the a theory of magnitude (atom) model in temporal perception and reproduction tasks

Acta Psychologica 139 (2012) 111–123

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Acta Psychologica

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The A Theory Of Magnitude (ATOM) model in temporal perception andreproduction tasks

Marco Fabbri ⁎, Jennifer Cancellieri, Vincenzo NataleDepartment of Psychology, University of Bologna, Italy

⁎ Corresponding author at: Department of Psychologerti Pichat, 5, 40127, Bologna, Italy. Tel.: +39 051 209

E-mail address: [email protected] (M. Fabbri

0001-6918/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.actpsy.2011.09.006

a b s t r a c t
a r t i c l e i n f o
Article history:Received 16 February 2011Received in revised form 7 September 2011Accepted 12 September 2011Available online 14 October 2011

PsycINFO classification:2340

Keywords:ATOM modelTimeSpaceNumberTime estimation taskTime reproduction task

According to the A Theory of Magnitude (ATOM) model, time, numbers and space are processed by a commonanalog magnitude system. The model proposes that time, numbers and space are influenced by each other.Indeed, spatial–temporal (STEARC effect), spatial–numerical (SNARC effect) and temporal–numerical(TiNARC effect) interactions have been observed. However, the processing of time, numbers and space hasnot yet been studied within the same experimental procedure. The goal of this study is to test the ATOMmodel using a procedure in which time, numbers and space are all present. The participants were asked toperform temporal estimation (Experiment 1) and reproduction (Experiment 2) tasks in two different condi-tions, with either numbers or letters as stimuli. In Experiment 1, significant STEARC, SNARC and TiNARC ef-fects were found in general and when numbers were presented. Moreover, a significant triple interactionbetween space, time and magnitude was observed, indicating associations between the left key, short dura-tion and small magnitudes, as well as between the right key, long duration and large magnitudes. These re-sults were similar in reaction times and accuracy. In Experiment 2, the results of reproduction times mirroredthe previous data but the triple interaction was not found on reproduction times. Considering the temporalaccuracy, the STEARC, SNARC and TiNARC effects as well as triple interaction were found. The results seemto partially confirm the ATOM model, even if differences between temporal tasks should be posited.

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1. Introduction

Time, space and quantity are important aspects of human and an-imal lives (Gallistel & Gelman, 2000). Recently, large volumes of re-search have examined the relationship between and among thethree aspects (Walsh, 2003; for a review see Bueti & Walsh, 2009;Fabbri & Natale, 2009). In particular the A Theory Of Magnitude(ATOM) model has been suggested, in which time, numbers andspace are computed according to a common metric and in a commonneural structure (Walsh, 2003). The model proposes that time, spaceand numbers are influenced by each other (see also, Conson, Cinque,Barbarulo, & Trojano, 2008).

The ATOM model has indeed received much behavioral evidence.For example, the interaction between time and space seems to conveya metaphoric representation of time along a line oriented from left toright (Casasanto & Boroditsky, 2008; Frassinetti, Magnani, & Oliveri,2009; Merritt, Casasanto, & Brannon, 2010; Vallesi, Binns, & Shallice,2008; Vicario, Caltagirone, & Oliveri, 2007; Vicario et al., 2008). Themain finding claims that short durations are represented on the leftof a temporal line while long durations are represented on the right.At the same time, it has been found that, in a temporal estimation

B

task, left stimuli tend to induce biases toward short durations whileright stimuli tend to induce biases toward long durations accordingto the position of durations along the line. More importantly, theidea of Spatial–Temporal Association of Response Codes (STEARC) effect(Ishihara, Keller, Rossetti, & Prinz, 2008) seems to further prove thespatial representation of time along the left–right dimension(Casasanto & Boroditsky, 2008). In the research carried out byIshihara et al., participants were asked to press one of two responsekeys depending on whether the timing of a given probe was earlieror later than expected based on the preceding clicks. The resultsshowed an association between left responses and early onset timingand between right responses and late onset timing (Ishihara et al.,2008). The time–space interaction seems to support the idea of amental time line (MTL), in which short temporal durations are repre-sented on the left side and long temporal durations on the right sideof a space in a left-to-right mapping (see also, Arzy, Adi-Japha, &Blanke, 2009; Arzy, Collette, Ionta, Fornari, & Blanke, 2009; Santiago,Lupiáñez, Perez, & Funes, 2007; Torralbo, Santiago, & Lupiáñez, 2006).

The metaphor of a MTL is similar to that of a mental number line(MNL) in which numbers seem to be represented along a horizontalleft-to-right line (Moyer & Landauer, 1967; Restle, 1970). The spatialnature of the MNL derives from the discovery of the Spatial–NumericalAssociation of Response Codes (SNARC) effect (Dehaene, Bossini, &Giraux, 1993). In a parity task, the small numbers are responded tofaster with the left key, whereas large numbers are responded to faster

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112 M. Fabbri et al. / Acta Psychologica 139 (2012) 111–123

with the right key (Dehaene et al., 1993; for a review see de Hevia,Vallar, & Girelli, 2008; Fias & Fischer, 2005; Gevers & Lammertyn,2005). The SNARC effect has been replicated using several cognitivetasks (for a meta-analysis see Wood, Nuerk, Willmes, & Fischer, 2008).Importantly, the SNARC effect arises even when the magnitudeprocessing is irrelevant for the task, suggesting the automatic activationof magnitude information (e.g., Fias, Brysbaert, Geypens, & d'Ydewalle,1996; Fischer, Castel, Dodd, & Pratt, 2003).

Recently, a connection between time and number magnitude hasbeen demonstrated. In a duration comparison task in which a Stroop-like interference paradigm was implemented, the larger stimuli (e.g.eight or nine dots) were judged to be temporally longer (Xuan, Zhang,He, & Chen, 2007). In line with this finding, it has been shown thatmerely looking at numbers causes a bias in temporal tasks that dependson the magnitude of the number (Cappelletti, Freeman, & Cipolotti,2009; Lu, Hodges, Zhang, & Zhang, 2009; Oliveri et al., 2008; Vicario,2007; Vicario et al., 2008). In the imaginative time bisection task, theparticipants were requested to stop an imaginarymental timer halfwaythrough a reference cue varying in duration andmagnitude (numbers 1,2, 8 and 9). The small numbers elicited a reduction in the bisectionreaction times while the large numbers caused the reaction timesto increase (Vicario, 2007). Similarly, in a time estimation task, theparticipants were required to judge whether the duration of a teststimulus was longer or shorter than that of a previous reference stimu-lus (a number 5 lasting 300 ms). The results showed that a low number(e.g. 1) led to underestimation and a high number (e.g. 9) led to overes-timation of the perceived duration (Cappelletti et al., 2009; Lu et al.,2009; Oliveri et al., 2008; Vicario et al., 2008). It is important to notethat this result was not obtained when presenting letters (Oliveri etal., 2008). Interestingly, Kiesel and Vierck (2009) used a parity judg-ment task within a classical SNARC paradigm, with the introduction ofdit–dah (i.e. Morse code, e.g. Klapp & Erwin, 1976) responses linkedto the temporal domain. Indeed, the dit response referred to a short du-rationwhereas the dah response referred to a long duration. The resultsshowed a Time–Numerical Association of Response Codes (TiNARC) effect,representing an association between small numbers and short responsedurations and between large numbers and long response durations(Kiesel & Vierck, 2009; see also Casarotti, Michielin, Zorzi, & Umiltà,2007; Müller & Schwarz, 2008; Turconi, Campbell, & Seron, 2006).

The STEARC, SNARC and TiNARC effects have usually been discussedwithin the framework of the ATOM model. In particular, they seem toreflect the idea that time, quantity and space are connected by a com-mon code for action. However, these three effects have been studiedseparately. Since there have been no studies, to our knowledge, consid-ering the processing of time, number and space in the same experimen-tal procedure, it is not completely clear whether there is onegeneralized system dealing with these three aspects or whether thereare separate modules that interact depending on current task require-ments. That is, it is not completely clear whether the time, numberand space systems are symmetrically or asymmetrically related. Theaim of this study is to investigate, for the first time, how time, spaceand numbers interact with each other when they are present withinthe same experimental procedure. In two experiments, the participantsperformed temporal estimation and reproduction tasks while numbersand letters were centrally presented. The spatial information wasprovided by two lateralized buttons. The participants were requestedto perform the tasks twice; the second time the assignment of instruc-tions to response keys was reversed. In the case of one generalizedsystem existing, STEARC, SNARC and TiNARC effects, as well as a tripleinteraction, were expected to appear in both tasks. If, on the otherhand, separate modules interacted depending on task requirements,the interactions among the three dimensions were expected to bevaried. In this latter prediction, for example, STEARC and/or TiNARCeffects were expected to appear in both tasks, with the SNARC effectand triple interaction absent, given that the temporal information wasrelevant for both tasks.

2. Experiment 1

The aim of Experiment 1 is to study the interaction between time,numbers and space in a time estimation task, such as that used byCappelletti et al. (2009). Moreover, a within-subjects design isapplied in order to detect the role of space in modulating time andnumber processing. The spatial information is provided by the spatialposition of the two response keys. The procedure used here has beendevised in order to detect STEARC and SNARC effects, the assignmentof the response keys to each task instruction being divided into twoseparate experimental blocks. The participants are tested in two con-ditions consisting of the presentations of either numbers or letters onthe computer screen. The choice of using both numbers and letters isbased on the results by Oliveri et al. (2008), who found that numbers,but not letters, influence temporal estimation. In a similar way,Casarotti et al. (2007) failed to report any automatic shift of attentionin a temporal order judgment task when letters (ranging from A to Ior from A to Z) were presented, while the attentional shift was dis-played with number presentation (for similar results with numberssee Fischer et al., 2003). However, a SNARC-like effect has beenreported when letters are presented (Gevers, Reynvoet, & Fias,2003; Nicholls & Loftus, 2007; for different results see Dehaeneet al., 1993; Zorzi, Priftis, Meneghello, Marenzi, & Umiltà, 2006); se-quentially ordered items (e.g. A and Z) also seem to influence the per-ception of temporal order (Schwarz & Eiselt, 2009). In Experiment 1we decided to test participants by presenting either numbers or let-ters in order to disambiguate the discrepancies reported in the litera-ture. It is important to note that the mental representation of ordinalsequences, such as numbers (Gevers, Verguts, Reynvoet, Caessens, &Fias, 2006), time (Santiago, Román, Ouellet, Rodríguez, & Pérez-Azor, 2010; Schwarz & Eiselt, 2009) and letters (Gevers et al., 2003;Nicholls & Loftus, 2007) seems to be spatially coded.

2.1. Participants

Thirty-six students from the University of Bologna participatedin this experiment as volunteers. The mean age was 23.50(SD=1.56) and there were 27 females (9 males). In order to assessthe handedness of the participants, they were asked to fill in the Ed-inburgh Inventory (Oldfield, 1971). There were 32 right-handed in-dividuals (mean=79.10, SD=18.83) and 4 left-handed individuals(mean=−84.47, SD=20.07). All participants had normal or corrected-to-normal vision.

2.2. Materials and procedure

In the task, the participants were required to judge whether theduration of a target stimulus was shorter or longer than that of a ref-erence stimulus (cue). Two conditions were created. In the experi-mental condition, the reference cue and target stimuli werenumbers. In the control condition, the reference cue and target stim-uli were letters. All participants were tested individually in a quietroom. They sat facing the computer screen. The viewing distancefrom the computer monitor was 60 cm. Stimulus presentation anddata collection were controlled using E-Prime 1.1. (Schneider,Eschman, & Zuccolotto, 2002). The display had a resolution of1072×960 pixels and was refreshed at a frequency of 72 Hz. The re-sponse keys were the “4” and “6” keys of the numerical keypad on anormal keyboard. They were covered by two green disks in order toavoid any numerical influence. The participants had to press the “4”key with their left hand and the “6” key with their right hand. Ineach condition, the task was performed twice, determining two dif-ferent blocks, in which the instruction-key assignment was counter-balanced. In one block the left key was pressed to judge the targetstimulus as shorter and the right key was pressed to judge the targetstimulus as longer. In the other block, the left key was pressed to

113M. Fabbri et al. / Acta Psychologica 139 (2012) 111–123

judge the target stimulus as longer and the right key was pressed tojudge the target stimulus as shorter. The order of key assignmentswas counterbalanced among subjects.

Similarly to the procedure carried out by Cappelletti et al. (2009),in both conditions a cross (+) sign, in white Courier New 120-pointtype, appeared on a black background at the center of the screen asa fixation point for 400 ms. After the fixation point a black screenappeared for 200 ms. Following this, a white reference fixed cue,also in Courier New 120-point type, appeared centrally on a blackbackground. It remained on the screen for 400 ms. Two referencecues were presented according to the conditions: a number 5 or letterE. Then a black screen appeared, as an inter-stimulus interval (ISI). Inorder to avoid any response strategy, this black screen could last for arandom interval selected from 700 to 800 ms. After this ISI, a targetstimulus, in white Courier New 120-point type on a black back-ground, appeared at the center of the screen (Vicario et al., 2008).One of four targets could appear according to the conditions (num-bers: 1, 2, 8, and 9 or letters: A, B, H, and I). The selection of the lettersfrom A to I was based on the fact that they corresponded to numbers1 to 9. The duration of the target could range from 200 to 600 ms insteps of 100 ms with the exception of 400 ms. The durations of 200or 300 ms were considered as short durations while the durationsof 500 or 600 ms were considered to be long durations. The partici-pants were instructed to refrain from making a judgment at this mo-ment. Indeed, after the presentation of the target stimulus, a whitequestion mark (?) on a black background appeared in the center ofthe screen. It was presented together with a BEEP sound in order toprompt participants to make a judgment by pressing one of the corre-sponding keys. The question mark remained on the screen for1500 ms or until the subject responded. It was stressed to the partic-ipants that accuracy as well as speed was fundamental. Finally, thenew fixation point appeared after a black screen of 400 ms. In eachblock, 128 trials were presented in a pseudo-random order. Thus, ineach condition, the participants judged 256 trials. Before the test, atraining sessionwas run, with 8 trials presenting all four target stim-uli lasting 200 or 600 ms. The training phase could be performed fora second time if requested by participants. The order of conditionswas counterbalanced across subjects. After each block, individualshad the opportunity to take a 1-minute break. The experiment lastedapproximately 60 min.

2.3. Data analysis

Means of reaction times (RT) of correct responses were calculatedfor both conditions. Moreover, RTs lower or greater than 3 SD wereexcluded from analysis because they were considered outliers(about 2%). A four-way repeated measures ANOVA was carried outon RTs, with Stimulus (2 levels: number vs. letter), Key (2 levels:left vs. right), Duration (2 levels: short vs. long) and Magnitude (2levels: small vs. large) as within-subjects factors. When numberswere presented, the numbers 1 and 2 represented a small magnitudewhile the numbers 8 and 9 represented a large magnitude. When let-ters were presented, the letters A and B represented a small magni-tude while the letters H and I represented a large magnitude. Thesame ANOVA was performed on accuracy, as the numbers of errors(NEs) after arcsine transformation. When a reliable significance wasfound, the Scheffè post-hoc test was run. Values with pb .05 wereconsidered significant.

2.4. Results and discussion

The mean RTs and numbers of errors for both number and letterconditions are summarized in Table 1. In order to describe the resultsin a more concise way, Table 1 also shows the results of ANOVAs onRTs and NEs.

Regarding RTs, there was a significant interaction between Stimulusand Magnitude, reflecting the fact that number stimuli determinedhigher RTs than when letters were presented for both small (1/2:516 ms vs. A/B: 501 ms) and large (8/9: 515 ms vs. H/I: 508 ms)magni-tudes (pb .005 for both comparisons). According to the aim of the study,theANOVA showed a significant Key×Duration interaction, indicating ageneral STEARC effect. The Scheffè post-hoc test revealed that the rightkey (485 ms) was faster than the left key (520 ms) in detecting longdurations while the left key (501 ms) was faster compared to theright key (534 ms) in detecting short durations (pb .05 for both com-parisons). In order to assess the strength and the direction of this inter-action, we applied the regression analysis method shown by Lorch andMyers (1990; see also Fias et al., 1996). Thismethod consists of comput-ing the dRT (RT right hand minus RT left hand) for each durationseparately. This score was then regressed on duration, yielding a non-standardized regression weight for each participant, and capturing thedirection and strength of the spatial mapping of time. These slope coef-ficients were tested against zero. A negative regression slope wasexpected, indicating that short durations were associated with the lefthand while long durations were associated with the right. The regres-sion weight was −0.19 ms/duration (SD=0.31 ms/duration), and itdeviated significantly from zero, t(35)=−3.63, pb .005, indicating areliable STEARC effect (Fig. 1A).

Moreover, the Key factor significantly interacted with Magnitudefactor, indicating a general SNARC effect. In the Scheffè post-hoc test,the right key (503 ms) was faster than the left key (520 ms) inresponding to largemagnitudes while the left key (501 ms)was fasterthan the right key (516 ms) in responding to small magnitudes(pb .005 for both comparisons). In order to quantify this SNARC effect,we applied the same regression analysis method adopted above, ex-cept for the fact that we considered magnitude as a predictor variable.The regressionweight was−4.42 ms/magnitude (SD=3.52 ms/mag-nitude), and it deviated from zero, t(35)=−7.53, pb .00001, indicat-ing a reliable SNARC effect (Fig. 1B).

A significant Duration×Magnitude interaction was also found, indi-cating a TiNARC effect. The post-hoc test showed that a largemagnitudewas more closely associated with long durations (501 ms) than shortdurations (521 ms), with pb .005. Within the short durations, the par-ticipants obtained higher RTs when a large magnitude (521 ms) waspresented compared to a small magnitude (513 ms), with pb .05. Inorder to quantify this TiNARC effect, we applied the same regressionanalysis method adopted above, except for the fact that we subtractedthe short duration from the long duration for each magnitude, this fac-tor being the predictor variable. The negative slope appeared to indicatethat short durations were associated with small magnitudes while longdurations were associated with large magnitudes. The regressionweight was −1.62 ms/magnitude (SD=3.37 ms/magnitude), andit deviated significantly from zero, t(35)=−2.88, pb .05, indicatinga reliable TiNARC effect (Fig. 1C).

In addition, the ANOVA showed three significant interactions:Stimulus×Key×Magnitude, Stimulus×Duration×Magnitude, andKey×Duration×Magnitude. The evaluation of these significantthree-way interactions was explained by a set of two-way ANOVAs asexplained below. On the contrary, the lack of statistical significance ofthe Stimulus×Key×Duration interaction seemed to indicate a similarSTEARC effect for numbers and letters. To corroborate this idea, theregression method was carried out. For numbers, the regressionweight was −0.20 ms/duration (SD=0.40 ms/duration), and it de-viated significantly from zero, t(35)=−2.94, pb .005, indicating areliable STEARC effect. For letters, the regression weight was −0.18 ms/duration (SD=0.32 ms/duration) and it deviated significantlyfrom zero, t(35)=−3.33, pb .005, indicating a reliable STEARC effect.Moreover, no significant difference between regression weights wasshown (t(35)=−0.30, p=.76) (Fig. 1A).

Two repeated measures ANOVAs with Key and Magnitude aswithin-subjects factors were carried out separately for numbers and

Table 1The mean RTs (and their SD) and NEs (and their SD) of Experiment 1 are presented for each magnitude, duration and response key. The table also summarizes the F, p and partialeta-squared (ηp2) values for both ANOVAs on RTs and NEs. In bold are the significant results.

1–2 8–9 A–B H–I

Left key Right key Left key Right key Left key Right key Left key Right key

RTsShort durations 478 552 531 523 495 528 499 532

(68.60) (95.66) (69.04) (92.34) (69.34) (57.60) (70.48) (57.17)Long durations 526 508 536 469 506 475 513 487

(136.22) (117.57) (141.74) (111.46) (112.12) (88.63) (107.94) (92.44)

NEsShort durations 0.35 0.99 1.09 1.03 0.64 1.27 0.68 1.36

(0.41) (0.61) (1.02) (1.04) (0.86) (0.91) (0.90) (0.97)Long durations 0.80 1.37 1.24 0.28 1.25 0.79 1.49 0.76

(0.93) (1.09) (1.03) (0.39) (0.90) (0.64) (0.92) (0.64)

ANOVA on RTs ANOVA on NEs

F values Degrees of freedom p values ηp2 F values Degrees of freedom p values ηp2

Stimulus 1.07 1,35 .31 .03 6.83 1,35 b.05 .16Key 0.12 1,35 .73 .003 0.29 1,35 .59 .09Duration 2.73 1,35 .11 .07 0.57 1,35 .45 .02Magnitude 2.50 1,35 .12 .07 1.42 1,35 .24 .04Stimulus×Key 1.23 1,35 .27 .03 0.01 1,35 .91 .000Stimulus×Duration 0.79 1,35 .38 .02 0.03 1,35 .86 .001Stimulus×Magnitude 5.43 1,35 b.05 .13 0.35 1,35 .55 .01Key×Durations 16.43 1,35 b.005 .32 23.38 1,35 b.00001 .40Key×Magnitude 54.92 1,35 b.00001 .61 31.41 1,35 b.00001 .47Duration×Magnitude 7.43 1,35 b.05 .17 10.56 1,35 b. 005 .23Stimulus×Key×Duration 0.38 1,35 .54 .01 17.56 1,35 b. 0005 .33Stimulus×Key×Magnitude 33.93 1,35 b.00001 .49 18.63 1,35 b.0001 .35Stimulus×Duration×Magnitude 17.09 1,35 b.0005 .33 11.10 1,35 b.005 .24Key×Duration×Magnitude 6.45 1,35 b.05 .16 8.72 1,35 b.005 .20Stimulus×Key×Duration×Magnitude 3.07 1,35 .09 .08 2.33 1,35 .14 .06


letters. As regards numbers, the ANOVA showed a significant interac-tion, indicating a significant SNARC effect (F(1,35)=60.59, pb .00001,ηp2=.63). Applying the regression method, we obtained a regressionweight of −9.14 ms/number (SD=7.08 ms/number), deviating sig-nificantly from zero, t(35)=−7.75, pb .00001, and indicating a reli-able SNARC effect (Fig. 1B). As regards letters, the analysis did notfind a significant SNARC-like effect (F(1,35)=0.21, p=.65), as fur-ther confirmed by regression analysis, given that the regressionweight was +0.30 ms/letter (SD=4.94 ms/letter), and it did not de-viate significantly from zero (t(35)=0.37, p=.71) (Fig. 1B).

As before, we performed two repeatedmeasures ANOVAs, with Du-ration and Magnitude as within-subjects factors, separately for num-bers and letters. As regards numbers, the interaction between the twofactors was significant (F(1,35)=44.68, pb .00001, ηp2=.56), indicatinga TiNARC effect. On performing the regression analysis, we found thatthe regression weight was −3.85 ms/number (SD=4.87 ms/number)and it deviated significantly from zero, t(35)=−4.75, pb .0005, indi-cating a reliable TiNARC effect (Fig. 1C). As regards letters, the analysisdid not reveal a significant interaction, indicating no TiNARC effectfor letter materials (F(1,35)=0.79, p=.38) as also shown by the re-gression analysis: the regression weight was +0.61 ms/letter(SD=4.49 ms/letter), which did not deviate from zero (t(35)=0.82,p=.42) (Fig. 1C).

Given the complexity of the three-ways interaction Key×Dura-tion×Magnitude, a set of two-way ANOVAs was performed, inorder to detect the modulation of STEARC, SNARC and TiNARC effectsaccording to the remaining magnitude, temporal or spatial factorsrespectively. All results are summarized in Fig. 1D.

As far as magnitude was concerned, the Key×Duration ANOVAshowed the STEARC effect for both large (F(1,35)=11.68, pb .005,ηp2=.25) and small (F(1,35)=20.31, pb .0001, ηp2=.37) magnitudes.These results were confirmed by both regression analyses (large: −0.17 ms/duration, SD=0.31 ms/duration, t(35)=−3.29, pb .005;

small: −0.20 ms/duration, SD=0.33 ms/duration, t(35)=−3.78,pb .0005). The comparison between both STEARC effects was non-significant (t(35)=1.39, p=.17).

Regarding duration, the Key×Magnitude ANOVA showed a sig-nificant SNARC effect for both long (F(1,35)=16.62, pb .0005,ηp2=.32) and short (F(1,35)=50.69, pb .00001, ηp2=.59) durations.As before, both regression analyses showed reliable SNARC effects(long: −3.08 ms/magnitude, SD=4.57 ms/magnitude, t(35)=−4.05, pb .005; short: −5.99 ms/magnitude, SD=4.66 ms/magni-tude, t(35)=−7.72, pb .00001). The comparison between bothSNARC effects showed that the SNARC effect for short durationswas stronger than that for long durations (t(35)=2.70, pb .05).

As far as key (space) was concerned, the Duration×MagnitudeANOVA revealed a significant TiNARC effect for left (F(1,35)=17.76,pb .0005, ηp2=.34) but not for right (F(1,35)=0.08, p=.78) keys. Thesame results were obtained by applying the regression method(left: −2.95 ms/magnitude, SD=4.04 ms/magnitude, t(35)=−4.39, pb .0005; right: −0.29 ms/magnitude, SD=5.19 ms/magni-tude, t(35)=−0.33, p=.74). The TiNARC effect for the left keywas stronger than that for the right key (t(35)=2.50, pb .05).

The ANOVA on NEs mirrored the results of RTs, confirming thepresence of general STEARC, SNARC and TiNARC effects (Table 1). Asbefore, the STEARC effect was found in conditions regarding bothnumbers and letters (number: F(1,35)=7.02, pb .05, ηp2=.17; letter:F(1,35)=33.05, pb .00001, ηp2=.49), while the SNARC (F(1,35)=32.26, pb .00001, ηp2=.48) and TiNARC (F(1,35)=19.11, pb .0001,ηp2=.35) effects were reliable with numbers but not with letters (F(1,35)=0.93, p=.34 and F(1,35)=0.10, p=.75 respectively). Asshown in Fig. 1E, the STEARC effect was found regarding magnitude(small: F(1,35)=11.77, pb .005, ηp2=.25; large: F(1,35)=24.20,pb .0001, ηp2=.41); the SNARC effect remained reliable for duration(short: F(1,35)=6.14, pb .05, ηp2=.15; long: F(1,35)=30.77,pb .0001, ηp2=.47); the TiNARC effect was shown for the right (F


(1,35)=23.16, pb .0001, ηp2=.40) but not for the left key (F(1,35)=0.09, p=.77).

The main results found here reinforced the idea of one generalizedsystem dealing with time, space and numbers. Indeed, general STEARC,SNARC and TiNARC effects were displayed, when both RTs and accuracywere analyzed. However, these effects were linked to experimentalma-terials in different ways. A significant STEARC effect (Ishihara et al.,2008; Vallesi et al., 2008) was found when both numbers and letterswere presented: the right space was linked to long durations and theleft space was linked to short durations. This spatial congruency be-tween time and response seems to give evidence for amental represen-tation of time along a spatial line (Casasanto & Boroditsky, 2008). Asregards the SNARC effect, the spatial–numerical association was onlyfound when numbers were presented on the screen (Dehaene et al.,1993). This result was in line with the study of Kiesel and Vierck(2009), given that a spatial–numerical association was found even if

Fig. 1. A) STEARC effect: observed RT difference (in ms) between right and left keys for geconditions, and the linear regressions of RT difference on the duration (black line for G, dashence (in ms) between right and left keys for general (G; black square) numerical (N; blackdifference on magnitude (black line for G, dashed line for N and dotted line for L) are displayfor general (G; black square) numerical (N; black dot) and alphabetical (L; black diamond) conline for N and dotted line for L) are displayed; D) dRT patterns of Key×Duration×Magnitude ininteraction is displayed for each factor separately.

the numerical processingwas irrelevant for the task.When alphabeticalmaterialwas presented, no SNARC-like effect was obtained (Casarotti etal., 2007; Dehaene et al., 1993; Oliveri et al., 2008; Zorzi et al., 2006). Inaccordance with Zorzi et al. (2006), we suggest that the associationbetween space and letters might be categorical. Indeed, we found amagnitude-like effect with letters, given that the RTs increased as afunction of the increase in letter order. This result could indicate, forinstance, that the participants processed the letters in a categoricalway. Specifically, the letters A and B were “lower” than reference Ewhile the letters H and I were greater than reference E. In the presentstudy, when numbers were presented and the resulting pattern of RTswas observed, a TiNARC effect was shown, albeit weakly. In particular,the TiNARC effect was shown in relation to the association betweenlong durations and large numbers, but it disappeared when the associ-ation between short durations and small numbers was considered.However, the time–number association was more consistent in the

neral (G; black square) numerical (N; black dot) and alphabetical (L; black diamond)ed line for N and dotted line for L) are displayed; B) SNARC effect: observed RT differ-dot) and alphabetical (L; black diamond) conditions, and the linear regressions of RTed; C) TiNARC effect: observed RT difference (in ms) between long and short responsesditions, and the linear regressions of RT difference onmagnitude (black line for G, dashedteraction displayed for each factor separately; E) NE pattern of Key×Duration×Magnitude

}

Fig. 1 (continued).


result pattern of NEs. Indeed, the participants committed lower NEswhen there was a congruency between temporal and numerical infor-mation. This result indicated an association, on one hand, betweenshort duration and small numbers, and on the other hand, betweenlong duration and large numbers (e.g. Cappelletti et al., 2009; Kiesel &Vierck, 2009; Oliveri et al., 2008; Vicario et al., 2008; Xuan et al.,2007). The TiNARC effect could suggest a common cognitive code fortime and numerical quantity (Kiesel & Vierck, 2009; Walsh, 2003).That is, the (irrelevant) numerical processing affects the time estima-tion, even if this influence hasmainly been shown in temporal accuracy.

Finally, a triple interaction between space, time and numbers wasobserved, in agreement with the idea of the ATOM model (Walsh,2003). This result seems to indicate that there is one generalized sys-tem dealing with space, time and magnitude, even if the modulationof STEARC, SNARC and TiNARC effects differed according to theremaining magnitude, temporal or spatial factors respectively. In-deed, the STEARC effect remained for both large and small magni-tudes. In a similar way, the SNARC effect was shown when long orshort durations were analyzed separately. The TiNARC effect was ob-served only when the left space was considered, disappearing whenthe right space was considered. This advantage of left space (i.e. leftkey or left hand) compared to right space (i.e. right key or righthand) is in line with recent results showing that the left hand is sen-sitive to visual context in reaching (Adam, Müskens, Hoonhorst, Pratt,& Fischer, 2010), and in finger lifting and manual aiming (Ishihara &Imanaka, 2007). Given that the role of the right parietal cortex inmodulating the interactions among time, space and numbers(Walsh, 2003), this advantage of the left key (hand) for the reliabilityof the TiNARC effect can be explained by the hemispheric asymmetryin motor domain (Adam et al., 2010; Ishihara & Imanaka, 2007). An-other possibility, however, is related to two findings. Firstly, theTiNARC effect remained reliable with the presentation of numericalstimuli but it disappeared with alphabetical stimuli. Secondly, we ob-served a magnitude-like effect for letters with increasing RTs fromletter A to letter I. As indicated by Gevers et al. (2003), alphabeticalmaterial can be ordered from left to right. Even if no SNARC-like effect

was found with letters, the association between A and B with the leftspace probably resulted as more salient than the association betweenH and I with the right space. Bearing in mind that the triple interac-tion was significant regardless of experimental stimuli, the TiNARC ef-fect in the left space could be due to additive effects of these twofindings.

However, alternative views could be taken into account, especial-ly observing Fig. 1. Indeed, the observation of dRTs seems to indicatea categorical fashion (for a similar result with numbers, see Gevers,Verguts, Reynvoet, Caessens, & Fias, 2006). Firstly, the SNARC effectseems to reflect a direct mapping between the position of a numberon an internal spatial representation and the corresponding re-sponse location in the space. In other words, the SNARC effect re-flects a spatial stimulus–response compatibility between themental number representation and the response position. Hypothe-sizing a mental time line, the STEARC effect, as well as the TiNARC ef-fect, could be provided by the same stimulus–responsecompatibility. Secondly, the SNARC effect has recently been consid-ered an instance of a polarity correspondence principle, assumingthat numbers (small [−] and large [+]) and response (left [−] andright [+]) are coded on a bipolar dimension and that correspondingpolarities induce a faster response selection (Proctor & Cho, 2006). Ina similar way, the STEARC and TiNARC effects could account for thepolarity correspondence between durations (short [−] and long[+]) and response, or numbers, respectively. Consequently, the tri-ple interaction found here could reflect a polarity correspondencebetween time, number and space (short [−], small [−] and left[−]; long [+], large [−] and right [+]), influencing performance re-garding both speed and accuracy. The third view explains the SNARC ef-fect according to a multiple-layers computational model with threelayers (Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006). The bottomlayer represents the mental number line. The middle layer receivesinput from the number line representation and thus the numbersare automatically categorized as either small or large (or categori-cally coded according to the task demands). The top layer refers tospatially defined responses, coding for a left or right response. As


before, this model could be applied in explaining both the STEARCand TiNARC effects, suggesting a mental time line on the bottomlayer. The triple interaction could arise from the activation of mentalnumber and time lines which interact with the intermediate layer,categorizing durations as short or long, and numbers as small orlarge. Consequently, this categorization is associated with spatiallydefined responses. The fourth view provides a working memory ac-count for spatial–numerical associations (van Dijck & Fias, 2011).The assumption is based on the fact that the SNARC effect disappearsunder a working memory load (Herrera, Macizo, & Semenza, 2008;van Dijck, Gevers, & Fias, 2009), suggesting the role of workingmemory resources for the spatial coding of numbers. It is plausiblethat the position within a sequence of numbers in working memorydetermines the association between numbers and space: items to-ward the beginning of the sequence are associated with the leftspace and those toward the end are linked to the right space. Thatis to say that the temporary associations between position andspace determine the SNARC effect rather than the long-term seman-tic representations of number. According to this view, the STEARC,SNARC and TiNARC effects could be explained taking into accountthat durations and numbers are ordered sequences. In our task pro-cedure, the participants were requested to estimate the durations ofboth reference and target but to respond only when a question markpresented together with a sound appeared on the screen. Thus, ourtemporal estimation task claims for the involvement of workingmemory resources, which appear to be crucial for the associationsbetween durations and/or numbers and space. Nevertheless, allthese alternative views should predict STEARC-, SNARC- andTiNARC-like effects when letters are presented. The letters seem tobe spatially ordered from left-to-right just as numbers or time are(Gevers et al., 2003). Consequently, an influence of non-numericalsequences (letters) on temporal estimation should also be found.This influence was, in part, shown in the general effects and interac-tion between triple Key×Duration×Magnitude, but the effectstended to disappear when analyzing the temporal performancewith letter presentation.

All together the ATOM model and alternative views seem to agreewith the idea that the STEARC, SNARC and TiNARC effects can arise ata response-related stage (Ishihara et al., 2008; Keus & Schwarz, 2005;Kiesel & Vierck, 2009), given that even the estimation task requires aresponse. Xuan et al. (2007) demonstrated a connection betweentime and several types of magnitudes (e.g., numbers of dot, size, lumi-nance and Arabic digits), confirming the existence of generalized andabstract components in the magnitude representations. The authorsclearly showed that larger magnitudes were judged to be temporallylonger. From Xuan et al.'s results, it seems that the effect might stemfrom a perceptual processing stage, because the perception of timeduration was judged.

Experiment 2 aims to study the interaction between time, spaceand numbers using a time reproduction task, with a strong motorcomponent. In this way, we should be able to deeply address wherethe ATOM model takes place. This is prompted by the fact that timeestimation and time reproduction tasks seem to be quite similar. Inboth tasks some motor preparation and/or execution is requiredand there is a visual presentation of the time durations. The differencebetween them is that the estimation of the whole duration is opera-tionalized by a motor act in the time reproduction task, whereas itis based entirely on a perceptual judgment in the time estimationtask. These two tasks are directly comparable as they share commonencoding and storing of temporal information, but differ in how par-ticipants use time information to make responses (Bueti, Walsh, Frith,& Rees, 2008). In the time estimation task, the motor response is re-quired in order to make an estimation (i.e., shorter-longer), while inthe time reproduction task, the motor response is required to repro-duce an estimated duration, and thus the response-related stageand response execution are probably involved.

3. Experiment 2

In literature, it has been found that the association between timeand space as well as time and number appears to be more pronouncedin a time bisection task than in a time reproduction task (Frassinetti etal., 2009; Oliveri et al., 2009; Vicario, 2007). Classically, in both tasksthe individuals are requested to press a non-lateralized response but-ton. This experimental procedure could reduce the stimulus–responsecongruency, limiting, in particular, the STEARC and TiNARC effects.

Here, a modified version of the time bisection task used by Vicario(2007) is applied. As before, a within-subjects design is applied inorder to detect the role of space in modulating the time and numberprocessing. Thus, the participants performed a time reproductiontask, pressing a right-hand side button in one condition and a left but-ton in the other, in order to stop an imaginary clock. As before, theparticipants performed the task both with numerical and alphabeticalmaterials. Since the STEARC, SNARC and TiNARC effects seem to ariseat the level of response initiation and execution (Ishihara et al., 2008;Keus & Schwarz, 2005; Kiesel & Vierck, 2009), these effects may alsobe found in a reproduction task.

3.1. Participants

Twenty-five students from the University of Bologna participatedin this experiment as volunteers. The mean age was 26.32(SD=5.75) and there were 18 females (7 males). In order to as-sess the handedness of the participants, they were asked to fill inthe Edinburgh Inventory (Oldfield, 1971). There were 23 right-handedindividuals (mean=84.08, SD=17.79) and 2 left-handed individuals(mean=−94.74, SD=7.44). All participants had normal or corrected-to-normal vision. None of them had participated in Experiment 1.

3.2. Materials and procedure

Except for the following changes, the materials and the procedurewere identical to the ones used in Experiment 1. First, the participantswere required to reproduce the duration of a presented stimulus. Thetask was repeated in four separate sessions: numerical condition andright key (in which subjects were instructed to press the “6” key of anumerical keypad on a normal keyboard with their right hand), nu-merical condition and left key (the “4” key was pressed with the lefthand), alphabetical condition and right key, alphabetical conditionand left key. The response buttons were covered by two green disksin order to avoid any numerical influence. The order of key assign-ments was counterbalanced among subjects. Then, the white refer-ence stimulus (cue) remained centrally on the black screen for oneof five different durations: 200, 300, 400, 500 or 600 ms (estimationphase). The durations of 200 and 300 ms were considered as shorttimes while those of 500 and 600 ms were considered as long times.Two reference cues were presented according to the conditions: anumber 5 or letter E. After this, five targets could appear accordingto the conditions (numbers: 1, 2, 5, 8, and 9, vs. letters: A, B, E, H,and I). The participants were then instructed to reproduce the dura-tion of the reference cue after the presentation of a BEEP sound(reproduction phase). The sound activated a virtual clock and the par-ticipants had to stop this clock by pressing the correct button when asimilar duration had passed. In each block, 125 trials were presentedin a pseudo-random order. Thus, in each condition, the participantsjudged 250 trials. Before the test, a training session was run with 10trials presenting all five target stimuli lasting 200 or 600 ms. Thetraining phase could be performed for a second time, if requestedby participants. The order of conditions was counterbalanced acrosssubjects. After each block, individuals had the opportunity to take a1-minute break. The experiment lasted approximately 60 min.


3.3. Data analysis

Means of reproduction times (ReT) were calculated for each taskin both conditions. A four-way repeated measures ANOVA wascarried out on ReTs with Stimulus (2 levels: numbers vs. letters),Key (2 levels: left vs. right), Duration (2 levels: short and long) andMagnitude (2 levels: small vs. large) as within-subjects factors.When numbers were presented, the numbers 1 and 2 represented asmall magnitude while the numbers 8 and 9 represented a large mag-nitude. When letters were presented, the letters A and B representeda small magnitude while the letters H and I represented a large mag-nitude. The same ANOVA was also performed on accuracy index (AI),defined by the ratio between mean reproduced durations and the ref-erence durations for each subject in each condition. Thus, valuesequal to 1 indicated a good performance, while values above 1 indi-cated overestimation and values below 1 indicated underestimation.When a reliable significance was found, the Scheffè post-hoc testwas run. Values with pb .05 were considered significant.

3.4. Results and discussion

The mean ReTs and AIs for both number and letter conditions aresummarized in Table 2. In order to describe the results in a more con-cise way, Table 2 also shows the results of ANOVAs on ReTs and AIs.

Regarding ReTs, there was a Duration effect, indicating that longdurations (514 ms) determined higher ReTs than short durations(383 ms). This duration effect was different for number and letterconditions. The Scheffè post-hoc test showed that the number condi-tion (540 ms) elicited higher ReTs than the letter condition (488 ms)for reproducing long durations (pb .0005). On the contrary, the num-ber condition (364 ms) induced lower ReTs than the letter condition(403 ms) for reproducing short durations (pb .0005).

Importantly, a significant STEARC effect was found. The post-hocindicated that the participants generated higher ReTs with the left(563 ms) compared to the right (465 ms) key when they reproducedlonger durations (pb .005). By contrast, the participants determined alower ReT with the left (346 ms) compared to the right (420 ms) keywhen they reproduced shorter durations (pb .005). As in Experiment1, the regression method (Lorch & Myers, 1990; see also Fias et al.,1996) was adopted. The regression weight was −0.51 ms/duration(SD=0.39 ms/duration), and it differed from zero significantly, t(24)=−6.47, pb .0001, indicating a reliable general STEARC effect (Fig. 2A).

The Key factor also interactedwith theMagnitude factor, reflecting aSNARC effect. The large magnitudes determined shorter reproductionswith the right (433 ms) compared to the left (472 ms) key, while thesmall magnitudes determined shorter reproductions with the left(437 ms) compared to the right (453 ms) key, with pb .005 for bothcomparisons. On applying the regression analysis it was found thatthe regressionweightwas−7.94 ms/magnitude (SD=4.06 ms/magni-tude) and it deviated from zero significantly, t(24)=−9.77, pb .00001,indicating a reliable general SNARC effect (Fig. 2B).

A TiNARC effect was also found. The post-hoc test showed that forshort durations only, small magnitudes (375 ms) elicited lower ReTsthan large magnitudes (392 ms), with pb .005. For long intervals nosignificant difference between large and small magnitudes wasfound, even if the ReTs were lower for large (513 ms) magnitudesthan small ones (515 ms). With the regression method, it wasshown that the regression weight was −2.72 ms/magnitude(SD=4.20 ms/magnitude) and it deviated significantly from zero, t(24)=−3.23, pb .005, indicating a reliable general TiNARC effect(Fig. 2C).

Finally, the significant triple-way interactions were: Stimulus×-Key×Duration, Stimulus×Key×Magnitude, and Stimulus×Dura-tion×Magnitude. In order to clarify the significant triple-wayinteractions, a set of repeated measures ANOVAs with two factorswas carried out on ReTs separately for number and letter conditions.

We performed two ANOVAs with Key and Duration as within-sub-jects factors separately for numerical and alphabetical materials. Re-garding numbers, the ANOVA displayed a Key×Duration interaction(F(1,24)=27.79, pb .00001, ηp2=.54), reflecting a STEARC effect. Theregression weight was −0.57 ms/duration (SD=0.60 ms/duration),and it differed from zero significantly, t(24)=−4.80, pb .0005. Regard-ing alphabetical materials, the ANOVA confirmed the same results, witha significant interaction (F(1,24)=92.12, pb .00001, ηp2=.79). The re-gression weight was −0.37 ms/duration (SD=0.22 ms/duration), andit differed from zero significantly, t(24)=−8.39, pb .0001. The STEARCeffect in the number condition was stronger than that in letter condi-tion, t(24)=−2.25, pb .05 (Fig. 2A).

Two repeatedmeasures ANOVAs,with Key andMagnitude aswithin-subjects factors, were carried out separately for numbers and letters. Asregards numbers, the ANOVA showed a significant SNARC effect (F(1,24)=222.91, pb .00001, ηp2=.90) and this was further confirmed bythe regression method (b=−14.44 ms/magnitude, SD=7.60 ms/mag-nitude; t(24)=−9.50, pb .00001). Regarding letters, the ANOVA did notreveal any significant interactions (F(1,24)=2.52, p=.12). The regres-sion weight was −1.43 ms/magnitude (SD=4.43 ms/magnitude) andit did not deviate from zero (t(24)=−1.61, p=.12). The comparisonbetween both regression weights showed a larger SNARC effect in thenumber condition (t(24)=−6.90, pb .00001), as shown in Fig. 2B.

Two repeated measures ANOVAs, with Duration and Magnitude aswithin-subjects factors, were performed on ReTs, separately fornumbers and letters. The analysis on numerical stimuli revealed areliable TiNARC effect (F(1,24)=19.50, pb .0005, ηp2=.45). The regres-sion analysis further supported this result: b=−5.06 ms/magnitude(SD=5.67 ms/magnitude), which was significantly different fromzero (t(24)=−4.46, pb .0005). The ANOVA did not find significantTiNARC effect regarding letters (F(1,24)=0.20, p=.66). The regressionanalysis showed a regression weight of −0.37 ms/magnitude(SD=6.20 ms/magnitude), which did not deviate from zero, t(24)=−0.30, p=.76. The TiNARC effect in the number conditionwas strongerthan that in the letter condition, t(24)=−2.79, pb .05 (Fig. 2C).

As regards AI, the analysis (Table 2) was similar to that of ReTs.Consequently, general STEARC, SNARC and TiNARC effects werefound. As shown from the post-hoc test, the STEARC effect reflecteda lower AI for short durations with the left (1.43) than with theright (1.74) keys (pb .005). Moreover, AI was lower for long durationsreproduced with the right key (0.85) than that elicited with the left(1.03) key (pb .05). The SNARC effect was only reliable comparingthe performance of the right (1.32) and left (1.17) keys for smallmagnitudes. Even if it was not significant, the right (1.27) key tendedto induce lower AI than the left (1.29) key for large magnitudes. Final-ly, the TiNARC effect indicated that Vierdordt's law (i.e., subjectsoverestimated short durations and underestimated long durations;see for example, Oliveri et al., 2009) was reduced for short durationswhen small magnitudes instead of large magnitudes were presented.

Moreover, the STEARC effect was significant in both number (F(1,24)=37.21, pb .00001, ηp2=.61) and letter (F(1,24)=30.82,pb .00001, ηp2=.56) conditions. The SNARC effect was found in thenumber (F(1,24)=110.19, pb .00001, ηp2=.82) but not in the letter(F(1,24)=0.31, p=.58) condition. In a similar way the TiNARC effectfollowed the same pattern (number: F(1,24)=35.66, pb .00001,ηp2=.60; letter: F(1,24)=0.49, p=.49).

Also the ANOVA on AI showed a Key×Duration×Magnitude inter-action. The STEARC effect was found for small (F(1,24)=45.00,pb .00001, ηp2=.65) and large (F(1,24)=32.26, pb .00001, ηp2=.57)magnitudes. The SNARC effect remained reliable for short (F(1,24)=48.34, pb .00001, ηp2=.67) and long (F(1,24)=94.29, pb .00001,ηp2=.80) durations. Regarding the left key (F(1,24)=37.44,pb .00001, ηp2=.61) but not the right key (F(1,24)=0.41, p=.53,ηp2=.02), the TiNARC effect was reliable (Fig. 2D).

Finally, main effects of Key, Duration and Magnitude were found.These effects reflected an overestimation in the following situations:

Table 2The mean ReTs (and their SD) and AIs (and their SD) of Experiment 2 are presented for each magnitude, duration and response key. The table also summarizes the F, p and partialeta-squared (ηp2) values for both ANOVAs on ReTs and AIs. In bold are the significant results.

1–2 8–9 A–B H–I

Left key Right key Left key Right key Left key Right key Left key Right key

ReTsShort durations 268 431 351 406 382 419 385 426

(21.74) (108.68) (50.02) (102.03) (79.63) (124.39) (71.53) (129.09)Long durations 573 513 616 460 524 450 538 438

(143.59) (26.65) (185.04) (34.24) (91.67) (88.15) (86.67) (103.48)

AIsShort durations 1.10 1.79 1.44 1.67 1.58 1.73 1.59 1.76

(0.10) (0.45) (0.21) (0.43) (0.33) (0.52) (0.30) (0.54)Long durations 1.04 0.94 1.13 0.84 0.96 0.82 0.98 0.80

(0.26) (0.05) (0.34) (0.06) (0.17) (0.16) (0.16) (0.19)

ANOVA on ReTs ANOVA on AIs

F values Degrees of freedom p values ηp2 F values Degrees of freedom p values ηp2

Stimulus 0.54 1,24 .47 .02 1.34 1,24 .26 .05Key 2.44 1,24 .13 .09 6.08 1,24 b.05 .20Duration 335.46 1,24 b.00001 .93 278.91 1,24 b.00001 .92Magnitude 6.84 1,24 b.05 .22 14.32 1,24 b.005 .37Stimulus×Key 1.69 1,24 .21 .07 7.22 1,24 b.05 .23Stimulus×Duration 77.00 1,24 b.00001 .76 41.26 1,24 b.00001 .63Stimulus×Magnitude 1.53 1,24 .23 .06 3.99 1,24 .057 .14Key×Durations 43.68 1,24 b.00001 .64 39.87 1,24 b.00001 .62Key×Magnitude 102.84 1,24 b.00001 .81 85.91 1,24 b.00001 .78Duration×Magnitude 9.92 1,24 b.005 .29 17.32 1,24 b.0005 .42Stimulus×Key×Duration 7.98 1,24 b.05 .25 18.31 1,24 b.0005 .43Stimulus×Key×Magnitude 48.16 1,24 b.00001 .67 64.07 1,24 b.00001 .73Stimulus×Duration×Magnitude 7.34 1,24 b.05 .23 9.33 1,24 b.005 .28Key×Duration×Magnitude 1.38 1,24 .25 .05 11.50 1,24 b.005 .32Stimulus×Key×Duration×Magnitude 2.40 1,24 .13 .09 17.72 1,24 b.005 .42


when the right key (1.29) was used, when short durations (1.58)were reproduced, reflecting Vierordt's law, and when large magni-tudes (1.28) were presented. Even if there was not a significant effectof Stimulus, this factor interacted with the remaining factors. Scheffèpost-hoc test did not reveal any significant comparisons of interactionbetween Stimulus and Key. For short durations, the letter condition(1.67) induced a higher overestimation than the number condition(1.50), with pb .005. For long durations, the letter condition (0.89) in-duced a higher underestimation than the number condition (0.99),with pb .05. In addition there were AI differences between larger(1.27) and smaller (1.22) numbers, while no difference was evidentbetween smaller and larger letters.

The results regarding ReTs from Experiment 2 mirrored those of Ex-periment 1. That is, general STEARC, SNARC and TiNARC effects wereobtained in a reproduction task. This result seems to confirm the ideaof one generalized system dealing with all threemagnitudes. As before,the three effects were modulated by the type of experimental materialpresented. These effectswere shownwhen numberswere used as stim-uli, even if the TiNARC effect was reliable only when short durationswere considered; there was a no significant trend for long durations.The STEARC effect was also found with the presentation of alphabeticalmaterials, as in Experiment 1. These results may be considered to ex-tend the previousfindings of spatial–temporal and temporal–numericalinteractions in a time bisection task (Frassinetti et al., 2009; Oliveriet al., 2009; Vicario, 2007), due to the fact that two lateralized responsekeys were used here. However, Experiment 2 differed from Experiment1 because the triple interaction between space, duration andmagnitude(i.e. Key×Duration×Magnitude) was not significant. This differencecould depend on the nature of the temporal reproduction task com-pared with the temporal perception task. The difference betweenthem is that the estimation of the whole interval is operationalized bya motor act in the time reproduction task, whereas it is based entirely

on a perceptual judgment in the time estimation task (Bueti et al.,2008). Previous studies claim for a sensorimotorial transformation pro-cess that mediates stimulus perception and response preparation inSTEARC, SNARC and TiNARC effects (Bueti & Walsh, 2009; Walsh,2003). In line with this assumption, there are studies demonstratingthat the three effects arise at response selection stage (Ishihara et al.,2008; Keus & Schwarz, 2005; Kiesel & Vierck, 2009), as was found in Ex-periment 1. The reproduction task engaging both the response selectionand response execution stage could mask or reduce the interaction be-tween the three magnitudes, according to the ATOM model. Recently,Lewis andMiall (2006) stated that tasks requiring replication of a dura-tion via an actionmay be especially reliant on themotor system. Conse-quently, including duration as a dimension of the response maynecessitate different encoding and memory processes that those en-gaged for temporal discrimination.

The results regarding AI were similar to those regarding NEs in Ex-periment 1, with reliable STEARC and SNARC effects, even for small-large magnitudes and short-long durations. On the contrary, theTiNARC effect was reliable in the left space but it disappeared in theright space. As before, the results could be explained consideringthe role of the right parietal area in the interaction of time, numberand space processing. In this case, however, our data seemed to fol-low Vierordt's law. This result could in part explain the inconsistencyof the TiNARC effect, given that, in general, participants tended tooverestimate short durations and to underestimate long durationsas indicated by AI. In their study, Kiesel and Vierck (2009) found amain effect of response duration when the error rate was analyzed.More specifically, the participants tended to respond erroneouslymore often when long responses were required. Thus, the linear re-gression model was applied only on positive error rate differencefor long and short responses to each number (for similar results seeFig. 2C). Given that similar results were obtained using unimanual

Fig. 2. A) STEARC effect: observed ReT difference (in ms) between right and left keys for general (G; black square) numerical (N; black dot) and alphabetical (L; black diamond) conditions, and the linear regressions of RT difference on theduration (black line for G, dashed line for N and dotted line for L) are displayed; B) SNARC effect: observed ReT difference (in ms) between right and left keys for general (G; black square) numerical (N; black dot) and alphabetical (L; blackdiamond) conditions, and the linear regressions of RT difference on magnitude (black line for G, dashed line for N and dotted line for L) are displayed; C) TiNARC effect: observed ReT difference (in ms) between long and short responses forgeneral (G; black square) numerical (N; black dot) and alphabetical (L; black diamond) conditions, and the linear regressions of RT difference on magnitude (black line for G, dashed line for N and dotted line for L) are displayed; D) AIpattern of Key×Duration×Magnitude interaction is displayed for each factor separately.

120M.Fabbriet

al./Acta

Psychologica139

(2012)111

–123

image of Fig.�2


(Kiesel & Vierck, 2009) or bimanual responses (in the present study),the TiNARC effect seems to be less consistent (or more task-related)compared to the SNARC or STEARC effect.

However, our data can be discussed considering the four views de-scribed at the end of the previous discussion. In particular, the com-putational model of Gevers, Verguts, Reynvoet, Caessens, and Fias(2006) seems to fit with the data, according to two assumptions.Firstly, the model assumes that magnitude information is coded auto-matically, reflecting here the SNARC and TiNARC effects. Secondly themodel assumes that the SNARC effect is resolved at the response-related stage, but not during the later response execution processes,as shown by psychophysiological studies (Gevers, Ratinckx, DeBaene, & Fias, 2006; Keus & Schwarz, 2005). Indeed, the model as-sumes that a response is emitted as soon as a certain threshold isreached and this does not imply any different response speeds oncethe threshold is reached. The same consideration could be made forboth STEARC and TiNARC effects as well as for the triple interaction.

4. General discussion

The aim of the present research is to study whether there is a sin-gle generalized system dealing with space, time and numbers, orwhether there are separate modules that interact depending on cur-rent task requirements. To our knowledge, this is the first study inwhich the relationship between time, space and numbers is directlyassessed in two temporal tasks.

The main results obtained in both experiments seem to partiallyposit that the ATOM model (Bueti & Walsh, 2009; Fabbri & Natale,2009; Walsh, 2003) reflects a single generalized system dealing withthese magnitudes. This assumption seems to derive from three piecesof evidence. The first regards the STEARC effect (Ishihara et al., 2008;Vallesi et al., 2008), reflecting the metaphor of a linear representationof time along a spatially-oriented line (Casasanto & Boroditsky, 2008;Ishihara et al., 2008; Vallesi et al., 2008). In the present research, thespatial information is provided by two lateralized keys while the num-bers or letters providing temporal information are centrally presented.The spatial congruency between time and response facilitates the per-formance according to the spatial position of temporal information onthe MTL. It is worth noting that this time–response congruency isfoundwhen presenting either numbers or letters because the time pro-cessing is relevant for the task. In Experiment 1, the participants per-formed the task better when short durations were judged by pressingthe left key, whereas long durations were judged in a more accurateway when the right key was pressed. In Experiment 2, the left key de-termined lower reproduction times when short durations were consid-ered; in a similar way, the right key determined lower ReTs when longdurations were considered (see also Vicario et al., 2008). These findingsindicate a left-to-right temporal representation which is a useful meta-phor inmapping time defined asmilliseconds (in the present research),life events (Arzy, Adi-Japha, & Blanke, 2009; Arzy, Collette, Ionta,Fornari, & Blanke, 2009), or past-future categorization (Santiago etal., 2007; Torralbo et al., 2006).

The second piece of evidence derives from the finding of a SNARC ef-fect (Cappelletti et al., 2009; Dehaene et al., 1993) in both experiments.Our data seem to indicate that the quantity information (i.e. irrelevantfor the tasks) is automatically processed and this processing is linkedto lateralized response codes (Dehaene & Akhavein, 1995; Dehaeneet al., 1993). In Experiment 1, small numbers were responded to fasterwith the left key while the right key facilitated a faster response to largenumbers. In Experiment 2, the ReTs of small numberswere shorterwiththe left rather than the right key, while those of large numbers wereshorter with the right rather than the left key. To our knowledge, thisis the first time that a classical SNARC effect has been observed in tem-poral tasks, extending the work by Kiesel and Vierck (2009) who founda SNARC-like effect using a response duration. On one hand, the lack ofSNARC effect in previous studies could be due to the fact that spatial

information is not given in a single way (Cappelletti et al., 2009; Oliveriet al., 2008; Vicario et al., 2008). The fact of introducing two differentsessions according to the mapping between response keys and task in-structions could influence the appearance of the SNARC effect. On theother hand, there is an association between the present study andthat of Kiesel and Vierck, because we obtained the SNARC effect usingtemporal tasks while Kiesel and Vierck obtained a SNARC-like effectusing a parity task with a response key for either a short or a long dura-tion. Both studies, thus, seem to indicate that spatial representation isaccessed automatically when numerical quantity is processed, reflect-ing the metaphor of a MNL (Dehaene et al., 1993; Moyer & Landauer,1967; Restle, 1970).

The third piece of evidence is linked to the TiNARC effect (Cappellettiet al., 2009; Kiesel & Vierck, 2009; Lu et al., 2009; Oliveri et al., 2008;Vicario et al., 2008; Xuan et al., 2007). In line with previous results,our data show that small magnitudes (1–2 or A–B) are associatedwith short durations, and large magnitudes (8–9 or H–I) are associatedwith long durations (Experiment 1); small magnitudes induce a timingunderestimation mainly when short durations are processed and largemagnitudes induce timing overestimation with long durations, even ifthis association fails to reach statistical significance (Experiment 2).These findings indicate the possible presence of a common system pro-cessing both quantity and time information. Importantly, the TiNARCeffect has been shown for the number condition but it disappearswhen the letter condition is analyzed. This result could reflect the factthat numerosity is automatically accessed even when it is task-irrelevant (e.g., Roitman, Brannon, Andrews, & Platt, 2007).

Here, one aspect of the data is worth commenting on. This aspectregards the fact that we found SNARC effect and its related effects inboth tasks. This finding could be discussed in term of response-related stage considering the role of the parietal cortex as assumedby the ATOM model. As pointed out by Bueti and Walsh (2009), the(right) parietal cortex “is equipped with an analogue system for ac-tion that computes ‘more than-less than’, ‘faster-slower’, ‘nearer-far-ther’, ‘bigger-smaller’, and it is on these abilities that discretenumerical abilities hitched an evolutionary ride” (p. 1832). In partic-ular the right inferior parietal cortex (IPC) seems to process severaltypes of information, including spatial, temporal and numerical infor-mation (Walsh, 2003). The IPC seems to play a role in a sensorimotortransformation and it is a generalized magnitude system for action.When a forced choice task (i.e., time estimation task) is employed,the STEARC, SNARC and TiNARC effects are essentially measuringhow much more efficiently responses can be planned and producedaccording to congruent (e.g., short durations and left key or smallmagnitude and left key) or incongruent (e.g., short durations andright key or small magnitude and right key) situations (see Ishiharaet al., 2008; Keus & Schwarz, 2005; Kiesel & Vierck, 2009, respective-ly). Thus, these effects are associated with a response selection and/ormotor programming. The time reproduction task not only hinges onresponse-related stage but also on response execution, given that amotor response is required to reproduce an estimated duration(Bueti et al., 2008). However, in the time reproduction task, it hasbeen found an activation of the right and left IPC, even if the activa-tion of right IPC was stronger (Bueti et al., 2008). The authors pro-posed that the right IPC used representational systems, perhapsspatially encoded, which are common to time, space and quantity asrelevant to action, and this representation was available to left hemi-sphere areas required for action selection and generation (Bueti &Walsh, 2009; Walsh, 2003). The choice of two lateralized responsebuttons in our time reproduction task could induce a stronger activa-tion of the right IPC resulting for the STEARC, SNARC and TiNARCeffects.

In the present research we also assumed that the triple interactionbetween time, space and magnitudes can account for direct evidenceof the ATOMmodel. It is worth noting that the triple interaction is notlimited to number representation but also to letter representation,


resulting from small (1–2 and A–B) and large magnitudes (8–9 andH–I). In Experiment 1 the significant interaction is found withinboth RTs and NEs, while in Experiment 2 this interaction is onlyfound in AIs, but not in ReTs. In Experiment 1, the data reveal thatcongruent (i.e. left key, short durations and small magnitudes; rightkey, long durations and large magnitudes) situations facilitate perfor-mance compared with incongruent (i.e. left key, long durations andlarge magnitudes; right key, short durations and small magnitudes)situations. Therefore, these data suggest that time, space, numberand other magnitudes are represented in a common representationalformat, according to the ATOM model (Bueti & Walsh, 2009; Xuanet al., 2007). However, the TiNARC effect seems to be less consistentcompared to STEARC and SNARC effects as reported by subsequentANOVAs analyzing each aspect of triple interaction. Indeed theTiNARC effect on RTs remained reliable for the left but not for theright space. We argue that a possible explanation may involve hemi-spheric asymmetry in motor domain (Adam et al., 2010; Ishihara &Imanaka, 2007). Regarding this point, the ATOM model suggests arole of the right parietal area in processing the three systems(Walsh, 2003), probably explaining the presence of the TiNARC effectfor the left space. Considering the accuracy pattern, the TiNARC effectobtains a higher consistency, suggesting that this effect seems to beinfluenced by latency and accuracy pattern. In line with this possibil-ity, Kiesel and Vierck (2009) found a significant TiNARC effect in RTsbut not when analyzing the error rate with the ANOVA model. TheTiNARC effect was reliable in both latency and accuracy using the re-gression model by Lorch and Myers (1990). These data are in linewith a recent work showing a dissociation between impaired timeprocessing and preserved numerical and spatial processing, claimingfor a partial independence among the three magnitudes (Cappellettiet al., 2009). Alternatively, considering two sources of evidence, thepresence of a TiNARC effect could be the result of a preference for as-sociating the short duration and small magnitudes with the left space;the TiNARC effect was obtained within the numerical condition butnot in the alphabetical condition, while a magnitude-like effect withletters was observed. The letters A and B are probably more greatlyassociated with the left space (Gevers et al., 2003) than are the lettersH and I with the right space. Bearing in mind that the triple interac-tion is significant regardless of experimental stimuli, the additive ef-fects of these findings could explain the inconsistency of the TiNARCeffect regarding the two spaces. Further studies should clarify thispoint more deeply.

In Experiment 2, the triple interaction is not significant regardingreproduction times, indicating that the STEARC, SNARC and TiNARCeffects are not modulated by other spatial, temporal or magnitudefactors. A possible interpretation could depend on the nature of thetemporal reproduction task compared with the temporal perceptiontask. The difference between the tasks can be expressed in the in-volvement of the motor act in the time reproduction task and the per-ceptual judgment in the time estimation task (Bueti et al., 2008). Asreported by Lewis and Miall (2006), the temporal reproduction taskinvolves different processes compared to those involved in the tem-poral discrimination task. In line with this assumption, it is possibleto hypothesize that the symmetrical interaction between andamong three systems arises at a response-related stage (Ishiharaet al., 2008; Keus & Schwarz, 2005; Kiesel & Vierck, 2009), but notduring later response execution processes as shown by psychophysi-ological studies of the SNARC effect (Gevers, Ratinckx, De Baene, &Fias, 2006; Keus & Schwarz, 2005). According to Walsh (2003),time, space and numbers are computed by a common metric andthey are connected by a common (visuomotor) code for action. Assuggested by Vidal, Bonnet, and Macar (1992), the action durationis coded as a part of a motor program and can be processed prior tomotor execution. In addition, the analysis of temporal accuracyshowed similar patterns, suggesting that the mechanisms responsiblefor timing in the two tasks led to similar representations of the standard

durations. Thus, the difference between the two experiments could berelated to task requirements, especially in the manner in which partic-ipants use time information to make responses (Bueti et al., 2008).

The data can, however, be discussed considering alternative views,differing from the assumption that number, time and space are repre-sented in common line representations. Within the framework of theSNARC effect, these alternative views consider the stimulus–responsecompatibility, the polarity correspondence principle (Proctor & Cho,2006), the multiple-layers computational model (Gevers, Verguts,Reynvoet, Caessens, & Fias, 2006) and the working memory account(van Dijck & Fias, 2011). However, both stimulus–response compati-bility and the polarity correspondence principle appear to fit with thedata resulting from the first experiment but not with the data fromthe second experiment. Indeed, the lack of triple interaction in Exper-iment 2 is not expected, considering these two points of view.According to the results obtained from both experiments, the compu-tational model seems to be more pertinent than other views. Indeed,this model assumes an automatic activation of magnitude informa-tion and its associated spatial code, a response-related origin, the rel-ative status of magnitude coding and a more categorical distributionof the SNARC effect in a magnitude comparison task. These four as-sumptions can be found in the present findings. The SNARC effecthas been found in two temporal tasks in which the number informa-tion is irrelevant for the task requirements. Moreover, the automaticprocessing of numerical information seems to affect the temporalprocessing resulting in a TiNARC effect. The STEARC, SNARC andTiNARC effects, as well as their triple interaction, seem to arise at aresponse-related stage, as observed in Experiment 1 and, partially,in Experiment 2. The third aspect regards the fact that number aswell as time is coded as either small or large; this, in turn, activatesleft or right responses, as shown in the SNARC and STEARC effects.Finally, the categorical distribution of STEARC, SNARC and TiNARC ef-fects is shown observing the dRT pattern of Fig. 1A, B and C, respec-tively. The temporal judgment task is very similar to a magnitudecomparison task given that in both tasks the target stimuli are distantfrom the reference stimulus. Since a distance effect is implemented inthe task, a categorical distribution of the effects is predicted andobtained. The working memory account may also explain the presentdata. The main assumption regards the fact that temporary associa-tions between the position of a determined stimulus in a sequenceand space, rather than the long-term semantic representation ofnumbers, account for the SNARC effect. In line with this consideration,in both experiments the STEARC, SNARC and TiNARC effects reflectthese temporary associations because of the fact that durations andnumbers are ordered sequences. From brain imaging studies, timingperception and reproduction seem to involve the dorsolateral pre-frontal cortex, supplementary motor area, cerebellum and basal gang-lia (for a review see Koch, Oliveri, & Caltagirone, 2009). Importantly,the dorsolateral prefrontal cortex seems to play an important role inboth cognitive timing and working memory. However, the discoveryof non-significant SNARC and TiNARC effects for alphabetical stimuli,notwithstanding the evidence that alphabetical letters are sequential-ly ordered, remains a serious limitation of these alternative views. Afuture research direction may be to investigate the possible reliabilityof these alternative views in explaining the present data.

In conclusion, the present study shows, in part, the reliability ofthe ATOM model using two temporal tasks. This reliability is mainlyshown by both individuating the STEARC, SNARC and TiNARC effects,and individuating a triple interaction between time, space and num-bers. The STEARC, SNARC and TiNARC effects can arise at a response-related stage (Ishihara et al., 2008; Keus & Schwarz, 2005; Kiesel &Vierck, 2009). The present data seem to confirm a symmetrical relation-ship between time, space and quantity, as has been recently observed inexperiments involving rhesus monkeys, which showed large bi-directional effects on spatial–temporal judgments (Merritt et al.,2010). Further studies should address the reliability of the ATOM


model using, for instance, numerical or spatial tasks in which time is anirrelevant factor for the performance of the task.

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