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JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR 2004, 82, 1–19 NUMBER 1 (JULY)

TRANSITIVE RESPONDING IN HOODED CROWS REQUIRES LINEARLYORDERED STIMULI

OLGA F. LAZAREVA, ANNA A. SMIRNOVA, MARIA S. BAGOZKAJA, ZOYA A. ZORINA,VLADIMIR V. RAYEVSKY, AND EDWARD A. WASSERMAN

INSTITUTE OF HIGHER NERVOUS ACTIVITY, MOSCOW STATE UNIVERSITY, AND UNIVERSITY OF IOWA

Eight crows were taught to discriminate overlapping pairs of visual stimuli (A1 B2, B1 C2, C1D2, and D1 E2). For 4 birds, the stimuli were colored cards with a circle of the same color on thereverse side whose diameter decreased from A to E (ordered feedback group). These circles weremade available for comparison to potentially help the crows order the stimuli along a physical di-mension. For the other 4 birds, the circles corresponding to the colored cards had the same diameter(constant feedback group). In later testing, a novel choice pair (BD) was presented. Reinforcementhistory involving stimuli B and D was controlled so that the reinforcement/nonreinforcement ratiosfor the latter would be greater than for the former. If, during the BD test, the crows chose betweenstimuli according to these reinforcement/nonreinforcement ratios, then they should prefer D; ifthey chose according to the diameter of the feedback stimuli, then they should prefer B. In theordered feedback group, the crows strongly preferred B over D; in the constant feedback group, thecrows’ choice did not differ significantly from chance. These results, plus simulations using associativemodels, suggest that the orderability of the postchoice feedback stimuli is important for crows’transitive responding.

Key Words: transitive inference, associative learning, cognition, visual discrimination, hooded crows

A relation between premises is said to betransitive if the relation Rl that links Stimulib and c and Stimuli c and d also links Stimulib and d. A transitively competent individualshould be able to deduce that if b Rl c and cRl d, then b Rl d. Moreover, such an individualshould reach this conclusion if and only if Rl

Olga Lazareva, formerly at the Institute of Higher Ner-vous Activity, Moscow, is now at the University of Iowa.Anna Smirnova, Maria Bagozkaja, and Zoya Zorina are atMoscow State University, Vladimir Rayevsky is at the In-stitute of Higher Nervous Activity, and Edward Wasser-man is at the University of Iowa.

The research was supported by the collaborative grantfrom NATO to Z.A. Zorina and J. D. Delius, by the NCCRNeural Plasticity and Repair, and by 7IP No 62645 (SwissNational Scientific Foundation). The results were par-tially presented at UFAW Symposium, ‘‘Consciousness,Cognition, and Animal Welfare’’ (London, 2000), at theFourth European Meeting for the Experimental Analysisof Behaviour (Amiens, 2000), and at the conference ofComparative Cognition Society (Florida, 2002). The ex-periment was a part of the doctoral thesis research ofOlga Lazareva while she was a Ph.D. student at the Insti-tute of Higher Nervous Activity (Moscow, Russia). Wethank J. D. Delius, M. Siemann, M. J. Acerbo, M. Cleave-land, B. Gibson, and I. A. Reyer for their helpful com-ments on an early version of the manuscript. We aregrateful to Clive Wynne, Tom Zentall, and Tom Critch-field who provided valuable theoretical feedback on themanuscript.

Correspondence concerning this article should be ad-dressed to Olga Lazareva, E11 Seashore Hall, Depart-ment of Psychology, University of Iowa, Iowa City, Iowa52242 (e-mail: [email protected]).

supports a linear order between the stimuli.In other words, from the premises b is biggerthan c and c is bigger than d, it follows that b isbigger than d. But, what follows from the prem-ises b stands near to c and c stands near to d?Because the spatial arrangement of the stim-uli is not fully specified, it is impossible toinfer the relation between b and d. The abilityto distinguish transitive relations from thosethat do not allow correct inferences isthought to be an important characteristic oftransitive inference (TI) formation in hu-mans, at least in verbal tests (Evans, New-stead, & Byrne, 1993; Wright, 2001).

Transitive inference could be beneficial foranimals as well. Suppose that a monkey entersa new group and must establish its own socialposition. One way to achieve this objective isto fight with every other group member andto chart the success or failure of these en-counters. A more economical way is to ob-serve the relations among the members ofthe group and, after a few strategic encoun-ters, to deduce one’s own position in the ex-isting hierarchy. Observe that an individualwho fails to distinguish transitive relationsfrom nontransitive relations can easily reachthe wrong conclusions about an existingdominance hierarchy that will lead to a largenumber of unprofitable encounters. Field ob-servations suggest that monkeys and apes ac-

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2 OLGA F. LAZAREVA et al.

tually use this kind of transitive informationwhen they enter into a new society (Altmann,1962; Kummer, 1982). This ability was docu-mented in the laboratory for two bird species(Hogue, Beaugrand, & Lague, 1996; Paz-y-Mino, Bond, Kamil, & Balda, in press).

A method for presenting semiverbal TItasks to very young children was developedby Bryant and Trabasso (1971); it was lateradapted into a fully nonverbal method for an-imals by many researchers. The method nor-mally involves presenting a series of overlap-ping pairs of stimuli: A1 B2, B1 C2, C1D2, and D1 E2 (where the letters stand fordifferent stimuli and the plus and minus sym-bols indicate that choices of the correspond-ing stimuli are either reinforced or nonrein-forced, respectively). Bryant and Trabassoused pairs of wooden sticks of different colorsextending an equal distance above the top ofa box. Only after a child had reported whichstick was longer or shorter did the investiga-tors show the child the full lengths of bothsticks by taking them out of the box. Thelengths of sticks, therefore, provided visualfeedback after the choice was made (in someexperiments, the children received verbal,rather than visual, postchoice feedback).Therefore, as in the verbal task, the choiceitems in this task could be ordered along aphysical dimension, which supported a tran-sitive relation between them.

To test for transitive inference, the novel BDtesting pair was presented. This pair involvedstimuli that were never before presented to-gether; additionally, the choice of the stimuliB and D had been reinforced in one pair andnonreinforced in another pair. A transitivelycompetent individual was expected to remem-ber that B . C and C . D, to infer that B .D, and to select B over D. Bryant and Trabasso(1971) found that 4-year-old children werequite able to select the transitively correct Bstimulus when the discrimination task was pre-sented in semiverbal form.

Several points must be emphasized aboutthe design devised by Bryant and Trabasso(1971). First, at least five stimuli have to bepresented; otherwise, there will be no newpair that does not contain end anchor stim-uli, the choice of which was always reinforcedor nonreinforced. Second, the stimuli have tobe orderable along a dimension that supportsa transitive relation between them. Third, any

dimensional comparison of the discrimina-tive stimuli can be allowed only after thechoice; otherwise, the successful choice be-tween the stimuli can be achieved by directdimensional comparison of the stimuli, with-out using associative information about therelations between different pairs. In the Bry-ant and Trabasso experiments, for example,the lengths of sticks provided visual feedbackafter the color choice was completed.

Of the three design principles describedabove, only the first has been followed in allof the prior animal studies. In some studies,the relation between the stimuli was explicit,just as in the Bryant and Trabasso (1971) ex-periments (McGonigle & Chalmers, 1977;Rapp, Kansky, & Eichenbaum, 1996; Roberts& Phelps, 1994; Treichler & Van Tilburg,1996). In others (Bond, Kamil, & Balda,2003; Boysen, Berntson, Shreyer, & Quigley,1993; Davis, 1992b; Siemann, Delius, &Wright, 1996; Steirn, Weaver, & Zentall, 1995;von Fersen, Wynne, Delius, & Staddon,1991), the stimuli were arbitrary, and the re-searchers believed that their subjects couldestablish such a transitive relation solely onthe basis of the reinforcement or nonrein-forcement of the stimuli. Note, on the onehand, that the nonordered TI task might berepresented as equivalent to the verbal prem-ises, ‘‘B is reinforced, C is not reinforced’’and ‘‘C is reinforced, D is not reinforced’’; ina verbal task, the transitive-like conclusion ‘‘Bis reinforced, D is not reinforced’’ may wellbe incorrect because the relation ‘‘rein-forced’’ is not transitive. On the other hand,a discrimination procedure such as this mayestablish a preference for one stimulus overthe other and such preferences may be transi-tive. In fact, value transfer theory, as well asother associative theories, suggests that therelation becomes transitive during training,as an ordered series of associative values aris-es. We will return to this idea later.

Still, many animals have been reported tobe able to select B in the new BD pair, evenin experiments in which the discriminativestimuli were not orderable with respect tosome physical characteristic. Unfortunately,only a few species have been studied in thesame experimental conditions both with andwithout transitive relations. Roberts andPhelps (1994) trained rats using wooden tun-nels marked with different odors as stimuli;

3TRANSITIVE INFERENCE IN CROWS

the tunnels were linearly arranged and tookthe same position throughout training. In thetest, the rats selected B over D at a high level.When, however, the tunnels were circularlyarranged during training, the rats selected Band D at a chance level, suggesting that thespatial information provided by the orderedplacement of the tunnels was crucial for tran-sitive responding.

It remains unclear whether the explicit or-derability of the stimuli is important for all var-iants of the TI task or only when spatial cuesare used. For example, in experiments by Da-vis (1992b) and Bunsey and Eichenbaum(1996), rats were found to respond transitively,although there was no transitive relationamong the stimuli (different odors). New ex-periments that use stimuli from a nonspatialdomain may help clarify whether the orderingof stimuli along a physical dimension may con-trol organisms’ transitive behavior.

The mechanisms underlying transitive re-sponding are in dispute. Some researchershave assumed that an organism integrates theindependently presented premises into an or-dered series of internal representations andthat these representations are spatial in na-ture (Davis, 1992a; Gillan, 1981; Rapp et al.,1996; Zorina, Kalinina, & Markina, 1996). Ex-perimental evidence has been interpreted tosupport this point of view. The first finding isthe ‘‘end-anchor’’ effect: higher performancein pairs that contain the first or the last stim-ulus (A or E) in the series compared to pairscontaining the middle stimuli (B, C, or D).According to the spatial representation hy-pothesis, comparisons including the enditems should be faster and easier to learn be-cause they enjoy the position of the largest orthe smallest object in the series. The secondfinding is the ‘‘symbolic distance’’ effect: thesmaller the separation between the items ina testing pair, the longer the response timeand the higher the probability of an error.Spatial representation hypothesis implies thatthe closer the representations of the items ina series, the more similar they are to one an-other, and hence the more difficult the dis-crimination between them should be.

Other theorists have proposed that thesame pattern of results can be explained byassociative models. Such models consider theorganism’s choice of the testing stimuli to bethe result of the difference in their relative

reinforcement history (see Couvillon & Bitter-man, 1992; Siemann & Delius, 1998; Wynne,1995). Value transfer theory, the first in thisclass of models, proposed that the associativevalues of the training stimuli produce an or-dered series, A . B . C . D . E, due tobidirectional transfer of associative value be-tween reinforced and nonreinforced stimuli(von Fersen et al., 1991). Later studies re-vealed, however, that an ordered series of as-sociative values arose even when no transfer ofvalues across stimuli was postulated. In otherwords, the common outcome of training withthe four premise pairs, A1 B2, B1 C2, C1D2, and D1 E2, was found to be the for-mation of an ordered series of the associativevalues of those stimuli (see Siemann & Delius,1998; Wynne, 1995, 1998 for more details).

Note that, in all current associative models,the ordered series of values arises from thetraining procedure, in which the premisepairs are presented in a given sequence andwith a given frequency. Thus we should beable to manipulate the outcome of a TI testby changing the frequency of the premisepairs. If we were to increase the frequency ofD1 E2 presentation, then we might expectthat, eventually, the value of D would becomegreater than the value of B and, consequently,an organism should prefer Stimulus D. Sucha test could provide strong support for an as-sociative model of transitive responding, be-cause the spatial representation hypothesis isunable to predict this outcome under anyconditions.

The spatial representation hypothesis, how-ever, suggests that the use of orderable train-ing stimuli is the prime reason for the emer-gence of an ordered series of stimulus valuesduring training. As argued by Markovits andDumas (1992), ‘‘what makes a relationshiptransitive is . . . the implicit placement of theitems along at least an ordinal scale’’ (p.311). (See Wright, 2001, pp. 386–387, for asimilar argument.) Therefore, if we were toshow that the use of orderable stimuli facili-tates transitive responding, then these resultswould speak in favor of the spatial represen-tation hypothesis, especially because stimulusorderability is deemed to be irrelevant to as-sociative models.

The present study examined transitive re-sponding in hooded crows (Corvus cornix L.),a member of the Corvidae family. Corvids are


characterized by a large telencephalon andrelatively high brain complexity (Rehkamper,Frahm, & Mann, 2001; Stingelin, 1958) as wellas by an ability to solve various types of com-plex cognitive tasks (Balda, Kamil, Bednekoff,& Hile, 1997; Heinrich, 2000; Jones, Antoniad-is, Shettleworth, & Kamil, 2002; Kohler, 1950;Krushinsky (1977/1990); Smirnova, Lazareva,& Zorina, 2000; Wilson, Mackintosh, & Boak-es, 1985). Indeed, some of these tasks, such astool making or complex numerical abilities,have been documented primarily in corvidsand primates (Weir, Chappell, & Kacelnik,2002; Zorina, 1997).

The present study compared transitive re-sponding in crows in ordered and nonorder-ed transitive inference tasks. In the orderedfeedback group of the current experiment, 4crows were shown colored choice stimuli fol-lowed by postchoice feedback stimuli of thesame color, but of different size. This methodis functionally equivalent to Bryant and Tra-basso’s (1971) task, in which children weregiven visual feedback after they had reportedwhich of two sticks was shorter or longer. Inthe constant feedback group of the currentexperiment, 4 birds were trained with feed-back stimuli that were of the same size; thosefeedback stimuli could not help to order thechoice stimuli along a common dimension.In both groups, discrimination training wasfollowed by extended D1 E2 training, whichled to a richer reinforcement history forStimulus D than Stimulus B. Under these cir-cumstances, the spatial representation hy-pothesis predicts transitive responding in theordered feedback group, but not in the con-stant feedback group, where the choice stim-uli cannot be ordered along any dimension;associative theories do not predict transitiveresponding in either group. In addition, weconducted post hoc simulations using severalassociative models including the value trans-fer model, to verify that associative modelsdid, indeed, predict no response to ‘‘logicallycorrect’’ Stimulus B in testing pair BD.

STUDY 1

METHOD

Subjects

Eight hooded crows (Corvus cornix L.),each more than 1 year old, were studied. The

crows were caught in the wild when they wereabout 6 months old and housed in outdooraviaries in small groups of 2 or 3 birds. All ofthe birds were experimentally naive. Beforethe start of the experiment, the birds werefood deprived for 2 days. Throughout the ex-periment, the birds had free access to waterand food, except when they refused to workduring training. In these cases, food withoutanimal protein was given for 1 or 2 days.

ApparatusA mesh wire cage (65 cm by 50 cm by 45

cm; mesh dimensions 4 cm by 4 cm) and aplastic tray (20 cm by 30 cm) with a handle(30 cm) were used. The left panel of Figure1 shows a schematic representation of the ex-perimental apparatus. Two cups (each 3.7 cmhigh and 5.0 cm in diameter) were placed onthe tray (13 cm from the base of the tray and1.5 cm from the side of the tray) with twoplastic blocks (4 cm by 4 cm by 5 cm) behindeach of them. Two transparent plastic pockets(7.1 cm by 7.1 cm) were attached to each ofthe blocks. During the experiment, colorcardboard cards that served as the choicestimuli (see below) were put into the pockets.The pockets were attached so that the cardscould serve as cup lids.

The tray was prepared for the trial out ofthe crow’s view. An opaque plastic screen (70cm by 40 cm) stood between the experimen-tal cage and the experimenter, so that whenthe crow made its choice, it could not see theexperimenter and vice versa.

StimuliColored cards (7 cm by 7 cm) were used

to represent the terms of the series A, B, C,D, and E. The colored upper surfaces of thecards (red, yellow, green, blue, or black)served as the choice stimuli. A circle of thesame color was drawn on the underside ofeach card. These circles served as the post-choice feedback stimuli. The diameter of thefeedback circles decreased from red (6.5 cm)to black (4.5 cm; right panel of Figure 1) inthe ordered feedback group. In the constantfeedback group, the diameter of all feedbackcircles was 6.5 cm. Additionally, two whitecards were used for pretraining (see below).

Experimental DesignThe stimuli and experimental design are

shown in the right panel of Figure 1. The


Fig. 1. Experimental setting (left) and design (right). Left top—bird in the experimental cage; left bottom—twocups with stimulus lids before and after opening. Right—choice and feedback stimuli for the constant and orderedfeedback groups.

birds were trained on a multiple discrimina-tion task involving overlapping pairs of visualchoice stimuli: A1 B2, B1 C2, C1 D2, andD1 E2 (where the letters stand for choicestimuli of different colors, and the plus andminus symbols indicate that choices of thecorresponding stimuli were either reinforcedor nonreinforced, respectively).

In order to minimize possible color pref-erences, the crows were randomly assigned totwo groups. The stimuli A, B, C, D, and Ewere represented by red, yellow, green, blue,black, respectively, for half of the crows withineach group (ordered feedback: Zosia and Ko-tia; constant feedback: Korsar and Solomon),and by black, blue, green, yellow, and red forthe other half of the crows (ordered feed-back: Dascha and Zelenaja; constant feed-back: Carolina and Kondrat). The postchoicefeedback stimuli had the same color as thecorresponding choice stimuli in both groups.Note that for 2 birds in the ordered feedbackgroup, the postchoice feedback stimulus forthe reinforced choice stimulus always had alarger diameter and the transitive series couldbe represented as A . B . C . D . E. Forthe other 2 birds in the ordered feedback

group, this postchoice feedback stimulus al-ways had a smaller diameter and the relationin transitive series was reversed: A , B , C, D , E. In the constant feedback group,the postchoice feedback stimuli were of equaldiameter (6.5 cm); therefore, these constant-size postchoice feedback stimuli could nothelp the pigeons order the choice stimulialong any dimension.

General Procedure

A trial began when the tray with the cupscovered by the cards was slid into the crow’scage. One of the cups contained two meal-worms as reinforcement. The bird could turnover either card and, if the choice was cor-rect, then it received mealworms. After a cor-rect choice, the birds usually also turned overthe card over the incorrect cup; if not, thenthe experimenter turned over this card forthe bird. Consequently, a bird could see bothfeedback stimuli and compare their diame-ters. After an incorrect choice, the tray wasimmediately angled so that the crow couldnot reach the correct cup. The experimenterthen turned over the correct card and left thetray in the bird’s field of vision for 3 to 5 s


before withdrawing it in order to give thebird an opportunity to see the postchoicefeedback stimuli.

During Phases 1 through 8 (see below), ifthe bird made three repeated choices of theleft or the right choice stimulus, then the trialwas repeated until a correct response wasmade (correction procedure). During thetesting phase, this correction procedure wasnot employed. The number of correction tri-als was recorded, but only the first responseon each trial was scored in the data analysis.

Training in Phases 1 through 7 continueduntil a criterion of 80% correct or better over30 consecutive trials was reached (p , 0.001;binomial probability test). Phases 8 and 9consisted of a fixed number of trials. Exper-imental sessions were conducted 6 days eachweek and involved 40 or 60 trials (correctiontrials were not included in these totals). Theexact number of trials per day depended ona bird’s performance on each particular day.

Preliminar y Training

The crows were trained to turn over thewhite cards to find food reinforcement. Dur-ing preliminary training, reinforcement wasplaced into both cups; therefore, any choicewas reinforced. Preliminary training contin-ued until the birds turned over the whitecards immediately after the tray was insertedinto the cage.

Training

Phase 1. Only the single choice pair, A1B2, was presented. The left–right location ofreinforced responding was counterbalancedwithin a block of 10 trials, under the restric-tion that S1 did not appear in the right orleft location on more than two successive tri-als. The birds were required to meet the 80%criterion during 30 consecutive trials.

Phase 2. The pair B1 C2 was presented;the counterbalancing and the criterion re-mained the same as during Phase 1.

Phase 3. Two pairs, A1 B2 and B1 C2,were presented during each session. Thecounterbalancing occurred in blocks of 16trials, and both choice pairs were presentedin a pseudorandom sequence so that no pairwas presented on more than two successivetrials. The crows were trained until they metthe 80% criterion to each pair during 32 con-secutive trials.

Phase 4. A pair C1 D2 was presented; thecounterbalancing and the criterion remainedthe same as during Phase 1.

Phase 5. Three pairs, A1 B2, B1 C2, andC1 D2, were presented during each session.The counterbalancing occurred in blocks of16 sessions, and each choice pair was pre-sented in a pseudorandom sequence so thatno pair was presented on more than two suc-cessive trials. The crows were trained untilthey met the 80% criterion to each pair dur-ing 32 consecutive trials.

Phase 6. The pair D1 E2 was presented;the counterbalancing and the criterion re-mained the same as during Phase 1.

Phase 7. Four pairs, A1 B2, B1 C2, C1D2, and D1 E2, were presented during eachsession. The counterbalancing occurred inblocks of 16 trials, and both choice pairs werepresented in a pseudorandom sequence sothat no pair was presented on more than twosuccessive trials. The crows were trained untilthey met the 80% criterion to each pair dur-ing 32 consecutive trials.

Bias Reversal Phase

It was conceivable that upon completion ofall training phases the reinforcement historyof stimulus B was richer when compared tostimulus D. If so, then the possible use of lin-early ordered series of the stimuli could beconfounded by the difference in their asso-ciative values. To disentangle the influence ofthe reinforcement history from the order-ability of the stimuli along a dimension, weevaluated the reinforcement history beforethe test on a bird-by-bird basis.

To estimate the possible influence of thereinforcement history of testing pair BD, thenumber of reinforcements (Nr) and nonre-inforcements (Nn) for selection of eachchoice stimulus during all of the trainingphases (including correction trials) was cal-culated. The reinforcement/nonreinforce-ment ratio R 5 Nr /Nn was used to comparethe value of the B and D choice stimuli. If theratio for D was larger than the ratio for B,then the testing phase was begun; if not, thenthe crows were exposed to a bias reversalphase.

The bias reversal phase involved the D1E2choice pair, with number of presentations ad-justed on the basis of the reinforcement/non-reinforcement ratios calculated earlier. This


Table 1

Number of correct and incorrect choices and reinforcement/nonreinforcement ratio bothbefore and after the bias reversal phase in the ordered feedback group. Nr and Nn stand fornumber of reinforcements and nonreinforcements, correspondingly; R stands for reinforce-ment/nonreinforcement ratio.

Stimulus

Bird

Zosia Kotia Dasha Zelenaia

Before bias reversalNr

Nn

R

BDBDBD

1245458522.11.0

11425683

1511.41.7

96145122500.82.9

1279896391.32.5

After bias reversalNr

Nn

R

BDBDBD

13115371601.82.6

bias reversal phase was conducted in order toaugment the reinforcement/nonreinforce-ment ratio of D, and it was designed to con-tinue until the reinforcement ratio of D wasat least 1.1 times greater than that of B. Afterthat, the birds were required to meet a cri-terion of 80% correct with each of the threeremaining training pairs during 32 consecu-tive trials.

For instance, for the crow Zosia the choiceof Stimulus B was reinforced 124 times andnot reinforced 58 times, whereas the choiceof Stimulus D was reinforced 54 times andnot reinforced 52 times, yielding R values of2.1 and 1.0, respectively. Therefore, the crowcould prefer B over D on the basis of unequalhistories of reinforcement. The bias reversalphase added 102 trials with D1 E2 and 10trials with each of the three remaining pairs:A1 B2, B1 C2, and C1 D2. This planwould have changed the ratio of D to 3.0 ifthe crow did not make any mistakes duringthat phase. Actually, the crow’s accuracy wasless than perfect (88.2%), yielding a final Dratio of 2.6. Consequently, if, during the sub-sequent BD test, this crow chose betweenstimuli according to these reinforcement/nonreinforcement ratios, then it should pre-fer D; if the crow chose according to the di-ameter of the feedback stimuli, then it shouldprefer B.

Testing

The testing phase consisted of the presen-tation of all four training pairs and the newtesting BD pair. On BD testing trials, the se-lection of either choice stimulus was rein-forced with a probability of 0.5. To preventthe birds from comparing the diameters ofthe postchoice feedback circles during thetesting pair trials, only one lid could beopened and the other remained closed. Thetesting pair was presented 40 times out of to-tal of 160 trials.

RESULTS

For the ordered feedback group, trainingwith all four pairs of choice stimuli took anaverage of 727 trials, with a minimum of 524trials and a maximum of 882 trials. For theconstant feedback group, training with allfour pairs of choice stimuli took an averageof 781 trials (range, 462 to 1,050). Therewere no obvious differences between birdswith different color assignments, so their re-sults were pooled.

Table 1 shows the number of reinforcedand unreinforced choices of Stimulus B andStimulus D by each bird in the ordered feed-back group before and after the bias reversalphase. The table also shows the final ratio, R,of reinforced and unreinforced choices. After


Table 2

Number of correct and incorrect choices and reinforcement/nonreinforcement ratio bothbefore and after the bias reversal phase in the constant feedback group. Nr and Nn stand fornumber of reinforcements and nonreinforcements, correspondingly; R stands for reinforce-ment/nonreinforcement ratio.

Stimulus

Bird

Korsar Solomon Carolina Kondrat

Before bias reversalNr

Nn

R

BDBDBD

15910172

2172.20.5

823570891.20.4

273269117862.33.1

34311561715.61.6

After bias reversalNr

Nn

R

BDBDBD

17059172

2172.42.7

9622476931.32.4

36462264895.77.0

training, R for Stimulus D was greater than Rfor Stimulus B for Kotia, Dasha, and Zelenaia,so they proceeded directly to the test. The4th crow, Zosia, was exposed to the bias re-versal phase. Originally, R for Stimuli B andD were 2.1 and 1.0, respectively, whereas afterthe bias reversal phase the values were 1.8and 2.6, respectively.

Table 2 shows the number of reinforcedand unreinforced choices of Stimulus B andStimulus D by each bird in the constant feed-back group before and after the bias reversalphase as well as the final reinforcement/non-reinforcement ratios. Carolina’s R values forStimuli B and D after training was completewere 2.3 and 3.1, respectively, so testing com-menced immediately after training. Threecrows—Korsar, Solomon, and Kondrat—hadto be given the bias reversal phase. Thereaf-ter, R was higher for Stimulus D than forStimulus B.

Figure 2 depicts discrimination accuracyduring testing for the four training pairs andpreference among the stimuli in the testingpair (BD). (Henceforth, ‘‘choice’’ is used torefer to performance involving the trainingpairs, and ‘‘preference’’ is used to refer toperformance involving the testing pairs.) Forthe constant feedback group (upper panel),training-pair accuracy was above chance (bi-nomial probability test, p , 0.05) in all casesexcept pairs BC and CD for Korsar and pairs

AB and CD for Kondrat. No crow in thisgroup strongly preferred Stimulus B over D.One (Korsar) preferred Stimulus D overStimulus B (80.0%), and the other 3 birds’performance was near chance (Solomon,Kondrat, and Carolina). For the orderedfeedback group (bottom panel), training-pairaccuracy was above chance (binomial test, p, 0.05) in all cases except pair DE for Kotiaand pairs CD and DE for Zelenaia. All crowsin this group strongly preferred Stimulus Bover Stimulus D (M 5 83.1%).

Analyses of variance (ANOVA) were con-ducted with group (ordered, constant) andpairs (A1 B2, B1 C2, C1 D2, D1 E2, BD)as factors and with accuracy as the dependentvariable. There were significant main effectsof group, F(1, 7) 5 9.19, p , 0.05, and pair,F(4, 28) 5 11.81, p , 0.05, as well as a sig-nificant pair 3 group interaction, F(4, 28) 515.48, p , 0.05. A follow-up Tukey test foundthat, in the ordered feedback group, prefer-ence involving testing pair BD did not differin magnitude from that of any of the fourtraining pairs. In the constant feedbackgroup, however, preference for Stimulus Bwas significantly lower than that for any of thereinforced stimuli in the training pairs. Mostimportantly, in the BD test trials, preferencefor Stimulus B was significantly higher in theordered feedback group than in the constantfeedback group.


Fig. 2. Choice accuracy in the training pairs and preference for Stimulus B in the testing BD pair in the constantfeedback group (upper panel) and the ordered feedback group (lower panel).

DISCUSSION

Strong transitive responding (i.e., prefer-ence of Stimulus B in new pair BD) emergedonly in the ordered feedback group. In theconstant feedback group, crows either select-ed the Stimuli B and D at chance levels orpreferred Stimulus D, which was more oftenreinforced during training. These results are

consistent with the spatial representation hy-pothesis (Davis, 1992a; Gillan, 1981; Rapp etal., 1996; Zorina et al., 1996), and support thenotion that orderability of the stimuli alonga physical dimension such as size may be anecessary and sufficient condition for non-verbal transitive responding in crows.

It is important to stress that, in the present


experiment, a crow could not directly comparethe sizes of the choice stimuli before a selec-tion was made; the choice stimuli themselveswere equal-sized squares of different colors.To establish an ordered series, a crow wouldneed to notice that each choice stimulus pre-ceded a feedback circle of the same color butof a different diameter, and that the diame-ters of the feedback circles could be dimen-sionally ordered.

One might note that 3 out of 4 birds in theconstant feedback group were exposed to thebias reversal phase, whereas only 1 out of 4birds required bias reversal in the orderedfeedback group. Could the difference in tran-sitive responding between groups be the re-sult of mere exposure to the bias reversalphase? If that were true, then the crows thatwent through bias reversal in both groupsshould demonstrate lower transitive respond-ing than the crows that proceeded directly tothe test. In the constant feedback group,mean preference for Stimulus B during BDtest trials was 36.7% for the crows exposed tothe bias reversal phase and 62.5% for thecrow that went directly to the test. In the or-dered feedback group, preference for Stim-ulus B was 82.5% for the crow exposed to thebias reversal phase and 83.3% for the crowsthat went directly to the test. It seems unlike-ly, therefore, that mere exposure to the biasreversal phase could affect BD performanceindependently of the other factors. Futurestudies could evaluate this experimentally byincluding a BD test both before and after thebias reversal phase.

Another possible concern involves the lowaccuracy on training trials in the test for someof the birds in the constant feedback group(for example, Korsar, see Figure 2). Couldthe low performance to test pair BD be ex-plained by low performance to the trainingpairs? Again, we found that poor perfor-mance to some training pairs during the testin the ordered feedback group (for example,Zelenaia) did not prevent the crow from se-lecting Stimulus B of the BD pair at a veryhigh level of accuracy. Moreover, the birdsthat were exposed to the bias reversal phasewere retrained with the training pairs beforethey went into the test (see Method section).A decrease in accuracy involving the trainingstimuli when novel testing stimuli are intro-

duced has been documented before for otherbird species (Pepperberg, 1987).

Although the ratio, R, of reinforced to un-reinforced choices was used as an index ofthe associative strength of the choice stimuli,the question arises whether the manipulationof this ratio is adequate to change the pre-dictions made by associative models for theBD test. An attempt was made, therefore, tosimulate the present results by using associa-tive models that previously have been foundto yield reasonably good accounts of transi-tive responding (see Siemann & Delius, 1998;Wynne, 1995, 1998).

STUDY 2

METHOD

Preliminary simulations disclosed that ele-mentary associative models (i.e., those thatdo not include a configural value compo-nent, such as the Luce, Bush-Mosteller, orRescorla-Wagner models) could not providea satisfactory fit to our data. (These simula-tions are available, upon request, from thecorresponding author.) Hence subsequentsimulations focused on two configural mod-els, one based on the Rescorla-Wagner equa-tions (Wynne, 1995) and the other based onthe Luce equations (Siemann & Delius,1998).

Also examined was a value transfer modelthat has been proposed as a possible accountfor nonverbal transitive responding (von Fer-sen et al., 1991). The value transfer modeldoes not specify how associative values are ac-crued; it simply states that the accrued valuescan be transferred between simultaneouslypresented stimuli. Therefore, the Siemann-Delius and Wynne models were modified toincorporate the value transfer mechanism.Short descriptions of each model, togetherwith the equations, are provided in the Ap-pendix.

The data of each bird were fitted individ-ually, using the full sequence of trials pre-sented during training (including correctiontrials) and employing the least-square errortechnique. Only the training data were fitted.The obtained associative values of the stimuliwere used to calculate choice probability fortesting pair BD according to the choice func-tion used by the model (see Appendix).


Fig. 3. The obtained percentage of correct choices in the training pairs and transitive responses in the testingBD pair compared with simulations using the Wynne and Siemann-Delius models.

RESULTS AND DISCUSSION

The results of the simulations are present-ed in Figures 3 and 4. The individual fits aregiven in Tables 3 and 4; Table 5 provides thevalues of the model parameters for the best-fitting result. Figure 3 (left panels) shows thatboth models provided a good fit for training-pair accuracy in the constant feedback group,although the Wynne model slightly overesti-mated accuracy for the D1 E2 pair. Themodels predicted mean preference betweentest Stimuli B and D to be at or below chance.Table 3 documents that the Siemann-Deliusmodel erroneously predicted preference ofStimulus B over D for 1 crow (Solomon), butaccurately predicted the other crows’ prefer-ence. The Wynne model either predicted toohigh (Carolina) or too low (Kondrat, Korsar,and Solomon) levels of transitive responding.

Both models also provided a reasonable fitfor training-pair accuracy in the orderedfeedback group (Figure 4, right panels). Im-portantly, the models again predicted meanpreference involving test Stimulus B to be ator below chance when in fact all of the birdsin this group strongly preferred Stimulus Bover D (83.1% on average). Table 3 showsthat the Siemann-Delius model predicted in-difference for all birds, whereas the Wynnemodel predicted preference for Stimulus Dfor 2 crows (Dasha and Zosia) and indiffer-ence for the other 2 crows.

Figure 4 (left panels) shows that the Sie-mann-Delius and Wynne models, when mod-ifed to incorporate the value-transfer mecha-nism, both produced excellent fits oftraining-pair accuracy for the constant feed-back group. The predictions of the modified


Fig. 4. The obtained percentage of correct choices in the training pairs and transitive responses in the testingBD pair compared with simulations using the modified Wynne and Siemann-Delius models that incorporated a valuetransfer mechanism.

Siemann-Delius model for preference involv-ing the BD pair matched the obtained results,whereas the modified Wynne model predict-ed less preference for Stimulus B than wasactually observed. Table 4 shows that themodified Siemann-Delius model erroneouslypredicted high transitive responding for 1 outof 4 crows (Carolina). The modified Wynnemodel predicted strong preference of Stim-ulus D over B for all 4 crows, instead of thechance responding that was observed.

Both models still predicted indifferent B-Dpreference in the ordered feedback group(Figure 4, right panel). Table 4 shows that themodified Siemann-Delius model predictedpreference of Stimulus B over D for 2 crows(Dasha and Zelenaia), chance responding for1 crow (Kotia), and preference of Stimulus D

over B for 1 crow (Zosia). The modifiedWynne model predicted preference of Stim-ulus B over D for 1 crow (Zelenaia), chanceresponding for 1 crow (Dasha), and prefer-ence of Stimulus D over B for the remaining2 crows (Kotia and Zosia). All birds, however,preferred Stimulus B over Stimulus D at a sta-tistically significant level. Thus neither confi-gural models alone nor configural modelsthat incorporated the value transfer mecha-nism provided a satisfactory account for thepresent data.

Both the Siemann-Delius and Wynne asso-ciative models predicted performance on BDtesting trials that was in clear disagreementwith the obtained data. Yet in previous studiesthose models have been used to simulate suc-cessfully transitive responding in both pi-


Table 3

Simulated accuracies for Siemann-Delium and Wynne models.

Bird A1 B2 B1 C2 C1 D2 D1 E2 BD

Constant feedback groupCarolina

Kondrat

ObtainedSiemann-DeliusWynneObtainedSiemann-DeliusWynne

70.0083.4662.2862.5062.2859.98

95.0088.4594.6380.0079.8676.44

75.0069.7878.3960.0060.1963.51

90.0086.26

100.0090.0089.7399.99

62.5050.0085.5550.0048.6628.99

Korsar

Solomon


80.0077.1382.0190.0092.6295.41

55.0056.8054.5190.0084.1689.05

37.5037.0742.9480.0089.4879.00

95.0094.39

100.0085.0084.47

100.00

20.0033.160.05

40.0097.846.90

Ordered feedback groupDasha

Kotia


87.5088.5583.5780.0064.2952.73

85.0085.2097.0380.0059.6957.99

80.0077.1178.1867.5068.6657.90

80.0084.8399.9955.0073.2673.11

82.5049.9821.4785.0050.0050.40

Zelenaia

Zosia


70.0077.8351.0090.0094.0886.33

90.0069.5960.5085.0096.8795.51

63.5064.4156.9395.0093.1590.50

50.0064.9073.1190.0085.83

100.00

82.5050.0052.5082.5050.0020.07

geons and people (Delius & Siemann, 1998;Siemann & Delius, 1998; Wynne, 1995, 1997,1998). What might be responsible for the dis-parity?

The associative value of a stimulus in theSiemann-Delius and Wynne models has twocomponents—an elemental value and a con-figural value (see Appendix). The elementalvalue is updated whenever a given stimulus ispresented, whereas the configural value is up-dated whenever a given stimulus is presentedin a given pair. Testing pair BD had neverbeen presented before, so the response tothis pair is determined solely by the elemen-tal values of Stimuli B and D. Thus, to predicttransitive responding to testing pair BD, theelemental value of Stimulus B must be higherthan the elemental value of Stimulus D.

The present experiment employed a biasreversal procedure that consisted of massivepresentation of the training pair D1 E2.From a model’s point of view, this proceduregreatly increased the elemental value of Stim-ulus D and left unchanged the elemental val-ue of Stimulus B, thereby assuring the pre-diction of choice of Stimulus D in the BD

testing pair. Unfortunately, transitive perfor-mance was not tested before the bias reversalphase. Such a test could show whether chang-es in predicted BD performance are due tothe bias reversal procedure or to peculiaritiesin the individual training history of each bird.

It was also found that including a valuetransfer mechanism in the associative modelsimproved their fit of the obtained trainingdata, but did not systematically improve theirpredictions of BD performance. Value trans-fer was first suggested as a hypothetical pro-cess that might help to explain transitive re-sponding in pigeons (von Fersen et al.,1991). Later, the transfer of associative valuebetween stimuli in a simultaneous discrimi-nation was experimentally documented(Clement & Zentall, 2000; Zentall & Sher-burne, 1994). Nevertheless, some experimen-tal data suggest that value transfer might notcontribute importantly to transitive respond-ing in pigeons (Siemann, Delius, Dombrow-ski, & Daniel, 1996; Steirn et al., 1995; Weav-er, Steirn, & Zentall, 1997). Our simulationsagree with these results: Table 5 shows thatboth associative models converged on a small


Table 4

Simulated accuracies for the modified Siemann-Delius and Wynne models that incorporateda value transfer mechanism.

Bird A1 B2 B1 C2 C1 D2 D1 E2 BD

Constant feedback groupCarolina

Kondrat


70.0069.3270.6362.5062.4557.28

95.0093.9092.3380.0079.8279.60

75.0075.2475.3360.0060.7759.55

90.0091.0988.8190.0090.2390.75

62.5080.633.30

50.0037.0929.81

Korsar

Solomon


80.0079.9578.9090.0087.9393.93

55.0055.5555.7090.0083.7192.63

37.5037.1935.4680.0089.2280.51

95.0095.8997.4885.0087.1586.76

20.0021.791.80

40.0049.9119.96

Ordered feedback groupDasha

Kotia


87.5087.7384.2080.0076.1184.74

85.0084.2387.0180.0082.4981.25

80.0079.2281.9067.5068.3069.91

80.0080.2080.0455.0048.1752.70

82.5062.4050.9885.0052.2721.40

Zelenaia

Zosia


70.0074.3774.0490.0097.6190.36

90.0084.6890.6785.0086.8485.29

62.5063.0160.7495.0089.3796.17

50.0045.9053.1790.0089.4888.50

82.5085.4572.9282.5029.0915.65

proportion of transferred value that did notimportantly change the predicted results. Wethus conclude that the associative modelsbased predominately on the reinforcementhistory that accrued during training cannotpredict transitive responding in the orderedfeedback group and the absence of transitiveresponding in the constant feedback group.

GENERAL DISCUSSION

In our experiment, crows selected StimulusB of novel testing pair BD only when thechoice stimuli in the transitive series were as-sociated with explicitly ordered postchoicefeedback stimuli (ordered feedback group).Without such ordered feedback stimuli (con-stant feedback group), crows either selectedStimuli B and D at chance levels or preferredStimulus D, which was more often reinforcedduring training. In short, the orderability ofthe choice stimuli along a physical dimensionlike size may be both a necessary and suffi-cient condition for nonverbal transitive re-sponding in crows.

The present study is not the first attemptto control the reinforcement history of Stim-uli B and D. Zorina et al. (1996) comparedthe number of reinforced choices for StimuliB and D and found that crows respondedtransitively even when the choice of StimulusB was reinforced less often than the choiceof Stimulus D during training. However, ex-perience with nonreinforced choices of thestimuli may be just as important as experi-ence with reinforced choices; thus, the rein-forcement/nonreinforcement ratio may be abetter index of associative strength than thenumber of reinforced and nonreinforcedchoices. Unfortunately, it is not possible tocalculate this ratio for Zorina et al. becauseraw numbers of reinforced and unreinforcedchoices were not reported.

In a series of studies, Zentall and his col-leagues (Steirn et al., 1995; Weaver et al.,1997) attempted to control their pigeons’ re-inforcement history by changing the proba-bility of reinforcement of the different train-ing stimuli. For example, Weaver et al.trained a group of pigeons to discriminate


Table 5

Best-fitting parameters for the original and modified Siemann-Delius and Wynne models. LSDstands for least-square difference between the proportion of correct responses predicted bythe model and obtained in the experiment; b, «, a, g, A1, and A2 are the model parameters(see Appendix).

Constant feedback group

Carolina Kondrat Korsar Solomon

Ordered feedback group

Dasha Kotia Zelenaia Zosia

Siemann-Deliusb1

b2

«LSD

0.0080990.0090000.06990.02652

0.0045000.0040000.05990.00002

0.0070000.0430000.92300.00120

0.0000010.1800000.99990.01312

0.0001000.0699990.00100.00328

0.0063090.0000010.00010.09940

0.0110000.0000100.00010.07034

0.0000100.2400000.00100.01784

WynneabgLSD

240.01560.10.01713

130.02400.20.01314

260.00800.30.00592

290.65900.20.02462

130.00920.60.05640

10.99990.90.16482

10.000010.90.17960

350.0230.20.02418

Siemann-Delius with value transferb1

b2

«A1

A2

LSD

0.0200.0790.8000.00700.0060.00029

0.0050.0200.4000.00100.0050.00007

0.0150.0500.9000.00900.0090.00004

0.0010.0990.0200.00010.0010.01449

0.0100.0790.5000.00600.0090.00013

0.0590.0990.9000.05990.0150.00686

0.0200.500.9000.01100.0100.00644

0.0200.0200.0010.00400.0080.00932

Wynne with value transferabgA1

A2

LSD

60.0100.800.02000.04000.00091

100.0300.200.00900.00900.00281

40.0050.500.02000.00200.00119

170.0400.160.05000.05000.00257

70.0700.200.09990.00800.00186

70.0600.200.06990.06990.00351

90.0800.110.03000.02000.00299

110.0990.410.06990.03790.00038

pairs A6 B2, B1 C6, C6 D2, and D1 E6(where 6 sign denotes the stimulus thatchoice was reinforced in 50% of the trials).This procedure was intended to eliminate thepossible effects of value transfer. Normally,Stimulus B is paired with always reinforcedStimulus A, and Stimulus D is paired withnever reinforced Stimulus E. Therefore, it isexpected that the cumulative transferred val-ue will be greater for Stimulus B than forStimulus D, thus explaining the preference ofStimulus B in the BD testing pair. More im-portantly, Weaver et al. also reported themean numbers of correct and incorrectchoices for all of the training stimuli. Usingthese numbers, we found that, for the groupdescribed above, the mean ratio of StimulusB was equal to 3.59 and the mean ratio ofStimulus D was equal to 10.98—but the birdsnevertheless selected Stimulus B over Stimu-lus D on 78% of the trials.

Although these findings cast doubt on re-

inforcement history as the basis of transitiveresponding in animals, they do not rule it outcompletely. Associative models based on a his-tory of reinforcement heavily depend on theexact sequence of reinforced and nonrein-forced responses in training (Couvillon & Bit-terman, 1992; Wynne, 1995, 1998). Global ra-tios of the number of reinforced andunreinforced choices cannot substitute forformal simulations. It may be noteworthythat, in several instances, the models exam-ined here predicted transitive respondingeven when the ratio of reinforced to unrein-forced training choices suggested otherwise(see, e.g., data for Carolina in Tables 2 and3). It would be interesting to see whether pi-geons would respond transitively even whenthe simulations using the exact sequence ofthe trials in training predicted the oppositeresult. So far, such simulations have agreedwith pigeons’ data (see Wynne, 1995, 1997,1998).


It might be tempting to conclude thatcrows solve the TI problem by using a learn-ing mechanism different from pigeons—amechanism that required the presence of atransitive relation between stimuli in a series.Thus pigeons may be able to respond transi-tively even when the stimuli are unorderable,whereas crows may require ordered feedbackstimuli. Such a conclusion would be prema-ture in the absence of data that documentpigeons’ preference for Stimulus B over Stim-ulus D when real-time simulations predict theopposite trend and when the stimuli areunorderable.

CONCLUDING COMMENTS

The failure of several associative models toaccount for the present data is not particu-larly surprising because none of these modelstake into account the ordering of the stimulialong a physical dimension (see Siemann &Delius, 1998; Wynne, 1995, for reviews),which appeared to play a vital role in ourstudy. Perhaps nonverbal transitive inferencecan be based on a history of past reinforce-ment when this history is the only cue avail-able, but also can be based on other cueswhen they are available? We suspect thatthere is more to the story than that. Markovitsand Dumas (1992) suggested that, if there isno scale underlying the relation ‘‘B is rein-forced, C is not reinforced’’ and ‘‘C is rein-forced, D is not reinforced,’’ then organisms’performance in such experiments could notby definition be based on transitive inference.In response, Wynne, von Fersen, and Stad-don (1992) proposed that the mere fact thattheir pigeons were able to select B over D inthe test is sufficient for showing that they ex-hibited transitive inference.

It is necessary to ask: What, exactly, aretransitive inference and transitivity? The termtransitive inference appears to require thattwo conditions be met: (a) there is a transitiverelation among the stimuli, and (b) the or-ganism performs an inferential operation. Al-though the importance of the second condi-tion is normally recognized, the importanceof the first condition is not.

In verbal syllogisms, transitivity is the prop-erty of a relation. It is the transitive relationthat makes transitive inference possible andvalid. For instance, given the nature of the

relation to like, no logical conclusion followsfrom the premises ‘’the boy likes the girl’’and ‘’the girl likes the dog.’’ The transitivityof a relation does not change with the train-ing schedule; no amount of exposure to thefirst and second premises renders the conclu-sion ‘‘the boy likes the dog’’ valid. Accordingto existing associative theories of nonverbaltransitive responding, however, transitive in-ference is possible and valid solely because ofreinforcement history. In fact, in associativeaccounts, a transitive inference becomes validas training proceeds. When the stimuli arenot orderable and untrained, there is no rea-son to prefer Stimulus B over Stimulus D.During training, an ordered series of stimuliemerges because the stimuli are presented ina given sequence and with a given frequency.In other words, transitivity is treated as aproperty of reinforcement history.

Although the sensitivity to different prob-abilities of reinforcement that led to transi-tive responding in the nonordered transitivetask is interesting per se, we suspect that itmight be largely unrelated to transitive infer-ence in the ordered task. This does not meanthat ‘‘true’’ transitive inference can never berevealed in a nonverbal task. Our data as wellas data obtained by Roberts and Phelps(1994) demonstrated that animals’ transitiveresponse could be controlled by transitive re-lations, not by the reinforcement history ofunorderable stimuli.

We do not imply, however, that an associa-tive approach is not useful for modeling non-verbal transitive inference in ordered tasks.Dynamic, reinforcement-based models havebeen shown to capture different effects of TIperformance (end-anchor, serial position,and symbolic distance effects) and the out-comes of different orders of training (Wynne,1995, 1998). It might be possible to developa reinforcement-based model that would in-corporate the relation between the stimuli—after all, this relation also needs to be learnedas training proceeds. Future research will seeif such an attempt can be successful. In con-clusion, we suggest that, for the sake of clarity,performance in the nonordered transitivetask should be called ‘‘transitive responding,’’as proposed by Delius and Siemann (1998).Nonordered tasks omit an essential definingfeature—the transitive relation among the


stimuli in a series—and should not be con-fused with ordered transitive tasks.

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Received March 21, 2003Final acceptance June 17, 2004

APPENDIX

A detailed description of the focal models,their advantages and their defects, can befound in the publications of Wynne (1995,1998) and Siemann and Delius (1998).

Siemann-Delius (or eta-kappa) model. In thismodel, the total associative value of each stim-ulus is a weighted average of its elementaland configural values. Assuming that pair X1Y2 is presented, the values of the stimuli areupdated according to the following equa-tions:

For elemental stimuli

V(X ) 5 V(X ) 1 b V(X ) p « (A1)i11 i 1 i X/XY

V(Y ) 5 V(Y ) 2 b V(Y ) (1 2 p )«i11 i 2 i X/XY

(A2)

where « is a weighting parameter for elemen-tal and configural values.

For configural values

V(X/XY )i11

5 V(X/XY ) 1 b V(X/XY ) p k (A3)i 1 i X/XY

V(Y/XY )i11

5 V(Y/XY ) 2 b V(Y/XY ) (1 2 p )ki 2 i X/XY

(A4)

where k 5 1 2 «, V(X/XY ) and V(Y/XY )are configural values for Stimulus X whenpaired with Y and for Stimulus Y when pairedwith X, respectively.

The probability of selecting Stimulus X ina given pair is

pX/XY

V(X )V(X/XY )5 . (A5)

V(X )V(X/XY ) 1 V(Y )V(Y/XY )

For the new testing pair that has neverbeen presented before, all configural valuesare zero. Thus the probability of selectingStimulus X in a new pair XA is

V(X )p 5 . (A6)X/XA V(X ) 1 V(A)

Siemann-Delius model with value transfer mod-ification. Here we report the results of simu-lations that incorporate value transfer onlyfor elemental values of the stimuli. The mod-el allows for different proportions of positiveand negative value transfer by using two dif-ferent parameters.

After the elementary values of the stimuliwere updated according to equations A1 andA2, the model further modifies those values

V(Y ) 5 V(Y ) 1 A V(X ) (A7)i11 i 1 i

V(X ) 5 V(X ) 2 A V(Y ) (A8)i11 i 2 i

where A1 and A2 are the parameters for pos-itive and negative value transfer, correspond-ingly.

All other values and probabilities are cal-culated as in the original Siemann-Deliusmodel.


Wynne model. Here stimulus value is updat-ed according to Rescorla-Wagner equations:for elemental values

V(X ) 5 V(X ) 1 b(1 2 [V(X ) 1 V(Z) ])i11 i i i

3 p (A9)X/XY

V(Y ) 5 V(Y ) 1 b(0 2 [V(Y ) 1 V(Z) ])i11 i i i

3 (1 2 p ) (A10)X/XY

and for configural values

V(X/XY )i11

5 V(X/XY )i

1 b[1 2 V(X/XY ) ]p (A11)i X/XY

V(Y/XY )i11

5 V(Y/XY )i

1 b[0 2 V(Y/XY ) ](1 2 p ) (A12)i X/XY

where V(X/XY ) and V(Y/XY ) are configuralvalues for Stimulus X when paired with Y andfor Stimulus Y when paired with X, respec-tively.

The probability of selecting X from a givenpair is

1p 5 (A13)X/XY 2a(2r21)1 1 e

where r is equal to

r 5 [V(X ) 1 V(Z ) 1 gV(X/XY )]

4 [V(X ) 1 V(Y ) 1 2V(Z )

1 gV(X/XY ) 1 gV(Y/XY )] (A14)

where g is a new parameter weighting config-ural stimulus values.

The probability of selecting Stimulus X ina new, first-presented pair (e.g. XA) is

1p 5 (A15)X/XA 2a(2r21)1 1 e

where r is equal to

V(X ) 1 V(Z )r 5 (A16)

V(X ) 1 V(A) 1 2V(Z )

because configural values for both X and Aare equal to zero.

Wynne model with value transfer. As before,only elemental values of the stimuli were sub-jected to value transfer, according to equa-tions A7 and A8. All other calculations re-main the same as in the original Wynnemodel.