nuerk et al (2004)

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Notational modulation of the SNARC and the MARC(linguistic markedness of response codes) effectHans‐Christoph Nuerk a , Wiebke Iversen b & Klaus Willmes a

a University Hospital of the RWTH Aachen, Aachen, Germanyb University of Cologne, Cologne, Germany

Available online: 13 May 2010

To cite this article: Hans‐Christoph Nuerk, Wiebke Iversen & Klaus Willmes (2004): Notational modulation of the SNARCand the MARC (linguistic markedness of response codes) effect , The Quarterly Journal of Experimental Psychology SectionA: Human Experimental Psychology, 57:5, 835-863

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Notational modulation of the SNARC and the

MARC (linguistic markedness of response

codes) effect

Hans-Christoph Nuerk

University Hospital of the RWTH Aachen, Aachen, Germany

Wiebke Iversen

University of Cologne, Cologne, Germany

Klaus Willmes

University Hospital of the RWTH Aachen, Aachen, Germany

Number magnitude and number parity representation are fundamental number representations.

However, the representation of parity is much less understood than that of magnitude: Therefore,

we investigated it by examining the (new) Linguistic Markedness of Response Codes (MARC)

effect: Responses are facilitated if stimuli and response codes both have the same (congruent)

linguistic markedness (even–right, odd–left) while incongruent conditions (even–left, odd–

right) lead to interference. We examined systematically the MARC (for parity) and the Spatial

Numerical Association of Response Codes (SNARC; for magnitude) effect for different number

notations (positive Arabic, negative Arabic, number words) and with different methods of data

analysis. In a parity judgement task, the SNARC effect indicating a magnitude representation was

replicated for all notations except for negative numerals. The MARC effect was found for number

words in all analyses, but less consistently for the other notations. In contrast, a correlational

analysis of the reaction time (RT) data, as suggested by Sternberg (1969) using a nonmetric multi-

dimensional scaling (MDS) procedure, produced a clear association of parity and response code

for all notations (MARC effect), but little evidence of the SNARC effect. We discuss the extent to

which these notation-specific MARC and SNARC effects constrain current models of number

processing and elaborate on the possible functional locus of the MARC effect.

Correspondence should be addressed to Hans-Christoph Nuerk or Klaus Willmes, University Hospital of the

RWTH Aachen, Neurology – Section Neuropsychology, Pauwelsstr. 30, D – 52057 Aachen, Germany. Email:

[email protected] or [email protected]

This research was supported by a DFG (German Research Foundation) grant to Klaus Willmes supporting Hans-

Christoph Nuerk. We would like to thank Stuart Fellows for checking the English grammar.

This article is an extended and a more elaborated version of a paper presented by Willmes and Iversen at the

Spring Annual Meeting of the British Neuropsychological Society in London on April 5–6, 1995.

2004 The Experimental Psychology Society

http://www.tandf.co.uk/journals/pp/02724987.html DOI:10.1080/02724980343000512

THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2004, 57A (5), 835–863

Q2166—QJEP(A)05701 / Jun 1, 04 (Tue)/ [29 pages – 3 Tables – 5 Figures – 2 Footnotes – 0 Appendices] .

Centre single caption • cf. [no comma] • Disk edited WTG.

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Parity and magnitude are the two most important features/properties of numbers when

judgements about the similarity of numbers have to be made (Miller & Gelman, 1983; Miller

& Stigler, 1987, 1991; Shepard, Kilpatric, & Cunningham, 1975). In these studies, nonmetric

multidimensional similarity scaling methods revealed that the spatial representations

obtained were all similar in that numbers were grouped according to magnitude and parity.

Miller and Gelman (1983) found that number magnitude was the most salient feature for

kindergarten children, whereas parity became more important for school children from Grade

6 on. Adults gave almost identical weight to both magnitude and parity.

These similarity judgement studies contrast remarkably with reaction time (RT) studies

yielding magnitude effects in tasks, in which magnitude was irrelevant for performance

(Berch, Foley, Hill, & McDonough Ryan, 1999, for children from Grade 2 on; for adults;

Brysbaert, 1995; Dehaene & Akhavein, 1995; Dehaene, Dupoux, & Mehler, 1990; Fias,

Brysbaert, Geypens, & d’Ydewalle, 1996). In contrast to magnitude, however, parity effects

seemed to occur inconsistently, even in parity judgement tasks (e.g., Dehaene, Bossini, &

Giraux, 1993, vs. Hines, 1990). This introduction is intended to provide a short survey

of magnitude and parity effects. In particular, it will be examined why huge parity effects

have been found in some experiments, but null effects in others. A possible account for this

divergence will be presented.

Magnitude effects and the SNARC effect

In RT studies, number magnitude seems to lead to the most reliable and stable effects in a

variety of tasks. Three effects are most often quoted as evidence for an internal magnitude

representation: the distance effect (Dehaene, 1989; Dehaene et al., 1990; Hinrichs, Yurko, &

Hu, 1981; Moyer & Landauer, 1967; for an overview, see Butterworth, 1999); the problem size

effect (Ashcraft, 1992; Brysbaert, 1995; Buckley & Gillman, 1974; Foltz, Poltrock, & Potts,

1984); and the Spatial Numerical Association of Response Codes (SNARC) effect (Dehaene et

al., 1993). The distance effect in number processing describes the finding that the time to

compare the magnitude of numbers decreases with the increase in numerical difference

between them. It can be accounted for by the assumption of an analogue magnitude represen-

tation for numbers, which serves as a medium for the comparison process on a “mental

number line” (Restle, 1970). The problem size effect denotes that performance in a variety of

number or calculation tasks usually worsens for relatively larger numbers. The most impor-

tant magnitude effect for this paper is the SNARC effect (Dehaene et al., 1993; Fias, 2001;

Fias et al., 1996). The SNARC effect consists of a systematic interaction between response

side and number magnitude: Small numbers in the range of numbers presented are responded

to faster with the left-hand key and large numbers faster with the right-hand key. Dehaene and

co-workers assumed that during parity judgements an analogue magnitude representation on

a left-to-right-oriented mental number line is invoked automatically. This oriented mental

number line is assumed to be associated with the response codes (left-hand and right-hand

keys) thereby producing the SNARC effect. The SNARC effect is most pronounced for one-

digit Arabic numerals and less so for number words in the range of 0–19. This pattern of

results may indicate a stronger relationship between Arabic code and magnitude representa-

tion than between verbal code and magnitude representation (triple code model of Dehaene,

1992; Dehaene & Cohen, 1995, 1997). Fias and colleagues (Fias, 2001, Fias et al., 1996)

836 NUERK, IVERSEN, WILLMES

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showed an interesting interaction between notation and task for the SNARC effect. While for

Arabic numerals the SNARC effect could be obtained for parity judgements and phoneme

detection, for number words a SNARC effect was only observed for parity judgement. This

result led Fias (2001) to postulate an additional asemantic route for processing number words

in asemantic tasks like phoneme detection for number words.

The SNARC effect could be generalized to two-digit numbers (Dehaene et al., 1993).

Thus, an analogue left-to-right-oriented compressed number line can be assumed for all

numbers from 0 to 99 (Brysbaert, 1995; Reynvoet & Brysbaert, 1999). However, for negative

numbers, it remains unclear what the orientation of the mental number line, as indexed by the

SNARC effect, might be. Recently, it has been shown that the direction of the SNARC effect

can be reversed with an appropriate visual anchor. Bächthold, Baumüller, and Brugger (1998)

found a reversal when they presented a clock and asked whether a number presented in the

centre of the clock was indicating a later or an earlier time than 6 o’clock. Since a reversal of the

SNARC effect is possible, it is an open question whether it occurs for negative numbers

because the absolute value of negative numbers increases from right to left. Vice versa, one

could also argue that the mental number line simply proceeds from left to right as in any graph

including positive and negative numerals. In this case, a nonreversed SNARC effect should be

obtained for negative numerals.

Parity effects: Their assessment and their account inreaction time tasks

In contrast to magnitude effects, parity effects are more controversial. When parity effects are

observed, they tend to be stronger for number words than for Arabic numerals. While Hines

(1990) observed an “odd” effect—that is, odd numbers being responded to more slowly than

even numbers—Dehaene et al. (1993) failed to find main odd effects. However, these differ-

ences can be explained by a closer look at the stimuli, the notation, and the task used in these

experiments.

Let us first consider the results reported by Hines (1990). In simple parity judgement tasks,

he obtained an odd effect for RT for number words (Experiment 5), but not for Arabic

numerals (Experiment 2). For Arabic numerals, there was only an odd effect for errors.

Dehaene et al.’s (1993) failure to find a main odd effect for RT in Arabic numerals in their

Experiment 1 is thus a replication of Hines’ finding. In their Experiment 9, Dehaene and co-

workers obtained an odd effect when number words and Arabic numerals were analysed

together for the range 0–9, thus again replicating the finding of Hines. In contrast to Hines,

however, they did not find an odd effect for number words in their Experiment 8 (nor in their

Experiment 9 for the range 0–19). The reason for these inconsistent results seems to be the

inclusion or exclusion of zero. While Hines employed stimuli from 2 to 9 to examine the odd

effect, Dehaene et al. (1993) used stimuli from 0 to 9. The 0 in Dehaene et al.’s Experiment 9

was about 60 ms slower than the average for other even numbers from 2 to 8 and still about 40

ms slower in Experiment 8 (see Figures 13 and 15). Given that Hines (1990) reported an odd

effect of 22 ms in number words for the range 2 to 9, it is quite reasonable to argue that the

inclusion of zero practically levels out any main odd effect: The inclusion of zero (1) slows

down the average response to even numbers and (2) greatly increases the mean square errors

contributing to the F statistics in a repeated measures analysis of variance (ANOVA). This

NOTATIONAL MODULATION OF SNARC AND MARC EFFECT 837

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argument is further confirmed by Dehaene et al.’s post hoc comparisons: Responses to zero

were slower than those to every other even number (except for number 6 in some experi-

ments). Most researchers seem to agree that zero is not a typical even number and should not

be investigated as part of the mental number line (see Brysbaert, 1995; Fias, 2001; see also

Armstrong, Gleitman, & Gleitman, 1983, who showed that not all numbers are subjectively

equally odd or even). Given that untrained participants are sometimes unsure about the status

of zero, rejection of the odd effect seems premature when based on a null effect that is only

obtained with inclusion of zero.

Sometimes Dehaene et al. (1993) nevertheless found an odd effect because they also

included the number 1, which is also responded to more slowly than other odd numbers.

However, inclusion of the number 1 does not compensate for inclusion of zero because (1) the

RT difference between zero and other even numbers is usually much larger than the RT

difference between “1” and other odd numbers and because (2) the inclusion of zero increases

the mean square error in the denominator of the F statistic even further. Whether or not zero is

similar to other even numbers could be assessed via a correlational analysis using multidimen-

sional scaling (which will be done in this study). If it is similar, it should be close in similarity

space. Otherwise, it should be far apart from the other even numbers. In sum, a main odd

effect seems to occur at least for one-digit number words when zero is not included. Thus, the

odd effect seems to be dependent on the range of the stimuli and the notation in the parity

judgement task.

However, there are huge alterations of the odd effect. In Hines (1990), Experiment 1,

participants had to decide whether the numbers had the “same” (unmarked) or a “different”

(marked) parity. In this experiment, the odd effect was very large compared to the 22 ms in the

simple parity judgement task above: about 150–200 ms for young participants and about 300–

400 ms for older participants.

So far, there is little theoretical background to account for the odd effect being more

pronounced in Hines’ (1990) same–different tasks than in the parity judgement task, as well as

being more pronounced for verbal notation. Hines explained the odd effect by pointing out

that the adjective “even” is linguistically non-marked as opposed to the marked adjective

“odd”. Nonmarked linguistic entities are assumed to be retrieved more quickly. However, this

explanation does in our opinion not suffice to explain why the odd effect is so much larger in

Hines’ task than in other parity judgement tasks.

As a possible hypothesis, we wish to put forward and to test in this paper a more general

version of the linguistic markedness hypothesis. In most languages there are pairs of comple-

mentary adjectives like “odd” and “even”. The nonmarked adjective is determined by

prefixing both adjectives with the syllable “un” or “in” or other prefixes negating the original

attribute. Only the nonmarked exemplar (“un-even”) can be negated this way (Zimmer,

1964). Apart from this formal type of markedness, Lyons (1969) mentions semantic (“right-

left”, “same–different”) and distributive markedness, which is related to word frequency

(“lion-lioness”). Markedness has been shown to affect response times in other domains.

Sherman (1973, 1976) has shown that marked adjectives lead to longer sentence comprehen-

sion times.

The more general version of the linguistic markedness hypothesis would not only account

for the main odd effect, but also for the huge enhancement of the odd effect in Hines’ (1990)

same–different parity judgement experiment in the following way: When people have to


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respond “same” (parity) to two “even” numbers, they have to respond with an unmarked

response (“same”) to an unmarked stimulus attribute (“even”). Stimulus and response are

thus congruent with respect to linguistic markedness. However, if they have to respond with

the unmarked response “same” to a marked stimuli attributes (“odd”), stimulus and response

are incongruent with respect to linguistic markedness. The markedness-incongruent odd–

same condition was much slower than the markedness-congruent even–same condition. This

difference was 5–20 times as large as the usual odd effect. We thus hypothesize that it is not

only the markedness of the stimulus that is important (as suggested by Hines), but also the

congruency of the markedness attributes of stimuli and responses.

EXPERIMENT 1Average-based ANOVA and regression analyses

If this linguistic markedness hypothesis is true, the predictions for a parity judgement task, in

which the hand-to-response relation is manipulated within participants, are straightforward:

the adjectives “right” and “even” are linguistically nonmarked (Zimmer, 1964). On the

contrary, the adjectives “left” and “odd” are linguistically marked. If the linguistic marked-

ness hypothesis holds, one should observe interference if unmarked responses are associated

with marked response keys and vice versa (markedness incongruent condition: odd–right,

even–left), while one should observe facilitation if the markedness association between stim-

ulus and response is congruent (even–right, odd–left). We would like to call this hypothesized

effect the Markedness Association of Response Codes (MARC) effect (cf. Willmes & Iversen,

1995; see also Berch et al., 1999; Reynvoet & Brysbaert, 1999) in analogy to the SNARC effect:

If the MARC effect can be demonstrated, then the two most important semantic features

(magnitude and parity) of numbers can be associated with certain response codes. We tested

this hypothesis with a typical odd–even judgement task in this study for different notations in

a within-participant design.

Parity effects have been shown to differ between notations. Mostly, they tend to be

stronger for verbal notation (“number words”). Linguistic markedness may also have a greater

effect for number words than for Arabic numerals, because of the verbal–linguistic attribute of

the concept (cf. Hines, 1990, for early indications of such a notation-specific effect and an early

suggestion that markedness may exert stronger influences on verbal notation). If this hypoth-

esis about notational modulation were true, the hypothesized MARC effect (even–right, odd–

left being faster than odd–right, even–left) should be observed particularly for those notations

that most likely activate verbal–linguistic concepts. Therefore, we expect a strong MARC

effect for verbal notation (number words) and less for positive and negative Arabic notation. If

linguistic markedness is an important dimension for the similarity of responses to Arabic

numerals, congruity between the markedness of number and response should be structuring

multidimensional similarity space as well. If the MARC effect only rests on responses to a few

numbers, linguistic markedness should not play such a role in multidimensional similarity

space.

With regard to current models of number processing, to our knowledge there is no model

that has yet linked number parity to linguistic concepts and properties. Neither the possible

stimulus–response markedness congruency nor the possible notation specificity of possible

MARC effects have yet been incorporated in any model.


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Besides verbal and positive Arabic notation, the goal of this study is to examine the SNARC

effect and its underlying representation more generally by also investigating negative Arabic

numbers in a within-participant design. As laid out above, the SNARC effect is interpreted

spatially as evidence for a left-to-right-oriented mental number line. Larger numbers (on the

right side) of the number line are responded to faster with the right hand in space. However, it

is interesting to question how the SNARC effect generalizes to negative numbers. Dehaene

and colleagues (Dehaene et al., 1993) postulated that the SNARC effect is due to reading

habits, as it was not found for Iranian participants living in France only for a short time. An

alternative account would be that the number line is oriented form left to right simply because

this is the way number lines are depicted in arithmetic graphs in Western culture. If arithmetic

graphs are so salient that they are the source of the SNARC effect, then the SNARC effect for

negative numerals should be reversed, in that negative numerals with higher absolute values

are farther left on the number line in mathematical function graphs. If reading habits are the

major source of the SNARC effect then there is no obvious reason why the minus sign before a

number should change the direction of the SNARC effect. To examine this question, positive

and negative Arabic integers were directly compared in a within-participant design.

Method

Participants

A total of 32 German participants, 11 female and 21 male, aged between 23 and 30 years (median 26

years) were tested individually. They were students from different faculties of the Technical University

of Aachen. A total of 31 were right-handed, and 1 was left-handed. The left-handed participant was

included since his performance pattern was not different from that of the right-handed participants.

Stimuli

Participants were presented with positive Arabic digits in the range 0 to 9, negative Arabic digits in

the range –9 to 0, German number words from “eins” (one) to “neun” (nine), as well as Roman numerals

from I to IX (which are not reported here). Each notation was presented in a separate block. In each block

a short instruction was given informing the participant about the notation and range of numbers to

follow.

Procedure

Participants had to decide whether each number was even or odd by pressing one of two response

keys. The experiment was subdivided into two parts. Half of the participants started with the even

response assigned to the right-hand key and the odd response assigned to the left-hand key. After

responding to the first part of the experiment there was a short break before the second part was started

with the same sequence of blocks as that in the first part but with reverse assignment of parity to response

key. Each of the two parts was preceded by a training list of 12 items (3 of each notation). In both parts of

the experiment, each participant had to respond to four blocks corresponding to the four notations.

Eight-block sequences were constructed such that presentations of positive and negative numerals in

consecutive blocks were avoided. The target numbers of one block were presented four times in random

order.

The experiment was controlled by an IBM-compatible PC (4/86-dx/66 Mhz) using the Experi-

mental Run-Time System (ERTS; Beringer, 1993). The targets were presented on a 17″-monitor screen


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using white symbols against a blue background. Responses were recorded by way of two response keys

placed before the participant at a distance of 60 cm. At the beginning of each trial, a cross was presented in

the centre of the screen for 500 ms. Then the target appeared and remained until response, but maxi-

mally 1500 ms. Afterwards the screen remained blank for 1000 ms before the next trial started. The

whole experiment lasted for approximately 30 minutes. The number words were given in small print

with a maximum height of 10 mm and a maximum length of 28 mm. All other targets also had a height of

10 mm.

Results

For comparability between the notations without zero, zero was also excluded in most analyses

for positive and negative Arabic numerals. In all analyses, only those 23 participants were

considered for whom at least one correct response was available for every numeral in every

presented notation. However, if one performs the analyses with the maximal number of

participants available for each single notation, the results do not change substantially.

1. RT analysis

ANOVA analyses

For all further analyses, the median RT for correct answers was computed for each target,

each side of response, and each participant. It was thus based on a maximum of four correct

responses. For negative numerals the labels of the target magnitude factor represented the

absolute values of the numbers presented in the analysis (for descriptive values, see Tables 1

and 2).

Positive Arabic numerals. We obtained significant main effects of magnitude and parity,

F(3, 66) = 6.58 and F(1, 22) = 29.80, respectively, both p < .001. For parity, the odd effect

(Hines, 1990) was replicated: Responses to even number responses were faster than to those

odd ones. For magnitude, a quadratic trend reached significance, F(1, 22) = 8.58, p < .01:

Responses to large and small magnitudes tended to be faster. Response hand, in contrast,

failed to reach significance (F < 1). Two 2-way interactions were (marginally) significant:

Parity × Magnitude, F(3, 66) = 4.14, p < .01 and Magnitude × Response hand, F(3, 66) = 2.48,

p = .07, indicating a SNARC effect, see Figure 1: The Parity × Hand interaction (MARC

effect) failed to reach significance, F(1, 22) = 1.93, p = .17. However, when the hypothesized

MARC effect is tested directly (MARC-congruent against MARC-incongruent conditions), a

marginally significant effect remains in that responses in congruent trials were 16 ms faster

than those in incongruent trials, t(22) = 1.41, p = .09, see Figure 2; Table 1. Finally, there was a

three-way interaction between magnitude, parity, and response hand for positive Arabic

numerals, F(3, 66) = 3.01, p = .05, Greenhouse–Geisser adjusted, which seemed to indicate a

greater SNARC effect for odd numbers.

Negative Arabic numerals. We only obtained one significant main effect: the odd effect,

F(1, 22) = 7.62, p < .001. No other main effect or interaction reached significance (all p ≥ .10).

Number words. For number words, the results were quite the opposite of the results for

the positive Arabic notation. All parity effects and interactions were clearly significant, while


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842

TABLE 1

Overview of the most important average median RT effects

Arabic Negative Arabic Number words

Overall 509 518 551

Odd 529 529 566

Even 489 508 535

Odd effect 40 21 31

Left hand 509 518 552

Right hand 508 519 549

Hand effect 1 –1 3

MARC-compatible 501 517 532

MARC-incompatible 517 520 569

MARC effect 16 3 37

MARC (Mult. Reg. ) 12 0 34

SNARC (Lin. Reg. ) –10.06 –5.70 –8.35

SNARC (Mult. Reg. ) –8.89 –5.68 –7.53

Note: For detailed RT differences regarding the SNARC effect see Table 2.

SNARC (Lin. Reg. ) denotes the mean SNARC effect as it is usually determined in

a simple linear regression. SNARC (Mult. Reg. ) and MARC (Mult. Reg.) denote

the SNARC effect and the MARC effect, respectively, when magnitude and parity

are both included in a multiple regression over left hand–right hand differences.

Although both effects decrease slightly in the multiple regression, the overall

pattern between notations is fairly stable.

TABLE 2

Average RT values for each number in each notation and for each response hand

Positive Arabic Negative Arabic Number words

————————————— ————————————— ————————————

Num. abs left right diff abs left right diff abs left right diff

1 542 518 565 46 531 507 555 48 589 554 623 69

2 478 483 473 –9 499 502 495 –7 535 537 533 –4

3 547 531 563 32 517 525 508 –17 562 554 569 15

4 484 477 490 12 494 482 506 24 522 547 497 –50

5 537 525 549 23 541 549 532 –17 549 539 559 20

6 517 545 489 –56 509 507 511 4 547 570 524 –46

7 491 510 471 –38 527 526 528 1 566 550 582 33

8 477 486 468 –18 531 546 516 –29 537 569 504 -64

Note: “Abs” refers to the overall response time, “left” and “right” refer to the left and right hand response times,

respectively, and “diff” refers to the difference between right-hand response and left-hand response computed for

analyses of the SNARC effect.

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843

Figure 1. SNARC and MARC effects for the RT data (to the left) and errors (to the right): A SNARC effect can be

observed when the negative regression slope (see the regression line) explains a large portion of the variance. The

MARC effect can be observed by comparison of the dotted line (odd numbers) and the dashed line (even numbers)

with each other and with the regression line. For a perfect MARC effect the dotted line for odd numbers should be

both above the regression line and above the dashed line for even numbers. Figure 1a (top left): RT positive Arabic

numerals. A SNARC effect (slope = –8.89 ms) can be observed, but no significant MARC effect because the

differences for the “4” and “7” are on the “wrong” side of the regression line, thus producing too much error variance.

Figure 1b (middle left): RT negative Arabic numerals. Neither a SNARC nor a MARC effect could be

observed. Figure 1c (bottom left): RT number words: A SNARC effect (slope = –7.53 ms) and a strong MARC

effect (b = –34.07) can be observed. Figure 1d (top right): Errors positive Arabic numerals. Figure 1e (middle right):

Errors negative Arabic numerals. Figure 1f (bottom right): Errors number words: For errors, a SNARC effect could

be observed for all notations. A significant MARC effect could be seen for number words.

Reaction Time Errors

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magnitude seemed to play a lesser role. We observed a highly significant odd effect, F(1, 22) =

19.36, p < .001, and a significant Parity × Hand interaction, F(1, 22) = 8.72, p < .01. MARC-

congruent trials were clearly faster (37 ms) than MARC-incongruent trials, t(22) = 2.95, p <

.01; see Figure 2. Finally, a Magnitude × Response Hand interaction was significant, F(3, 66)

= 3.01, p < .05, indicating a SNARC effect.

Regression analyses

The ANOVA may be too conservative because it tests all possible differences between

conditions and not only between those specified by the SNARC and MARC hypotheses.

Therefore, we tested the SNARC and MARC effect more specifically: In an analysis


Figure 2. MARC effects for RT (top) and errors (bottom) for different notations. A MARC effect is an interaction

between parity and hand in this graph. Clear MARC effects were obtained for number words, (marginally) significant

effects for positive Arabic numerals, and null effects for negative Arabic numerals.

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suggested by Lorch and Myers (1990) and applied to the SNARC effect by Fias et al. (1996),

we performed a multiple regression over the average difference in RT (dRT: right–left hand)

for each number for each individual participant. Then we tested whether the individual

nonstandardized regression weights for magnitude and parity differed from zero across

participants. A significant SNARC effect was obtained for all notations, all t(22) > 2.30, all p <

.02, except for negative Arabic numerals: only marginally significant, t(22) = 1.51, p = .07. A

MARC effect was again found for number words, t(22) = 2.69, p < .01. For Arabic integers,

once again, no MARC effect could be observed, both t(22) < 1.10, p > .14. The comparison of

the regression slopes of the SNARC effect revealed some surprising results. All comparisons

between the slopes from different notations were not significant. In particular, the slope for

negative Arabic numerals was also not significantly flatter than that for positive Arabic

numerals, t(22) = 0.545, p = .30. Thus, previous differences in significance between positive

and negative Arabic numerals must be interpreted with caution because the two notations do

not significantly differ in the direct comparison of their SNARC effect.

In contrast, the MARC beta weights were larger for number words than for negative Arabic

numerals, t(22) = 1.85, p < .05, and marginally larger than those for positive Arabic numerals,

t(22) = 1.47, p = .08. Thus, the greater MARC effect for verbal notation tended to be

confirmed in this analysis.

2. Error analysis

Because of the rather high proportion of errors, we also performed the same analyses for

errors to examine whether SNARC and MARC effects are due to speed–accuracy trade-off

effects.1Contrary to our expectations, we found significant effects in the error analyses. They

were almost always in the same direction as in the RT analysis: Slower conditions tended to be

more error prone. Since to our knowledge SNARC effects have rarely been reported for error

data (and MARC effects not at all), we report the error SNARC and MARC effects but—due

to space restrictions—we do not discuss other effects in detail.

ANOVA analyses

Positive Arabic numerals. No main effects were observed; however, all two-way interac-

tions—and in particular the SNARC- and MARC-type interactions—reached significance:

Parity × Magnitude, F(3, 66) = 8.89, p < .001; Magnitude × Response Hand, F(3, 66) = 3.48,

p < .05; indicating a SNARC effect; and Parity × Hand interaction, MARC effect, F(1, 22) =

5.42, p < .05. With regard to the Magnitude × Hand Interaction, a clear SNARC effect was

observed, F(1, 22) = 7.20, p = .01: For the right hand, error rates decreased with increasing

magnitude while for the left hand the effect was reversed. The Parity × Hand interaction

revealed a clear MARC effect in that MARC-compatible trials were more accurate than

MARC-incompatible ones, t(22) = 2.33, p = .01. The Parity × Magnitude interaction revealed

a similar pattern as that in the RT analyses: Finally, there was a marginally significant three-

way interaction between magnitude, parity, and response hand for positive Arabic numerals,

F(3, 66) = 2.07, p = .05.


1We thank Robert Logie for suggesting this analysis.

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Negative Arabic numerals. The three-factorial ANOVA over errors for negative Arabic

numerals revealed a main effect of parity, F(1, 22) = 6.47, p < .05, indicating a reversed odd

effect. Even numbers produced more errors than odd numbers. Most importantly, there was a

marginally significant SNARC–like Magnitude × Hand interaction, which was not found in

the RT analysis for negative numerals. As indicated by a significant linear-by-linear contrast,

F(1, 22) = 6.48, p < .05, larger numbers tended to produce more errors with the left hand while

for smaller numbers the opposite occurred. No other interaction reached significance.

Number words. The error analyses revealed interaction effects both of magnitude and of

parity in the same direction as that of the RT data. First, however, right-hand trials produced

again more errors than left-hand trials, F(1, 22) = 8.44, p < .01. Most importantly, all two-way

interactions reached significance. The MARC effect was also found for the error data, F(1, 22)

= 8.53, p < .01: MARC-compatible trials were less error prone than MARC-incompatible

trials. A SNARC-like Magnitude × Hand interaction was observed, F(3, 66) = 3.60, p < .05.

For right hand responses, smaller numbers tended to produce more errors, while for left-hand

responses, larger numbers tended to produce more errors, F(3, 66) = 10.65, p < .01. Finally, a

Magnitude × Parity interaction was significant, indicating that the different numbers

produced a different number of errors, F(3, 66) = 12.71, p < .001. The 3-way interaction did

not reach significance, F < 1.

Error ANOVA summary. The only indications of SATOs were main effects of hand

(right-hand responses tended to be more error prone in some notations) and a reversed odd

effect for negative numerals. SNARC-like interactions were observed in all notations, and

MARC effects were observed for number words and positive Arabic numerals. Whenever

SNARC- or MARC-like interactions were found, they were always in the same direction as

that for the RT data.

Regression analyses

Many participants were error free in many conditions. Therefore, individual regression

analyses do not, in our opinion, make much sense. However, the average error rates can be

analysed in a regression analysis because there are enough data points for n = 23 participants to

produce sufficient (and more than zero) errors for each condition. In a multiple regression

with the predictors number and parity over the average percentage of error difference

between right-hand and left-hand responses, we replicated the SNARC effect for positive

Arabic numerals, t(5) = 2.46, p < .05, b = –1.75, negative Arabic numerals, t(5) = 3.34, p <

.01, b = –1.29, and number words, t(5) = 3.67, p < .01, b = - 1.52. Thus, as in the error

ANOVA, a SNARC effect could be observed for all notations, even for negative Arabic

numerals for which the SNARC analyses only produced marginally significant effects in the

same direction or null effects (see Figure 3). We only found a MARC effect for number words,

t(5) = 3.20, p = .01, b = –6.09; there was no significant effect for all other notations. For posi-

tive Arabic numerals, for which a MARC effect was obtained in the error ANOVA, the

predictor parity failed to reach significance in this analysis, t(5) = 1.32, p = .12, b = –0.37.

In sum, we observed a significant SNARC effect for all notations in the error analysis, and a

significant MARC effect in the error analyses for number words. For positive Arabic numerals


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there was an indication of a MARC effect in the ANOVA, but not in the regression analysis.

Except for the obtained SNARC effect for negative Arabic numerals, the error analyses

mirrored those of the RT analyses. In particular, no indication of a speed–accuracy trade-off

was observed for any SNARC or MARC effect.

Discussion of average-based ANOVA and regressionanalyses

The foremost aim of the RT and error analyses was to investigate the linguistic markedness

association of response codes (MARC) effect. A MARC effect was consistently observed in all

analyses for verbal notation, but not for Arabic integers. Only a trend for positive Arabic

integers was obtained when markedness compatibility was directly compared and in the error

analyses. These results seem to confirm that linguistic markedness plays a role for parity


Figure 3. Smallest space analysis (SSA) for different notations for the numbers 1 to 8 responded to with the right

hand (squares) and the left hand (diamonds). For convenience, MARC-compatible conditions (solid) and MARC-

incompatible conditions (open) are depicted differently. For all notations parity, magnitude, and response hand did

not lead to a separate grouping of different conditions, but only MARC compatibility or incompatibility (see the line

that separates every single compatible from every single incompatible condition; no such line could be drawn with

respect to parity, magnitude or hand of response). Figure 3a (top left): positive Arabic numerals. Figure 3b (bottom

left): negative Arabic numerals. Figure 3c (top right): number words.

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decisions. This seems particularly to be the case for number words, indicating that (1) the

verbal notation may particularly trigger linguistic–semantic concepts such as markedness and

that, vice versa (2) the MARC effect is indeed verbal–linguistic in nature. In summary, the

differences in the MARC and the odd effect between different notations are hard to reconcile

with the assumption that parity is retrieved from one single abstract semantic code, which is

the same for all notations.

With regard to magnitude, we mostly obtained statistical trends of a SNARC effect for

negative Arabic numerals, while for positive Arabic integers we observed a SNARC effect in

all analyses. These statistical trends, however, should not be totally neglected, because in

direct comparisons positive and negative Arabic numerals did not differ significantly with

respect to the slope of the regression line. However, when a SNARC trend for negative Arabic

numerals was observed, it corresponded to the spatial position of the absolute value of these

negative Arabic numerals rather than to the spatial position of their raw (negative) values.

Thus, the spatial representation of the mental number line does not simply continue for nega-

tive numerals to the left of zero, but it becomes reversed. Smaller negative numerals (with a

larger absolute value) tend to be met with faster responses with the right-hand key, but the

results are less consistent and conclusive than for positive Arabic integers. Although the

spatial representation of the raw negative values in mathematical function graphs may

produce some interference in some participants, this does not determine the SNARC effect.

Rather, the data are in line with Dehaene’s reading habit hypothesis about the SNARC effect

(Dehaene et al., 1993).

EXPERIMENT 1Non-metric multidimensional scaling analyses

While for the RT difference analysis magnitude affected performance in every notation, the

effects for parity were less consistent. However, in several studies asking for judgements about

the similarity of numbers, nonmetric multidimensional scaling (MDS) methods have revealed

that the spatial representations obtained were all similar in that numbers were grouped

according to magnitude and parity (Miller & Gelman, 1983; Miller & Stigler, 1987, 1991;

Shepard et al., 1975). In particular, Miller and Gelman (1983) found that adults gave almost

identical weight to magnitude and parity. Thus, there is a difference between the results of RT

experiments and similarity judgements. We wanted to investigate whether this difference is a

matter of the task (e.g., speeded parity judgement vs. similarity judgement) or whether it is

rather a matter of the analysis method chosen (i.e., investigating mean differences vs.

correlational patterns).

The idea to analyse the structure of intercorrelations among experimental conditions is not

new. Its usefulness is laid out in the last part of the famous additive factor logic article of Stern-

berg (1969). While the first part concerning the interaction of factors has often guided RT

analyses in cognitive psychology (see Sternberg, 1998, for a variety of applications in different

areas), the last part about intercorrelations seems to have been considered less often.

According to the logic of the additive-factor method (Sternberg, 1969, pp. 308-309) patterns

of correlations in addition to interactions among factors reveal which experimental factors

influence the same processing stage. The most strongly interacting factors should be most


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highly correlated. In number processing research with RT as dependent variable, as far as we

know, differences in mean RT among conditions have been analysed exclusively.

In the following, we report on nonmetric MDS analyses of patterns of correlations among

RT data across all numbers for both response sides within each notation as well as of the corre-

lations between dRT differences (right–left) for each number in each notation. For these data,

a measure of monotone relationship (monotonicity coefficient µ2; Borg and Lingoes, 1987,

p. 79) was computed between all pairs of stimuli. The resulting correlation matrix was entered

into a nonmetric multidimensional similarity scaling procedure (Smallest Space Analysis,

SSA-I; Guttman, 1968; Lingoes, 1973). Points that are close together in the SSA configura-

tion represent stimuli that are highly correlated.

With regard to a multidimensional similarity scaling procedure, the hypotheses are clear. If

similarity in parity and markedness of response is important, odd and even numbers should be

grouped together. In particular, for even as well as for odd numbers, all compatible conditions

(e.g., 2–right, 5–left) should be grouped close together, as should all incompatible conditions

(e.g., 2–left, 5–right). However, if the linguistic markedness hypothesis is correct, compatible

and incompatible responses should not be very similar and should be located relatively far

apart from each other. If magnitude is an important similarity factor, numbers of similar

magnitude should be grouped together, while numbers with a large distance in magnitude

should be relatively far apart. Thus, if an internal mental number line exerts a strong influence

on response similarly in all participants, the multidimensional scaling procedure should group

the numbers according to magnitude. Finally, the multidimensional scaling procedure will

help us to provide an answer about the status of zero in different notations. If zero is not part of

the mental number line (e.g., Brysbaert, 1995) it should be located farther apart from the other

numbers. In contrast, if it is part of the number line and if it is just an ordinary even number, it

should be located close to other small and to other even numbers.

As in the average RT analysis, we have carried out the SSA-I procedure for both absolute

RT values and the RT difference between right-hand and left-hand responses. Clearly, with

respect to the RT difference, numbers of the same magnitude should again be grouped

together (SNARC), and numbers of the same parity should be grouped together (MARC).

Again, we hypothesized that the MARC effect should be strongest for the verbal notation.

Results

In this study, nMDS is used to map the number stimuli (different numbers for different

response hands) in a space of low dimensionality such that stimuli with high (positive) correla-

tions are mapped close to each other while stimuli with low (negative) correlations are placed

far apart. In nonmetric MDS only the pattern of ordinal relations among correlations (larger

vs. smaller) is mapped into the corresponding pattern of ordinal relations among distances

between stimulus points (Borg & Lingoes, 1987). If no specific a priori hypothesis about the

dimensionality of the MDS solution has been formulated, one usually chooses the lowest

dimensional solution with acceptable fit as indicated, for example, by the coefficient of alien-

ation (see Borg & Lingoes, 1987, for mathematical definition). A coefficient of alienation for 2-

D solutions is usually considered adequate in the range of about [.10, .25], particularly, if a

theoretically relevant interpretation can be given. If the coefficient of alienation is substan-

tially higher then solutions for higher dimensionality are sought. If the coefficient of alienation


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is very close to zero, then the solution is very often degenerate—for example, all individual

stimulus points are very close together, virtually mapped onto one or two data points of the

solution space. In contrast to metric MDS or other metric multivariate dimension-reducing

procedures, the dimensions of the solution space themselves are not interpreted. It is the

configuration of stimulus points in space that matters (Borg & Lingoes, 1987).

The coefficient of alienation as a measure of monotone fit for the spatial representation

was acceptably low in the 2-D solution (Arabic numerals, .180; number words, .225; nega-

tive numerals, .100; see Figure 4). For all notations the SSA solutions were very similar in

that a straight line could be drawn, dividing the space into two regions. On one side of the

straight line all points corresponding to the compatible (right–even and left–odd) condi-

tions were located, whereas on the other side all points corresponding to the incompatible

(right–odd and left–even) condition could be found. Concerning number magnitude, no

clear-cut pattern could be detected, although in the ANOVAs magnitude had a main effect

in all conditions.

To investigate the special role of zero in the mental number line, we also computed an SSA

including zero for positive and negative Arabic numerals: The points representing zero were

well separated from all the other numbers (see Figure 5). Thus, the results of the multidimen-

sional scaling analysis are consistent with the idea that zero is represented differently than

other numbers in a parity judgement task.

As for the RT analysis, we also explored the results of the multidimensional scaling analysis

for the difference in response times with the right and left hand (SNARC and MARC effects).

The coefficient of alienation as a measure of monotone fit for the representation was

acceptable in two dimensions for positive Arabic integers (.117), while for negative Arabic

numerals (.001) and for number words a degenerate solution was obtained (alienation: .001

and .003, respectively). Both degenerate solutions were such that odd trials were mapped

virtually onto the point (100, –100) in the MDS space, while the even trials were mapped

onto the point (–100, –100). While the exact coordinates are meaningless in the MDS, it is

very meaningful that now the MARC effect alone determined mapping in the MDS, which

was virtually independent of the identity of number and response hand. Thus, these degen-

erate solutions provide even stronger evidence for the determination by the MARC effect.

For positive Arabic integers, the SSA solution exhibited two aspects for right–left RT

difference similarity: A parity aspect (the MARC effect) and a magnitude aspect: While all

even numbers were located on one side and all odd numbers on the other (see Figure 5),

numbers of greater magnitude were located more in the top part and numbers of smaller

magnitude more in the lower part (thus possibly indicating the SNARC effect). But there were

two exceptions from a perfect ordering by number magnitude: The number 2 was located

between 4 and 6, and the number 7 was located very close to 5 but a little less distant from the

number 3 (see Figure 5). Thus, for the similarity in RT right–left differences, a parity dimen-

sion and a magnitude dimension seem to determine how similar RT patterns to different

numbers are. For negative Arabic numerals and number words, the degenerate solutions were

due to the fact that every single correlation of numbers of the same parity was positive, and

almost every single correlation of numbers of different parities was negative (except r = +.04

for the number words “seven” and “eight”). Given such a consistent correlation pattern, the

SSA simply groups all even and all odd numbers closely together with a maximal distance

between the groups. The second dimension played no role in the similarity structure any


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851

Figure 4. SSA for positive (top) and negative (bottom) Arabic numerals 0 to 8 responded to with the right hand

(squares) and the left hand (diamonds). MARC-compatible conditions (solid) and MARC-incompatible conditions

(open) are depicted differently. Again, parity, magnitude, and response hand did not lead to separate grouping of

different conditions, but only MARC compatibility or incompatibility. It can also be seen that zero strongly differs

from all other numbers regardless of whether it is responded to with the left or the right hand. (See the line that sepa-

rates zero from the other numbers.)

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852

Figure 5. SSA for differences between right-hand and the left-hand responses for positive (top) and negative

(bottom) Arabic numerals, either even (solid diamonds) or odd (open diamonds). It can be seen that the numbers are

grouped according to parity. However, this is the only SSA in which magnitude also may play a role (with greater

numbers being in the top part rather than in the bottom part) for positive Arabic numerals.

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more. So, if there was any magnitude effect, it could not be observed because the scaling result

was completely determined by parity (respectively the MARC effect) for those two notations.

In sum, three results are important: The interaction of parity and response hand (MARC

congruity) predominantly determined the SSA similarity structure for all notations, while

parity and magnitude themselves did not seem to shape similarity structure much. Second, for

the analysis of similarity of right–left-hand RT differences, the parity dimension (i.e., in its

MARC interaction with hand) again determined similarity for all notations, while magnitude

only determined similarity for positive Arabic integers. Finally, zero could be clearly sepa-

rated from all other positive and negative Arabic integers in the SSA analysis.

Discussion of nonmetric multidimensional scaling analyses

Summary and interpretation of nonmetric MDS results. In line with Sternberg (1969), we

can conclude on the basis of the SSA that the markedness of response hand and the marked-

ness of parity affect the same processing stage. In all analyses of the RT data, markedness-

congruent and incongruent markedness stimulus–response mapping for both even and odd

numbers were grouped together according to markedness congruency: All markedness-

congruent and markedness incongruent conditions could be separated by one straight line in

all analyses. No other single factor such as magnitude, parity, or an interaction with response

hand was effective in determining the structure in any analysis. Thus, the nonmetric MDS

strongly supports the idea that congruency of the markedness of the parity of a given number

and the markedness of response codes determine similarity of response patterns in this

analysis. The SSA of the right–left RT difference further confirms these conclusions. For all

notations, even numbers (for which right–left RT difference means compatible–incompatible

condition) and odd numbers (for which right–left RT difference means the reverse: incompat-

ible–compatible condition) are fully separated. Further, with respect to positive numbers,

magnitude seemed to determine similarity structure, while for other notations no magnitude

effect could be observed in the SSA structure. In the case of the degenerate solutions for nega-

tive Arabic numerals and number words, it can nevertheless be stated that magnitude did not

play a role in the SSA structure for these notations within the current range of stimuli. In sum,

it is remarkable, how much the results of the SSA differ from the results of the conventional

ANOVA and regression analyses.

Finally, the current SSA seems to confirm the idea of Brysbaert (1995) and others that zero is

not part of the mental number line. For both negative and positive Arabic numerals in both

response codes, right- and left-hand key, zero could be separated easily from all other numbers.

For no other number could such an exceptional location could be consistently observed.

On the use of multidimensional scaling techniques with RT data as compared to ordinary RT

analyses. Average-based analyses (i.e., ANOVA and regression analyses) have been used

mostly for RT or error data, while multidimensional scaling techniques have been applied to

other off-line data (Miller & Gelman, 1983; Miller & Stigler, 1987, 1991; Shepard et al., 1975).

Although Sternberg (1969) suggested analysing the correlational pattern of RT data, this has

rarely since been done. The regression analysis of the RT and error data provided evidence for

a SNARC effect in all notations (at least in the analyses of Lorch & Myers, 1990, and of the

error data); but a MARC effect could only be observed for number words in all analyses and


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(marginally) for positive Arabic numerals in the ANOVA analyses. In striking contrast, the

MARC effect was obtained for all notations in the SSA, while—except for a possible tendency

in the dRT MDS for positive Arabic numerals—no SNARC effect could be observed. This

discrepancy can have two possible sources. First, the MDS with a regional interpretation of

the spatial solution obtained is probably better suited to detect categorical effects such as the

MARC effect than gradual effects like the SNARC effect.2

Second, the MARC effect may

vary in a more systematic fashion between subjects than the SNARC effect. This may lead to a

more stable pattern of correlation in the MDS while in the general linear model this

interindividual variation contributes to error variance.

First, the MARC effect is a categorical effect, which is supposed to be homogeneous for

all members within one category (i.e., all within-category correlations should be higher than

between-category correlations). Thus, the optimal correlational pattern is clear and may not

be very sensitive to measurement errors. For the SNARC effect, which is a linear effect, the

situation is different. For example, how should the answer to 5 (apart from the MARC

effect) be correlated with 1 or 8 for the dRT data according to the SNARC effect? The

numbers 1 and 5 should not be as highly (positively) correlated as 1 and 2, but still higher

than 1 and 8. Thus, if one would expect medium size correlations in medium distance ranges

between numbers, the MDS result for the SNARC effect may be more prone to measure-

ment errors than the MDS results for the MARC effect. In summary, the MDS may be

better suited to detect the reliability of the categorical MARC effect, while RT analysis may

be better suited to examine the linear SNARC effect. Support for this argument comes from

the average RT analyses. While a SNARC effect was obtained for all notations in the regres-

sion analysis with the Lorch and Myers (1990) method, the categorical interaction in the

ANOVA between response hand and magnitude only reached significance for number

words, and not for positive and negative Arabic numerals. The linear trend that is revealed

in the regression analysis is not specifically tested in the categorical ANOVA, making the

latter too conservative.

The second argument may, conceptually, be equally important. Individuals may differ

systematically with respect to their use of linguistic markedness. If such systematic differences

between participants equally affect the markedness congruency for all numbers and both hand

responses, a stable correlational pattern will be observed: for participants prone to the MARC

effect, all MARC-incongruent trials may be slow as compared to MARC-congruent trials. For

other participants this effect may play a lesser role. Thus, for the correlations over participants

between, for example, two MARC incongruent trials, these systematic differences would

result in quite high and stable correlations. Consequently, a stable MDS structure will be

obtained. For an ANOVA (and the regression analysis), these individual differences in the

MARC effect are conceptualized as error variance, because the ANOVA assumes that an inde-

pendent factor should affect all replications in a cell of the experimental design (i.e., all


2We wish to thank Wim Fias for many helpful suggestions and discussions concerning this article. In particular, he

suggested performing a median split into a fast and slow half of participants. We also had a fruitful discussion about

the issue of discrepancy between nMDS and ANOVA/regression analysis, in which he suggested the first argument

with regard to that discrepancy. Finally, his comments were extremely helpful in strengthening the theoretical impact

of the paper by relating the format-specificity of the MARC effect to current models of number processing and

describing its functional locus more explicitly.

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participants) equally. Thus, the same individual differences that may lead to stable correla-

tions in the MDS may increase error variance in the ANOVA and regression analysis and

therefore lead to the different pattern of results in the two analyses.

GENERAL DISCUSSION

This study had three goals: First, we wished to examine the notational dependency of the

SNARC and the hypothesized MARC effect in a within-participant design. We highlight the

most important results of this study with regard to parity (MARC effect) and magnitude

(SNARC effect; for an overview, see Table 3). Since the hypothesized MARC effect was

found in all analyses for number words and still in some analyses for positive and negative

numbers, we elaborate on a possible account for the MARC effect and its notation depend-

ency. Additionally, we discuss how the results of this study constrain current models of

number processing. With regard to the SNARC effect, we discuss particularly the results for

negative numbers for which the SNARC effect tends to follow the absolute rather than the

relative (negative) values. Finally, the role of the number zero is not clear in number

processing research. Sometimes it is included in parity judgement tasks and sometimes it is

excluded because of its “special” status. The nonmetric MDS results in this study provide

convincing evidence that zero is indeed different from all other numbers in the parity judge-

ment task.

On the notation dependency of the SNARC effect and theMARC effect

SNARC effect. For RT, significant SNARC effects were obtained for positive Arabic

numerals and number words, but only a marginally significant SNARC effect for negative


TABLE 3

Overview of the SNARC and MARC effects obtained in this within-participant design for

the different notations in the different analyses

SNARC effect MARC effect

—————————————– —————————————–

Positive Negative Number Positive Negative Number

Type of analysis Arabic Arabic words Arabic Arabic words

RT ANOVA � — � (�) — �

RT regression � (�) � — — �

Error ANOVA � (�) � � — �

Error regression � � � — — �

SSA (abs. RT) — — — � � �

SSA(dRT left - right) (�) — — � � �

Note: With regard to notation, the SNARC effect was most consistently found for positive Arabic

numerals and number words; however, positive Arabic numerals was the only notation that tended to

show a SSA-scaling pattern in accord with a SNARC-like ordering of numerals corresponding to

magnitude. The MARC effect was most consistently obtained for number words. With regard to analyses

methods, SNARC effects were obtained in the conventional ANOVA and regression analyses while

MARC effects were reliably obtained for all notations in the SSA (see text for discussion).

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Arabic numbers. In the error analyses, however, SNARC-like interactions (specifically tested

in a linear-by-linear contrast) were observed for all three notations. For the first time, SNARC

effects could be shown in the error analyses also for all notations (however, in a fixed-effects

regression model, because many participants had zero errors). In the SSA, however, a

tendency towards a SNARC scaling could only be observed for positive Arabic numbers in the

RT difference SSA. For all other SSAs, there was no indication of a SNARC effect. So, in

sum, the most reliable effects were found for positive Arabic digits: They were the only nota-

tion in which SNARC-like scaling effects could be observed. For number words, consistent

SNARC effects were observed in the average RT and error ANOVAs and regression, but no

indication of the SNARC effect could be obtained in the SSAs. The least reliable SNARC

effects were obtained for negative numbers, which only reached conventional significance

level in the error analyses, and not in any other analyses.

However, although the SNARC effect for positive Arabic integers was more pronounced

and more stable, the common RT SNARC effects for positive and negative Arabic integers did

not differ significantly (as indexed by the regression weights). This was the case because the

SNARC effect for negative numbers was marginally significant. The tendency of a SNARC

effect, which was observed for negative Arabic numerals, had the same orientation with

respect to their absolute value rather than their raw negative value. Had we computed the indi-

vidual SNARC slopes with respect to the negative rather than the absolute value of negative

numbers we would have obtained significant differences, t(22) = 3.00, p < .01. These patterns

of results imply that the mental number line does not simply continue into negative Arabic

numerals as in standard mathematical function graphs (at least for block-wise presentation as

in this study).

Two possible reasons for the less consistent SNARC effects for the negative Arabic

numerals can be suggested. First, the minus sign, which is irrelevant for the parity decision,

may nevertheless require additional processing capacity that masks the SNARC effect. The

parity decision for negative numerals indeed tended to be slightly slower than that for positive

Arabic integers (see Table 1). This hypothesis is, however, not consistent with the data of Fias,

Lauwereyns, and Lammertyn (2001). They even obtained SNARC effects for irrelevant

nonattended digits when stimuli other than these digits had to be processed in some adequate

primary tasks. Given this SNARC effect for irrelevant digits , it seems unlikely that nonspe-

cific interference from the irrelevant minus sign masks the SNARC effect when the digit

themselves are still relevant. The second hypothesis is that negative Arabic numerals might

evoke two spatial–numerical associations: one number line corresponding to the absolute

values of the negative numerals, with –1 being on the far left and –8 being on the far right,

which is—according to our observed trend—predominant in most participants. A second less

dominant number line might correspond to the spatial–numerical association of the raw nega-

tive numerals with –8 placed far left and –1 far right. The hypothesis that access to multiple

number lines (or multiple access to one number line) is responsible for the less reliable results

for negative Arabic numerals is consistent with recent data of Nuerk, Weger, and Willmes

(2001, 2002) with regard to two-digit magnitude comparison. Nuerk and colleagues could

show that the magnitude of the decade and the unit digits of two-digit numbers separately acti-

vate their respective magnitude representations. Thus, if multiple activation of different

magnitude representations is possible, negative numbers may well activate both their absolute

and their raw negative value to a certain extent.


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MARC effect. MARC effects have been obtained for every notation in the SSA in this

study. However, the notational differences in the RT and error analyses revealed a surpris-

ingly different picture: The MARC effect was most reliably and strongly found for the verbal

notation in all analyses. For positive Arabic numerals, MARC effects could only be signifi-

cantly obtained in the error ANOVA and marginally in the RT ANOVA (besides the SSA).

For negative Arabic integers, no MARC effect was observed in the average-based analyses.

The nonmetric MDS results confirming the MARC effect in all analyses for all notations

are surprising, as the partitioning of space according to linguistic markedness congruency was

obtained even for positive and negative Arabic numerals for which no MARC effect was

observed in the RT difference analyses. Following the logic of Sternberg’s (1969) additive-

factor method for correlational analysis, it is reasonable to assume that the markedness of the

response codes and the markedness of parity influence the same processing stage because the

association between the markedness of response codes and parity determines the scaling struc-

ture for all notations. We had not hypothesized that stimulus–response markedness congru-

ency would determine scaling structure for Arabic integers, and we were surprised by the

clarity of this result (see in particular Figure 5 for the RT differences). In our opinion this

pattern of results clearly indicates that the results obtained with one analysis method may not

be conclusive for another analysis method.

Finally, as one alternative explanation (see Footnote 2) for the MARC effect one could

assume that it is “just a peripheral instruction–response compatibility effect: Press left if

response is even is incompatible whatever the type of numbers”. Indeed, this explanation

could account for our pattern of results. However, this account is in our view inconsistent with

the data reported by Hines (1990): In his same–different parity judgement task, participants

had to decide whether or not two presented numbers had the same or different parities. The

“same” responses should be made with the right hand and the “different” responses with the

left hand. The critical difference with regard to the MARC effect is the difference between

“same” responses to two odd versus two even numbers. The peripheral response (right hand)

as well as the response decision “same” was identical in both conditions. Nevertheless, partici-

pants took much longer to respond “same” to two odd numbers than to two even numbers. In

our opinion, a peripheral instruction–response account of the MARC effect is incongruent

with these data. However, a markedness congruency account would just predict that the mark-

edness-congruent condition “same”–“right” (both unmarked) is slower than the markedness-

incongruent condition (“different”–“right”; marked–unmarked).

On the functional locus of the MARC effect with regard to current models of number processing

(see Footnote 2).

Markedness may be a property more strongly associated with some internal linguistic representa-

tion(s) than with more conceptual, nonlinguistic representations such as digits. (Hines, 1990,

p. 46).

No model of number processing has yet incorporated an explanation for the MARC effect

obtained in this study. The only explanation available so far is that the congruency of linguistic

markedness of the parity (odd–even) of a given number stimulus and of the linguistic marked-

ness of response (left–right) facilitates performance as compared to markedness-incongruent


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stimulus–response mappings. In line with Hines (see above) we have argued that linguistic

markedness as an abstract verbal concept may influence verbal notations more strongly (in a

similar way that the positive Arabic notation seems to produce stronger SNARC effects). We

elaborate on this assumption a little more below.

The notational modulation of the MARC effects was quite clear in our study: The MARC

was consistently much stronger and more reliable for verbal notation than for other notations.

Any model trying to explain the MARC effect should also account for this notational modula-

tion. The notation dependence of the MARC effect is hard to reconcile with the assumption

that the underlying semantic representation (i.e., possibly linguistic markedness) exerts its

influence on one central semantic parity representation equally for all notations. We now

discuss briefly the notation specificity of the MARC effect with respect to three different

models: the triple code model and its successors (Dehaene, 1992; Dehaene & Cohen, 1995,

1997), the modular model of McCloskey and colleagues (McCloskey, 1992; McCloskey,

Macaruso, & Whetstone, 1992) and the encoding complex model of Campbell and Clark

(Campbell, 1992, 1994; Campbell & Clark, 1988, 1992; Clark & Campbell, 1991).

First, in the model of Dehaene, parity is retrieved from the visual Arabic number form. If

this were the case, number words must be transformed to an Arabic number from which their

parity, which is linked to markedness, could be retrieved. In this case, however, we should

have observed identical parity (markedness) effects for Arabic numerals and for number

words or even larger effects for Arabic numerals because Arabic integers are assumed to be

closer linked to parity (cf. Dehaene et al., 1993). This was not the case: MARC effects were

smaller for Arabic notation.

In the modular model of McCloskey, parity retrieval is also supposed to rely on one single

format: the abstract semantic representation (e.g., McCloskey et al., 1992). However,

McCloskey and colleagues to our knowledge have not been explicit about the retrieval of

parity. For example, Sokol, Goodman-Schulman, and McCloskey (1989, p. 108) point out

that “our model does not require that effects of odd–even status may be limited to calculation

tasks. For example, effects of this variable could even be taken to suggest that the internal

numerical representations posited by our model reflect odd–even status as well as numerical

proximity (although we do not find this interpretation particularly appealing).” However, the

MARC effect as an interaction between numerical comprehension and output must neces-

sarily be transformed into the abstract semantic number representation according to

McCloskey. Even if parity is then retrieved via the number fact system or the calculation

system, parity cannot be modulated by format, specifically because it has already been repre-

sented independent of notation. Format specificity could occur with main effects (words are

slower than Arabic numerals), and with sophisticated interpretations even some interactions

with notation may be explained (see, e.g., McCloskey et al., 1992). However, we are unable to

see how format-specific interactions with the output such as the MARC effect are consistent

with the model’s assumption of a central abstract semantic number representation that is

obligatorily processed before any output is created. In short, the format specificity of the

MARC effect is not well accounted for by the modular model.

The only one among these three models that can in our opinion incorporate the MARC

effect is the encoding complex theory of Campbell et al. “The basic assumption of [their]

approach is that number concepts and skills are based on modality and format-specific mental

codes that are interconnected in a complex and highly integrated associative structure”


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(Campbell & Clark, 1992). Campbell and colleagues normally use the interaction effect of

notation with another variable (e.g., type of errors) in calculation tasks to argue for their theory

and as counter-arguments against the modular theory (Campbell, 1992, 1998, 1999; Campbell

& Clark, 1992). In contrast to calculation tasks, the parity judgement task is much simpler to

perform and easier to interpret. The notational modulation of the MARC effect is—as are

other interactions with notation—consistent with the assumption that format-specific codes

and skills are used throughout a given task. In combination with our assumption that the

MARC effect is modulated by a linguistic (verbal) attribute, the linguistic markedness, it is

consistent with the encoding complex perspective that the MARC effect is more pronounced

and more stable for verbal numerals—that is, number words.

Finally, we wish to elaborate a little more on how our account of the MARC effect may

incorporate its notation dependency. Above, we have argued that linguistic markedness as a

verbal–linguistic concept may activate verbal representations more strongly than nonverbal

representations. What does verbal in this context mean? Is it the same verbal code that is used,

for example, for stored multiplication facts in the model of Dehaene (see Cohen & Dehaene,

2000)? Furthermore, does it derive from a necessary processing stage or is it merely a side

effect deriving from subvocal verbalization?

The answers to these questions are speculative and call for further empirical investigation.

First, with regard to the question of verbal codes, one needs a more fine-grain verbal-

processing account than most number-processing models possess. Most models simply

assume that if numerals are presented in verbal form (number words), verbal-processing

routes or modules must or can be used (e.g., Cipolotti & Butterworth, 1995; Fias, 2001, for

recent discussions). However, the nature of this verbal processing is not very well specified.

To achieve this, one needs to explore the possible source of the MARC effect in language-

processing models. However, many language-processing models lack conceptual levels and

have a lexical level as the highest implemented level. A model that incorporates a conceptual

level is the language production model of Levelt and co-workers (e.g., Levelt, Roelofs, &

Meyer, 1999). The concept of linguistic markedness is that most adjectives possesses an oppo-

site form, with one of them being the basic form (the unmarked form) and the other being the

derived form (the marked form). One would probably localize this selection at the highest level

of the Levelt model, the conceptual preparation of lexical concepts (see Levelt et al., 1999, p.

3). If one tried to relate multiplication facts stored by verbal rote learning, one would probably

locate this verbal process much lower in the Levelt model, possibly at the level of phonological

encoding in the mental lexicon. Thus, although one might believe that the sources of the

MARC effect and of the learned multiplication facts are both verbal they appear to be on very

different levels of verbal processing.

These considerations also allow more specific hypotheses about the derivation of the

MARC effect. As linguistic markedness is a high-level concept, it is unlikely that it is strongly

activated by a relatively low level process such as subvocal articulation. Rather, we believe that

two lexical concepts are activated: those of even-ness (or odd-ness) and right-ness (or left-

ness) in this study as well as those of same-ness (or different-ness) and right-ness (or left-ness)

in the Hines’ (1990) study. If two associated concepts share the same attribute (marked,

unmarked) of linguistic markedness in the conceptual level of Levelt et al.’s (1999) model, a

response is facilitated while different markedness concepts may lead to interference. The

markedness effect for number words may be stronger because number words activate the


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verbal–lexical entry associated with them rather strongly while Arabic numerals may do so to a

lesser extent (note that the nMDS also revealed MARC effects for Arabic numerals). Arabic

numerals may rather activate a representation (be it the visual number form as suggested by

Dehaene or an abstract semantic—magnitude and parity—representation) that is not verbal–

lexical and helps the participant to decide whether the number is odd or even. MARC effects

for Arabic notation may therefore rely on the coactivation of the verbal representation of this

number. In contrast, number words also seem to activate their verbal lexical concepts that are

associated with a certain markedness.

If this linguistic markedness account is valid, markedness-related parity effects should

therefore always be stronger for number words than for Arabic numbers. For instance, in the

same–different-parity judgement tasks, even stronger odd effects should be obtained for

number words than for digits. However, even for digits, the markedness effect can be tested.

Attributes like symmetry/asymmetry have a clear markedness distinction. Although the

markedness representation of parity may not totally disappear in a vertical symmetry judge-

ment task about Arabic digits (as magnitude representation does not disappear in parity judge-

ment task), the judgement “symmetric” may be faster for the (unmarked) right hand, and the

judgement “asymmetric” may be faster for the (marked) left hand.

In sum, the linguistic markedness account of the MARC effect and its notational modula-

tion is currently the only account for the MARC effect that is also consistent with the results

obtained by Hines (1990) in a same–different parity judgement task. Furthermore, the

linguistic markedness account provides testable and falsifiable hypotheses for future research.

On the special status of zero and five. Zero has sometimes been included in the analyses of

parity judgement tasks and sometimes not (see introduction and Hines, 1990, vs. Dehaene et

al., 1993), and the question whether or not to include zero is still controversial. Based on the

magnitude effect obtained, Brysbaert (1995) took a clear standpoint on this controversy and

claimed that zero is not part of the mental number line. In a parity judgement task, participants

often are not sure about the parity status of zero and need to be introduced to the concept that it

is an even number (which is true in terms of mathematical group theory).

We investigated the role of zero with multidimensional scaling techniques for the first time

in a parity judgement task. Our SSA results clearly showed that zero is separated from any

other positive and negative Arabic numeral for both right- and left-hand responses while the

other odd and even numbers (in association with the congruent or incongruent response hand)

were always grouped together. This result is inconsistent with the assumption that the parity

status of zero is that of other even numbers. Therefore, we suggest that zero should no longer

be included in parity judgement experiments. At least, it should be controlled whether the

effects and null effects obtained in a given parity judgement task do still prevail if zero is

excluded. SNARC, MARC, or odd effects whose significance only rests on the inclusion of

zero can—when our SSA results are considered—hardly be generalized to magnitude or

parity representation in general.

In physical and numerical magnitude comparison studies, it has been claimed that the

number 5 has a special status (Tzelgov, Meyer, & Henik, 1992). Tzelgov et al. suggested that

the digits 1–4 or 2–4 are automatically considered to be small, while the digits 6–9 or 6–8 are

automatically considered to be large. Indeed, they could show that in a physical judgement

task, the size congruity effect differed for trials with and without five, perhaps because of the


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special status of five as the “neutral” middle in the one-digit number range of the mental

number line. In our study, we have found no evidence that five is represented differently than

other odd numbers. The RT and error SNARC effects revealed no special status for number 5

and in the nonmetric MDS; it was always grouped with the other MARC-congruent or

MARC–incongruent odd numbers. Thus, the special status that the number 5 may well have

in magnitude comparison tasks was not observed for parity judgement in any notation in this

study.

CONCLUSIONS

First, this study has established that parity is associated with response code (even–right, odd–

left). This MARC effect is notation dependent: It is most reliably observed for verbal notation

and less for Arabic nonlinguistic notation. Our account of this pattern of results is the

linguistic markedness hypothesis: If the parity of a number has a specific markedness (even–

unmarked, odd–marked), responses tend to be faster for some notations when the response

code has the same markedness (right–unmarked, left–marked). The notation specificity of the

MARC effect in the different analyses is hard to reconcile with models that assume a central

abstract representation of parity for all notations alike. Second, multidimensional scaling

techniques can be helpful in exploring the similarity structure of numbers. Markedness

association determined the correlational grouping structure also for positive and negative

Arabic integers, while no respective RT differences were found. For magnitude, the opposite

was true: The SNARC effect was obtained in the analysis of mean RT differences and in the

error analyses. However, in the nonmetric MDS, magnitude only influenced the pattern of

similarities for positive Arabic integers. Finally, multidimensional scaling indicated that the

zero is represented differently from all other numbers when participants are asked about

parity. Therefore, it is recommended that effects obtained in parity judgement tasks (e.g.,

SNARC, MARC, odd effect) should at least be also tested without being checked for the

impact of zero. Effects that rely on the inclusion of zero can—in our opinion—not easily be

generalized to number representation in general.

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Original manuscript received 15 May 2001

Accepted revision received 24 April 2003

PrEview proof published online 3 September 2003


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