw the center of this bem ilesûongly attended, whereas mo¡e distant puts of the lmdmilkreceive lss attention. This yields a distribution of attention acrossthe iandmrk object. in addition, ât each point of the lmdlrrük, wedefrne a vecto¡, rooted at that position ild pointing towtrd thetajector. This yields a population of vectoff. The model thencomputes the vector sum over this population, weighted by atten-tion (Moran & Dsimone, 1985), fotlowing Equation 9. In thisinstmce, however, ¿¡ is the amount of attention cuffently beingpaid to landmuk point i, and õi is the vector rooted at lildmilkpoint r', pointing towrd the t¡ajector. The rsulting vectot-sumdirection is then compæed with upright vertical.

Figure 5 illustrates this idea. Figure 5a shows u attentionalbeam that has bæn focused on the landmrk, as described above,This attentional beam mdiates out to illuminate different pafis ofthe landmek at different strengths, depending on distmce fromthis focus, Figure 5b shows the vecto$ rooted at the ludmilkpoints, pointing towõd the tajector. These vectors üe weightedby attention at the root of each vector. The result is shown inFigure 5c: Vætore rooted neuby ue more heavily weighted thmdistilt otrs. These weighted vectors de then sumed. The direc-tion of the resulting vector sum, shown in Figure 5d, is thencompued with upright vefiical, shown in Figure 5e.

The interaction of attention md vecto¡ sumation can yieldeffects of the proximal æd center-of-mass orientations. This is

277

3 Singlc-factor models consisting of only one orientation f4tuÉ (e.9.,

the prcximal model ând the center-of-mass model) werc consistently

ou(perfomed by the PC model and hcnce de not plesented.

278 RECIER AND CARIJON

n , @

M

of-mass ud proximal orientations. This dependence of modelbehavior on attentional width implicirly raises a qustion: rühatatlentional drop.off ñ¡nction do humils actually possss? In theirempirical studies of spatial anention, Downing md Pinker (1985)have found m attentional drcp-off thar is roughly exponential indistmce (sæ âlso LaBeÌge & Brcwn, 1989). Thus, we use mexponential decay ñrnction in the AVS model. T'his is intemediarebetwæn the two extremes outlined above; thus, we exp€ct bothproximal ild center-of-mass effects. The specific fi¡nction isshown in Equation 10:

4r = exp[-(Euclidean distance from focus to Doint i

of landmark)/(Àø)1. (10)

Here, lhe stcepness of the exponential drop-off is govemed bytwo valu6, À md oì À is a free pumeter, md o is the Euclidemdistmæ between the Eajæror and the focus point, that is, thelandmilk point on which the attentional bem is centered. Thus,the farther the kajector is from the lmdmrk, the wider theattentional bem will be. This is natural, because the perceivermust encompâss both the Eajæror and a part of rhe lmdm{kwithin the attentional bam.a Thus, a neuby trajector requires onlya fairly næw attentional bem, but a distant trajector requires awider one.

The AVS model as a whole detemines the alignment of theattentional vætor sum with upright verticâl and then gats theresulting quiltity by tmjector heighr. The fom of the completemodel is shown in Equation ll:

AVS rcde| øov" : e(\. ai) x IEiSht. (l l)

Here, g( ) is the mgulr aligment fr¡nction used in the PC-BBhy'orid model. The value åeþår is the heighr fi¡nction used in boththe BB ild PC-BB models. This qumtity indicats whether thehajætor is strictly higher than (1.0), srrictly lower than (0.0), or atthe sme height æ (intemediate yalues) the top of the lmdmuk.The model hæ four free pdmeteß: the slope and y-intercept ofthe lineil aligment function; À, which controls the width of theattentional field; and the gain of the upper sigmoid in the heightfunction.

Contrasting Predictions

These models make a number of contr¿sting predictions. Somepredictions æe shred by more thm one model, wheræs othen ueunique. For ach prediction, we indicate which models give rise toit &d why.

Proximal ud cenrer-oÍ-Ms orientøtíorc, The PC, PC-BB,md AVS models all predict effæts of the proximal and center-of-mæs orientations. Specifically, acæptabitity ratings should drcpoff with deviation of either orientation from upright vertical. ThePC md PC-BB models predict this b€cause they æ explicitlybased on thse features. In the AVS model, these effæts emergefrom the attentional vætor-sum opeRtion. The BB model does notmake these predictions, because it is not orientation bæed, Instød,

Figure 5. The attentional vætor-sum model. panel a, illùshts theattentionâl field, focused on the landnùk (LM), nø the hjætor (fR).Differcnt pft of the landmdk receive differcnr mounb of attention.Pmel b illusmres the vecrore rcoted ar*ch point of the lddfrùk, pointing(owùd the tr¿jector. Panel c illusmts the arentionally weighted vætoß.Pmcl d illustútes the dircction of the attentiona.lly wei3hted væior suñ.Pmel e illushtes the orienhtion of the vector sum, relative to venicaluprighr.

seen most easily by considering what happens as the attentionalfield vuies in width.

Narrov bean, At one exheme, the anentional field is sonffiow that only the landmæk point at the center of the bemreceivs any attention. This will be that point on the lmdmtrk topthat is vertically aligned with the trajæto¡ or closst ro being soaligned. This point often coincides with the landmilk point that isneilest the trajector in Euclidem distance, æ in Figre 5. Theorientation of the vector rooted at that point is, by definition, theproximal orientation, Thus, under such circumstanæs, thc vætorsum reduces to the proximal orientation, ud the modcl's responsesüe bæed on that fqture.

Wíde beam. At the other exùeme, the attentional field isinfinitely wide, exhibiting no drop-off in attention with distance.Unde¡ such circumstances, the vætor sum always yields thecenter-of-mass odentation, as shown in Apprcndix A, Thus, with aunifom attentional field, the AVS model ¡educes to one that bæesits predictions on the center-of-mass orientation.

Attentional field widths betwæn theæ two exhems cm therc-fore produce vætor-sum dirætions that æ intemediate betwænthe center-of-mass orientation md fhe prcximal orientation. Themodel's reponses will then exhibit inftuences of both rhe center-

iÌ predicts that acceptability ratings will drop off with horizontal or

vertical distance from the lines defining the bounding box. These

predictions tre tested in Experiments l-4.

Grazhry line. The BB, PC-BB, æd AVS ¡nodels all p¡edict

that the horizontal line grazing the top of the landmark will affect

acceptability ratings. Speciltcally, a point just above this line

should receive a higher acceptability rating than a conesponding

point jusr below it. This is predicted because these models explic-

itly gate their acceptability ratings by the grazing line. In cont¡ast,

the PC model contains no grazing line, and therefore does not

make this prediction. These predictions are tested in Experiments

5-6.

D6rarc¿. Imagine holding a small marble half an inch above

a large book lying on a table and moving the muble uound over

the surface of the book but maintaining the half-jnch height. The

AVS model predicts that at low heights such as this, øáore ratings

will be relatively insensitive to the extent to which the muble is

centered above the book. However, ratings should be more sensi-

tive to centeredness if the muble is much higher above the book.

Why? Because at low heights, the attentional field will be natow,

and not much of the book will receive appreciable amounts of

attention-and critically. the edges of the book will not receive

much attention. This means that for all intents and purposes, the

muble is located above a limitless plane. Vy'hen the milble is

higher above the book, the attentional beam grows large¡ and

begins to take in the edges of the book, not just the surface, and

this will lead to a greater sensitivity to centeredness. The PC and

PC-BB models make exactly the opposite prediction. In both of

these models, acceptability dec¡eases with deviation of the center-

of-mass orientation from uprigbt venical. This deviation covers a

greater range in the case of the low trajector than in the case of the

high one. Therefore, there should be greater sensitivity to centered-

ness in the low case. The BB model does not predict a difference

between the two cases. These predictions are tested in Experi-

ment 7.

These thræ predictions guide our experimental and computa-

tional work and allow us to discriminate amons the models.

Initial Model Fits

As an initial test of model feasibility, we evaluated each model's

fit to the aborre data of Logan and Sadler (1996, Experiment 2).

Once the models were fit to these data, the resulting parameter

settings were saved and used in testing the models on out own

data. This was done for two reasons. The fißt reason was to ensure

compatibility of data across reseüchers; a model that fit data from

two laboratories was deemed stronger than a model that only fit

data from one, The second reason was to rewud models that

generalized rvell to novel Iandmuk shapes. The stimuli in the

Logan and Sadler (henceforth LS) data set consist of a very small

tmjector located relatiye to a very small landma¡k. Our stimuli, in

conlrast, contain landmuks with appreciable two-dimensional ex-

tension. Thus, ifa model generalizes well from the LS data to ours,

that will suggest that it caprures morc than just ùe pa.ticulils of

the I.s data set: It might also capture the general procsses under-

lying spatial ianguage, as applied to a variety of landmuks.

The models were fit to the LS data by an inverse hill-climbing

procedure, inverse in that it perfoms descent rather than ascent in

its objective function. The algorithm incrementally adjusts the

- - " 1

GROUNDINC SPATIAL LANGUAGE

parameters of the model so as to minimize the etor-the discrep-ancy between the empirically observed and model-predicted val-ues, The ptocedure teminates when no further pilameter changesyield a reduction in etror. ìn all cases, the step size for parameteradjustment was 0.001. Põameteß were initialized as follows: 1.0

for sigmoid exponents (BB model), alignment function

-!-intercepts (all other models), top si8moid gains (BB. PC-BB'AvS), and À, controlling the width of the attentional field (AVS);

0.5 for d, the relative weight of the proximal and center-of-massorientations (PC, PC-BB); and 0.0 for all remaining sigmoid gains(BB, PC) and alignment tunction slopes (PC, PC-BB, AVS). Theseinitialization values led to tight fìts for all models. No tighter fìts

on the LS data we¡e obtained with a wide range of other initial-ization values.

The resulting pilameter settings üe showo in Table I Lineilregression was perfomed to measure how well the models' outputpredicted the LS ratings, given these parameter settings. Table 2

shows the results of this exercise, For each model, the tablepresents R2, a meæure of the goodness of fit, with a value of 1.0

indicating a perfect fit. The table also presents adjusted À2, an

adjustment of the original R2 value that supports compilisonsarnong models with diffe¡ent nunbers of free pilameters. Alsoshown are the slope and '-intercept of the regression lines for eachmodel.

All four models fit the LS dâta fairly well, paficularly the PC,

PC-BB, and AVS nodels. This established the initial feasibility ofall models, To more effectively dislinguish among the models, we

tested the conhasting predictions in a series of experiments. The

experiments üe Brouped in three sections, each testing one of the

tfuee major predictions made by the models Thus, Experiments

Table I,

aWe assume that the mjætor itself must eeivê a fair amoùnt ofanention; otheNise ib læation would be uncler.

279

BBCain on left-right sigmoidsGain on top sigmoidExponent for left-right sigmoids

PCd, relat¡ve wei8ht of P and C

/-intercept of alignment functionSlope of alignment functionGain on sigmoid

PC.BBd, relative weight of P and C

)-intercept of alignment functionSlope of alignment ñ¡nctionGain on Ìop sigmoid

AVS^, attentional field width

)"intercepl of alignment functionSlop€ of alignment tunctionCain on top sigmoid

Model püameterLogan & Sadlcr

(1996) Experimen¡ 7

0.1090.0660.062

0.5000.969

-0.0050.1 12

No¡¿. Pmmeter values were from Logan Ând Sadler's (1996, Experiment

2) aàoee data and fiom our Experiment ?. The Logan and Sadler pdåmeter

values werc used for all simulâtions unlss otheNise noled in text. BB :

bounding box; p¡ : proximal and center of mass; PC-BB = hyb¡id;

AVS = attentional vecto¡ sum.

0.0650.3730.220

0.t740.929

-0.0063.265

0. r l50.909

-0.0056.1 14

0.5121.224

-0.00?0.002

0.s001.00?

-0.0060 . 1 3 1

1.0001.007

-0.0060.13 1

280

Table 2Model Fits to Above Data Fron lngan and Sadler (1996)

BB .9M .897 0.907rc .959 .955 1.0ilPC-BB -963 .959 1.030AVS .963 .959 1.030

P

Notg. ^dj: adjusted; slope 4d /-interccpt de from regression lincs ofmodels. BB = bounding box; PC : prcximâl and center of mass; pC-BB : hybrid; AVS = ânenrional væto¡ sum.

l-4 test for thc proximal and center-of-mass orientâtions: ExDer-iments 5 ild 6 test for the groing line; md Ex¡reriment ? tests form effect of distanæ. Within each section, we also detemine howwell the models fit the data collected for each experiment mdwhether the models exhibit the qualitative effæs thát each exper-iment was dsigned to test.

Testing for Proximal and Center-of-Mass Orientâtion

Experiments 1-2

The PC, PC-BB, md AVS models predict that acceprabiliryratings wilì exhibit effæts of the prcximal md center-of-massorientations. The BB model does not predict these effects. To testfor independent effects of proximal and center-of-mass orientation,in Experiment 1, we milipulated proximal orientation while hold-ing center-of-mæs orientation constmt, whereas in Experiment 2,we manipulated center-of-mass orientâtion while holding proximalorientation constant. In both experiments, we used horircntallyextended (i.e., wide) md vetically extended (i.e., rall) recranglesas lmdmilks md smaìle¡ squæs as tuajectoß. Lmdmuks ap-peued centered in the middle ofm invisible 5 X 5 grid, æd acrosstrials, the trajector was placed in each of the empty cells of the

Table 3

Adj F

RTGIER AND CART.SON

Slope ){nrercepl

grid. Following Logan md Sadle¡ (1996), paficipmrs rated theacceptability of the vertical rclations (dåov¿ and åelow) md hor-izontal relations (leÍr and right). The only difference betweenExperiments I and 2 was in the placement of the trajectoß.

In Experiment l, trajecloß were always placed itr the samelocation within a cell, so that the center-of-mæs orientation wasequated across the tall and wide lmdmilks. This næcsæily re-sulted in different proximal orientations for the tall md widelandmrks within the oblique region. Table 3 givæ the proximalorientations for velical ild horizontal relations for tall md widelmdmuks. If proximal orientation influences spatial tem accept-ability in the mmner suggesred by some models, then acceptabilitywould declitre as this orientation deviated from the cildinal axis(updght vertical for above). Fo¡ vertical relations. the Droximalorientations for the tall lmdmæk were luger thm they weie for thewide lmdmùk however, for horizontal relations, the proximalorientations for the wide landmuk were ldger than for the talllmdmilk. Therefote, m influence of proximal orientation wouldbe obserued æ m inte¡action between spatiat relation (vertical vs.horizontal) md landmilk (tall vs. wide).

In Expedment 2, trajectors were placed in different locationswithin a cell for the tall and wide lmdmaks, thereby equadngproximal orientation and næessaily varying center-of-mass ori-entation. Table 4 gives the center-of-mæs orientation for verticaland horizontal relations for taII æd wide lmdmaks. If cente¡-of-mass o¡ientation influences the acceptability of a spatial telation,then the grearer the orientation, the less acceptable the spatialrelation would be. For veftical relations, the center-of-mæs orien-tation for the wide lmdmrk was ltrger thm the center-of-massorientation for the tall lmdmilk; however, for horizontal relations,the center-of-mass orientation fo¡ the tall landmrk was læger thanthat for the wide landmilk. Therefore, an influence of center-of-mæs orientation would be observed as an interaction betwænspatial relation (vertical vs. horizontal) ffid lildmilk (tall vs.wide).

0.038-0.024-0.036-0.036

ProxímL Orìentations Jor Tall and. Wide landmarks for ø Veú¡cal Relatiorrqnd a Hoilzontal Relation Jor Experiment I

RIR2R3R4R5

c2

Tdl

90r06.8130.5

CROUNDING SPATIAL LANGUAGE

Table 4Center-of-Mass Ofientations for Tall and Wid.e landmrks for ø Veftical Rela,ionatd a Hoúzontal Relation

RIR2R3R4K)

90t24t )¿ . i

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180r80

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ø1--4rs tt l-4os 163l| 56 73.2 | | 62.s 34 |90

124r52.5

90 I l7-5 139.590 146 t63.2

E 180 18090 146 t63.290 1t7.5 139.5

R IR2

R4R5

90106.8130.5

c l

c2

Ho¡izonbl

| 40.2 ærl o fãß----n2l l-4rs =ã|I 54.8 36.s I 0 | 36.s s4.8 | | 66.2 53.s I

90 90I 17.5 t46139.s t63.2

Horizontal

Rl f4e.J-¡-----6-6.t1 e0 r l3.B 130.2 f?õ5-----54t1 e0 t2s.2 139.8R2 | ¡s.z s¡.s I so t26.s t44.8 90 143.s 156.2R3 0-------T- E l8O t80

--------õ:- E l8o 180R4 F5i

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R5 | 49.8 66.21 90 113.8 130.2 I 40.2 54.8 I 90 125.2 139.8

torr. *of lhe 5 X 5 grid that defined placement of the læated objæt.

Tall

c3

90 90125.2 t43.5139.8 t56.2

Method

Pañicipan¡s. Thiny-six University of Note Dme undergûduats pd-ticipated in Experiment l, dd 39 pmicipared in Experiment 2. All pr-ticipanß ¡n th6e and subsequent expedmenb gave i¡fomcd consent.Pdicipmts werc compensated with pffiial credit in an undergrad!âtepsychology couße,

St¿aú¡i. The displays were coishcted on the basis of a 5 x 5 mahixthat was invisible to pdicipanb. Thc matrix measurcd 95 mm along a side,and qch æll within the maFix møsured 19 mm along a side. Displayswerc presented on a computer monitor; viêwing distæce was uncon-shained, but w6 approximately 5l cm, The landmdk was always placedin the middle ofthe mahix (i.e., in cell 3, 3). The landmdk was eirher a ult¡ed rechgle (5 mm wide X l5 mm tall) or a wide green EcÞ¡gle (15 mw i d e X 5 m b l l ) . 5

In addition to the lmdmük, each picture d¡splay conhined a mjætotwhich was a 3-mm blue squde. In Experiment l, the tajætor was placedi¡ the same location within eâch cell of a 5 X 5 glid surounding thelâDdmdk, thereby equâting center-of-mass orientation acrcss the hll edwide lodmùks but va¡ying proximal orientation. In Experiment 2, thcposition of the squæ within sch cell ofthe marix was v4ied (i.e., it wænot always centered in the æll) across the hll dd wide Mhngles, therebyequating proximal orienhtion but varying center-of-mass orienþtion. Pixellocations for all rajætors and lædm{ks for Experimenb l-7 ild empir-i€l dd model-predicted Étings for all tajector plaæmenß in thseexperiñents æ provided at htp://www.æp.uchicago.edry'-ßgier/avs,

Beforc each picrure display, a sentenæ ãppæd of the fom'"The sqùdeis _ the rel/grecn block." The blmk was filled in with one of thefollowing spatial relations: "above," "below," "to t¡e left of," or "(o theright oi" The spatiâl rclãtion wæ €pitalized ând centered for emphasis.Accompilying the sentence wæ a reminder of the acæphbility raringscale (sæ below) and an insmction to prcss the retum kcy to sæ thepicbrc displây.

Procedure. Paficipants were told that they would be râtin8 theacceptability of sentences as descriptions of pictures using a lo-pointscâle, ranging from 0 (not at all acceptable\ to9 (pefectly accepþble).They we¡e encouraged to use intemediate values if they wished. Eåchpdicipant completed 5 practice kials, followed by 480 experimentaltrials (5 landmdks X 4 spatial relations X 24 unoccupied marrixlocations). The tdals were self-paced and were presented in a unique

00

E

r80180

c4

i t6r---403I a L Á 1 4

90 90146 I 17.5163.2 139.5

E 9 0180 143.5180 t56.2

9090

E

9090

06.8tÁ802406.8

90 90125.2 l r3.8139.8 130.2

c2

130.5152.5r80t52.5r30.5

90t26.5144.8

c4

00

E

180180

281

90126.5lM.8

mndom order fo¡ each pmicipant. Eâch experimetrt took about 50 minto complete.

Results and. Discussíon

Empírical døta: Experiment L Meanacceptability ratings forall placements of the trajector for each spatial relation ileprovided in Appendix B. An influence of proximal orientationwould result in the following pattem: For vertical relations(averaging acrass above and below), wide ludmuks wouldreceive higher acceptability ratings thatr the tall landmüksbecause the proximal orientation for the tall lÐdmdks wasgreater than the proximal orientation fo¡ the wide landmilks;however, for horizontal relations (ayeraging across l¿f and¡¿gl¿f), tall landmarks would receive higher acceptability ratingsthan wide landmilks because the proximal orientation for widelmdmilks was g¡eater than the proximal orientation fo¡ tallobjects. To test these predictions, a 2 (spatial relation: verticalvs. horizontal) X 2 (landmùk shape: tall alone vs. wide alone)repeated measures analysis of viliance (ANOVA) was con-ducted on the mean acceptability ratings calculated across cellsin the oblique region.6 Unless otheruise noted, the p valueadopted for significace fo¡ this and all other anaÌyses was .05.

There was a main effect of spatial relation, F(1, 35) : 5.0, p <.032, MSE : 0.33, with vefical relations (M : 6.6) mted signif-

90113.8t30.2

s Pdicipilts also gave ratings to th@ other lmdmüks, ach one madeftom superimposing the hll dd wide Mhgl6. On soñe riâls, pdici-panb were asked to interpret the Elation with rcspæt to the wide blæk; otrsome, with Epæt to the tall blocki dd on some, with respe.l lo bothblocks. Dah for Fials with rcference to the tall and wide blæks rcplicatethe effæb reported when the wide dd tall ìÐdnrks werc prcsentedindividually. Because the individual tall ild wide lmdMks werc the ßtc6s for t¡e models, only thse dah æ rcported.

6 Similu malyses werc also conducted on the mm accepbbility Étingsfor the dircct ùd other rcgions for both Experiments I md 2. All Fs < l

282

icantly higher than horizontal relations (M = 6.4). Superior per-fomance for vertical tems relative to horizontal tems has beenconsistetrtly found and has bæn attributed to the fact that yerticalrelations ae aligned with salient perceptual and environmentalcues (e.9., gravity, the top and bottom of the display) whe¡eas leftmd right ue not explicitly muked (e.g., Clrk, 1973; Franklin &Tversky, 1990; Levelt, 1984). There was no effect of Imdmukshape (p > . l2). Most criticalty, there was a significant interadionbetween spatial relation æd landmuk shape, F(I, 35) = 16.8, p <.001, MSE = 0.13. This interaction took the expecred fom; Fo¡vertical relations, wide objects (M : 6.'l) received higher ratingsthan tall objects (Ms = 6.5); however, for horizontal relations, tallobjects (M = 6.6) ræeived higher ratings than wide objects(Ms = 6.3\. 495?o confidence interval constructed on the basis ofthe eFor tem for the interaction revealed a critical differenceof,17 for significance; thus, the tall veruus wide comparisons weresignihcant for both vertica¡ and hodzontal relations. These dataindicate that proximal orientation signihcantly influenced the pus-ing ofspace æound an object when center-of-mass orientation washeld constant.

Enpirical data: Erperiment 2. Mean acceptability ratings forall placements ofthe trajector for each spatial relation æ providedin Appendix C, An intluence of centeÍ-of-mass orientation wouldresult in the following pattem: For vertical relations, tall landmukswould receive higher ratings than wide landmuks because thecenter-of-mass orientation was greater for wide than for tall land-muks; for horizontal relations, wide landmdks would receivehigher ratings than tall landmilks because the center-of-massorientation was greater for tall than for wide. To test this predic-tion, a 2 (spatial relation: vertical vs. horizontal) x 2 (landmukshape: tall vs. wide) repeated measures ANOVA was conducted onthe mean acceptability ratings calculated across cells in the obliqueregion. There was a main effect ofspatial relation, F(1, 38) = 9.2,p < .004, MSE = 0-17, with vertical relaúons (M = 6.7) ratedsignificmtly higher than horizontal relations (M = 6.5). There wasno effect of landmark shape (F < l). Most imponant, there was asignincant interaction between spatial relation and landmukshape, F(1, 38) = 12.4, O < .001, MSE : 0.18. This interactiontook the expected fom: For vertical relations, tall landmuks(M : 6.8) received higher ratings than rvide landmuks (M = 6.5);however, for horizontal relations, wide landmrks (M : 6-6)received higher mtings than tall landmilks (M = 6.3). A 95Eoconfidence interual constructed on the basis of the e[or tem forthe interaction revealed a critical difference of.20 for significance.Thus, the tall versus wide compæisons were significant for bothvertical and horizontal relations, These data indicate a significantinfluence of center-of-mass orientation when proximal orientationwas held constant, Taken together, Experiments I and 2 provide

REGIER AND CARLSON

empirical verifìcation of the impotance of the center-of-mass andproximal orientations in spatial tem ratings, as predicted by thePC. PC-BB. and AVS models.

Model rtß and símulatíons, To furthe¡ probe the models, wetsted them on the above data collected for Experiments 1 and 2.We were interested in two issues. Fist, we investigated, quiltita-tively, how tightly the models fit the data. Second, we deteminedwhether each model exhibited qualitative effects of the proximaland center-of-mass o¡ientations. This was detemined using thelogic of the experiments, applied to model outputs ¡ather thanhuman responses.

We retained the pilmeter settings obtained from ñtting Loganand Sadler's (1996) data and measured the models' generalizationto the new data. This was done by presenting the models with ourexperimental stimuli, recording the model's output, md detemin-ing tfuough regression how well the model output predicted theempidcally obtained âcceptability rating. This was done sepilatelyfor the two landmüks within each experiment, givitrg us a total offour data sets for Experiments l-2. The results ue presented inTable 5. This table shows R2, adjusted R2, and the slope and

-v-intercept of the regression line. All four models perfomed weÌlon the data collected from these experiments. The PC-BB mdAVS models in particulu provided very good fits.

To determine whether the models exhibited qualitative effects ofthe proximal and cenrer-of-mass orientations, we presetrted thecritical stimuli fo¡ the vuious experiments to the models andrecorded the output. By the logic of Experiment 1, il effect of theproximal orientation would be ¡eflected in a¿ov¿ ratiDgs in theoblique regions. Specifically, these model-predicted mtings wouldbe higher fo¡ the wide lmdmilk thæ for the tall lmdmük.Therefore, given each model's predicted ratings, we measured theaverage oblique rating relative to the wide lüdmilk, minus theaverage oblique Éting relative to the tall landmilk. If this numberwas positive, the model would show an effect of the proximalorientation. By the logic of Experiment 2, an effect of the center-of-mæs orientation would be reflected in higher aåove ratings forthe tall latrdmark, as compued with the wide landmark. Therefore,we measured the average oblique rating relative to the tall lând-mdk minus the average oblique rating relative to the wide lmd-milk. lf this number was positive, the model would show an effectof the center-of-mæs orietrtation.

Table 6 contains the results of these tests. All fou¡ modelsexhibited an effect of the proximal orientation, ild all modelsexcept for BB exhibited an effect ofthe center-of-mass orientation.It is cleil why the PC, PC-BB, and AVS models exhibited theseeffecß: They predicted them from the outset.T The BB modelexhibited an effect of proximal orientation, but not center-of-massorientation, due to the specific nature of the trajector placements.lo Experiment [, the placements were uanged so as to hold thecenter-of-mass orientation constant while manipulating the proxi-mal orientation, However, these manipulations also affected theGiven that prox¡mal and center-of-mass models did not va¡y across these

regions, the expected interaction betwæn relation and lmdmdk shape wasnot significant (F < l), except for the ânalysis of the direct region inExperiment 2, F(1, 38) : 6.0, p < .019, MSE = 0.4. The forñ of thisinteraction w6 directly opposite from the pattem of data in t¡e obliquercgion, Specifically, venical relations with wide landmtrks (M = 8.9) wererated higher than with tall landmdks (M = 8.5), whereas hodzontalrelations with tall landmùks (M = 8,9) were mted higher than with widelandmilks (M = 8.8).

GROI.JNDING SPÀTIAL LANCUAGE

Table 5Model Fits to Datøfrom Experiments l-7, Brokcn Down by landmark Shøpe

ìì¡.:liiì:l

iìl

Exp€riment 1Tall @ta¡gle (24 data points)

BB

PC.BBAVS

Wide æchngle (24 dstã poìnts)BBrcPC-BBAVS

Experiment 2Tall @tangle (24 dah poinb)

BBPCPC-BBAVS

Wide ræhgle (24 dâh poinb)BBPCPC.BBAVS

Experiment 3Tãll ptangle (56 data points)

BBPCPC-BBAVS

Wide Etilgle (56 data points)BBPCPC-BBAVS

Expedment 4Upright triilgle (4 dûh poirß)

BBPCPC-BBAVS

Invened rimgle (4 dah poitrb)BBPCPC.BBAVS

19 slope )-intercept

.982 .979 0.945

.963 .955 0.975

.995 .994 1.075

.996 .995 1.088

.97t .966 0.981

.954 .944 0.989

.993 .992 1.075

.994 .993 1.060

.995 .994 0.998

.973 .968 0.998

.992 .991 r.085

.993 .992 1.098

.984 .981 1.010

.960 .952 0.991

.994 .993 1.070

.995 .993 1.056

t The AvS nìodel predicts an cffect of thc p¡oximal oricntation underpúicultr conditions. The focus ofthe atentional bmm at the landmilk toppoint that is vcdcally âligred with the mjector or closest to being soaligned must coincide with ûe landmdk point that is closest to the lrajector

in euclidean distmce. This was the case for the miætor placemenb

examined in this experiment,

-0.3870.073

-0.6r6-0.614

-0.0760.202

-o.32s-0.323

-0.3910.007

-0.635-0.637

-0.372

-0.728-o.121

-0 .51 I0.936

-o.626-0.596

-0.3100.187

-0.43r-o.47

0.2t2u al l

-2.459

-0.329-0.1ó1-0.909-0.907

Experiment 5L shape (65 dah poin6)

BBPCPC-BBAVS

Experiment 6Tall bimgle (31 data poinß)

BBPCPC-BBAVS

Experiment 7Wide shngle (25 dab points)

BBPCPC-BBAVS

Critical points only(14 data poinß)

BBPCPC-BBAVS

Fit distly to dah(25 dah poinß)

BBPCPC-BBAVS

Fit distly to dsta,critical points only(t4 data points)

BBPCPC-BBAVS

Composite (337 dab poi¡b)BBPCPC-BBAVS

Model

.979 .917

.883 .873

.983 .982

.984 .983

.988 .987

.968 .965

.992 .99¿

.993 .992

aÁ?.967.993.991

.999

.987

.986

.990

283

1.0130.9031.0?91.060

1.062o.9311.0241.017

.941 .941

.862 .853

.943 .940

.976 .915

.750 .'123

.770 .734

.898 .882

.930 .9r9

01, o11

.957 .949

.916 .900

.965 .958

Nore. Adj = adjusted; slope dd )-inÞrcept de ftom rcgression lin6 of modcl. Unl6s nokd otheNise, model pameter settings were obþined by fifting

the modcls to Logd md Sadle¡'s (1996) data. BB = bound¡ng box; PC : p¡oximal æd center of mass; PC-BB = hyb¡id; AVS = attentional vætor sùm.

vertical md ho¡izontal distmce betwæn the trajector ild thelædmuk. For the wide lmdmuk, venical distance wæ greater ildhorizontal distance wæ less thm for the tall lmdmuk. Both ofthese factom contributed to higher mtings for the wide lmdnuk.Thus, for thse placements, the BB model produced results thatmatched our empirical findings.

However, the BB model's failue to exhibit a center-of-masseffect in Experiment 2 ügued against this model. In this experi-ment, the trajætor placements werc mged so as to hold theprcximal orientation constant while mmipulating the centercf-mass orientation. These tr-¿jæto¡ placements also held constant

0.9600.8510.8700.944

0.9210.9951.009l .168

-0 . l4 l0,546

-0.417-0.462

0.6650.9760.no0.169

1.048o.697t.4851.402

1.070 -0.56?1.015 -0.2201.043 -0.8191.068 -0.836

0.925 0.389t.174 -1.6661.496 -4.1261.820 -7.t66

1.130 -0.9481.054 -0.44'l1.102 -0.8¿161.028 -0.236

1.635 -5.1030.428 4.5520.243 6.01sr.138 - t.28'l

1.007 -0.u20.926 0.M11.01 -0.4501.031 -0.439

.097

.û7 .t43

.113

.555 .357

.983 .981

.918 .914

.978 .914

.985 982

.184 .7r9

.400 .133

.367 .086

.888 .838

.9r0 .909

.959 .958

.970 .970

both the vertical md the horiæntal distance to the neilest edge of

the lmdmuk. Because these BB-relevilt features did not varywithin Experiment 2, the model did not exhibit a cente¡-of-masseffæt. (In fact, the BB model exhibited a small effect in theopposite direction. This wæ the iDfluence of the sigmoid thatmuked the fa edge of the landmuk, opposite the trâjector. Be-cause the trajector was farther ftom this sigmoid in the cæe of the

wide lmdmilk, the model prcduced a slighdy higher rating for thislmdmrk.) A cente¡-of-mæs effect was found empirically, withthe sme stimuli. This disconfmed the BB model Criticauy, tltisdisconfimation holds not just for these Pilmeter settings of the

284

Table 6Efects of Proxíru| and. Center-of-Mass Or¡entations

Experiment

Note, BB = bounding box: PC = prcximal ild center of mass; PC-BB =hybrid; AVS = attentional vætor sum. Experimenr I tsted for rhe prox-imal orientation; Experiment 2 tested for the cente¡-of-ñass orienhtion. Apositive value indicatB that lhe model exhibited the effæt in question.

BB model, nor just for the particula implementation of the BBmodel tsted here. Rather, it holds for any pæmeter setting of anyimplementation of the model, as long æ the model is fundmen-tally based on vertical and horizontal distance.

Exper¡ment 3

In Experiments I md 2, multiple locations within the obliqueregion were probed to assess the influence of proximal and center-of-mæs orientation. However, only two locations were probedwithin the direct region, æd at those locations the proximal udcenter-of-mass orientatíons were confounded. Thus, the experi-ments did not offer an opporüìnity to exmine the effects oforientation within the direct rcgion. This limitation also held forLogm æd Sadler's (1996) spatial templates, as well as the tem-plates consEucted in other studies (e.g., Cdlson-Radvffisky &Logm, 1997; Haywæd & Tm, 1995). Thus, one purpose ofExperiment 3 was to exmine the possibility of orientatiotr effectswithin the diræt region. A second purpose was to replicate thecenter-of-mass effect obtained in Experiment 2. This was impor-tânt given that the only other reseucher to exmine orientationeffects on spatial tem acceptability sæmed to suggest that onlyproximal orientation played a significant role (Gapp, 1995). Fi-nally, it was not clø that the exact locaúon of the center of masswæ relevant, particulæly for highly elongated landmilk objects,for exmple, a lmp above a long conference table.s To exminethis, in Experiment 3, we used a luger wide rectanguld lmdmilkthat pemitted vrious placements within its dircct region. Thisvüied center-of-mass orientation while hotding proximal orienta-tion constant.

Method

Pan¡cipants. Thifty-six Univereity of Noue Damc undergraduates p{-ticipâted in exchuge for partial coume credit in il undergraduate psychol-ogy class. None had pdicipated in Experimenß I or 2.

Sriødl¡. Two lddmüks we¡e used: a ull @hgle and a wide Mtsdgle. Thselândmùks (25 mm X 75 m) were five times rhe size (in borhdimensions) of the lmdmùks used in Experiments I and 2 and pemittedmultiple placements of the tÉjector along the sùrfa6 ofthe lmdmdk (sæTable 7). In all, the @jecþr was placed in 56 læations douÍd eachlandmùk. The kajector was a circle with â radius of I pixel (0.42 mm). AsiD Experimenb I and 2, a text d¡splay præeded the onset of the picBre.This display contained the mting scâle, a sentence of the fom '"Ihe circleis above/beloøto the left of/to the right of the ræhgle," ud instsùctio¡sto press tìe rctum key to see the picturc.

0.596-0.003

0.347 0.3610.329 0.333

RECIER AND CARI.SON

AVS

0.0920.583

Empiricaldata

Procedure. '[he ptæedw frcm Experiments I ùd 2 was used. Pd-ticipilß mkd the âcceptability of a sentence as a description of a picturecontaining a landmdk ud a hjæto¡. In all, pmicipmß perfomed ¿148trials (2 Imdmùks X 56 placemenrs X 4 rclarions).

Results and Discussion

Empirical datø. The possible influence of the ænter-of-massorientation can be æsessed by exmining acceptabílity mtings fora single spatial tem. Accordingly, in the interest of space we focüson the data for aúov¿ while noting that similil effæts hold fo¡ theother relations. This is reæonable given that the ¡elations øáove,below, lei, md ¡igåf all belong to the sme class of relations(Logm & Sadler, 1996), as illustrated by the simililiry across thethree regions in the spatial templates for these relations in Exper-iments I md 2.

Mem acceptability ratings for above for placements of thetrajector ilound the wide and tall lildmilks de giyen in Table 7.For the wide rætangle, the cells of interst for æsessing contri-butions of center-of-mass orientation ae in rows l-2, colums3-9. Mean acceptabitity mtings for these cells were regressed onthe absolute value of the center-of-mass orientations using lineuregression. The regræsion equation revealed a significÐt influ-ence of canter of mass (betâ weight = -.78, p < .001, ovenltR' = .61). This indicatq that as the ceoter-of-mæs orientationbecomes more deviant from upright, the relation becomes lessacceptâble.

A similil pattem of rsults wæ found with the tall rectangle.The cells of interest ue in rows l-2, columns 5-7. Mean accept-abiliry ratings for these cells were regressed on the absolute valueof the center-of-mæs orientations. There was a signilicmt influ-ence of ænter of mass (beta weight = -.9ó, p < .004, overallR2 : .92). These ræuits fuither support the findings ftom Exper-iment 2 in showing a significant influence of the center-of-massorientation when proximal orientation was held constânt, contra¡yto Gapp (1995). Moreover, this influence was found in multipleregions across the spatial template: in the direct region in thecurent experiment ild in the oblique region in Experiment 2.

Model Jits ønd simulations. The results of the models' fits tothe data æ shown in Table 5. Each model wæ t€sted for aænter-of-mass effæt using the stimuli of Experiment 3 md ex-amining the model's predictions for the points in the top row ofthediræt region above the wide lmdmilk (points 3-9, in colum 3-9in Table 7). We used the wide lmdmuk because its width allowedmo¡e opportunity fo¡ a cente¡-of-nass effæt to reveal itself. Thersults ee displayed in Figure ó.

The empirical data ue displayed in Figure 64, ud the predic-tions of the different models ile shown in Figure 68. The empir-ical data show a cled peaft at the ænte¡ of the landmãk. All fourmodels also show such a peak. When compiled wit¡ the empiricaldata, the peak of the PC, PC-BB, and AVS models ile somewhattoo sharp, md the peak of the (already disconfimed) BB model issomewhat too flat, But all models do exhibit a peak æd thereforepass this qualiøtive test. Thus, for the model competition, thisexperiment did not eliminate my further models. The remainingmodels were still the PC. PC-BB. and AVS models.

--:-" We tha¡k Gordon L¡8il fo¡ this obseflation.

0.0930.1 14

isi+1F:rtritff;l'È',:s,,ìi

GROTJNDING SPATIAL LANGUAGE

Table 7Mean Ratings for Above by landrurk ønd Pldcerunt lor Experiment 3

' I

s

Rl 6.9 6.1R2 5.9 6.7R3 2.7 3.2R4 0.8 0.9R5 0.5 0.2R6 0.2 0.2R7 0 , t 0 . t

ì!l'.

.tlìtit!::a;:

R I

R4R5R6R7R8Þ o

R 1 0R I I

E.0 7.9' i i

* * 0 . 6 0 i0.3 0.3 0.5 0.4

Wide Ræhgle

E,7 8.9E.0 8.8

Experirunt 4

In Experiment 3, acceptability ratings peaked for the centemostpoint within the direct region and dropped off with distmce fromthat center point. The vilious models cm all account for such apeak, but they attribute it to difïerent factoÃ. According to the PC,PC-BB, md AVS models, acceptability will peak above or ner thecenter of mass, because all th¡ee models predict a center-of-mæseffect. In contrast, according to the (already disconfmed) BBmodel, acceptability wiu peak at the midpoint: the point equidis-tant from the two edges of the rectangle. In Experiment 3, mid-point and center of mass were at the sme location. The goal ofExperiment4 was to dissociate t-hese points, to provide mother testof the BB model against the other models. To this end, accept-abitity ratings were collected for four placements uound a widetrimgulu landmæk, as indicated in Figure 7. This lmdmak couldeither be upríght or inverted. Fo¡ both triangles, the critical poinsue A, B, md C. Point A is above the center of mass of the trimgle,Point B is above the midpoitrt of the base of the trimgle, md PointC is placed so that its distmce from B equals the distance betwænA æd B. The BB model p¡edicts no difference betwætr Points Amd C, because both ile e4uidistant ftom B and therefore equidis-tant from the edges. However, the PC, PC-BB, md AVS modelsall predict that Point A (center-of-mass point) will be rated signif-icantly higher than Point C. This is because these models uesensiúve to the center of mass rathe¡ thm the midDoint of thelandmuk object.

Method

Participanls. Thifty Univesity of Nohe Dde undergraduates pmic-ipated in exchange for either exh credit in il undergnduate psychology

c6

't.2 '1.5

6.7 6.53.0 3.0l.ó r.4t.0 1.30.9 0.60.5 0.20.5 0.40.3 0.20.2 0.20.5 0.3

Nat¿. Critical cclls are in boldface. R ild C rcfeÍ to rows dd colums, spectively, that comprise the marixùat defines the placements of the loøted objæt. Dash6 indicâte the læation of the rcference objæt.

c8

ai 8.5 7,4 7.2 6.98:t 8.3 7.4 6.4 6.4

2.9 2.70.6 1.30.4 0.3

0.3 0.2 0.1 0.2 0.30.5 0.2 0.s 1.0 0.2

Tall R4hgle

8.4 8.98.4 8.9

ao

0.1 0.30.3 0.4

cl0

8.2t:

-u.00.4

285

1.1 1.01 .2 1 .13 .1 4 .11 .5 1 .4l . t t .40.9 0.10.5 0.60.6 0.20.3 0.60.3 0.20.4 0.1

class or a payment of $6 per hou¡ of pffiicipation. None had pdicipatedin Experimenß 1-3.

Stimuli. A wide riqgle wEs cr4ted by cufting a diûgonal (top leftcomer to botom right) through wide ledmrks from Experimeût 3. lt wæprcsentel both upright aod invened, âs shown in Figure 7. The sañe fourlocations werc probed æu¡d both t¡idgles.

P¡ocedure. T\e p@edurc ftom Experimenß l-3 wâs used. In all,pdicipùts perfomed eight rials (two lddmdks x foùr placemenß).

Results and Discussion

Empirical data. MeN a€eptability mtings as a function oflandmdk (upright vs. inverted) md trajætor placement ue shownin Figue 7. The critical points (4, B, md C) ue in boldface.Acceptability ratings for thse critical points were submitted to a 2(lmdmuk upright vs. inverted) X 3 (points: A, B, C) repeatedmeæurcs ANOVA. There was no effæt of lmdmilk (F < l), amain effect of points, F(2, 58) = 4.5, p < -O15, MSE : 0.84, anda significilt interaction between lildmtrk md points, F(2,58) : 3.9, p < .O26, MSE : 0.34. To follow up the intemction, a957o confidence interyal ws constructed: a critical diffe¡enceof .30 wæ required for significmce. Thus, for the upright trimgle,PointC (M = 7.4) was signifrcmdy lower thil Points A (M = 8.1)md B (M : 8.0), which did not differ. For the inverted trimgle,Point C (M : 7.8) wæ significmt¡y lower thil Point B (M = 8.1)but not Point A (M : 8.0), ild Points A md B did not differ. Note,however, that the mems show the sme patt€m as for the uprighttimgle. These r6ults challenge the prediction made by the BBmodel that mtings will not differ betwæn points equidistant ftomthc midpoinq rather they support the importmce of ænter of massæ a feature for defining spatial relations.

286

9,5

9

ã

ë g

7.5

7

AÉsñmont3: EmÞlddt Ëutb

RECIER AND CARLSON

sensitive to the midpoint mther thm the center of mass. The BBmodel's failure to exhibit â ænter-of-mæs effect here seryes as afurther disconfimation of that model.

3 4 5 8 7Cotumn

B

g r.5

i '

Apof,msnl3: Modola

Sumrury of Experircnts 1-4

Taken as a whole, these fmt four experiments ild the accom-panying model simulations provide evidence of the proximal mdcenter-of-mass odentations in spatial tem mtings, They also rgueagainst models based on horizontal md vertical distilce from theedges of the ludmak. Thus, this first set of exlEriments supportsthe PC, PC-BB, md AVS models over the BB model.

Testing for the Effect of the Grazing Line

ID Experiments 5-6, we attempt to funher discriminate mongthe remaining models: PC, PC-BB, æd AVS. rüe do this by tstinga prediction of the PC-BB md AVS models: Points above thehoriæntal græing line will receive Ngher ratings thÐ pointsbelow it. This prediction is shæd with the already-disconfimedBB model. But criticalty, the PC modet, unlike thse otheß, has noexplicit græingline mechanism ud therefore might not be able toshow such m effect. Thus, testing for the effect might furthernffiow the mge of viable models.

Experiment 5

Above acceptability judgments for 65 hajector plaæmentsdound m upúght L-shaped lmdmuk were collected, æ shown in

8.1 8.0 7.4

3 1 5 8 7 8 9Column

F¡eilre 6. (A) Empiricâl data úd (B) ñodel results for Experiment 3.BB : bounding box; PC = proximal md cenþr ofmæs: PC-BB = hybrid;AVS : atlentional v@tor sum.

Model Jits end simul¿t¡bns. The results of the models' fits tothe data ile shown in Table 5. To detemine which models wouldexhibit the asymetry about the midpoint obtained in the empir-ical data, we examined the models' predictions for Points A, B,md C rctative to the upright triangle. (These predicrions werequalitatively the sme for the invened triargle.) Figure 8 displays(A) the empirically detemined acceptability ratings for thesepoints md (B) the predictions of the vrious models.

As discussed above, the empirical data show that Point A ismted considerably higher thm Point C. Tbe PC, PC-BB, and AVSmodels all capture this differenæ. This is to be expected, bæauseall thfee models re sensitive to the center-of-mass orientation. mdPoint A is located directly above the center of mass. The BBmodel, however, does not capture this aspect of the empidcal datå.It produces ratings that appeu to be entirely flat. However, closeexmination of the BB-predicted ratings reveals that they ile veryslightly symetrically peaked about the midpoint, so that Points Aæd C ræeive the same rating (A : 8.982, B = 8.995, C : 8.982).This midpoint peak is to be expæted, because the BB model is

A

8.5

Ë 8ã

7

6.5

E&odmort4: EnpliEl E0lb

GROIJNDING SPATIAL LANGUAGE

B

8,0

@

A

8.1

Fígure 7, The rpnght dd invefred tiagles used as lmdmrks in Ex-periment 4. Tmjætor Point A coæsponds to the center of müs of theIedmdk; Point B coæ$ponds to the midpoinl Point C is the smedishce from B 4 A is from Bi Point D is within boundinE box of theIandmark but below the $uing line.

9 s

7.8

@ @

B C

BPolnl

ãÞsrlment4: Mod€lg

.2 .6 .2 .2 .l . l .t .3 .4€ o o o ø

.2 .2 .2 .t .t .2 .t .2 .2o o o o o o

Figurc 8. (A) Empi¡ical data and (B) model ßulß for Experiment 4.BB = bounding box: PC = proxiñal ed center ofmassi PC-BB : hybrid;AVS = atlentional væto¡ sum.

Figure 9. Of these 65 placements, we were particuhly interestedin compuing the ratings of points that straddled the gru ing line.Thus, we compaed the 6 points that were above the bæe of thelandmuk and above the greing line with the 6 points that wereabove the base of the lmdm{k but the sme distance below thegroing line. These points ile shown in boldface in Figure 9. If thegræing line played a ¡ole in definitrg spatial relations, then pointsabove the grøing line would be rated significmdy higher thmpoints below the grzing line, with distmce from the line equated.

Method

Parl¡cipØts. Tweniy-four Univereity of Noæ Dame u¡dergÉduat$participated in exchilge for either exm credit in m undergÉduate psy-chology cl6s or a payment of $6 per hour of pdicipation. No¡e hadpdicipated in Expêriments l-4.

Sførr¡ì. Upright dd invefed L-shrpel srimuli werc used as lúd-mùks. The height of the bæe md the pidth of t¡e m werc both 25 mm,and the height of the m and the length of the base werc 63 ffi, The

8.0 't.5

8.0 7.7

5.2 4.5 5.0 3.0o o

FiB¡¡e 9. Placements dd m@ accepÞbility ratings ùound L-shapedlmdmùk in Experiñent 5. Groing line is indica(ed. Critical placementsde outlined ud appeû in boldfaæ.

tajætor from Experiment 3 vas used; it wæ placed in the sme 65læations æund Éch lildmdk s showd in Figure 9. Only the relationdåoy¿ wæ tsted.

Procedure, The p@edurc frcm Experimenb l-4 wæ used. In all,pa¡ticipub perfomed 130 tsials (2 landmrks X 65 placements),

Resulrs and Discussion

Empirical data. The isverted L-shaped stimulus did not haveplaæments of the located objæt that werc diagnostic of a gr¿ing-line effect. As such, these trials were conside¡ed Frllers. Memacceptability mtings for placements of the t¡ajætor ilound theupright L-shape æ shown in Figure 9, with critical points frmed.There wæ a significmt difference between the points above thegræing line (M : 8.7 acmss the six poinb) md the points belowthe gf¿ing line (M = 5.1), t(23) = 8.8, p < .001. These ¡æultssuggst that the græing line is m importæt feature for defìningspatial relations.

Model fts ød simuløtiorc. We presented the models withtrajector placements relative to the L-shaped lildmük. The rsultsof the fits æ shown in Table 5. rrVe then focused in padiculr onthe critica¡ plaæments above ild below the græing line. For eachmodel, we calculated the average predicted rating above the græ-ing line minus the avenge predicted rating below the græing line.A positive resulting qumtity indicated il effect of the gmzing line.The results ile shown in Tablc 8.

Table 8Efecß of the Grøzing Líne Relative to an L-Shaped IandmarkFollowing Experiment 5

A Spo¡nt

287

G@ing line

t.7

rJ

3.050

N¿l¿. BB = bounding box; PC = prcximal dd cen(er of ñass; PC-BB =

hybrid; AVS = aftentional vætor sum. Values shown æ the avemgemodel-predicted ratings for selæted points higher thM the greing lineminus the avemge for poiDts below the g@ing line. A positive valueindicates m effæt of the g@ing line, All models exhibit the effæt,

A.VS

3.738

Empi¡icaldah

3.528

288

All four models exhibit m eff€ct of the grazing line. For thePC-BB and AVS (md the discoRfimed BB) models, this is to beexpected, because the græing line is m integml pilt of the ilchi-tæture of these models. The PC model, in conkast, has no suchelement in its design, but it too exhibits the eff@t. This musr beattributei to the fact that the proximal ild center-of-mass orien-tations also vuied ac¡oss the critical points of the experiment.Thus, this simulation fails to qualitatively discriminare among themodels. Qumtitatively however, the overall measure of fit sug-gests that the PC model does not account as well for these data asthe PC-BB and AVS models.

Experiment 6

Experiment 5 provided initial evidence of the impofance of thegræing line. However, in that experiment, center-of-mass mdprcximal orientations were vuied ac¡oss the critical poins. Be-cause these features were not conaolled, the effect obsewed inExperiment 5 might be due to these features ud not to placamentrelative to the græing line. To address this concem, in Experi-mûa 6, above acceptability judgments were collæted for 3l tra-jector placements ilou¡d a tall trimgulu landmuk, as shown inFigure 10. Of thqe 3l locations, there were two critical pairs ofpoints (frmed in Figure l0) that satisFred the following criteria.First, the center-of-mass and proximal orientations ofpoints withina pair were equal. Sæond, one member of the pair was locatedabove the grazing line, ud the othe¡ was located below the grazingline.e Given t¡se criteria, my differences between points within a

Llne I Llns 2

REGIER AND CARIION

line could be att¡ibuted to an influence of the grazing line, but notto the center-of-mass or proximal orientations.

Method

Panicipants. Twenty-six Univeßity of Note Dame undergradualespdicipated in exchange for either exm crcdit in an undergraduate psy-chology clæs or a pâyment of $6 per hour of pdicipation. None hadpalicipÂted in Experifrents l-3.

St¡ñul¡, T^ll and wide ùiilgles were used as landmarks; thse werecreated by cùtting a diagonal (top lcft comer to bottom ¡ight) thrcugh thetall and wide lmdmdks frcm Experiment 3. The Þjec(or from Experi-ment 3 wâs uscd; ¡t was placed in the same 3l læations aroùnd %chlandmdk. These ùe shown suÍounding ihc tall landmtrk in Figurc 10.Only the rclation dóov¿ was tested.

Procedure. The prccedue from Expcrimeob l-3 was uscd. In all,palicipants periomed 62 úials (2 landmdks X 3l placements).

Resuhs and Discwsion

Empiricøl data. the wide triangle landmilk did not haveplacements of the located object that were diagnostic of a grazing-line effect. As such, these trials were conside¡ed frllere. Meanacceptability ratings for placements of the tajecto¡ üound the talltriangle ae shown in Figure 10. The critical points ile ftamed. Todetemine whether therc was a gr¿ing-line effect, we submittedthe acceptability ratings for these 4 cells to a 2 (gwing line: abovevs. below) X 2 (line: 1 vs. 2) repeated measures ANOVA. Therewæ a main effect of grazing line, F(1, 25) = 25.8, p < .001,MSE = 7.94, a main effect of linc, F(1, 25) : 4-3, p < .049,MSE = 3.61, md a significant interaction, F(1,25) = 19.2, p <.00t, M.çE = 1.57. To follow up the interaction, we constructed a957¿ confidence interval on the basis of the eÍor tem for theinteraction; a critical difference of,70 was required for signifucilce. Thus, ratings were signif,rcudy higher when the point wasaboye the græing line than when the point was below the grazingline, both for the points on Line I (Ms : 5.6 vs. 3.9, respectively)md for those on Line 2 (Ms : 5.9 and 2.0, respectively)- Thiseffect was present despite the fact tbat the pairs of poins alongthse lines shred the sme center-of-mass and proximal orienta-tions, Thus, the sufficiency of these orientations to predict spatialrelational use is questionable; rather, a grazing{ine feature mightalso be necessary.

Model fts and simulations. T\e results of the models' fits tothe data ile shown in Table 5. We tested the models using thestimuli of Experiment 6, focusing in particuld on the two criticalradiating lines, collæting model-predicted ratings at both pointsalong each line-one point above the grazing litre and one below.The value of interest was the raúng for the higher point minus therating for the lower point, with a positive value indicating agræingJine effect. The results ue displayed in Table 9.

The PC model failed to exhibit a græing-line effect. This was tobe expected. The PC model relied only on the center-of-mass andproximal orientations, ild these fqturcs were held constant alongthe radiating lines of the experiment. Thus, we expected no dif-ference in predicted rating between poitrts along a radiating line:The difference scores would be ærc. The difference scores actu-

(a) s.z@

4,2@

G@ing line

.2 .3 .4 .4 .3

FiBurc 10. Plãceñents and ñøn accephbility raúngs ùound the hllhiùgle ledñdk used in Expe¡iment 6. The four critical po¡nts on the twomdiating lins ae oudincd ild apped in boldface. Solid radiating linesind¡cate center-of-mass orienÞtion. Ddhed lines indicate Droximal o¡ien-tation.

Table 9Grazittg-Line EÍfect Followirtg Experínent 6

@_3 @

@

Line I 1.975 *0.003 2.541Line2 -0.078 0.048 298i

Nor¿. BB = bounding box; PC = proximûl and centcr of mass: PC-BB =hybrid; AVS = atrenrional vecror sum, Each value shown is thc model_prcdicted rating for the h¡gher point along a radiating line. minus rhenrodcl-predicted rating for rhe lowe¡ point along the sâme line. A Dositivenumbcr indicütes a graz¡ng-linc effect.

ally obtained from the model were nedly. but not exactly, zero.The minor deviation resulted from the model's intemal reDresen-tation of the triangle as a set of poinrs in the two-dimeìsionalplane. This representation required that the continuous hypotenusebe discretizcd into a finite set of points. This discretiation gave¡ise to minor diffe¡ences across points in the calculation of theproximal orientation. hence, the minor difference in mtings. Thecriticâl point was that the PC model failed ro show an effect ofgrazing line when orientational features were held constant. Thisdisconfirrned the PC model. Moreover, it disconfimed æy imple-mentâtion of rhe model, as long as it was based solely on thecenter-of{nass and proximal orientations.

The PC-BB and AVS models both showed a cleil effect of thegrazing line. Again, thjs was to be expected. Both modeÌs explic-itly relied on two components: (a) a height function bonowed fromthe BB model that highlighted rhe grazing line 4d (b) m orien-tational element. The proximal and center-of-mass orientationswere controlled for in this experiment, so that the influence of thegrazing line was isolated. This influence was captured in thesimulat ions.

(Thc already-disconfinned BB model showed the effect forLine l, but not for Line 2. This failure might seem surprising,because the BB model was by definition sensitive to the grazingline. However. it was also sensitive to ho¡izontal distance. InLine 2, the higher point was ho¡izontally quite fu from thelandmark bounding box. Thus, this point received a very lowrating, even lower than the rating for the other point on the line,which was below the grazing line but neil the bounding box.Because the higher point received a lower rating, the model failedlo exhibit the effect. We did not take this as a further stronedisconfimation of the model. however. The BB model would noiencounter this problem, given other pammeter settings. Thus, weviewed th¡s result as a minor implementational problem rather thana clear falsiircation. However, Experiments 2 and 4 had alreadydisconfirmed this model in its generality, not merely in thisimplementation.)

Testing for the Effect of Distance: Experiment 7

Across six cxperiments, only two models have passed all qual-itative empirical tests and have also provided tight quantitative fitsto the empirical data: AVS md PC-BB. The goal of Experiment 7was to discriminate between these two remaining models by fo-cusing on the contrasting predictions that the models make con-ceming the effec( of distance on acceptability rarings. The AVS

@ @

GROUNDING SPÀTIAI LANGUAGE

AVS

3.O31

Empiricaldata

model predicts that if one werc to move a trajector back ud forthover the sudace of il object, but at a low elevation, ratings wouldnot be very sensitive to the centeredness of the trajector above thelandmdk. However, at a higher elevation, there would be greatersensitivity to centeredness. The PC-BB model makes exactly theopposite p¡ediction: Ratings itr the lower row would be moresensitive to centerednqs. Thus, these two models cil be distin-guished by compring ratings for placements of the t¡ajector thatæe very close to the lmdmuk with ratings for placements that defr from the landmuk.

Method

Pañicipants. Fofty-seven Univeßity of Nore Dme undergmduâtespdicipaled in exchmge fo¡ either extra credit in s ùudergüduaþ psy-chology class or a payment of $6 per hour of pa¡ticipation. None hadpdicipated in Experimenb l-6.

Stiûuli and prucedurc. The wide @hglc lddma¡k ftom Experiment 3was used. The tsajætor wð placrd at 25 l@tions smudi¡g ttE lûdmÀk(sæ positions indi@bd by t¡e aæephbitiry Erings in Tabte l0). Only rherelation d¿ov¿ w4 tsted. The prccedw fiom Experimenb l-4 was used.

Results and Discussion

Empirical deta. Meú acceptability mtings for above forplacements of the trajecto¡ õound the wide landmuk ue given inTable 10. The cells of interest ile in rows l-2, colums 2-8. MeÐacceptability ratings for thæe cells were ¡egressed on the absolutevalue of the center-of-mass orientations and distance (explicitlycoded as I or 2, conesponding to rows l-2) for the 14 cells usinglineil reg¡ession. The ¡egrcsion equation revealed a significmtinfluence of center of mass (beta weight = -4.09, p < .002) mda significilt interaction betwæn center of mass md rcw (betaweiêht : 3.87, p < .008). There wæ no cffect of distaace (p ).15, ovemll model Â2 = .76). As shown in Table 10, the effect ofcente¡ of mass was greater for trajætor placements that were filfrom the lmdmak (row l) thu for trajector placements that wereclose to the landmrk (row 2), as predicted by AVS.

Model ftts and simulation& 'lhe model fis for this ex¡rcrimentæ shown in Table 5. under 'Experiment 7, Wide Htangle." Toqualitatively tst the models for a distÂnce effæt, we eorded æchmodel's predicted valus fo¡ the critical stimuli used h the experi-menl As in the experimenE we fæu$d on poinls in the diræt region,in colms 2-8, rows I md 2. We læked for ætrteredns effætswithin thse two rcws. The Hulrs æ displayed in Figæ ll. The

Table l0Mean Ratings for Above by Placemefi for Experiment 7

Colum

1.7301.88s

eThese displays werc inspi¡ed by obseflations made by AnnetteHeßkovits.

289

R IR2

R4

6.3 7.8 8,3 8J 8.E 8.4 8.2 7.7 6.24.'1 7,7 8.1 83 83 8.3 8.3 7.7 4.31 .3 t .40.8 0.9 0.7 0.1 0.8

¡,fore. Critical cells æ in boldface, The spacing of the rcws reflæb thedishnce milipulation. R and C refer to rows dd colums, rcspectively,ú¡at comprise úe mat¡ix that defines the placement of the loÉted objæt.Dæh6 indi@te the location of the reference obiæt.

290

A

I

8.8

> E.4

Ë e.zå 8

ã t €

< 7.6

7.4

7.2

7

Exp€dmont 7: Emplrløl @ultD

REGIER AND CARTJON

c

B

4 5 6 7 8

Column

l=TÞr€wì I1...¡...t-mrql I- 5

! " E

< 7

EerlmontT: BBmod€l

ExporlmontT: PC nod6l

D

4 5 8 7

Colúmn

Expodm€nt7: Pc€Bmodsl

7

E

empirical data ue displayed in A, md the predictions of thedifferent models æ shown in subsequent frmes. The empiricaldata show a cleil peåk in the rarings of the upper row, replicatingExperiment 3. But the ratings in the lower ¡ow ãe relatively flat.Thus, as the AVS model læds us to expæt, there is â greatereffectof centeredness for high points thæ for low points. The PC-BBmodel fails to exhibit this effect. Insrqd, it predicts flaner ratingsfor the higher row thm for the lower ¡ow, the opposite of what wefind experimentally. The already-disconfimed BB ild pC modelsalso fail to exhibit the effect. The PC model behaves as the PC-BBmodel does, and the BB model shows lugely flat respoqses at bothelevations.

The AVS model dæs exhibit the predicred qualitatíve effect:The ratings æe more peaked in the upper row thm in the lowerrow. And æ shown in Table 5, its fit to rhe data is ber¡er thm thatof the other models. This is kue when all 25 data points ãeconsidered ("Experiment 7, wide Htangle") md also when werestrict our attention to the 14 critical points shown in the Frgure.This suppots the AVS model over irs competitom.

A Expsdmnt 7: BB modsl, fü to exporlmntat data

3 z

5.5

Êxparlm.ntT: ÀVS modol

2 3 4 5 e 7 8

Column

Fí84¡¿ JI. Resulß ofExperiment ?. Püel A shows emPi¡ical data; Pmels B-E show prcdicþd outPut ftom

bounding box, pÌoximal md center-of-mæs. hybrid, and attentionål vætor-sum models' rspstively.

GROUNDING SPATIAL LANCUAGE

However, the AVS model dos not capture the details of thsedata perfætly. Itr the AVS simulation, the ratings for the upper rowæ all considerably higher thil the mtings for the conespondingpoints in the lower row. ln the empirical dara, in conhast, this isnot always the @se. This nisæ a quction: Is the AVS modelcapable of providing a closer fit to the Experiment 7 datâ, underother pmeter settings? If not, its status as the suryivo¡ of ourcompetition could potentially be called into question. To ûest this,we adjusted pmeteß to optimize the fit of each model dirætlyto the Experiment 7 data. The resulting pilmeter values ueshown in Table I ild the fits ile shown in Table 5. As before,numben æe given for the fit to the Expedment 7 data set æ awhole ("Fit directly to data") ild fo¡ the critical points only.Again, the AVS model provides the best ove¡all fit. The BB fit isvery neãly as good, but that model has al¡ady bæn disconfrmedon other grounds. Significutly, when we exmine the criticâ¡points only, the AVS model clwly provides the best fit-a muchtighter fit than obtained with the original pilmeter sertings mdtighter than that of my other model. Figure 12 displays each

" Exporlmont 7: PC-BB modol, fit to exPsrlmental data

= ö.Þ

E 8

7

B ExperimentT: PC modâl,fittoexporlnentel dsta

291

4.2

ã! 7.8

7.4

7.2

8.4

a.2

È 8ã 7,8

È - ^8 , . "< 7.4

^--1-^

"---=f ' *----o_

o. . . . . - . . Í 1 . . . . , . . , .

¡ ¡ _ ^ ^

7.2

7

c o l u m n z i 4 5 6 7 8

Column

Frþ¡¿ 12. Rsulß of Expeliment 7, with model pâmeteß set spæificålly to fit these daÞ. Puels A-Þ showpEdicted outpùt ftom bounding box, prcximal úd cenþr-of-mæs, hybrid, &d attentional vætor-sum modelsHpætively.

pücnøo1[ ¡ Løerrøl

DExFsdmsnt 7: AVS modol, ñt fo oxperlmontal dsts

8.88.68.4

Ð 8.2

E 8

å 7.67.47,2

7

292

model's new fit to the critical points from this exp€riment. Asbefo¡e, the AVS model exhibits the qùalitative effæt of peakcd-ness as a ñrnction of distance while also prcviding a fairly close fitto t¡ese points overall. The other models again fail to exhibit theeffect, even when fit direcdy to thse data. Critically for ourpurposs, these fìndings support the AVS model md also discon-fim the PC-BB model, the only remaining competiior undcrconsidemtion.

Composite Model Fits

As a final test of the models, we exmined their Frts to all of owexperimental above datÂ, pooled together. Herc, we again usedmodel pümeter valus obtained ftom fitting Logm ud Sadler's(1996) data, not our own. This yielded a set of overall compositeñts, shown in Table 5. All four models perfomed monably well,with the AVS model providing the best fit. Figw 13 shows thesecomposite fits. A-ll empiricaly obtained above data from Expcri-ments l-7, with the predictions of each model md descriptions of

REGIER ÀND CARLSON

the stimuli, ue ayailâble at htÞ://www.æp.uchicago.edu,/-rcgier/avs.

Individual Differences

The AVS model also accoutrts for some individual differcncsin pattems of above nlings. We exmined thc responss of indi-vidual pa¡ticipmts md found tbat a good prcportion of them(22Er58%, dépending on the experimeqt) gave a muimal rc-sponse as long æ the trajælor ws diMdy above some part of thelmdmuk. Thus, thæe individuals gave respons that werc flatacross the diræt rcgion; however, other individuals did not prcvidesuch flat rcsponses. The AVS modcl accounts for both Hponsepattems, with different pffieter settings. If the attentiotral bem-width ftee pmeter is exFemely nmw, therc is only negligibleintegmtion over the lmdmuk rcgardl$ of height. Under thæecircustancs, the vætor sum wiu be perfædy aligned with up-right vertical æ long æ the Fajætor is above some part of thelildmilk-yielding flat mtings of the son obseryed. But when the

C PC-BB: R2 = 0-959

ConñftrcS

AConÞ6bfÈ BB

BB: R2: 0.953

I

E

7

c 3

2

1

0

úpodm¡blt obhh.d nüng

CompollbflÈ rc

4 6

atteDtional bm is broader, the AVS model integrates over thelmdmuk, prcdicting nonflat mtings of the solt obseñed. Withdifferent pilmeter settings, the AVS model exhibits the qualita-tive effects shown in both the flat md nonflat responses mdprovids good qumtitative fits to both response types.

No model other thm AVS accoutrts for both ¡esponse pâttems.Howeve¡, the BB ud PC-BB models cu account for the flatHponses, with apprópriate settings of their pmeten. Thus,dspitc their eliminatiòn'in ou model competition, there is partialsupport for thse two moSels: They account for a subclass ofindividuals. The PC model, in conbast, cmnot consistently ac-count for either clæs of response. Details may be found at http://www,ccp.uchicago.edu/-regier/avs/diffs-results.htnl. Thus, bothaggregate rcsponses and individual differenca in response provídesupport for the AVS model,ro

General Discussion

Ou studies confm the three central predictions of the AVSmodel. Fißt, spatial tem mtings ue influenced by the proximalmd center-of-mæs orieDtations. Second, mtings de sensitive tothe groing line. Third, mtings re affected by distance. The AVSmodel also provids a ræonably good fit to ourexperimental data.In contrast, none of its competitoß pass all empirical tests. mdnone fit the data ö well.

What is the significilce of this finding? What does ir rell usabout the relation betwæn lmguage ild peræprion? The AVSmodel is motivated in part by the role of attention in spatialperæption md by vector-sm coding of direction. Thus, to theextent that its success stems from these independently motivatedsorcs, it provides a preliminary grcunding of linguistic spatialcategoris in nonlinguistic perception

As it happens, its succss siems onìy itr part from these sources.Recall that the AVS model consists of two components: theattentional vætor sum, which is independently motivated, md theheight functiotr, which is not. This height function was bonowedfrom the BB model for purely data d¡iven ¡easons. It explicitlyloweß mtings for poitrts below the græing line, yielding a gruing-line effect. Thus, we cmor attribute rhe AVS gruing{ine effectto independently motivated aspects of the model.

However, the effects of proximal md center-of-mæs orientationmd the distæce effect æ attributable to the attentional vætorsum. As we have sæn, thae effects emerge from thc interaction ofattention md vætor-sum coding. In addition, much of the model'ssuccess in quiltitative fits to the data may also be attributed to theattentional vector sum. Spæifically, the attentional vætor sumaccounts for most of the viliation in mtitrgs for bajector positionsthat õe well above the groing line. We know this bæaùse theheight function is alrnost unifom across these positions. Thecdtical point is that significmt aspæts of the AVS model's successmay be haced to its independently motivated components. Thus, itgrounds some æpects of spatial lmguage in perception.

We de not the fißt to put forth such a proposal. Lmdau ildJackendoff (1993) noted that spatial tems tend to encode thenatue of the related objæts in a sketchy or schematic fashion,specirying only their gross perceptual chæcteristics. They pro-posed that this fact reflæts the underlying treural representation ofspatial relations. In particulù, they noted that the visual system isdivided into two cortical pathways, one capturing where an object

PC: R2 = 0.910

kFdmãnhl¡y oÞblnod ãüng

Figu¡¿ ,J. Composiûe model fits to pooled dah ftom all experiments (l-?). (A) Bouding box model (R2 =

.953). (B) Prcxinâl ùd cente¡of-mas model (À2 = .910) (C) Hyb¡id model (P = '959). (D) Arentionai

vætor-sum model (R'? = .970).

a

GROUNDING SPATIAL LANGUAGE

0 2 4 6 0tFfr.últo$lnd nüñg

is md the other capturing what it is (Ungerleider & Mishkin,1982). They suggæted that the schematic nature of objæt repre-sentation in spatial tem semmtics might rsult from this neuraldivision of labor. Because spatial tems code location, they trepresumably underpimed by the wåe¡e pathway, ild this pathwaycodes the objæts themselves in less deøil thu dos the wl¡¿fpathway.

Our ugument bm simililitis to this one md differencs fromit. The most importmt similility is that our overall point isessentially the sme: Linguistic spatial categories cm be explainedin tems of underlyitrg structures thât re not linguistic in chæacter.We fæl there is il importilt difference, however, conceminglevel of detail. A stength of the AVS account is that it is quitefine-grained, relying on qumtitative matches to empidcal datamther thm perceived conceptual comonality between il aspectof spatial lmguage md il aspæt of visual sys¡em æhitecture. Itis this fine grain that allowed us to discriminate betwæn the AVSaccount ild other plausible-sæming accounts against which it waspitted. Of course, we readily acknowledge that we sacrifice thebreadth of coverage that Lædau md Jackendoff (1993) haveachieved.

The work of Haywad md Tffi (1995) is aother natuml com-prison. They collæted spatial tem acceptability judgments in thesame mmner that we did md noted participmts' perfomilce ona relat€d but nonlinguistic spatial memory task. They found thatthe best perfomæce on the spatiâl memory tâsk occured at lheprototypes, or best exmples, of the linguistic categoris (i.e., inthe direct region above, below, to the lcft, or to the right of thelmdmilk). They concluded that becaùse perfomace peaked atthe same locations for both their linguistic æd nonlinguistic tasks,the sme structures might subserye both tasks. Cmwford, Regier,md Huttenlocher (2000) further exmined this issue, using similutasks md additional malyses. Although theù finditrgs ilgueagainst some of Haywüd ild Tu's (1995) claims, they furthersuppofed the geneËl notion that skucture is shæed across linguis-tic æd nonlinguistic spatial categoris. Again, the over¿ll point isa similu one: the possible grounding of spatial language in spatialperceptron.

It is also natural to compile this fesult with ølier work in thesemantic domain of color. Berlin md Kay (1969) fouud thatdifferent lilguages have different color categories. For exmple,not every læguage has a word directly translatable as Englishpürp¿e. More inter$tingly, they also found substantive constraintson this crosslinguistic vdiation. They found that the bst exuplesof color tems, across lmguages, occuned at tbe sme 11 focialong the color spætrum, regildless of the color categoriætion

AVS: R2 = 0.970

293

¡o There is a posible qualificâtion to our claim thêt ú¡e AVS modelaccounts well fo¡ our data, both individual md aggfegaþ. Like the olhermodels. AvS prcdicb left-right symeky of Étings rclative to symetsrical lmdmdks. It is not clø whether this prediction is entircly valid. InExperiñenß 2 dd 3, poinb on the right side ofspûæ @eived very sl¡ghdyhigher râtings the coEspondin8 points on the left. The mm left-¡ighldiffercnæ wæ -0.1236, t(7l) : -3.2186, p = .0016. We found no ñ¡ftherevidencc of such a asymetry, however. In fact Experiment 7 shoved ânonsigoificot üend i¡ the opposite direction (M = 0.08), (10) = 1.70,p = .12. Given the inconsistent naturc of these findings ùd the ext¡emelysmall siæ ofthe asymmery we did frnd, we æ comfortâble advilcing thesymmety-predictiûg AVS model.

294

scheme of the lilguage in questiou; Kay and McDmiel (1978)presented a reduction of these linguistic color foci to the neuro-physiology of the visual system. In particulù, they noted a phys-iologically obseñable neural coding for the 4 color categories red,yellow, green, md blue in the opponent response cells of the latera.lgeniculate nucleus and suggested a shaighdoruæd mechanism fo¡building the other color foci up out of these. Thus, their worksuggested that the obseiled conskaints on crosslinguistic viliationresult from the fact that all humms shæe essentially the smeperceptual appüatus.

Althoùgh we have not yet explored the crosslinguistic applica-bility of the AVS model, we ue eager to do so. Lmguages vary intheir structuring of space, but there ùe limits to this vüiation(Boweman, 1996; Talmy, 1983), just as there ãe in the domain ofcolo¡. Because the AVS model is grounded in putâtively univenalperceph¡al prffiss, it would bolster our ilgùment if the spatialtem acceptability judgments of speakers of other læguages alsoappeüed to implicate such processes. For instmce, the AVS modelappem to be consistent with Geman data collected by Gapp(1995) but witl¡ different pmeter settings from those dictâted byour English data. We intend to investigate this furthe¡ in the neufuture.

Limitations and Extensions

The work æ it studs is limited in several imDortmtresDects andcalls for extension.

Trajector Shape

The shape of the trajector is limited to a single point in thecu[ent model. A simple extension to this model would allow forextended tajectors, Such a new model would use vectors rooted atthe lædmilk, pointing to the closest point on the trajector. This isclerly equivalent to the cu[ent model in the point-trajector casemd may be adequate for other trajætor shapes as well. We intendto invstigate this in future reseilch

Object Function

It has been shown that the functional relationship between thelmdmak and trajector affects the mæner in which spatial conftg-umtions ue described (Calson-Radvmsky, Lattaui, & Covey,1999; Crlson-Radvmsky & Tmg, 2000; Culson-Radvmsky &Radvilsky, 1996; Coventy, Cmichael, & Guod, 1994; Cov-entry & Prat-Sala, 1998). For exmple, Cdlson-Radvansky et al.eliciteÅ above mtings for a coin relative to a píggy banl{. Theyfound that people gave the highest ratings when the coin wasplaced above a ñrnctionally importat part of the piggy bmk: theslot through which it was memt to drop. This was found evenwhen this functionally impofant paft wæ dissociated from thecenter of mæs of the piggy bank. More specifically, across pü-ticipmts, the slot of the piggy bank was moved, so that sometimesit was at the back of the pig, sometimw it was in the middle,coinciding with its ænter of mass, md sometimes it was at thefront of the pig. The criticat finding was that the peak in theacceptability ratings for placements of the coin above the piggy

bank shifted in accordmce with the position of the slot.

REGIER AND CARIION

Functional featuÍes ile notably absent in our displays, md thusthe AVS model does not cuffendy address function. We considerthis a serious limitation. However, we also feel that a naturalextension to the model might be able to account for functionaleffæts. Spæifically, the functional p¿rt of an object might receivegreater mounts of attention thm the remainder of the object.There is some support for this notion: Lin ud Murphy (1997) haveshown that people more quickly md accumtely detect that a Part ismissing from an object when it is a funct¡onal part as opposed toa nonfr¡nctional part. Thus, the funct¡on of m object part wouldbias the attentional yector sum, which would retum higher ratingsfor positions neu the functional part of the object. In this milner,some functional effects may be explicable in tems of the AvSmodel. A fomal version ofthis extension of the model is cunentlybeing explored.

Izxical Competílion

Finally, another issue concems the height function used by the

AVS model. As we have sæn, this part of the model is somewhatad hoc, in that it was chosen to dirætly reflect the data rather thm

explain it. Thus, it is the "unmotivated" comPonent of the modelOne possibly rclevilt obseryaúon in this regild concems leícal

competition (MacWhinney, 1987). This is the notion that theapplicability of one tem in the lexicon is âffected by the aPplica-bility of other related tems. For exmple, a trajector that is belowa lmdmtrk provides a poor exmple of aåave precisely bæause it

is below: TIte two tems õe opposites. Humans do have to select

mong contending lexemes, md there is curently no locus for

such a competition in the AVS model, We speculate that at least

some aspects of the height function we ile cunently using may beexpiicâble in tems of leÃical competition between oPPosing terrrs.This is mother issue for future studies to address.

Effects and Mechanisms

One general, broadly applicable conclusion that may be drawnfrom this work is distinctly cautionary in chmcter. This conclu-

sion is that effects næd not reflect underlying mechmisms in a

straightfovæd, transpüent fashion. This point has been made

several tim* recendy, particulilly iu the connect¡on¡st liteÉture(Etmm et al., 1996; Rumelhart & McClellmd, 1986; Seidenberg &

McCleltmd, 1989). We add our voices to this chorus' In our case,

the PC and PC-BB models account for the effects of proximal mdcenter-of-mass orientations in a straightfovüd fæhion: The ef-

fects ile ssentially built into the üchitecture. Ultimately, how-

ever, we have concluded that this easy answer is the wrong one.

One reæon for this decision is that the easy models ile not tight;

they do not frt ou data as well as does the comPeting AVS model.

Another reason is that the easy model is not particululy illumi-

nating. To enshrine effects as mæhmisms would be to make do

with a descriptive nodel of the phenomenon, without æy explm-

alory links to other known phenomena. The AVS model, in con-

trast, does provide such links while accounting fo¡ the sme

empi¡ical effects and others. In so doing, this latter model gives us

an initial grounding of some asPects of sPatial lilguage in the

nonlinguisúc structures and processes of visual Perception.

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Appendix A

Demonstration That the Attentional Vector-Sum Model Produces the Cenþr-of-MassOrientation When the Attentional Field Is of Uniform Suength

LÆt d be the ûnifom attentional shength, (r,, ),,) be the loøtion of the Ìl/e may divide by ø, bæause that will not affæt the di@tion of the

mjæto¡, (x/, )r) be the læation of point i of the lûdmùk objæt, ed n be vecror. This yields the following:the nuñber of poinß in the lmdmdk. Then the aftentional væro¡-sumñodel's vætor sum is æ follows:

),¿,= ) "ð,: " ) ô,: a ) k,, - x, t, - tìt ¡

= d[> (¡,,- ¡), ) ty"- y'¡l = "(*"

- ) x¡ ¡y, - 2 yì)

=*(.- l ; ,"- l i)

GROUNDING SPATIAL LANCUAGE

Appendix B

Mean Acceptability Ratings for Each Placement of the Trajector for EachSpatial Relation for Wide and TaII Landmarks in Experiment 1

The di¡ætion of this vector is the center-of-mðs o¡ientation, bequse the

lúdndk's center of mass is at (>¡ 4/n, >r yrlr).

/ " - s r . , - s ¿ \\ ' ' i , ' " ' i " )

6.7 - 1.4

s.6 "6.60.9 0.90.6 0.30.4 0.4

0.4 0.5 0.3 0.3 0.30.3 0.3 0.4 0.4 0.30.8 1.0 0.9 0.95.9 6.2 8.6 6.4 5.46.4 7.2 8_6 6.9 6.6

8.98.9

o¡0.3

7.4 6.86.2 6.01 .0 1 .30.4 0.60.6 0.3

6.7 5.8 0.6 0.4 0.36.9 6.5 0.9 0.3 0.28.4 8.6 0.4 0.47.0 6.4 0.8 0.6 0.36.8 6.0 0.8 0.3 0.3

0.6 0.3 0.8 5.9 6-90.5 0.7 0.8 6.9 6.90.4 0.3 8.7 8.60.3 0.3 1.2 6.8 7.20.4 0.2 0.1 6.0 6.7

6.5 7.3 8.9 7.O 6.96.2 6.4 8.4 6.9 6.20.7 0.8 0.1 0.80.4 0.5 0.3 0.4 0.30.4 0.4 0.4 0.3 0.3

N¿re, Dæhs indicate rhe [Þsition ofthe rcfercnæ object.

(Append¡xes conlînue)

0.s 0.6 0.2 0.4 0.90.5 0.8 0.6 0.3 0.40.8 0.8 0.7 0.95.9 6.8 8.8 6.4 6.r6.8 7.2 8.8 7.5 6.5

29'l

6.1 5.5 0.9 0.7 0.77-O 6.4 0.8 0.5 0.38.7 8.9 0.4 0.47.t 6.3 0.8 0.3 0.36.3 5.5 0.8 0.3 0.3

Right

0.4 0.5 0.9 6.0 6.70.4 0.5 0.9 6.3 6.60.4 0.4 8.6 8.80.5 0.3 r.0 6.3 6.90.3 0.6 Ll 5.4 6.2

298 REGIER AND CARLSON

Appendix C

Mean Acceptability Ratings for Each Placement of the Trajector for EachSpatial Relation for rtry'ide ild Tall Landmarks in Experiment 2

6.6 7.1 8.7 1.7 6.96.3 6.7 8.6 't.O 6.3t .2 l . l 1 .5 1 .20.3 0.4 0.5 0.4 0.40.5 0.4 0.3 0.3 0.5

Tall

0.9 0.6 0.3 0.7 0.60.6 0.6 0.3 0.4 0.8r . t t .3 1 .2 1 .25.8 6.8 8.8 6.9 6.06.5 7.1 ',t.g 7.6 6.8

6.3 5.8 l. l 0.8 0.66.8 6.2 1.3 0.6 0.89.0 8.7 t.2 0.46.ó 6.1 1.0 0.6 0.66.s 5.8 0.9 0.3 0.7

0.1 0.4 t.2 5.8 6.50.5 0.3 1.1 6.7 '1.1

0.3 0.6 9.0 8.9o.'7 0.5 0.9 6.3 6.60.3 0.5 t. l 5.8 6.4

6.7 7.O 9.0 7.4 7.15.9 6.8 8.9 6.7 6.4l . l 1 .2 1 .2 1 .60.6 0.6 0.4 0.1 0.10.6 0.5 0.9 0.9 0.9

Wide

lvofe. D6hes indicate the position of tÌ¡e rcfercnce objæt.

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Cognitive Arithmetic Across Cultures

Jamie I. D. Campbell and Qilin XueUniversity of Saskatchewm

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Reæived October 5, 1998Revision received Much 28, 2000

Accepted May 15,2000 I

(C3nadian university students either of Chinese origin (CC) or non-Asian origin (NAC) ed Chineseunìvèrsity studen$ educûted in Asia (AC) solved simple-dithmeric problems in the 4 basic opeEtions(e.9.,.,3 + 4, 1 - 3,3 Y, 4, lZ + 3) ând rcported r.t¡ei¡ solution smtegis. They also completed astandùdized tes( ofmore complex multistep dithmetic. For complèx dithmetic, ACs outperfomed bothCCs and NACs. For simple üithmetic, howevet ACs úd CCs wcre equal ild both perfomed befte¡ thmNACS. The superior simple-dithmetic skills of CCs relative to NACS implis that exmcuriculdcùlmre-specific factoß Éther ùan differences in fomal education explaiû tle simple-rithmetic advù-þge for Chinse relative to non-Asiân Nonh American adulb. NAC'S rcÌâtively poor simple-üithmet¡cperfomrnce rcsulted both from less efficient rcrieval skills ûd g@ate. use of proceduÉl sfrtegis,None¡¡elss, al¡ 3 groups repofed using procedures fo¡ rhe l{ger simple subfrction and divisionproblcms, confiming the iñpofrance of præcduEl knowledge ir skilled adulß' perfomùce of ele-mentary matheñatics.

Knowledge of elementa¡y ilithmetic (i.e., simple additioû, mul-tiplication, subtraction, and division) is â pervasive re4uirement ofeveryday modem life, providing the essential mems for dealingwith a widely diverse vuiety of numerical-problem-solving situ-ations. Basic ilithmetic also provids the foundation for the mo¡eadvmced mathematical skills that õe central to all modem scien-tific disciplines. CoNequently, understanding this fundamenralintellectual skill is an imponmt goal for cognitive science (Ash-craft, 1995; Geary, 1994). In this study, Chinese adults educated inthe People's Republic of China md Canadian adults educated inCmada, either of Chinese or non-Asim origin, solved simpleæithmetic problems involving the four basic operations (e.g.,3 + 4,7 - 3,3 x 4, 12 + 3). Participants also reported theirstrategy (i.e., direct memory retrieval vs. præedural strategies)after each problem. The purpose was to address three importantquestions of cu[ent rese{ch in cognitive ilithmetic. Fißt, whenadults solve simple-üithmetic p¡oblems, what is the relative bal-ancc of direct memory retrieval vemus use of prccedural sEategiessuch æ counúng ortransfomation (e.g., 6 + 7 = 6 + 6 +'l :13)? Recent evidence suggsts that even skilled adults make sub-stantial use of procedures (e.g., LeFevre, Sadesþ, & Bisanz,1996), but no study has attempted to assess this fo¡ the entiredomain of elementa¡y æithmetic. Second, what detemines theproblem-siæ effect (PSE) in cogtritive uithmetic? The PSE is thevirtually ubiquitous phenomenon that the difficulty of simple-mithmetic prcblems increases as problem size increases. The PSEhas been recognized and studied systematically for over 75 yøs(e.g., Ctapp, 1924), bùt our study was the frrst to estimate the

relative contributions of retrieval md notretrieval strategi* to thePSE for all four operations. Third, what ile the sourcs of differ-ences in the simple-uithmetic perfommce of No¡th Americm mdAsian adults? Several studies have found that Asian adults gener-ally outperfom Nonh Americm adults on simple ilithmetic(Geary, 1996b; Geary et al., 1997; L€Fevre & Liu, 1997). Byexmining perfomæce æ a function of the type of strategyreported, we hoped to identiry whethe¡ differences in the effi-ciency of retrieval processes, proceduml strategies, or both under-lie the overall perfomance difference.

Direct Retrieval Veßus Procedural Strategies in SkilledPerformance of Simple Arithmetic

It is widely accepted that young children's perfommce ofuithmetic is often based on counting or other proædural strate-gies, although some retrieval is evident evetr during the preschælyeffi, especially for small problems such as I + 2 (Siegler &Sh¡ager, 1984). What happens as ilithmetic skill develops, how-ever, is more uncertain md conEoveßial (Bdoody, 1994). Oneview is that with accumulating exposure to the bæic sithmeticcombinations, proceduml strategies æ gmdually replaced by di-rect memory rctrieval. The switch to retrieval is assumed to beginduring the eilly public schæl grades ild to prc@d gmduallyover ensuing ym as more md morc specific facts ile comittedto memory (Koshmider & Ashcmfr, 1991; Siegler, 1988; Siegler &Sb¡ager, 1984). By the time the rypical lemer reaches college age,most of the basic uithmetic facts presumably have been encoun-tered so frequently that memory retieval is the predominant strat-egy. Indeed, mæy reaction time (RT) md enor phenomena pro-duced by skilted adults perfoming simple addition ildmultiplication problems æe well explained by retieval-bæedmodels (Ashcraft, 1992; Cmpbell, 1995; G¡ahm & Cmpbell,1992; McCloskey, Huþ, & Sokol, l99l).

Altbough the development of simple uithmetic skill generallyproceeds from reliânce on procedures to relimce on retrieval, itmay be ræ to achieve exclusive relimce on direct retrieval, NorthAmerican univeßity students do not report exclusive reliilce on

Jamie I. D. Cmpbell dd Qilin Xue, Depdment of Psychology, Uni-ve$ity of Sdkatchewe, Sõkatoon, Saskatchewù, Canadâ.

This resørch was supponed by a grant fiom the NatuEl Sciencs andEnginæring Reseæh Council of Cmãdâ. lie thmk Mdk AshcÉft forhelpñ¡l coments on a previous veFion of this micle.

CoFespondence conceming this micle should be addressed to JmieI. D. Cmpbell, Depdment ofPsychology, Univeßity of Saskãrchewan,9Campus D¡ive, Saskatoon, Saskatchewar S?N 5A.5, Canada, Elecronicmail may be sent to jmie.campbell@us4k,ca,