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An association of numbers and space (SNARC effect) has been examined in an ever growing literature. In the present quantitative meta-analysis, 46 studies with a total of 106 experiments and 2,206 participants were examined. Deeper number magnitude processing determined by task, stimulus and participants characteristics was associated with a stronger SNARC effect. In magnitude classification tasks the SNARC effect assumed consistently a categorical shape. Furthermore, the SNARC effect was found to increase with age from childhood to elderly age. No specific difference in the size of the SNARC effect was observed due to the explicit use of imagery strategies that could not be explained by increased reaction times. In general, these results corroborate the predictions by the dual-route model of the SNARC effect regarding the activation of number magnitude representation and suggest that automaticity may play a role in the development of the association of numbers and space across the lifespan.
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Psychology Science Quarterly, Volume 50, 2008 (4), pp. 489-525
On the cognitive link between space and number:
a meta-analysis of the SNARC effect
GUILHERME WOOD1, KLAUS WILLMES2, HANS-CHRISTOPH NUERK1 & MARTIN H. FISCHER3
Abstract
An association of numbers and space (SNARC effect) has been examined in an ever growing litera-
ture. In the present quantitative meta-analysis, 46 studies with a total of 106 experiments and 2,206
participants were examined. Deeper number magnitude processing determined by task, stimulus and
participants characteristics was associated with a stronger SNARC effect. In magnitude classification
tasks the SNARC effect assumed consistently a categorical shape. Furthermore, the SNARC effect was
found to increase with age from childhood to elderly age. No specific difference in the size of the
SNARC effect was observed due to the explicit use of imagery strategies that could not be explained by
increased reaction times. In general, these results corroborate the predictions by the dual-route model of
the SNARC effect regarding the activation of number magnitude representation and suggest that auto-
maticity may play a role in the development of the association of numbers and space across the life-
span.
Key words: SNARC, mental number line, aging, imagery, meta-analysis
1 Dr. Guilherme Wood, Department of Psychology, Paris-Lodron University of Salzburg, Hellbrunnerstrasse 34,
5020 Salzburg, Austria; Tel.: 0043 662 8044 5163, Fax: 0043 662 8044 188, email: guilherme.wood@sbg.ac.at
2 Department of Neurology/Section Neuropsychology, University Hospital RWTH-Aachen
3 School of Psychology, University of Dundee, Scotland UK
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
490
In 1993, Dehaene, Bossini, and Giraux showed that small numbers (e.g., 0 or 1) were as-
sociated with faster left hand responses, and larger numbers (e.g., 8 or 9) with faster right
hand responses (Figure 1A). This result held for single- (Experiment 1) as well as two-digit
numbers (Experiment 2) and was not affected by whether participants were left-hand domi-
nant (Experiment 5), or crossed their hands on the response buttons (Experiment 6). The
effect did, however, depend on the relative magnitude of numbers in the stimulus set (Ex-
periment 3) and was reduced in participants with right-to-left reading habits (Experiment 7).
A similar spatial association was absent for letters (Experiment 4) but present when partici-
pants categorized number words (Experiments 8 and 9). Dehaene et al. (1993) concluded
that numbers are systematically associated with space, and that this association reflects the
orientation of a “mental number line”. Specifically, the authors proposed that
“…the representation of number magnitude is automatically accessed during parity
judgment of Arabic digits. This representation may be linked to a mental number line
[…], because it bears a natural and seemingly irrepressible correspondence with the
natural left – right coordinates of external space.” (p. 394).
Recently, Proctor and Cho (2006) proposed an alternative explanation for the SNARC
effect based on polarity correspondence that contests the necessity of assuming the existence
of a mental number line. The polarity correspondence account is a general theory of com-
patibility effects and does not require a spatially oriented mental number line to explain the
SNARC effect. Proctor and Cho (2006) argue that the polarity assigned to each stimulus and
response depends on the relative saliency of their dimensions. Instead of perceptual or con-
ceptual similarity, the polarity categorization suffices to produce the mapping of stimuli onto
responses. This account is based on the activation of positive or negative polarities for dif-
ferent dimensions and is applicable not only to the SNARC effect but also to non-numerical
experimental set-ups. When the same polarity (“+” or “-”) is assigned to different dimen-
sions of the experimental set-up (e.g., stimuli and responses), they become associated. When
applied to the domain of numerical processing, both “right” and “large” have polarity “+”
while both “left” and “small” have polarity “–“. When the polarities of stimulus and re-
sponse overlap, a link is established regardless of perceptual, spatial or conceptual overlap,
improving performance.
Although the polarity correspondence account offers a parsimonious explanation for the
SNARC effect in speeded binary classification tasks, some new evidence from cognitive
neuropsychology lends support to the existence and relevance of a mental number line for
the formation of spatial numerical associations (Zorzi et al., 2002). Zorzi et al. (2002)
showed that hemineglect patients manifest a specific deficit in the representation of those
number magnitudes which should be located on the left side of the mental number line.
Furthermore, Zorzi et al. (2006) recently reported that this deficit is specific for the represen-
tation of number magnitudes and does not generalize to other ordinal sequences such as
letters of the alphabet or months of the year.
A further theoretical account of the SNARC effect was proposed by Gevers et al. (2005a,
2006b) who developed a dual-route cognitive model of the SNARC effect. It describes how
number magnitude may gain control over motor responses through different routes of infor-
mation processing, namely a conditional and an unconditional route. The conditional route
allows for controlling motor responses, e.g. by verbal instruction, in a very flexible way. The
A meta-analysis of SNARC 491
unconditional route, on the contrary, conveys the automatic activation of pre-existing asso-
ciations between stimuli and responses. The association between number magnitudes and
response codes observed in SNARC experiments, even when these associations are irrele-
vant for the task, is attributed by Gevers et al. (2005a, 2006b) to the automatic activation of
the unconditional- in parallel to the conditional route. Both routes of information processing
are activated in parallel until a threshold for the production of a response is reached and both
may have an influence on response speed and accuracy. In congruent trials both conditional
and unconditional routes lead to the same associations between stimulus and response. This
reduces the latency of responses and contributes to an increase in response accuracy. In
contrast, in incongruent trials conditional and unconditional routes activate distinct associa-
tions between stimulus and response. Since in incongruent trials the correct response is acti-
vated in the conditional route and a concurrent response in the unconditional route, response
latencies become longer and accuracy decreases. The longer the time necessary to activate
the conditional route, the stronger is the influence of the unconditional route on motor re-
sponses. Accordingly, when the activation of the unconditional route is strong enough to
reach the response threshold, an incorrect response may be triggered.
While the dual-route model describes the general cognitive architecture of the SNARC
effect, some authors have also asked for interindividual variability in the strength of the
SNARC effect. Recently, Wood et al. (2006a, b) have reported that across different studies,
the proportion of participants showing a negative SNARC slope varies between 65% and
75%. Accordingly, Piazza, Pinel and Dehaene (2006) showed that idiosyncratic associations
between number and space may coexist with the usual SNARC effect. Finally, Cohen Ka-
dosh and Henik, (2007) suggest that the implicit mental representation of numbers may
differ across individuals and may deviate from the standard left-to-right representation typi-
cally described (Dehaene et al., 1993), what may result in higher interindividual variability
in the SNARC effect.
This multiplicity of empirical findings leads to the current debate over the nature of the
association between numbers and space. Reflecting an increasing scientific interest in the
SNARC effect in its various forms, four qualitative reviews have appeared in the recent past
(Fias and Fischer, 2005; Gevers and Lammertyn, 2005; Hubbard et al., 2005; Cohen Kadosh
et al. 2008). The first two of these reviews concluded that the SNARC effect probably re-
flects an inherent spatial attribute of the mental representation of number magnitude, the
third review focused on the neuroanatomical underpinnings of this representation whereas
the latter concluded that non-numerical magnitude is also spatially coded. Neither of these
previous reviews has provided a quantitative assessment of the literature or encompassed
recent theoretical proposals such as polarity correspondence or the dual route model. This
was the motivation for carrying on the present meta-analytic review.
1. Overview
Our review of the SNARC literature is subdivided into three parts. In section one, a set
of hypotheses will be derived from the polarity correspondence account as well as the dual-
route model of the SNARC effect and from the general literature. In section two, these hy-
potheses will be tested meta-analytically and an overview of our database will characterize
the state of the field in terms of the most popular paradigms, materials, and tasks. The final
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
492
section evaluates the impact of tasks, stimuli, responses, participant characteristics on the
size of the SNARC effect and discusses these results in light of the current accounts of the
SNARC effect.
2. Some hypotheses about the SNARC effect
The SNARC effect is commonly defined as an association between number magnitude
and response codes. Therefore, the SNARC effect should become stronger when magnitude
processing is activated more intensively (Gevers et al., 2006b).
2.1 Size of the SNARC effect and the strength of number magnitude activation
According to Gevers et al. (2006b), more intense number magnitude processing should
be observed in slower responses because the unconditional route carrying up information on
the association between number magnitude and response codes has more time to interfere
with the selection of a response button. Furthermore, a differential effect of task on the
strength of the SNARC effect is also plausible. The SNARC effect might be stronger in
those tasks requiring deeper semantic number processing such as magnitude classification
and parity judgement than in tasks requiring the analysis of superficial features of numerical
stimuli such as colour- or orientation discrimination. Therefore we would expect that
(i) the size of the SNARC effect depends on response latencies and on the strength of
number magnitude activation.
and (i b) the size of the SNARC effect depends on the amount of semantic number proc-
essing required in each task
2.2 The shape of the SNARC effect in magnitude classification tasks
Most of the time, the association between number magnitude and spatial coordinates is
stronger for numbers in the extremes of the numerical interval tested but less strong for
numbers in the intermediate positions giving origin to a continuous slope of reaction time
differences on number magnitude (Figure 1A). However, there seems to be an exception for
this pattern in magnitude classification tasks (i.e., “Is the number larger or smaller than 5?”).
In magnitude classification tasks the shape of the SNARC effect seems to be categorical
(Figure 1B, Gevers et al., 2006b, Nuerk et al., 2005a). It is known that comparing numbers
against a fixed numerical standard leads to relatively slow reaction times for numbers close
to the standard (i.e. 4 or 6) in comparison with numbers far from the standard (i.e. 1 or 9,
Moyer and Landauer, 1967) because the magnitude of numbers close to the standard needs
to be processed more intensively when discriminating their position in comparison to the
standard (Gevers et al., 2006b). Since the SNARC effect should be stronger when number
magnitude is processed more intensively, in magnitude classification tasks the SNARC
effect should be stronger for numbers close to the standard (e.g. in the interval between 1
and 9, 4 and 6 are close to the standard 5 while 2 and 8 are far from the standard). A direct
A meta-analysis of SNARC 493
consequence is that in the magnitude classification task the SNARC effect observed in num-
bers in the intermediate range of the interval should be almost as large as for numbers in the
extremes of the numerical interval i.e. the SNARC effect should assume a categorical shape
(Gevers et al., 2006b; Nuerk et al., 2005a). This is not expected to happen in tasks such as a
parity decision task because in these cases the time to judge the parity of numbers remains
approximately constant across the numerical interval. Therefore we predict
(ii) in tasks of magnitude classification the SNARC effect should assume a categorical
shape.
A)
B)
y = -5,48x + 15,63
R
2
= 69%
-40
-30
-20
-10
0
10
20
30
012345678910
RT right hand - RT left hand (ms)
y = -7,95x + 34,53
R
2
= 79%
-40
-30
-20
-10
0
10
20
30
012345678910
RT right hand - RT left hand (ms)
Figure 1:
A) Typical representation of the SNARC effect. The difference between the reaction times of
right and left hand plotted against number magnitude produces a negative regression slope.
B) Shape of the SNARC effect in magnitude classification tasks. The magnitude numbers near the
standard (i.e. near 5 in the present example) is processed more deeply than that of numbers far
from standard. For this reason the SNARC effect for numbers near the numerical standard
becomes comparable to the SNARC effect observed for numbers far from the standard.
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
494
2.3 The effect of imagery on the SNARC effect
Instructions underscoring the use of visual imagery have been shown to determine the
orientation of the SNARC effect. Bächthold et al. (1998) told participants to judge the mag-
nitude of numbers 1 to 5 and 7 to 11 in relation to the standard 6. In the first part of the ex-
periment, participants were instructed to represent numbers as indicators of distances on a
ruler. In the second part of the experiment, participants were instructed to represent numbers
as the time displayed on a clock-face. A regular SNARC effect was found when numbers
were conceived as distances on a ruler, but an inverted SNARC effect was observed when
numbers were conceived as the time read on a clock-face. Therefore, the manipulation of
strategies determines the relative size of the SNARC effect: while the SNARC effect is
negative when participants are instructed to represent numbers on a ruler, it becomes posi-
tive when representing numbers as the time on a clock face.
Proctor and Cho (2006) attributed the results obtained by Bächthold et al. (1998) to two
different types of association between number and space. On the one side, (i) categorical
spatial relations are according to Proctor and Cho (2006) the main determinant of the
SNARC effect and reflect the polarity correspondence between number magnitude and re-
sponse codes (e.g. small-left/large-right). On the other side, according to these authors (ii)
coordinate spatial relations that can modulate and even invert the SNARC effect when in-
structions induce the activation of mental imagery, such as in the study by Bächthold et al.
(1998). Therein, Proctor and Cho (2006) attribute the activation of coordinate spatial rela-
tions to task instructions, which may more or less emphasize the use of mental imagery
when solving numerical tasks. Importantly, these authors do not specify whether the activa-
tion of coordinate spatial relations should lead to a change in the absolute size of the
SNARC effect or only on its direction. Interestingly, while categorical spatial relations are
directly associated with the use of verbal categories which are processed primarily by the left
hemisphere, coordinate spatial relations are more associated with right hemisphere functions
(Laeng et al., 2003). Therefore, it is not improbable that the activation of coordinate spatial
relations, which are subserved by specific brain regions in the right hemisphere, might im-
pact on the absolute size of the SNARC effect and modulate its distribution across response
time ranges. For these reasons we tentatively formulate the hypothesis that
(iii) imagery strategies might have a specific impact on the absolute size of the SNARC
effect.
2.4 The SNARC effect and response discrimination
Beside bimanual responses, the SNARC effect has been reported for many other re-
sponse parameters such as pointing (Fischer, 2003a), eye-movements (Fischer et al. 2004)
and grasping (Andres et al. 2008). Interestingly, Fischer, (2003a) and Fischer et al. (2004)
found consistently a SNARC effect in response latencies but no SNARC effect in response
amplitudes. In contrast, Andres et al. (2008) found a robust SNARC effect in grasping aper-
ture, which is a form of response amplitude. These discrepant results suggest that not all
response parameters are sensitive to the SNARC effect; however, they do not answer the
question about why some but not every response parameter should be sensitive to the
A meta-analysis of SNARC 495
SNARC effect. Recently, Ansorge and Wühr (2004) proposed a response discrimination
account for performance in manual choice reaction tasks. According to this account, only
those response parameters allowing for an association with stimulus categories can generate
spatial biases. In accord with this hypothesis, Gevers et al., (2005b) have shown that the
SNARC effect can be found in different spatial aspects of the stimulus-response set when
these aspects discriminate between the alternative. This can explain why in the studies by
Fischer (2003a) and Fischer et al. (2004) response amplitude was not sensitive to the
SNARC effect. In those studies amplitude did not discriminate responses but was held con-
stant. In contrast, in the study by Andres et al. (2008) response amplitudes discriminated
between responses. Therefore,
(iv) a SNARC effect should be found in those spatial dimensions of the experimental de-
sign which discriminate responses.
2.5 The impact of age on the size of the SNARC effect
Berch et al. (1999) have shown that in a parity decision task the SNARC effect can be
found not earlier than in 9-years old children. Recently, van Galen and Reitsma (in press)
found a SNARC effect in 7-years old children in a magnitude comparison task. Moreover,
Vuilleumier et al. (2004), Castronovo and Seron (2006) and Priftis et al. (2006) reported a
SNARC effect for participants far older than common college students. Is the SNARC effect
size comparable in all age groups or is there an association between the SNARC effect size
and age? In case that age has an impact on the size of the SNARC effect, there are at least
two cognitive factors which may induce age-related variability in mental associations such
as the SNARC effect: long-term practice (Knoch et al., 2005; Brigman and Cherry, 2002),
and inhibition capacity (Hasher and Zacks, 1988). Practice effects cumulate across the life-
span and may produce an age-related increase in the SNARC effect. Moreover, inhibition
may be associated with an increase in the SNARC effect in children and elderly participants
in comparison with young adults. Therefore, practice effects should increase the SNARC
effect in function of age while inhibition should increase the SNARC effect in very young
and very old groups in comparison with intermediary age groups. Therefore, we can tenta-
tively formulate the hypothesis that
(v) age should have an impact on the size of the SNARC effect.
2.6 The interaction between stimulus format and the SNARC effect: number words
Finally, we consider effects of number formatting. Fias (2001) reported that both magni-
tude classification and parity judgments for written number words resulted in a SNARC
effect, whereas a phoneme detection task with the same materials did not. In contrast, Arabic
digits always activated the number’s magnitude representation in all these tasks. A contrast-
ing result was reported by Nuerk et al. (2005b) who found no difference in the SNARC
slopes for Arabic digits, number words, auditorily presented number words and dice pat-
terns. Fias (2001) argued that number words can be processed without access to their mean-
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
496
ing through an non-semantic route while Arabic numbers are putatively processed through a
semantic route. In light of these conflicting results it will be interesting to determine meta-
analytically whether
(vi) the SNARC effect for number words should be small or absent when the task at hand
can be solved without putative access to the magnitude meaning of those words.
A quantitative assessment of these six hypotheses formulated above has never been con-
ducted with meta-analytic techniques. A meta-analytic re-analysis characterizes the strength
and consistency of the empirical findings of the literature on the SNARC effect.
3. The present study
Our meta-analysis of SNARC studies has several interesting features. First, given the an-
nually increasing output of papers (see Wood and Fischer, 2008), our report includes a sub-
stantial body of recent work that was not yet available to previous reviewers of the field.
Second, our meta-analysis uses quantitative methods to identify and investigate patterns
across studies that have remained undiscovered in previous reviews. Third, a quantitative
meta-analysis of hypotheses already tested in the literature may help to evaluate the consis-
tency of these single findings across different studies. Finally, the quantitative features of our
review also allow us to test the novel predictions we have derived from recent theoretical
advances and debates inference-statistically.
3.1 Inclusion and exclusion criteria for the present meta-analysis
In order to be considered for the subsequent meta-analysis, a study had to investigate the
spatial representation of numbers with the SNARC effect, and had to be in press or pub-
lished between 1993 and August 20064. To be included, a study also had to provide a suffi-
ciently detailed description of the stimuli and task used, as well as some participant charac-
teristics. Most importantly, we required the study to contain a quantitative description of the
association between numbers and space from which an estimate of the statistical strength of
the SNARC effect could be obtained. We excluded work in which stimulus dimensions other
than numbers were studied, such as days of the week or months of the year (e.g., Gevers et
al., 2003; 2004) or tone height (Rusconi et al., 2005, 2006, Ishihara et al., 2008). Our review
also excludes studies using the number bisection task in which participants either produce or
verify the midpoint of an experimenter-defined temporal (Casaroti et al., 2007) or numerical
interval (e.g. Calabria and Rosseti, 2005, Goebel et al., 2006, Nuerk et al., 2002, Zorzi et al.,
2002). Although results obtained with this task are currently used to infer spatial properties
of the mental number line (see for instance, Hoeckner et al., 2008), we decided to exclude
4 We contacted several colleagues, their web sites, and electronic preview sites of journals to obtain in-press
papers, including also conference posters. We wish to thank those colleagues who responded and supplied
information. We also apologize if we have inadvertently neglected any relevant papers and would appreciate
to receive updates from their authors for the purpose of future reviews.
A meta-analysis of SNARC 497
them from our analysis because many different number representations become activated in
the number bisection task, in addition to magnitude (see Table 1 in Nuerk et al., 2002, p.
702; see also Wood et al., 2008). The selection of a measure of number magnitude in the
number bisection task is far from trivial, since there are several different ways to compute
the effect of number magnitude on performance (e.g., numerical distance to the correct re-
sponse, number range, numerical distance between first and second number, problem size,
etc.). However, in the end, we decided that above measures are not typical SNARC-like
measures. This latter decision could – of course – be further discussed, given that interval
estimation tasks (de Hevia et al. 2006) and bisection tasks with flanker digits (Fischer, 2001)
were included in the present analysis to better capture the range of available methods. In the
end, we included 46 studies, several of which contained more than one experiment. A list of
studies as well as a general description of them can be found in Appendix II. The number of
experiments considered for our separate analyses as reported below may, however, differ
between analyses because many experimental factors were varied in any given study.
3.2 Method for determining the size of the SNARC effect
The formula used to calculate the effect size d (Cohen, 1988) for each experiment was
taken from Hedges and Olkin (1985). Specifically, we calculated the statistic d as follows:
2/dtdf= (1)
where t is the value of the t-test statistic reported for the strength of the association between
numbers and space, and df reflects the degrees of freedom associated with the test (usually
the number of participants in the sample minus 1). Before entering a statistical analysis, any
sampling bias on d was corrected according to the procedure suggested by Hedges and Olkin
(1985, p. 81). Confidence intervals for the point estimates of size of the SNARC effects were
obtained following the procedure described by Hedges and Olkin (1985, p. 86). For studies
with multiple experiments or with multiple experimental conditions for which a SNARC
effect could be estimated, separate effect sizes were calculated for each experiment or ex-
perimental condition. When only an F-value but no t-value was given in a study, the effect
size d was estimated directly from the associated p-value (Fricke and Treinies, 1985, p. 94).
Often more than one estimate of the size of the SNARC effect could be obtained from a
single study or even from a single experiment. For instance, Priftis et al. (2006, Experiments
2 and 3) tested hemineglect patients and young healthy controls. In this case we determined
the size of the SNARC effect for each sample separately (see Appendix II). Moreover, when
two non-independent estimates of the size of the SNARC effect could be subsumed into a
single category, the arithmetic mean of effect size estimates was taken. For instance, the
study by Wood et al. (2006a) examined the SNARC effect in four different notations (Arabic
digits, visual number words, auditory number words, dice patterns) using a parity judgment
task and always the same participants. To estimate the size of the SNARC effect for the
factor “parity”, the effect sizes obtained for these four different notations were averaged.
However, to estimate the size of the SNARC effect for a specific format, only results from
that format were considered.
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
498
Before reporting effect sizes as a function of experimental factors, we examined whether
the effect sizes obtained in the different studies were homogeneous. The overall effect size
estimate for the SNARC effect was calculated and pooled together from a total sample size of
2206 participants examined in 46 different studies. The statistic Q testing for homogeneity of
effect sizes was calculated according to Hedges and Olkin (1985). This statistic for the overall
size of the SNARC effect was highly significant (χ2(45) = 147.0; p < 0.05), pointing to signifi-
cant variability in the effect sizes across the 46 studies. As a result of this heterogeneity, the
impact of different experimental factors should be examined, which will be done below.
3.3 The size of the SNARC effect depends on response latencies and on the strength of
number magnitude activation5
According to the dual route model (Gevers et al., 2006b), the size of the SNARC effect
should increase together with the average RT (prediction (i), see above), since the magnitude
representation has more time to interfere with response selection through the unconditional
route. Average RT could be obtained from 43 independent experiments and was a significant
predictor of size of the SNARC effect. A weighted least squares regression of the size of the
SNARC effects on reaction times revealed a significant coefficient (b = 0.001, t(42) = -2.50,
p < 0.05, = 13 %, see Hedges and Olkin 1985, p. 110, for more details on the weighting
procedure). This evidence supports the view that the size of the SNARC effect increases
with average RT. This result is in line with our prediction (i) as derived from the dual-route
model of the SNARC effect (Gevers et al. 2006b, see also the Discussion below).
In analogy to results on the association between SNARC and RT, tasks involving seman-
tic number processing such as parity decision or magnitude classification also were associ-
ated with a stronger SNARC effect than tasks in which semantic number processing is not
required (Figure 2). Out of a total of 106 SNARC experiments that were included in this
meta-analysis, just over half (number of experiments Ne = 54; 50.9 % of the total number of
experiments) used a parity decision task, Ne = 18 (17 %) studied a magnitude classification
task, and only Ne = 5 (4.7 %) used a magnitude comparison task (Bull et al., 2006, testing
deaf and hearing participants; Fischer, 2003b; Nuerk et al. 2005a; Shaki and Petrusic,
20056). The remaining Ne = 28 (27.4 %) experiments used non-semantic tasks which did not
require any manipulation of number meaning. A large variety of tasks was subsumed in this
group, such as naming or discriminating the colours of coloured numerals (Caessens et al.,
2001, Keus and Schwarz, 2005, Lammertyn et al., 2002), simple detection of Arabic digits
(Fischer et al., 2003), arbitrary mapping of numbers to response keys (Gevers et al. 2006b),
bisection of digits strings or lines flanked by Arabic digits (Fischer, 2001), judgement about
the spatial orientation of Arabic digits, arrows or geometrical figures (Caessens et al., 2004,
5 A large variety of response modalities has been investigated, with some studies even comparing the SNARC
effect across response modalities. When this was the case we selected data only from the less common re-
sponse modality for analysis. From the study by Schwarz and Keus (2004, Experiment 1), for instance, only
data from saccadic latencies were used in the present analysis. For the same reason, only naming latencies
from the study by Keus and Schwarz (2005, Experiment 2) and only bipedal responses from the study by
Schwarz and Müller (2006, Experiment 2) were included in this analysis.
6 In some cases, the dRT values were drawn from the figures of the published studies and should be consid-
ered as approximate values.
A meta-analysis of SNARC 499
Fias et al., 2001, Lammertyn et al. 2002, Notebaert et al., 2006), and discrimination of visual
forms (Fias et al., 2001; Zebian, 2005), as well as phoneme detection and number naming
tasks (Caessens et al., 2004; Fias, 2001; Fias et al., 1996).
Table 1 shows that the size of the SNARC effect was heterogeneous in all tasks, suggest-
ing the presence of significant variance in the size of the SNARC effects. Average effect
sizes, tests for homogeneity as well as the number of participants (Np), the number of inde-
pendent experiments (Ne), and average reaction times are listed in Table 1. Average effect
sizes and 95 % confidence intervals are shown in Figure 2.
Table 1:
Pooled size of the SNARC effects and tests for homogeneity for different tasks
Task d χ² df Np Ne
RT
(ms)
Parity -0.99 197.36 a 53 1198 54 536
Magnitude classification (fixed standard) -1.04 77.43 a 17 415 18
632
Magnitude comparison (variable standard) -0.59 19.95 a 4 100 5
604
Tasks without semantic manipulation -0.60 155.54 a 28 513 29 647
df: Degrees of freedom; Np: number of participants; Ne: number of experiments; d: pooled size of the
SNARC effect; a: p<0.05
Figure 2:
Average size of the SNARC effects +/- 95% confidence intervals estimated for different
experimental parameters.
-4,5 -4,0 -3,5 -3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0
Over all Ne=46 Ns=2 206
parity Ne=53 Ns=1174
magnitude classification Ne=18 Ns=415
magnitude comparison Ne=5 Ns=100
non-semantic Ne=29 Ns=513
non-semantic without number words Ne=28 Ns=493
Arabic Ne=90 Ns=1929
number words semantic tasks Ne=5 Ns=123
number words non-semantic tasks Ne=1 Ns=20
auditory Ne=5 Ns=82
non-crossed Ne=2 Ns=104
crossed Ne=2 Ns=104
crossed without Wood et al. 2006 Ne=3 Ns=72
bimanual response Ne=73 Ns=1646
eye saccades latency N e=3 Ns=47
eye saccade am plitudes Ne=1 Ns =15
manual bisection Ne=2 Ns=31
pointing RT Ne=2 Ns=39
pointing MT Ne=3 Ns=39
unimanual finger response Ne=10 Ns=145
naming Ne=8 Ns=126
bipedal response Ne=2 Ns=50
grip aperture Ne=1 Ns=26
SNARC effect size d
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
500
The average size of the SNARC effects estimated for parity judgment tasks and magni-
tude classification tasks were comparable, as well as the average RTs. The effect size in the
parity judgment task was slightly larger than that obtained for non-semantic tasks (Figure 2).
Interestingly, the larger SNARC effect in the parity task cannot be accounted for by slower
speed of responding, since in non-semantic tasks the RT was marginally slower (average m
= 647 ms, standard deviation SD = 197 ms) than in the parity task (m = 573 ms, SD =
122ms).
In summary, both average RT and the nature of numerical tasks have an impact on the
size of the SNARC effect. In general, the longer the time needed to reach a motor response,
the stronger is the size of the SNARC effect. Furthermore, in tasks involving semantic num-
ber processing such as parity decision task, the SNARC effect tended to be larger than in
non-semantic tasks.
3.4 In tasks of magnitude classification the SNARC effect should assume a categorical
shape
In order to test hypothesis (ii) concerning the shape of the SNARC effect in magnitude
comparison tasks, we compared the fit obtained when modelling data with categorical and
linear predictors using regression models. Categorical and continuous predictors of dRT
were entered stepwise in regression models calculated separately for each of the 15 studies
analysed. Results are shown in Table 2.
In nine out of 15 studies (60 %) the SNARC effect could be accounted for by the cate-
gorical predictor alone. In only 1 study (7 %) the continuous predictor alone was a signifi-
cant predictor of dRT (Bachot et al., 2005). In another study (Nuerk et al. 2005a7), both
categorical and continuous predictors entered in the regression model (7 %). In the four
remaining studies neither a categorical nor continuous predictors accounted for variance in
the dRTs (26 %).
These results confirm prediction (ii) about the shape of the SNARC effect in magnitude
comparison tasks. Gevers et al.’s (2006b) account of the categorical shape of the SNARC
effect is based on the expectation of a stronger SNARC effect in slower responses. Impor-
tantly, 40% of the studies examined did not show a categorical shape. However, as will be
discussed below, this may be explained by characteristics of the task, stimuli employed and
participants tested.
7 We classified the “Eriksen flanker” task used by Nuerk et al. (2005a) as a task with variable standard be-
cause, although targets should be compared against the fixed standard 5, they were flanked by other Arabic
numbers.
A meta-analysis of SNARC 501
Table 2:
Stepwise regression models of the SNARC effect. Continuous and categorical predictors were
contrasted.
Study Predictor
remaining
in the
model
(adjusted)
F-test
(df1, df2)
Increment§
Shaki and Petrusic (2005) intermixed continuous .45 10.1 a (1, 10) <1%
Shaki and Petrusic (2005) negative blocked continuous .94 75.3 a (1, 4) 4%
Shaki and Petrusic (2005) positive blocked continuous .94 85.4 a (1, 4) 4%
Shaki et al. (2006) categorical .92 92.5 a (1, 6) 13%
Bachot et al. (2005) control children continuous .42 6.2 a (1, 6) 20%
Bachot et al. (2005) VSD children† (continuous) .24 2.1 (2, 5) 15%
Gevers et al. (2006b) categorical .82 32.9 a (1, 6) 7%
Castronovo & Seron (2006) blind participants categorical .92 84.4 a (1, 6) 16%
Castronovo & Seron (2006) sighted
participants
categorical .93 91.6 a (1, 6) 23%
Nuerk et al. (2005) categorical
continuous
.96 7.0 a (1, 5) 7%
Fischer and Rottmann (2005) whole interval categorical .69 40.2 a (1, 16) 10%
Fischer and Rottmann (2005) negative
interval†
(categorical) 0.01 .24 (2, 5) <1%
Fischer and Rottmann (2005) positive
interval†
(categorical
continuous)
.52 4.9 (2, 5) 14%
Bull et al. (2005) deaf participants categorical .94 126.4 a (1, 6) 21%
Bull et al. (2005) hearing participants categorical .60 11.6 a (1, 6) 2%
Ito and Hatta (2004) † (categorical
continuous)
.16 .52 (2, 5) 2%
Bächthold et al. (1998) ruler task categorical .96 221.1 a (1, 8) 22%
Bächthold et al. (1998) clock-face task categorical .97 292.9 a (1, 8) 22%
df: Degrees of freedom; a: p<0.05; § The increment refers to the extra proportion of variance accounted by
the predictor kept in the model. † Both predictors were excluded from the stepwise model. Numerical values
shown refer to a regression model computed using the enter method.
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
502
3.5 Imagery strategies might have a specific impact on the size of the SNARC effect
In section 2.3 above, we have tentatively postulated that the absolute size of the SNARC
effect may vary in function of imagery instructions. In order to test this hypothesis, we com-
pared the slopes of size of the SNARC effect on RT obtained separately in experiments
explicitly requiring the use of mental imagery (i.e., Bächthold et al., 1998; Vuilleumier et al.,
2004) and experiments with instructions neutral about mental imagery. If imagery instruc-
tions lead to the activation of coordinate spatial relations, possibly the association between
size of the SNARC effect and response latencies may be stronger than when only categorical
spatial relations are activated. To test that we regressed separately the size of the SNARC
effects on average RT for tasks requiring the use of mental imagery and those tasks with
neutral instructions. For the interpretation of eventual differences in intercepts and slopes
between the two types of tasks we employed the rationale presented by Verhaegen and
Cerella (2002). Verhaegen and Cerella (2002) distinguish between additive and multiplica-
tive processes interfering with the size of behavioural effects. Additive processes impact
only on the intercept but not on the regression slopes, and are believed to be effective upon
one single stage of cognitive processing. In contrast, multiplicative processes produce
steeper slopes in the more complex conditions (i.e. in the present case the imagery condition)
and are believed to interfere with two or more different stages of cognitive processing.
When instructed to use mental imagery to guide magnitude classification, participants
were significantly slower (m = 832 ms; SD = 212 ms) than when the instructions were neu-
tral (m = 548 ms; SD = 126 ms; t(15) = 2.60; p < 0.05). A weighted least squares regression
analysis revealed for tasks requiring the use of mental imagery the following intercepts and
slopes of the size of the SNARC effects on RT (intercept = -0.770, standard error SE = 1.64;
b = -0.002, SE = 0.002). For tasks with neutral instructions, the following intercepts and
slopes were obtained: intercept = -0.474, SE = 0.913; b = -0.002, SE = 0.002. Interestingly,
neither the intercept (t(15) = -0.55, p>0.05) nor the slope of the size of the SNARC effect on
RT (t(15) = 0) differed between conditions. These results suggest that although mental im-
agery has an impact on the average time necessary to solve SNARC tasks, the function de-
scribing the association between the size of the SNARC effect and RT remains constant.
Therefore, no evidence for additive or multiplicative increase in task complexity was evident
in the present analysis. These results will be further discussed in section 4.3, below.
3.6 A SNARC effect should only be found in those spatial dimensions of the
experimental design which discriminate responses
The SNARC effect is often interpreted as a response bias resulting from spontaneous
magnitude processing. Therefore it is of theoretical importance to determine the generality of
this bias across different response modalities. Information from 105 independent experi-
ments was classified into the following categories: Bimanual responses (Ne = 73, 70 %),
saccadic latencies (Ne = 3, 2.9 %; Fischer et al., 2004; Schwarz and Keus, 2004), saccade
amplitudes (Ne = 1, 1 %; Fischer et al., 2004), motor bias in number string bisection (Ne =
2, 1.9 %; Fischer, 2001), pointing latencies (Ne = 2, 1.9 %; Fischer, 2003a), movement times
in pointing (Ne = 3, 2.9 % Fischer, 2003a), unimanual finger responses (Ne = 10, 10 %; e.g.
Priftis et al., 2006), naming latencies (Ne = 8, 7.6 %; e.g., Keus and Schwarz, 2005), foot
A meta-analysis of SNARC 503
responses (Ne = 2, 1.9 %; Müller and Schwarz, 2006), and grip apertures (Ne = 1, 1 %;
Andres et al., 2004). With the exception of saccadic amplitudes and naming latencies all
response modalities showed a large SNARC effect thus speaking for the ubiquity of the
effect and its importance for studies of human cognition more generally.
The size of the SNARC effects for bimanual responses, saccadic latencies, manual bisec-
tion, reaction time (RT) and movement time (MT) from pointing, as well as unimanual fin-
ger responses and bipedal responses were all comparable although average RT in the differ-
ent tasks differed between response modalities (Figure 2, Table 3). The null effect for nam-
ing tasks is mainly due to the results of Zebian (2005), who found no SNARC effect for
Arabic speaking participants. For this reason the present results may be interpreted with
caution. For saccadic amplitudes (Fischer et al., 2004) a null effect was found as well. Fi-
nally, a very strong size of the SNARC effect was found for grip apertures.
Importantly, a SNARC effect was not observed in saccade amplitudes and in naming
tasks. The lack of a SNARC effect in saccade amplitudes can be explained by the experi-
mental setup adopted by Fischer et al. (2004), in which saccades from a fixation point to
eccentric and equidistant squares should be carried on. In that experiment, the amplitudes of
saccadic movements did not discriminate between responses. Interestingly, in the same
experiment the direction of saccades was associated with a reliable SNARC effect. These
results are in line with the account proposed by Gevers et al. (2005b) based on the response
discrimination. The direction of saccades was highly discriminative of responses while am-
plitudes did not distinguish between responses. The null effect observed in naming tasks
may be due to many different factors and cannot be easily interpreted. The populations tested
by Zebian (2005), as well as the task used and the response modality may be responsible for
the null effect. For this reason we refrain from interpreting this result in more detail.
In summary, the SNARC effect can be observed in many different motor responses. Cru-
cially, the dimensions of response which discriminated between different responses were
associated with the SNARC effect. These results corroborate the role of response discrimina-
tion in the association between number magnitudes and response codes.
Table 3:
Pooled size of the SNARC effects and tests for homogeneity for response modality.
Response d χ² df Np Ne RT
bimanual response -0.79 320.44 a 72 1646 73
561
eye saccades latency -1.20 0.30 2 47 3 450
eye saccade amplitudes -0.07 - - 15 1 -
manual bisection -1.08 4.54 a 1 31 2
-
pointing RT -1.02 1.93 a 1 39 2
497
pointing MT -0.94 0.87 2 39 3 404
unimanual finger response -1.69 20.13 a 9 145 10
660
naming 0.09 0.00 7 126 8
724
foot response -1.59 0.32 1 50 2 592
grip aperture -3.29 - - 26 1 492
df: degrees of freedom; Np: number of participants; Ne: number of experiments; d: pooled size of the SNARC
effect; a: p<0.05
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
504
3.7 Age should have an impact on the SNARC effect
From Ne=106 independent experiments included, information about the age of partici-
pants was available only for 95 samples (89 %). The average age of participants in those
studies was 25.56 years (SD = 12.45, range = 9 to 66 years). In order to investigate the im-
pact of age on the size of the SNARC effect we selected only those studies reporting average
RT and average age of participants and entered both variables as predictors of the size of the
SNARC effect in a weighted least squares regression model. Seventeen studies met the se-
lection criteria for this analysis. The weighted least squares regression model including age
and RT as predictors of the size of the SNARC effects explained (adjusted) = 65% of
variance. The higher the mean age of participants the stronger was the effect size. While age
was a highly significant predictor of the size of the SNARC effect (b = -0.036; SE = 0.007;
t(14) = -5.29; p < 0.05), response speed was only marginally significant (b = -0.001; SE =
0.001; t(14) = -2.04; p = 0.06, but see Footnote8). Figure 3 shows the association between
size of the SNARC effects and age estimated after partialling out the effect of RT.
Our results strongly suggest that the size of the SNARC effect increases with age. More-
over, they suggest that the SNARC effect may not be found easily in children much younger
than 10 years old. Inspection of the confidence intervals on the estimated regression slope of
size of the SNARC effect on age shows that the SNARC does not differ from 0 unless chil-
dren are approximately 9.5 years old (lower 95% confidence interval CI = -0.050, upper
95% CI = -0.021). This conclusion converges with empirical results obtained by Berch et al.,
(1999) who assessed the SNARC effect in children aged 7.8, 9.2, 9.8, 11.7, and 13.6 years
old on average: Only from age 9.2 onwards was the SNARC effect reliably present (Bachot
et al., 2005, but see van Galen & Reitsma, in press).
3.8 The SNARC effect for number words should be small or absent when the task at
hand can be solved without putative access to the magnitude meaning of those words
Numerical information can be conveyed in several different formats. In addition to the
ubiquitous Arabic digit format, we frequently encounter visual and auditory number words,
fingers (Bull et al., 2006), dot patterns (Nuerk et al., 2005b; Wood et al., 2006a) and tally
marks. Most studies of the SNARC effect used Arabic numbers as stimuli (Ne = 90, 79 %).
Both visual and auditory number words were each employed in 4.4 % of the studies (both Ne
= 5; these were Dehaene et al., 1993; Fias, 2001; Nuerk et al., 2004; Nuerk et al., 2005b; and
Wood et al., 2006a, for visual number words; and Castronovo and Seron, 2006; Fischer and
Hill, 2004; Nuerk et al., 2005b and Wood et al., 2006a, for auditory number words). Other
stimulus formats, such as fingers (Atmaca et al., 2006; Bull et al., 2006), dot patterns (Nuerk
et al., 2005b; Wood et al., 2006a), East Arabic numbers (Dehaene et al., 1993), and arrows
(Caessens et al., 2004) were used in the remaining 12.3 % of experiments.
8 To examine the effect of age on the size of the SNARC effects we have averaged the RT for each of the
seventeen studies, because the average age of participants almost did not vary within studies. Note that for the
analysis of the association between RT and size of the SNARC effects we examined the 43 independent
experiments separately, because RT did vary considerably between different samples.
A meta-analysis of SNARC 505
0 102030 405060
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
age (years)
SNARC effect size d
y = -0.36*age + 0.917, R²=65%
Figure 3:
The size of the SNARC effect plotted against the average age of participants. The bold line
describes a negative slope of the size of the SNARC effect on age, reflecting the increase in the
size of the SNARC effect due to the increase in age.
Table 4:
Pooled size of the SNARC effects and tests for homogeneity for different stimuli.
Format d χ² df Np Ne
Arabic -0.83 395.81 a 89 1929 90
Number words with semantic -0.82 20.00 a 6 153 7
Number words without semantic 0.12 - - 20 1
Auditory number words -0.98 16.29 a 4 82 5
df: degrees of freedom; Np: number of participants; Ne: number of experiments; d: pooled size of the SNARC
effect; a: p<0.05
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
506
Concentrating on Arabic digits, auditory and visual number words, we present size of the
SNARC effects and 95 % confidence intervals in Figure 2. The size of the SNARC effect
was highly heterogeneous for all types of stimuli. Statistical tests for homogeneity, the num-
ber of studies and of participants are summarized in Table 4.
Note that the effect sizes for number words were split into two categories: number words
in semantic tasks (Dehaene et al., 1993; Fias, 2001, Experiment 1; Nuerk et al, 2004; Nuerk
et al, 2005b and Wood et al., 2006a) and number words in non-semantic tasks (Fias, 2001,
Experiment 2). As predicted by Fias (2001), the size of the SNARC effect for number words
in non-semantic tasks did not differ from zero (i.e. the confidence interval contains zero, see
Figure 2), suggesting that magnitude information was not activated. Furthermore, as can be
seen in Figure 2, the size of the SNARC effect for number words in non-semantic tasks
differed from the size of the SNARC effect for Arabic digits and auditory number words,
since the 95% confidence intervals for the average effect in these two conditions did not
overlap. Critically, the small overlap between the confidence intervals obtained for number
words in semantic and non-semantic tasks was due to the null effect reported by Wood et al.
(2006a). When excluding this study from the sample, the overlap disappears (Figure 2).
These data corroborates the findings originally reported and shows that these results are
robust and not specific for experimental conditions of the original study by Fias (2001). In
summary, comparable size of the SNARC effects were obtained for Arabic numbers, audi-
tory and written number word. This corroborates an amodal association between number and
space, as proposed by Nuerk et al. (2005b). Furthermore, a typical size of the SNARC effect
can be obtained for number words in semantic tasks but not in non-semantic ones. Therefore,
the association between number and space seems to be mediated by the specific materials
used to test it.
4. General Discussion
The size of the SNARC effect is modulated by a broad scope of experimental factors in-
cluding type of task, stimuli, motor response and participant-related factors. In general,
predictions derived from the literature and from the dual-route model were confirmed by the
different analyses presented above: the SNARC effect depends heavily on the strength of
number magnitude activation (hypothesis i). In tasks of magnitude classification (e.g., “Is the
number larger or smaller than 5?”), the SNARC effect clearly assumed a categorical shape
(Gevers et al. 2006b, hypothesis ii). However, evidence does not seem to corroborate hy-
pothesis (iii) on the impact of imaging strategies on the absolute size of the SNARC effect.
Furthermore, a SNARC effect was found in spatial dimensions of the experimental design
whenever they discriminated responses (prediction iv) as well as an effect of age on the size
of the SNARC effect (hypothesis v). Finally, data lent further support to the view that the
SNARC effect for number words is absent when the task at hand can be solved without
putative access to the magnitude meaning of those words (vi). In the following, the most
relevant of these results will be discussed.
A meta-analysis of SNARC 507
4.1 The strength of number magnitude activation
Very robust evidence that the strength of number magnitude activation determined the
size of the SNARC effect has been provided in the literature. The impact of number magni-
tude activation could be observed in response latencies and in the nature of the experimental
task and task demands. In general, the longer the time needed to reach a motor response the
stronger was the size of the SNARC effect. The association between response latencies and
the SNARC effect was depicted as a significant regression slope which was estimated from a
broad variety of studies differing in many aspects such as task, population, stimulus modality
and stimulus format, as well as response modality. Therefore, since a significant regression
slope was obtained, which can explain about 13 % of the variance across all SNARC ex-
periments, one may conclude that response latencies are one determinant of the SNARC
effect. An increase in the SNARC effect due to increased response latencies is predicted by
the dual-route model (Gevers et al. 2006a): the longer the time to select a response, the
stronger is the impact of number magnitude on responses and stronger is the SNARC effect.
Although the present meta-analytical finding is not new, it illustrates the generality of the
impact of response latencies on the SNARC effect. Moreover, it is necessary to distinguish
the influence of response latencies from other determinants of the SNARC effect, such as
task demands. The SNARC effect was larger in tasks involving semantic number processing
such as the parity decision task than in non-semantic tasks such as colour discrimination.
Importantly, this effect was independent from response latencies, since in non-semantic tasks
RTs were even slower than in the parity task. Therefore, response latencies alone cannot
explain the effect task on the size of the SNARC effect.
To our knowledge this is the first evidence for dissociation between the effects of re-
sponse latency and task demands on the size of the SNARC effect. The effect of task de-
mands can be interpreted as a consequence of deeper number magnitude processing induced
by specific task characteristics or demands. In line with the polarity correspondence account
(Proctor and Cho, 2006), this effect can be seen as an increase in the saliency of magnitude
in semantic tasks. Different causes for this increase in the saliency of number magnitude can
be enumerated: first, task instructions can induce the processing style of stimuli to be deeper
or more superficial. In the case of semantic tasks, deeper number processing is necessary due
to abstract properties of stimuli such as parity of magnitude. In non-semantic tasks periph-
eral features of stimuli should be processed such as phoneme detection, colour or even arbi-
trary criteria (see Gevers et al. 2005b) which are quite independent from semantic properties
of numbers.
Not only task demands seem to determine the SNARC effect but also stimulus materials.
In the special case of number words, the size of the SNARC effect obtained in tasks without
semantic manipulation even did not differ from 0, but differed from the SNARC slopes
obtained for written number words in semantic tasks. These results are a meta-analytical
corroboration of the empirical study by Fias (2001). In the case of number words, different
routes can be activated, one of which is a semantic route and leads to the activation of num-
ber magnitude while the other one converts a pre-semantic input directly in an output format.
In semantic tasks involving the comparison of two or more numbers (e.g. Nuerk et al. 2005a)
size of the SNARC effects was smaller than in other semantic tasks. A possible explanation
for this fact is the interference between the different stimuli presented in each trial. While in
a magnitude classification task or in a parity task only one association between number
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
508
magnitude and response codes is activated in each trial, in magnitude comparison tasks as
many spatio-numerical associations are potentially activated as there are numbers being
presented. Since these responses and their spatial associations may differ depending on the
magnitude of numbers being presented, they can counteract each other and lead to a smaller
effect size.
In summary, the present results corroborate the dual route model by Gevers et al.
(2006b) that the strength of magnitude activation is a pervasive determinant of the SNARC
effect. Magnitude activation influences the SNARC effect through the response latencies
specific for different experimental designs as well as through task demands which may re-
quire more or less magnitude activation.
4.2 Categorical vs. continuous SNARC slopes
In 60 % of the experiments examined for the shape of the SNARC effect, a categorical
shape was observed. However, in 40 % of the experiments a more complex pattern of results
arose. In only one experiment the SNARC effect was clearly continuous (Bachot et al., 2005,
Table 2). In three experiments neither a categorical nor a continuous predictor explained the
results, while in a last experiment both continuous and categorical predictors explained a
significant proportion of variance. According to the dual route model of the SNARC effect
(Gevers et al., 2006a), the categorical shape of the SNARC effect is due to the stronger
activation of the magnitude representation typically associated with longer response laten-
cies to numbers near the numerical standard. Accordingly, in those studies in which the
SNARC effect presents a continuous shape, the number magnitude representation should
have been activated to a lesser extent than in usual SNARC experiments. As discussed be-
low, it seems to be the case at least in part of these studies. In the only study reporting a
continuous slope participants were children in the age range of 7 to 12 years. As will be
discussed in more detail in Section 4.5 below, at this age participants may still not activate
number magnitude to the same extent as adults. Furthermore, the mapping of numbers onto
space also may be less automatic at this age (Wood et al. submitted). Consequently, the
impact of the distance effect on the shape of the SNARC effect may be reduced in children
in comparison with adults, and the shape of the SNARC effect may resemble more a con-
tinuous slope, even in magnitude classification tasks.
Altogether, the impact of magnitude classification on the shape of the SNARC effect
may depend on the exact nature of stimuli and characteristics of participants in the studies.
More importantly, the present results confirm predictions (i) and (ii) of the dual-route model
(Gevers et al. 2006b) that the strength of magnitude representation determines the amount of
SNARC effect observed and that in magnitude classification tasks the SNARC effect may
assume a categorical shape whenever the magnitude representation gets activated in a more
precise and automatic fashion.
A meta-analysis of SNARC 509
4.3 The impact of imaging
The manipulation of strategies has an important effect on the relative size of the SNARC
effect: the SNARC effect is negative when participants are instructed to represent numbers
on a ruler and becomes positive when representing numbers as the time on a clock face
(Bächthold et al., 1998). Here we have investigated whether the effect of instructing partici-
pants to use imagery might also impact on the association between the size of the SNARC
effect and RT. The findings of the present review revealed that experiments requiring the
explicit use of imagery strategies produced much slower RT than experiments which did not
require the use of imagery. Interestingly, both intercepts and slopes of size of the SNARC
effects on RT were similar when participants were instructed to use imagery strategies and
when they were not. Therefore, the stronger SNARC effect found in experiments involving
the explicit use of imagery strategies can be parsimoniously attributed to the slower reaction
times required to implement the strategy of using a mental image according to the given
instructions. These results suggest that the use of imagery demands more effort and takes
longer to be processed but they do not indicate a qualitative change in the association be-
tween the SNARC effect and response latencies. Regardless of the preferred interpretation of
the effect of mental imagery on the size of the SNARC effect, one may conclude that the
impact of these instructions is restricted to slowing down the response latencies and has no
impact on the association between the SNARC effect and response latencies. These results
can be interpreted in two ways: the first one implies that coordinate spatial relations are
activated in tasks explicitly requiring the use of imagery strategies as well as in neutral tasks.
In this case, participants may have used mental imagery resources in an implicit way even in
those experiments not requiring this explicitly. Moreover, it is also possible that both coordi-
nate and categorical spatial relations are associated to number magnitude to the same extent.
In this case the slope of size of the SNARC effect on RT should not differ between tasks
depending on imagery instructions. Unfortunately we cannot decide between these two ex-
planations based on our meta-analytical data alone. Further empirical studies addressing this
point are necessary for disclosing the role of categorical and coordinate spatial relations on
the association between numbers and space. Finally, due to the small number of studies
examined in this analysis, the negative results obtained should be interpreted with some
caution.
4.4 Response discrimination
A robust SNARC effect was found for almost all response modalities which discriminate
responses (Gevers et al., 2005b). The only dimensions which were not associated with the
SNARC effect were naming latencies and saccadic amplitudes. The results from naming
experiments are difficult to interpret because of massive differences in the stimulus, popula-
tions and tasks employed. For this reason one may refrain to discuss them. In contrary, the
null effect observed for saccade amplitudes can be accounted by response discrimination
(Ansorge and Wühr, 2004).
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
510
4.5 Age
For the first time we have disclosed a robust association between size of SNARC and
age. Recently an empirical study confirmed the present finding in a large sample of indi-
viduals with ages between 9 and 79 years (Wood, et al., submitted). The SNARC effect was
present in every age group, but was stronger in older participants. The association of larger
(smaller) numbers and the right (left) hemi-space seems to be a habit acquired in and rein-
forced by the cultural environment and the linguistic background (Dehaene et al., 1993,
Hung et al., 2007; Zebian, 2005). Although 9 years old children already show a significant
SNARC effect, it is plausible that the amount of practice accumulated until they become
adult increases. Interestingly, the SNARC effect further increases in adulthood and is largest
in elderly. Therefore, it is possible that the habit of associating numbers and space becomes
stronger along the whole life span. In line with this view, Velanova et al., (2006) report that
older participants perform cognitive tasks in a less flexible fashion than younger participants
and that their performance may depend more on well-learned habits. Therefore, it is possible
that the habit of associating numbers and space consolidate along the life span. This can
explain the data observed in this review.
An increase in the SNARC effect could still occur since older adults are less efficient in
suppressing task-irrelevant information, thus leading to more interference of the automatic
association on performance than younger adults. Supporting evidence for this claim comes
from studies reporting a decrease in inhibitory control in elderly in comparison with younger
adults (Hasher and Zacks, 1988; Lustig et al., in press; Martin et al., 2006) and from priming
studies showing that the performance of elderly but not of younger participants is facilitated
by irrelevant priming information (Kim et al., in press).
Therefore, the present empirical evidence lends stronger support to the lifelong devel-
opment of the habit of associating numbers with specific regions of space and possibly also
to a reduction of inhibitory capacity as the cause of the increase in the size of the SNARC
effect. In summary, although the association between age and the SNARC effect reported
here was highly dependent on the few studies including older participants, the results re-
ported by Wood et al. (submitted) allow us the conclusion that an effect of age on the
SNARC effect is robust.
4.6 The dual-route model of the SNARC effect
The dual-route model is a powerful cognitive framework which can account for a large
diversity of features of the SNARC effect, such as the association between response speed
and accuracy and the size of the SNARC effect and its shape in magnitude comparison tasks,
as well as the association between the Simon and the SNARC effect (Gevers et al. 2005a).
The Simon effect describes an automatic trend to respond in the direction of stimuli (Simon,
1969), and was shown to interact with the SNARC effect (Gevers et al. 2005a, Keus and
Schwarz, 2005). In the present section we will discuss some challenging aspects of present
meta-analysis of the SNARC effect which may lead to constraints helpful for further devel-
opments of the dual-route model by Gevers et al. (2006b).
One of the most interesting new developments is the finding that the SNARC effect is
associated with the different spatial dimensions of the task that discriminate between the
A meta-analysis of SNARC 511
different responses (Gevers et al. 2005b). Another important development are studies show-
ing that the SNARC effect may be associated with hand-based and space-based frames of
reference (Müller and Schwarz, 2006; Wood et al. 2006a, b; but see also Fischer, 2006).
Together, these pieces of evidence suggest that saliency of stimuli and responses, as well as
strategic components may be important determinants of the SNARC effect.
Solid evidence has been presented supporting the view that the SNARC effect appears at
the stage of response selection (Daar and Pratt, 2008, Keus, Jenks and Schwarz, 2005, Keus
and Schwarz, 2005). Nonetheless, a few studies now suggest that this may not be the whole
story about the association between numbers and space. Fischer et al. (2003) found a
SNARC effect in a detection task not involving response selection at all (Salillas et al.,
2008). This result suggests that the computation of congruity between spatial and numerical
features characteristic of the SNARC effect may not be restricted to the stage of response
selection – as typically the case with the SNARC effect – but may extend to earlier cognitive
processing stages.
In summary, the present version of the dual-route model captures the most important fea-
tures of the SNARC effect with regard to spatial response selection. However, in our view,
the existing data indicate that potentially a component should be added to the model to ac-
count for SNARC effect at cognitive stages earlier than the response selection stage. Fur-
thermore, the saliency of different spatial features of experimental setups and for the strate-
gic use of space to discriminate magnitudes (Wood et al., submitted) should be considered.
4.7 The polarity correspondence account of the SNARC effect vs. the mental
number line
In our view, the data suggest that currently there is no need for a decision between the
polarity correspondence account and the mental number line account; rather the contribution
of both of them to the SNARC effect should be identified. To distinguish between the con-
tributions of coordinated and categorical spatial relations for the SNARC effect is difficult
and still an under explored terrain. Therefore, it is in our view premature to reject the num-
ber line account at this point as contributions of coordinated as well as categorical spatial
relations have been repeatedly presented in the literature. While in most tasks involving
speedy binary classification the polarity correspondence framework can account for more
data than the mental number line, in the context of neuropsychological research very clear
evidence for the use of a mental image is available (Hoeckner et al., 2008; Vuilleumier et al.,
2004; Zorzi et al., 2002, 2006). The results from these studies suggest that both coordinated
and categorical spatial relations may be important for the SNARC effect, in particular, and
for the association between numbers and space, in general.
5. Conclusions
Overall, it is clear that the strength of magnitude activation is a pervasive determinant of
the SNARC effect. Magnitude activation influences the SNARC effect through the response
latencies specific for different experimental designs as well as through task demands which
may require more or less magnitude activation. The impact of imagery instructions is re-
G. Wood, K. Willmes, H.-C. Nuerk & M. H. Fischer
512
stricted to slowing down response latencies and seems to have no impact on the quality of
spatial representations recruited. Moreover, an important effect of age on the SNARC effect
was found. Two possible, not mutually exclusive origins for this effect are the development
of the habit of associating numbers with specific regions of space and, especially in elderly,
the capacity to inhibit it. The association of number magnitude and space has been shown to
be very flexible and reflects the activation of a more abstract magnitude representation.
Recent studies present evidence that counting habits are relevant for the SNARC effect
(Fischer, 2008, see also Sato and Lalain, 2008). Future research must explore whether the
polarity account might suffice to explain the occurrence of the SNARC effect or whether the
number line account is still needed as suggested by recent neuropsychological (neglect)
studies.
In sum, this meta-analysis has suggested some determinants of the SNARC effect which
have previously been not systematically investigated in one controlled study, such as the
effect of lifetime development on the SNARC effect. We hope that some of the questions
and hypotheses raised by the present meta-analysis will be experimentally addressed in the
future.
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Acknowledgments
The authors are indebted to Dr. Anton Kuehberger and Matthias Unterhuber for their
help with the choice of the meta-analytic methods for this manuscript. GW received travel
support from the Experimental Psychology Society (UK) and from the University of Salz-
burg.
Appendix I
The following letters were used to characterize the studies included in each one of the
different analyses conducted in the present meta-analysis.
A: Overall SNARC slope
B: Parity decision task
C: Magnitude classification task
D: Magnitude comparison task
E: Non-semantic tasks
F: Non-semantic tasks without number words
G: Arabic numbers
H: Number words in semantic tasks
I: Number words in non-semantic tasks
J: Auditorily presented number words
K: Hands in non-crossed position
L: Hands in crossed position
M: Hands in crossed position without the study by Wood et al., 2006
N: Bimanual response
O: Eye saccade latencies
P: Eye saccade amplitudes
R: Manual bisection
S: Pointing RT
T: Pointing MT
U: Unimanual finger response
V: Naming
W: Bipedal response
X: Grip aperture
Shape: categorical vs. continuous shape of the SNARC effect
Size: Sample size
Age: Effect of age
RT: average RT
Errors: average error rates
Hand: Effect of the proportion of right-handed participants
Gender: Effect of gender
Language: Effect of language
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... This effect reflects an association between small numbers and the left-hand side and large numbers and the right-hand side (Dehaene et al., 1993). The SNARC effect is commonly seen as support for the concept of a mental number line (MNL) oriented from left to right or right to left (Fabbri & Guarini, 2016;Wood et al., 2008). In addition, the above-mentioned SNARC effect has been observed for the vertical axis with a preference for large numbers at the top and small numbers at the bottom (Aleotti et al., 2020;Cappelletti et al., 2007;Fabbri, 2011Fabbri, , 2013Gevers et al., 2006;Hesse & Bremmer, 2017;Holmes & Lourenco, 2011, 2012Ito & Hatta, 2004;Müller & Schwarz, 2007;Schwarz & Keus, 2004;Shaki & Fischer, 2012;Sixtus et al., 2019;Winter & Matlock, 2013;Winter, Marghetis, et al., 2015;. ...
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... Wendt et al. (2013; for similar results see also Pfister et al. 2013) found a reduced SNARC effect in switch trials. The SNARC effect reflects faster responses to small numbers with the left hand and large numbers with their right hand (Dehaene et al. 1993;Wood et al., 2008). Wendt and colleagues suggested that the reduced SNARC effect might be accounted for by additional control processes due to increase between-task interference in switch trials. ...
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A growing body of research suggests that basic numerical abilities such as number magnitude processing are influenced by cognitive control processes. So far, evidence for number processing being affected by cognitive control processes stems primarily from observed adaptations of numerical effects to stimulus set characteristics (e.g. order or ratio of specific stimulus types). Complementing previous research on adaptation to stimulus set characteristics as an index of influences of cognitive control, the present study employed a task-switching paradigm to examine how cognitive control processes influence number processing. Participants were presented with a two-digit number and had to either judge its parity or compare its magnitude to a standard depending on a preceding cue. We expected numerical congruency effects (i.e. the unit-decade compatibility effect for magnitude comparisons and the parity congruity effect for parity judgements) to be larger in switch trials, as persisting activation of the task set of the preceding trial should increase interference. In contrast to our expectations, both numerical congruity effects were reduced following task switches as compared to repetitions. This interaction of task-switching with numerical congruency effects suggests an influence of cognitive control on basic number processing in form of persisting inhibition of previously abandoned task sets, so that these exert less influence on current number processing demands.
... Here we focus on its subtype, directional SNAs: phenomena revealing the links between numbers and directions in space. Among them, the SNARC effect (Spatial-Numerical Association of Response Codes; Dehaene, Bossini, & Giraux, 1993) is the most thoroughly studied (see Wood, Willmes, Nuerk, & Fischer, 2008 for a meta-analysis, and Cipora, Soltanlou, Reips, & Nuerk, 2019 for a large-scale online replication). In speeded bimanual setups (e.g., parity judgment task), faster reactions are observed to small/large numbers with left/right hand, respectively. ...
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The Spatial-Numerical Association of Response Codes (SNARC) effect (i.e., faster left/right side responses to small/large magnitude numbers, respectively) is considered as strong evidence for the link between numbers and space. The studies have shown considerable variation in this effect. Among the factors determining individual differences in the SNARC effect is the hand an individual uses to start the finger counting sequence. Left-starters show a stronger and less variable SNARC effect than right-starters. This observation has been used as an argument for the embodied nature of the SNARC effect. For this to be the case, one must assume that the finger counting sequence (especially the starting hand) is stable over time. Subsequent studies challenged the view that the SNARC differs depending on the finger counting starting hand. At the same time, it has been pointed out that the temporal stability of finger counting starting hand should not be taken for granted. Thus, in this preregistered study, we aimed to replicate the difference in the SNARC between left- and right-starters and explore the relationship between the temporal stability of finger counting starting hand and the SNARC effect. We expected that higher stability should be associated with a stronger SNARC effect. Results of the preregistered analysis did not show the difference between left- and right-starters. However, further exploratory analysis provided weak evidence that this might be the case. Lastly, we found no evidence for the relationship between finger counting starting hand stability and the SNARC effect. Overall, these results challenge the view on the embodied nature of the SNARC effect.
... Numerical cognition research shows that small numbers are associated with the left peripersonal space, and large numbers are associated with the right peripersonal space (Spatial Numerical Association of Response Codes, or SNARC effect, see Dehaene et al., 1993). This effect has been demonstrated with button press responses, finger movements (Fischer, 2003), eye movements (Myachykov et al., 2016), foot responses (Schwarz & Müller, 2006), and even full-body movements (see for reviews Fischer & Shaki, 2014;Toomarian & Hubbard, 2018; see for meta-analysis Wood et al., 2008). Despite this overwhelming evidence for the presence of spatial-numerical associations, we found no SNARC effect in grip force (Miklashevsky et al., 2021), either because no attentional shifts appeared without explicit motor responses to numeric stimuli (cf. ...
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Previous research demonstrated a close bidirectional relationship between spatial attention and the manual motor system. However, it is unclear whether an explicit hand movement is necessary for this relationship to appear. A novel method with high temporal resolution – bimanual grip force registration – sheds light on this issue. Participants held two grip force sensors while being presented with lateralized stimuli (exogenous attentional shifts, Experiment 1), left- or right-pointing central arrows (endogenous attentional shifts, Experiment 2), or the words "left" or "right" (endogenous attentional shifts, Experiment 3). There was an early interaction between the presentation side or arrow direction and grip force: lateralized objects and central arrows led to an increase of the ipsilateral force and a decrease of the contralateral force. Surprisingly, words led to the opposite pattern: increased force in the contralateral hand and decreased force in the ipsilateral hand. The effect was stronger and appeared earlier for lateralized objects (60 ms after stimulus presentation) than for arrows (100 ms) or words (250 ms). Thus, processing visuospatial information automatically activates the manual motor system, but the timing and direction of this effect vary depending on the type of stimulus.
... About thirty years ago, Dehaene, Bossini, and Giraux (1993) showed that participants executing parity judgments responded faster with their left hand to small numbers and with the right hand to larger numbers. This effect is known as SNARC (Spatial Numerical Association of Response Codes; for reviews, see Wood, Willmes, Nuerk, & Fischer, 2008;Toomarian & Hubbard, 2018) and it has been taken as evidence for the human natural tendency to spatialize numbers and numerical magnitudes. The SNARC effect is considered to reflect an analogue, leftto-right oriented internal representation for number magnitudes, i.e. a mental number line (MNL; Restle, 1970), though this interpretation has been debated and alternative accounts have been proposed, based on working memory (Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006) or polarity correspondence (Proctor & Cho, 2006). ...
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... Two additional aspects of our findings are worth noting: RTs were significantly faster, and hand/side biases were generally weaker, on small and cross-range pairs as compared to large pairs. These outcomes are consistent with previous findings showing a link between slower RTs and stronger SNARC effects [78,79]. The faster RTs observed on small and cross-range pairs in this study are attributable to the rapid enumeration process associated with the subitizing range, which generally refers to numerosities 1-4 and is not culture-specific [16,80,81]. ...
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Directional response biases due to a conceptual link between space and number, such as a left-to-right hand bias for increasing numerical magnitude, are known as the SNARC (Spatial-Numerical Association of Response Codes) effect. We investigated how the SNARC effect for numerosities would be influenced by reading-writing direction, task instructions, and ambient visual environment in four literate populations exemplifying opposite reading-writing cultures -- namely, Arabic (right-to-left script) and English (left-to-right script). Monoliterates and biliterates in Jordan and the U.S. completed a speeded numerosity comparison task to assess the directionality and magnitude of a SNARC effect in their numerosity processing. Monoliterates' results replicated previously documented effects of reading-writing direction and task instructions: the SNARC effect found in left-to-right readers was weakened in right-to-left readers, and the left-to-right group exhibited a task-dependency effect (SNARC effect in the smaller condition, reverse SNARC effect in the larger condition). Biliterates' results did not show a clear effect of environment; instead, both biliterate groups resembled English monoliterates in showing a left-to-right, task-dependent SNARC effect, albeit weaker than English monoliterates'. The absence of significant biases in all Arabic-reading groups (biliterates and Arabic monoliterates) points to a potential conflict between distinct spatial-numerical mapping codes. This view is explained in terms of the proposed Multiple Competing Codes Theory (MCCT), which posits three distinct spatial-numerical mapping codes (innate, cardinal, ordinal) during numerical processing -- each involved at varying levels depending on individual and task factors.
... First systematic evidence for a left-to-right oriented association of number and physical space came from the Spatial Numerical Association of Response Codes (SNARC) effect (Dehaene et al., 1993; see also Wood et al., 2008, for a metaanalysis). This effect describes the phenomenon that in Western cultures, partici-pants tend to react faster to smaller numbers with their left hand and to larger numbers with their right hand (Dehaene et al., 1993). ...
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SNARC effect refers to space-number association, which has been observed in recent years. Various studies have shown that people associate the left and right side with relatively small and big numbers respectively. Traditionally, it was assumed that the observed effect is caused by the existence of the mental number line stored in long term memory. However, later research, which explored how the manipulation of different parameters influences the SNARC effect, suggests that working memory is also responsible for the phenomenon. This chapter presents a review of the most recent studies on the influence of long-term and working memory on the SNARC effect.
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Abstract The SNARC effect specifically relates small magnitudes,to the,left hand,side and larger magnitudes to the right hand side (e.g. Dehaene et al., 1990; Dehaene et al., 1993). It is certain that cultural characteristics define the SNARC effect: In western,cultures small and large numbers,are coded,in a,left-right direction while in Arabic countries magnitude,infor- mation is coded from right to left (Dehaene et al., 1993; Zebian, in press). In this sense, reading,and ,writing direction have ,been ,considered ,to be ,the main ,determinants ,of the SNARC effect. Indeed, a number of recent studies support the idea that the mastering of a language,(and thus reading and writing direction) biases scanning,habit in a,favourable,di- rection (e.g. Chatterjee et al., 1999; Padakanaya et al., 2002). If this is indeed the case, it is not surprising that the SNARC effect has been ,found ,with stimuli other than numbers (Gevers et al., 2004). Related to this, future research could address the question if magnitude and ordinal information ,are processed ,by the ,same ,mechanism,or by ,different mechanisms with similar properties. However, reading and writing direction alone are not sufficient to explain all SNARC related findings. For instance, it does not allow for an explanation of a vertical SNARC effect as observed by Schwarz and Keus (2004). Apparently, other variables are important in the onset of an association between numbers and space. Furthermore, the existence of a ,SNARC effect in the ,vertical dimension ,could indicate that the number ,line can be conceptually extended,to a number,map. Another puzzling,question is to what,reference frame the SNARC effect is based. Based onthe finding of a SNARC effect with both unimanual (Fischer, 2003a) and oculomotor responses (Fischer et al., 2004; Schwarz & Keus, 2004) at this moment an egocentric alloca- tion of the SNARC effect does not seem,feasible. Onthe other hand, both behavioural (e.g. Bächtold et al., 1998) and neuropsychological studies (Veuilleumier et al., 2004) point out that the mental representation of numerical information,in itself is of major importance,to the SNARC effect. ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
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