Cognition 118 (2011) 97–113
Contents lists available at ScienceDirect
Cognition
journal homepage: www.elsevier.com/locate/COGNIT
The size congruity effect: Is bigger always more?
Seppe Santens ⇑, Tom Verguts
Ghent University, Department of Experimental Psychology, H. Dunantlaan 2, B-9000 Ghent, Belgium
a r t i c l e
i n f o
Article history:
Received 1 October 2009
Revised 19 October 2010
Accepted 20 October 2010
Keywords:
Size congruity
Numerical Stroop
Numerical cognition
Number processing
a b s t r a c t
When comparing digits of different physical sizes, numerical and physical size interact. For
example, in a numerical comparison task, people are faster to compare two digits when
their numerical size (the relevant dimension) and physical size (the irrelevant dimension)
are congruent than when they are incongruent. Two main accounts have been put forward
to explain this size congruity effect. According to the shared representation account, both
numerical and physical size are mapped onto a shared analog magnitude representation.
In contrast, the shared decisions account assumes that numerical size and physical size
are initially processed separately, but interact at the decision level. We implement the
shared decisions account in a computational model with a dual route framework and show
that this model can simulate the modulation of the size congruity effect by numerical and
physical distance. Using other tasks than comparison, we show that the model can simulate novel findings that cannot be explained by the shared representation account.
! 2010 Elsevier B.V. All rights reserved.
1. Introduction
In order to act adaptively in a constantly changing environment, animals need to keep track of different quantities, including time, length, size, area, and volume. In
recent evolutionary history, also abstract quantities (numbers) have been added to this list. Research in different disciplines has recently focused on understanding the
cognitive foundations of these different quantities. A still
unresolved issue, however, concerns the relation between
them (for reviews, see Cantlon, Platt, and Brannon
(2009), Cohen Kadosh, Lammertyn, and Izard (2008)).
An effect that plays a pivotal role in this discussion is
the size congruity effect (Besner & Coltheart, 1979; Henik
& Tzelgov, 1982). In the original size congruity paradigm
(Besner & Coltheart, 1979), two numbers are presented in
different physical sizes, and the participant’s task is to
choose the numerically larger (or smaller) stimulus. The
size congruity effect entails that it is easier to pick the
numerically larger (or smaller) of two digits when this is
⇑ Corresponding author.
E-mail address: [email protected] (S. Santens).
0010-0277/$ - see front matter ! 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.cognition.2010.10.014
also the physically larger (smaller) stimulus. Similarly,
when the task is to pick the physically larger (or smaller)
of the two digits, the task is easier if the physically larger
stimulus is also numerically larger (Henik & Tzelgov,
1982). Size congruity effects are also found when only
one stimulus is presented per trial. In this paradigm, the
task is to judge whether the presented digit is larger or
smaller than a fixed standard (e.g., standard 5) (Schwarz
& Heinze, 1998; Schwarz & Ischebeck, 2003). The current
paper addresses the origins of this effect using both computational and behavioral studies.
The size congruity effect shows that numerical and
physical (size) dimensions are not processed in complete
independence. An interaction between different quantities
must be assumed somewhere along the processing stream.
This interaction can take place at the input, representation,
decision, or output level (Verguts & Fias, 2008). It is still debated at which of these levels the size congruity effect
originates. In the literature, two opposing accounts have
been put forward (e.g., Cohen Kadosh et al., 2007; Schwarz
& Heinze, 1998). Schwarz and Heinze (1998) have clearly
identified both positions, so we take their definitions as
examples. In the first account, the size congruity effect is
taken as evidence for representational overlap between
98
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
the dimensions of number and physical size. According to
this account, it is the case that ‘‘[. . .] both, the digit’s physical size and its numerical value are first mapped onto an
integrated internal analog representation, which is then
processed further to activate the appropriate response’’
(Schwarz & Heinze, 1998, p. 1168). We will call this the
shared representation account (Fig. 1A). In this account, size
congruity originates at the representational level, i.e. a level where numerical and physical size would be jointly
represented in an analog format. A stimulus configuration
in which numerical size correlates positively with physical
size is congruent (e.g., in the stimulus configuration 2 8,
the numerical and physical size are both small for the left
stimulus and both large for the right stimulus). A stimulus
configuration in which the two dimensions correlate negatively, is incongruent (e.g., in the stimulus configuration
8 2). An instance of this shared representation theory is
Walsh’s influential theory of magnitude (ATOM), according
to which there are common metrics for different quantitative dimensions, including space, time, and number (Bueti
& Walsh, 2009; Walsh, 2003). Many recent studies on congruity effects between numerical size and other quantities
have been interpreted in terms of ATOM (e.g., Chiou,
Chang, Tzeng, & Wu, 2009; Cohen Kadosh, Cohen Kadosh,
& Henik, 2008; Cohen Kadosh & Henik, 2006; Cohen
Kadosh et al., 2007; Kaufmann et al., 2005; Pinel, Piazza,
Le Bihan, & Dehaene, 2004; Xuan, Zhang, He, & Chen,
2007). A theoretical problem with this proposal, and with
the shared representation account more generally, is that
the common metric for the shared dimensions has remained unspecified. Put simply, it is unclear whether 1 s
(quantity in the time dimension) corresponds to 1 cm or
1 m (quantity in the length dimension); or to 1 g or 1 kg
(quantity in the weight dimension); and so on. One exception is the proposal by Meck and Church (1983) for counting and time representations in rats. In their mode-control
model, both counting and timing rely on a pacemaker that
sends pulses to an accumulator. Depending on the task, the
accumulator fills up to represent either time or the number
of discrete elements (i.e., counting). They even proposed
that a count corresponds to a passage of time of 200 ms
(Meck & Church, Experiment 4). However, it is unclear
how general this count-to-time mapping is. Furthermore,
the model cannot easily be extended to other quantitative
dimensions.
The second account for explaining the size congruity effect suggests that it results from the fact that, although
numerical and physical size are initially processed separately, the two separate processing pathways interact at
the decision level. According to this second account it is
the case that ‘‘[. . .] size and numerical information are first
processed in parallel, functionally independent channels
and that both of them can activate a specific ‘‘subresponse’’
’’ (Schwarz & Heinze, 1998, p. 1168). We can think of these
subresponses as the decision alternatives that are imposed
by the task and we therefore call this account the shared
decisions account (Fig. 1B). In a standard size congruity
paradigm, the decision alternatives (e.g. left larger or right
larger) have a one-to-one mapping to the responses (e.g.
left hand or right hand). Therefore, dissociating between
the decision level and the output level is neither possible
nor relevant here. Importantly, the shared decisions
account assumes that the size congruity effect is due to
the fact that the two dimensions activate the same (in
the congruent case) or different (in the incongruent case)
task-relevant, discrete codes. Hence, according to this
account, congruity can be defined only relative to the task
at hand. For example, in a magnitude comparison task (but
not necessarily in other tasks), the stimulus configuration
2 8 is congruent because the numerical and physical size
both activate the right larger code. In contrast, the stimulus
configuration 8 2 is incongruent because the numerical
dimension activates the left larger code, whereas the physical dimension activates the right larger code. In a sense,
the activation of the decision units also establishes a representation, namely of the decision that is made. This could
cause confusion about what is exactly meant by a shared
representation account. However, the definitions of
Schwarz and Heinze (1998) are clear: In the shared representation account, the size congruity effect originates at
the level where numerical and physical magnitude are
jointly coded in an analog format. This information about
Fig. 1. Schematic illustration of the shared representation account (A) and the shared decisions account (B). The origin of the size congruity effect according
to each account is marked.
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
numerical and physical size is lost at the decision level. We
reserve the term ‘‘shared representation’’ for a representation that codes numerical and physical size in a shared
analog format, before task-specific operations are applied
to it.
Although many studies have discussed the interaction
between numerical and physical size (e.g. Algom, Dekel,
& Pansky, 1996; Ansari, Fugelsang, Dhital, & Venkatraman,
2006; Henik & Tzelgov, 1982; Kaufmann et al., 2005; Pinel
et al., 2004; Schwarz & Ischebeck, 2003), few have been designed to directly oppose the shared representation and
the shared decisions account. Schwarz and Heinze (1998)
recorded event-related potentials (ERPs) from six electrodes to determine the locus of the size congruity effect.
In their results, the first reliable effects of congruity at
the frontal electrodes occur at 280 ms after stimulus presentation for the physical comparison task and at 368 ms
for the numerical comparison task. The authors interpreted
this finding as evidence for early interactions and hence for
a shared representation. They also examined the lateralized readiness potential (LRP), an ERP component that is
thought to reflect motor preparation and execution (Coles,
1989; Hackley & Miller, 1995; Masaki, Wild-Wall, Sangals,
& Sommer, 2004). An initial deflection of the LRP towards
the incorrect response in incongruent trials compared to
congruent trials is considered as evidence for response
competition between the correct response activated by
the relevant dimension and the incorrect response automatically activated by the irrelevant dimension (e.g.,
Gevers, Ratinckx, De Baene, & Fias, 2006). Schwarz and
Heinze (1998) interpreted the absence of such a deflection
in their data as additional support for a shared representation. However, as also argued by Szücs and Soltész (2008),
the absence of an initial LRP deflection cannot be taken as
evidence in favor of a shared representation. If the irrelevant dimension is processed more slowly than the relevant
dimension (e.g. because the relevant dimension is emphasized by task instructions), then it is possible that the irrelevant dimension does not cause an initial LRP deflection
and yet slows down selecting the correct response. Szücs
and Soltész (2008) also recorded ERPs in a size congruity
paradigm. They found both early (between 150 and
250 ms after stimulus presentation) and late (between
300 and 430 ms) facilitation and interference effects of
numerical and physical dimensions. They suggested that
the size congruity effect appears at multiple levels of stimulus processing and response preparation. Cohen Kadosh
et al. (2007) combined ERP measures with functional magnetic resonance imaging (fMRI) in both a numerical and a
physical comparison task. A region of interest analysis of
the fMRI data from the primary motor cortex showed an
interference effect in the hemisphere ipsilateral to the
hand for the correct response. It was concluded that congruity is not completely resolved until response initiation.
In their ERP analyses, the authors found an initial deflection of the LRP when numerical distance was large, but
not when numerical distance was small. It was argued that
response competition plays a role only when cognitive load
is low. Under a high cognitive load, a shared representation
would cause the congruity effect. However, as discussed
above, the absence of an LRP effect does not justify this
99
conclusion. A more general problem with the available
ERP evidence is that there is no consensus on what constitutes an ‘‘early’’ or ‘‘late’’ ERP component. For example,
both Schwarz and Heinze (1998) and Cohen Kadosh et al.
(2007) categorize the congruity effect around 300 ms after
stimulus onset as early, whereas Szücs and Soltész (2008)
categorize this as a late component.
In sum, the available behavioral and neuroimaging results are not conclusive in dissociating between the shared
representation and shared decisions accounts. A crucial
point that blurs the discussion is the fact that both accounts so far have remained underspecified. Only when
both are specified in sufficient detail will it be possible to
resolve this debate. As mentioned above, specifying the
shared representation account has an inherent indeterminacy problem (what is the common metric for quantities?).
On the other hand, the shared decisions account can be
straightforwardly modeled with a dual route architecture.
This approach has proven successful in congruity
paradigms similar to numerical and physical size congruity. Cohen, Dunbar, and McClelland (1990) for example
simulated the well-known Stroop effect (Stroop, 1935)
with a computational model implementing a dual route
architecture. In their model, an ‘‘ink color’’ and a ‘‘word’’
route both activate one of the two possible responses
(‘‘red’’ or ‘‘green’’). An attentional bias strengthens the
task-relevant route (i.e. ‘‘ink color’’ in a classic Stroop task)
to ensure that the correct response is selected (e.g.
response ‘‘red’’ when the word the ink color is ‘‘red’’).
The automatically activated task-irrelevant route (‘‘word’’)
causes facilitation in congruent trials (e.g. the word ‘‘red’’
printed in red) by activating the correct response unit
and it causes interference in incongruent trials (e.g. the
word ‘‘green’’ printed in red) by activating the incorrect
response unit. Later, also other congruity effects were
successfully modeled with a dual route architecture, including the Eriksen flanker effect (Cohen, Servan-Schreiber, &
McClelland, 1992; Yeung, Botvinick, & Cohen, 2004), the
Simon effect (Zorzi & Umiltà, 1995) and the SNARC effect
(Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006).
In the current study, we took a similar approach to
model the size congruity effect. For this purpose, we
started from our earlier computational model of number
processing (Verguts, Fias, & Stevens, 2005). This model
simulates behavioral performance in a number of tasks,
but here we focus on number comparison. The model uses
place coding (i.e. number-selective coding) to represent
numerical magnitude on two input layers, one for each of
the to be compared numbers. These input layers then activate the output layer, which codes for the decision that has
to be made. If the instruction is to choose the larger number, the units of the output layer code for ‘‘left digit is larger’’ and ‘‘right digit is larger’’ (if the responses are
arranged left and right). After training the model on the
comparison task, the pattern of weights between the input
and output layers produces a comparison distance effect
(for details, see Verguts et al., 2005). Here, we implement
only the comparison mechanism of the Verguts, Fias, and
Stevens model in a simplified way. To simulate the size
congruity paradigm, we add to this a similar mechanism
for physical size comparison. Like the model of Cohen
100
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
et al. (1990), the task-relevant route (e.g. numerical size
comparison) and the task-irrelevant route (e.g. physical
size comparison) interact only at the decision level. At this
level, the units that code for the decision alternatives (e.g.
‘‘left digit is larger’’ and ‘‘right digit is larger’’) are shared
between the numerical and the physical dimension. Hence,
the resulting model is an instance of the shared decisions
account. Notably, the model is always trained on numerical
and physical size comparison separately. Only in the test
phase, the model is confronted with the size congruity paradigm. The distinction between the relevant dimension
and the irrelevant dimension is made by simply attenuating the activation from the irrelevant dimension (for details, see Appendix B). With four behavioral experiments
and four simulations, we show that the principle of physical and numerical routes operating independently but converging at the decision level can account for many aspects
of the size congruity effect. In Experiment 1, we first replicate the typically observed interactions of numerical and
physical distance with size congruity and test whether
the shared decisions model can simulate these effects. In
Experiments 2–4, we directly oppose the two accounts.
In Experiment 2, it is tested whether a size congruity effect
is present in both a magnitude judgment task and a parity
judgment task because only the shared representation account predicts that it will be observed in both tasks. We
further investigate the influence of the task on the size
congruity effect in Experiment 3. Here, we use a close/far
task (Santens & Gevers, 2008) to test whether modulations
of the size congruity effect follow the predictions of the
shared representation account or of the shared decisions
account. In the first three experiments, numerical size is
the relevant dimension and physical size the irrelevant
dimension. In Experiment 4, we investigate whether the
conclusions based on Experiment 3 still hold when changing the relevant and the irrelevant dimension. We use the
same close/far task as in Experiment 3, but now with physical size as the relevant dimension and numerical size as
the irrelevant dimension.
2. Experiment 1
The size congruity effect can be affected by manipulating the discriminability between the values on the relevant
and irrelevant dimension (Pansky & Algom, 1999). Therefore, we calibrated the physical sizes of the stimuli used
in this study in order to guarantee equal discriminability
on the two dimensions. In Experiment 1, this was done separately for each participant on the first day of the experiment (see Appendix A). On the second day, we used the
calibrated stimuli in a standard size congruity paradigm.
2.1. Method
2.1.1. Participants
Eighteen students from Ghent University were paid to
participate in an experiment of two times 1 h (for the calibration phase and the experimental phase, respectively),
spread over two days within the same week. 14
participants were female, all but one were right-handed.
Their mean age was 21 years (range: 18–24 years). All participants had normal or corrected to normal vision. They
signed an informed consent prior to the experiment.
2.1.2. Stimuli and apparatus
The experimental procedure was implemented using
the Tscope library for the C programming language (Stevens, Lammertyn, Verbruggen, & Vandierendonck, 2006).
Button presses were registered using a response box (RB730 with modified buttons, Cedrus Corp., San Pedro, CA).
Stimulus pairs were presented on a 17 in. screen, white
on a black background. The viewing distance was approximately 50 cm. Stimuli consisted of the Arabic numerals
1, 2, 7 and 8, printed in the Courier New font. These digits
appeared in 4 different heights (a < b < g < h), determined
in the calibration phase (see Appendix A). The mean
heights in visual degrees for a, b, g and h were about 2.7",
5.4", 9.4", and 13.3", respectively. On each trial, two stimuli
were presented, centered around the vertical midpoint and
7.4 visual degrees left and right from the fixation point,
respectively.
2.1.3. Experimental procedure
Participants had to perform a number comparison task
on digits that varied in numerical and physical size. They
were asked to ignore the physical size of the digits. The
specific task instruction (‘‘choose the smaller’’ or ‘‘choose
the larger’’) was the same as in the calibration phase and
thus counterbalanced between subjects. A trial started
with the presentation of a small square in the center of
the screen that served as the fixation mark. There was a
variable time interval before the stimulus appeared, with
a duration randomly drawn from a uniform distribution
with a minimum of 500 ms and a maximum of 1000 ms.
After this interval, the stimuli appeared, one on the right
and one on the left. There was a response deadline of
1500 ms. When the response was given or when 1500 ms
had elapsed, there was a blank screen for 500 ms until
the next trial started. All combinations of numerical (1, 2,
7, 8) and physical sizes (a, b, g, h) were presented, except
for the combinations in which the numerical or the physical size was equal for the left and right stimulus. In this
way, 12 ! 12 = 144 different stimulus pairs could be constructed. Each pair was presented once in each of the eight
blocks in the experiment. This gave a total of 1152 trials.
Between blocks, participants could take a short pause.
2.2. Results and discussion
All correct trials entered the RT analyses. The results of
these analyses are presented in Fig. 2. Mean RTs were calculated for each condition, collapsing over response
instructions (choose the numerically larger or smaller).
An ANOVA was run with a 2 (congruity: congruent or
incongruent) ! 2 (distance on the (relevant) numerical
dimension: small or large) ! 2 (distance on the (irrelevant)
physical dimension: small or large) design with all factors
treated as within subjects variables. To obtain an equal
number of observations at each level of numerical distance, the two smallest distances (distances 1 and 5) were
treated as small and the two largest distances (distances 6
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
101
Fig. 2. Experiment 1: observed (panels A and C) and simulated (panels B and D) RT. Panels A and B: congruity plotted against distance on the relevant
dimension. Panels C and D: congruity plotted against distance on the irrelevant dimension (Error bars represent 95% confidence intervals based on the
guidelines by Loftus and Masson (1994). Error bars for the model data were too small to depict. To aid visual inspection of the effect sizes, the model data of
all simulations are presented on the same scale and so are the behavioral data of all experiments. The range of values on the Y axis is the same for all
simulations, whereas it changes between experiments for the behavioral data. Note that the data points in these graphs are marginal means. Therefore the
triple interaction cannot be evaluated from this figure).
and 7) were treated as large. A similar categorization was
used for physical distance: Small distances were |a–b|,
|b–g| and |g–h|; large distances were |a–g|, |a–h|, and |b–
h|. There was a main effect of congruity [F(1, 17) = 45.42;
p < .001; g2p ¼ :73]: Participants were faster to pick the
numerically larger or smaller of two stimuli when their
numerical and physical size were congruent than when
they were incongruent. The main effect of numerical distance was also significant [F(1, 17) = 188.27; p < .001;
g2p ¼ :92]: When the numerical distance was large, responses were faster than when the numerical distance
was small. Importantly, there was a significant interaction
between congruity and numerical distance [F(1, 17) =
15.64; p < .01; g2p ¼ :48]: The congruity effect (i.e. the absolute difference between RT for congruent pairs and RT for
incongruent pairs) was larger for a small numerical distance than for a large numerical distance (see Fig. 2A).
There was also a main effect of physical distance
[F(1, 17) = 26.93; p < .001; g2p ¼ :61]: Overall, participants
were faster to respond when physical distance was small
than when it was large. The interaction between congruity
and physical distance was also significant [F(1, 17) = 6.50;
p < .05; g2p ¼ :28] and in the opposite direction as for relevant distance (see Fig. 2C). The interaction between
numerical distance and physical distance and the three-
way interaction were not significant (both F(1, 17) < 1;
g2p < :05).
On average, participants did not respond in time or
made an error on only 1.8% of all trials. The same ANOVA
used for the RT analysis was used for the analysis of the
proportion of errors after arcsine transformation. The pattern of the RT analysis was reflected in the error rates analysis: in all cases, more errors were made in the slower
conditions. There were significant main effects of congruity
[F(1, 17) = 23.72; p < .001; g2p ¼ :58], numerical distance
[F(1, 17) = 67.51; p < .001; g2p ¼ :80], and physical distance
[F(1, 17) = 48.45; p < .001; g2p ¼ :74]. There were significant
two-way interactions between congruity and numerical
distance [F(1, 17) = 18.16; p < .01; g2p ¼ :52] and between
congruity and physical distance [F(1, 17) = 5.40; p < .05;
g2p ¼ :24]. In contrast to the RT analysis, there was also a
two-way interaction between numerical and physical distance [F(1, 17) = 19.64; p < .001; g2p ¼ :54]: the numerical
distance effect was larger when physical distance was
large than when it was small.
To sum up, we replicated the main findings reported by
Schwarz and Ischebeck (2003) for the numerical comparison
task. In particular, the size congruity effect is larger for small
relevant distances than for large relevant distances; but it is
larger for large irrelevant (physical dimension) distances
102
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
than for small irrelevant distances. An interpretation of this
difference will be provided in the next section.
3. Simulation 1
In Simulation 1, we show that the shared decisions account can be implemented in a computational model with
a dual route architecture. We evaluate whether the model
can simulate the effects of Experiment 1 and explain how
these effects originate from the model.
3.1. Method
As outlined in the introduction, our implementation of
the shared decisions account has a numerical and a physical
processing route that interact only at the decision level.
These routes both implement the comparison mechanism
of the Verguts et al. (2005) model. The model was trained
separately for numerical size comparison and physical size
comparison. During this training, the weights between the
representation layers and the right larger and left larger
decision units were adjusted in order to improve performance of the model on the numerical and the physical comparison tasks. After training, the model was tested using the
144 stimulus pairs from Experiment 1 that varied in
numerical and physical size. To show that the dual route
architecture of the model is sufficient to simulate the
behavioral effects, we used a basic steady state model and
did not implement the specificities of a response selection
mechanism (for a similar approach, see Van Opstal, Gevers,
De Moor, & Verguts, 2008). The amount of activation of a
decision unit was interpreted as the amount of evidence
in favor of the corresponding decision. We formalized the
intuitive notion that the more evidence there is for one
decision relative to the other one, the faster the response
time (RT) will be (e.g., Ratcliff, 1978; Usher & McClelland,
2001) by defining simulated RT as an inverse function of
the absolute difference in activation of the two decision
units. See Fig. 3 for a graphical illustration of the model
and Appendix B for details of the implementation.
3.2. Results and discussion
As illustrated in Fig. 2B and D, the model simulated all
the significant effects that were observed in the behavioral
data. Overall, the simulated RT was faster for congruent
pairs than for incongruent pairs. Like in the behavioral
data, the congruity effect was modulated by distance on
the (relevant) numerical dimension: it was larger when
the numerical distance was small than when the numerical
distance was large (Fig. 2B). As also observed in the behavioral data, the opposite pattern was found for distance on
the (irrelevant) physical dimension: when physical distance was small, the congruity effect was smaller than
when it was large (Fig. 2D). The model activated the incorrect decision on 1.5% of all test trials. These errors were all
made in the slowest condition (i.e. incongruent, small
numerical distance, large physical distance).
It is straightforward to understand how the size congruity effect emerges from the model by examining the
examples in Fig. 3. In the case of a congruent stimulus pair
like 2 8 (Fig. 3A), the numerical size route will activate the
correct right larger decision unit, and so will the physical
size route. This results in a fast RT. In the case of an incongruent stimulus pair like 8 2 (Fig. 3B), the physical size
route will still activate the correct right larger decision unit.
However, the numerical size route will activate the left larger decision, resulting in a slower RT.
We now consider why our dual route model simulates
the effects of distance. First, because the numerical route
in the shared decisions model is based on the model of Verguts et al. (2005), an effect of numerical distance is also observed in the current model. Because the physical route is
formally equivalent to the numerical route, it also produces a distance effect. The specific pattern of interactions
between the two routes is caused by the non-linear function that transforms the output activation of the decision
units into response time. This is similar to the non-linear
function that maps drift rate onto RT in the diffusion model
of Schwarz and Ischebeck (2003). Intuitively, this can be
understood as follows: When distance on the relevant
(numerical) dimension is large, there will be a stronger
activation induced by the numerical route on the decision
units than when the numerical distance is small. Hence, in
the former case the irrelevant (physical) dimension will
have relatively less influence, and so there will be a smaller
congruity effect when numerical distance is large. On the
other hand, when distance on the irrelevant dimension is
large, there will be a stronger influence on the decision
units than when the physical distance is small. It follows
that there will be a larger congruity effect when physical
distance is large. Importantly, the effects observed in Simulation 1 are caused by the dual route architecture of our
shared decisions model and not by specific parameter settings in the model (see Appendix B).
4. Experiment 2
According to the shared decisions account, the size congruity effect originates from the fact that the numerical
size route and the physical size route activate shared decision alternatives. These alternatives are set by the task
instructions. In Experiment 1 and Simulation 1, the decision alternatives were ‘‘left larger’’ and ‘‘right larger’’ or
‘‘left smaller’’ and ‘‘right smaller’’. In the current experiment, we investigate what happens when the physical
and numerical magnitude of all stimuli stays exactly the
same, but the decision alternatives are changed by the task
instructions. According to the shared representation account, the size congruity effect originates at the level of
the shared analog representation of numerical and physical magnitude. If no change is made to the stimulus material, then the representation of numerical and physical
magnitude remains unchanged. Hence, the shared representation account does not predict a change in the size
congruity effect in this case. In contrast, the size congruity
effect in our implementation of the shared decisions account originates from the pattern of weights between the
representation of magnitude and the decision. If changing
the decision alternatives leads to a change in this pattern
of weights during training, then the size congruity effect
will also change. To sum up, keeping the stimulus material
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
103
Fig. 3. Shared decisions model for solving the number comparison task in a size congruity paradigm. Numerical size is the relevant dimension, physical size
the irrelevant dimension. Panel A: congruent stimulus pair. The numerical and the physical size processing routes both activate the same decision (right
larger). Panel B: incongruent stimulus pair. Both dimensions activate different decisions: numerical size activates the (correct) left larger decision, but
physical size activates the (incorrect) right larger decision. Because numerical size is the relevant dimension, the activation of the left larger decision will be
higher than the activation of the right larger decision.
the same, but changing the decision alternatives can affect
the size congruity effect according to the shared decisions
account, but not according to the shared representation account. We test these predictions in Experiment 2.
We compare the performance on two different tasks in
a size congruity paradigm. The first task is magnitude judgment with one stimulus per trial (e.g., Schwarz & Heinze,
1998; Schwarz & Ischebeck, 2003). The participant has to
judge if a number is larger or smaller than the mean
numerical size, in this case 5. In this task, the two accounts
make the same predictions about the congruity of each
stimulus. For example, the stimulus 2 is numerically small,
but physically large, so it is incongruent according to the
shared representation account. If the physical size of 2 is
larger than the standard size, then this stimulus is also
incongruent according to the shared decisions account, because the physical dimension activates the ‘‘larger than the
standard’’ decision unit, whereas the numerical dimension
activates the ‘‘smaller than the standard’’ decision unit (see
Fig. 5A). The second task is parity judgment. Here, the
participants have to judge whether a stimulus is odd or
even. For the shared representation account, 2 is still an
incongruent stimulus for the same reason as before. However, at the decision level, the irrelevant dimension no
longer causes competition between the two decision alternatives (‘‘odd’’ and ‘‘even’’). The numerical dimension will
activate the ‘‘even’’ decision unit for 2, but the physical
dimension will not activate any of the two alternatives
(see Fig. 5B), because parity cannot be assigned to physical
size values. Hence, there will be no congruity effect. In
sum, the shared representation account predicts a congruity effect for both the magnitude judgment and the parity
judgment task, whereas the shared decisions account only
predicts a congruity effect in the magnitude judgment task.
4.1. Method
4.1.1. Participants
Sixteen students from Ghent University received course
credits to participate in an experiment of 1 h. Three
104
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
Fig. 4. Experiment 2: observed (panels A and C) and simulated (panels B and D) RT. Panels A and B: magnitude judgment task. Panels C and D: parity
judgment task (Error bars represent 95% confidence intervals based on the guidelines by Loftus and Masson (1994). Error bars for the model data were too
small to depict. To aid visual inspection of the effect sizes, the model data of all simulations are presented on the same scale and so are the behavioral data
of all experiments. The range of values on the Y axis is the same for all simulations, whereas it changes between experiments for the behavioral data.)
participants were female, 13 were right-handed. Their
mean age was 20 years (range: 18–27 years). All participants had normal or corrected to normal vision. They
signed an informed consent prior to the experiment.
4.1.2. Stimuli and apparatus
The experimental setting was the same as in Experiment 1. Stimuli were now the Arabic numerals 1, 4, 6
and 9. The four different physical sizes of the stimuli were
based on the mean values obtained from the calibration
procedure in Experiment 1.
4.1.3. Experimental procedure
A trial started with the presentation of a small square in
the center of the screen that served as the fixation mark.
There was a variable time interval before the stimulus appeared, with a duration randomly drawn from a uniform
distribution with a minimum of 500 ms and a maximum
of 1000 ms. After this interval, the stimulus appeared centrally. Participants had to press the left or right button
according to the numerical size of the stimulus (larger or
smaller than the mean size, i.e. 5) or according to the parity
of the stimulus (odd or even). They were instructed to
ignore the physical size of the stimuli. There was a response deadline of 1500 ms. When the response was given
or when 1500 ms had elapsed, there was a blank screen for
500 ms until the next trial started.
All combinations of numerical and physical sizes were
presented (16 stimuli). Each stimulus was repeated four
times in each of the 20 blocks in the experiment. This gave
a total of 1280 trials. Half of the participants started with
the magnitude judgment task for 10 blocks, the other half
started with the parity judgment task for 10 blocks. In the
magnitude judgment task, smaller than 5 was associated
with a left response and larger than 5 with a right response
for five consecutive blocks. In the other five blocks, this
mapping was reversed. Similarly for the parity judgment
task, odd was associated with a left response and even with
a right response for five blocks; for the other five blocks
this mapping was reversed. The order of tasks and of response mappings was completely counterbalanced between subjects.
4.2. Results and discussion
All correct trials entered the RT analysis. Mean RTs per
condition were analyzed with an ANOVA with a 2 (task:
magnitude judgment or parity judgment) ! 2 (congruity:
congruent or incongruent) design. Both variables were treated as within subjects factors. Congruity was defined as in
other studies that used one stimulus per trial (e.g., Schwarz
& Heinze, 1998; Schwarz & Ischebeck, 2003), similarly for
the two tasks. Stimuli were treated as congruent when their
numerical and physical sizes were both smaller or both larger than the mean numerical and physical size (e.g., a stimulus with the numerical size 2 and the physical size a) and
incongruent otherwise (e.g. a stimulus with the numerical
size 2 and the physical size h). There was a main effect of
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
105
Fig. 5. (A, B) Neural network model for Experiment 2: the magnitude judgment task (panel a) and the parity judgment task (panel b). (C) Model for the
close/far task of Experiment 3.
task [F(1, 15) = 13.31; p < .01; g2p ¼ :47]: participants were
faster to judge the magnitude of the stimuli than to judge
their parity. The main effect of congruity was also significant
[F(1, 15) = 7.18; p < .05; g2p ¼ :32]. Importantly, task interacted with congruity [F(1, 15) = 27.36; p < .001; g2p ¼ :65].
Post hoc contrasts revealed that there was a size congruity
effect in the magnitude judgment task [F(1, 15) = 30.29;
p < .001; g2p ¼ :67] (see Fig. 4A), but not in the parity judgment task [F(1, 15) < 1; g2p ¼ :04] (see Fig. 4B).
On average, participants did not respond in time or
made an error on 4.3% of all trials. The same ANOVA as
for the RT analysis was run for the arcsine transformed error rates. There was no main effect of task [F(1, 15) = 2.17;
p = .16; g2p = .13]. There was a significant main effect of
congruity [F(1, 15) = 5.31; p < .05; g2p = .26]: Less errors
were made on congruent trials than on incongruent trials.
The interaction between task and congruity was not significant [F(1, 15) = 1.76; p = .21; g2p ¼ :11]. In sum, these results confirm the predictions derived from the shared
decisions account.
5. Simulation 2
5.1. Method
The models used in Simulation 2 are presented in
Fig. 5A and B. They are similar to the model of Simulation
1, but now only 1 stimulus is present for both dimensions.
The models were trained on both magnitude judgment and
parity judgment. After training, the performance of the
models was tested using the 16 stimuli from Experiment
2 that varied in numerical and physical size.
5.2. Results and discussion
After training, the model responded correctly to all
stimuli. Simulated RT is plotted in Fig. 4. The model
showed a congruity effect for the magnitude judgment task
(Fig. 4B), but not for the parity judgment task (Fig. 4D). This
is in agreement with the interaction between congruity
and task observed in the behavioral data. In the magnitude
judgment task, the model exhibits a size congruity effect
for the same reason as in Simulation 1. Because a parity
judgment on physical sizes is not possible, the model
was trained to neither select an odd or an even decision
when a physical size from a to i was presented at the input.
Therefore, only the numerical size input had an influence
on the activation of the decision units and there is no congruity effect in this case.
In the model, there is no difference in average RT between the magnitude judgment and the parity judgment
task, as was the case in the behavioral data. A possible
explanation of this discrepancy could be differential
amounts of training on the two tasks in real life, but not
106
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
in the model. However, this difference is beyond the scope
of the current study.
6. Experiment 3
In Experiment 2, we showed that the size congruity effect is absent when the decision alternatives required for
the task are not automatically activated by the irrelevant
dimension. However, caution should be taken in making
strong conclusion based on this null effect. It could be argued that in the parity task of Experiment 2, participants
were able to completely ignore the physical dimension, because parity does not apply to this dimension. Therefore, in
Experiment 3, we further explore the influence of task
instructions on the size congruity effect by using a task
in which the required decision alternatives can be activated by both the relevant and the irrelevant stimulus
dimension. Participants are now instructed to judge the
numerical distance of the target digit to the standard as
being either close to or far from 5 (close/far task; Santens
& Gevers, 2008). In particular, 4 and 6 are considered to
be close and 1 and 9 are considered to be far. Importantly,
in this task some stimuli will be congruent according to the
shared representation account, but incongruent for
the shared decisions account and vice versa. According to
the shared representation account, congruity is defined
by the relation between the numerical and physical magnitude of the stimuli. Stimuli in which numerical and physical size are both large (e.g. 9) or both small (e.g. 1) will
always be congruent, independent of the required decision.
For the shared decisions account however, congruity originates from competition between the decision alternatives.
This competition will be present only if the physical size
route automatically activates one of the decisions that
are needed to perform the task. For example, in the
close/far task, the numerical size route will activate the
far (from the standard) decision unit when a stimulus like
1 is presented. If the physical size of 1 is far from the mean
physical size, then the physical size can automatically activate the same far (from the standard) decision unit (see
Fig. 5C). In other words, if the physical size activates the
close/far decision units, then this stimulus is congruent
according to the shared decisions account as it does not
lead to competition between decisions. However, because
numerical size is small and physical size is large for this
stimulus, it is incongruent according to the shared representation account (see Fig. 6 for additional examples). In
Experiment 3, we will examine whether there is a congruity effect as defined by the shared representation account
or as defined by the shared decisions accounts in a close/
far task.
6.1. Method
6.1.1. Participants
Seventeen students from Ghent University received
course credits to participate in an experiment of 30 min.
14 participants were female, 12 were right-handed. Their
mean age was 19 years (range: 18–23 years). All participants had normal or corrected to normal vision. They
signed an informed consent prior to the experiment.
6.1.2. Stimuli and apparatus
The experimental setting was the same as in Experiments 1 and 2. The stimuli were exactly the same as in
Experiment 2.
6.1.3. Experimental procedure
Trials were presented as in Experiment 2, only the task
differed. Participants now had to press the left or the right
button according to the numerical distance (close/far) between the target stimulus and the standard 5. All combinations of numerical and physical sizes were presented. In
this way, 16 different stimuli could be constructed. Each
stimulus was repeated four times in each of the 10 blocks
in the experiment (640 trials). In one half of the experiment, participants were instructed to press the left key
for a stimulus that was close to 5 (4 or 6) and the right
key for a stimulus that was far from 5 (1 or 9). In the other
five blocks, this response mapping was reversed. The order
of the response mappings was counterbalanced between
subjects. Like in Experiments 1 and 2, no details about
the physical sizes were given. Participants were instructed
to ignore the differences in physical size.
6.2. Results and discussion
Mean RTs of all correctly answered trials entered the RT
analysis. In the first ANOVA, congruity was defined according to the shared representation account (i.e. in the same
way as in the magnitude judgment task). Stimuli that were
both numerically and physically larger (or smaller) than
the mean size were regarded as congruent (e.g. 9), stimuli
that were not were regarded as incongruent (e.g. 9). For
this analysis, the main effect of congruity was not significant [F(1, 16) = 1.23; p = .28; g2p ¼ :07] (see Fig. 7A).
In the second ANOVA, congruity was defined according
to the shared decisions account: if both the numerical and
the physical dimension activated the same decision unit,
then the stimulus was regarded as congruent (e.g. 9:
numerical and physical dimension activate the far decision
unit). If not, it was regarded as incongruent (e.g. 6: numerical dimension activates the close decision unit, but physical dimension activates the far decision unit). For this
analysis, the main effect of congruity was significant
[F(1, 16) = 5.42; p < .05; g2p ¼ :25] (see Fig. 7C). In contrast
with the parity task of Experiment 2, the presence of a congruity effect here implies that the irrelevant physical size
was automatically categorized as close or far from the
mean size and that this influenced the decision process.
This was the case even though the mean physical size
had never been presented, nor were any details given
about the physical sizes. Note that the null effect of congruity according to the shared representation account cannot be due to a lack of power, as it was computed on
exactly the same data as the (significant) congruity effect
according to the shared decisions account.
On average, participants did not respond in time or
made an error on 4,7% of all trials. The ANOVA on the transformed error rates in which congruity was defined according to the shared representation account did not show a
significant effect of congruity [F(1, 16) = 2.36; p = .14;
g2p ¼ :13], neither did the ANOVA in which congruity was
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
107
Fig. 6. Examples of stimuli in Experiments 3 and 4, categorized as congruent or incongruent according to the shared representation and the shared
decisions accounts. The four different physical sizes are ordered from left to right and top to bottom within the shared representation panel and within the
shared decisions panel.
defined according to the shared decisions account
[F(1, 16) < 1; g2p ¼ :02].
7. Simulation 3
7.1. Method
The model used for the magnitude judgment task in
Simulation 2 was adapted to give the decision close (to
the mean size) or far (from the mean size) for both numerical
size and physical size. After training, simulated RT was obtained for the same 16 stimuli as used in Experiment 3.
7.2. Results and discussion
The model gave the correct answer on all stimuli. Simulated RT is plotted in Fig. 7. The results from the model
are in accordance with the behavioral results. When congruity is defined according to the shared representation account, there is no congruity effect (Fig. 7B). However, when
defined according to the shared decisions account, there is
a clear congruity effect (Fig. 7D). More specifically, simulated RT was faster to stimuli for which the numerical size
route and the physical size route activated the same decision unit (close or far), whereas it was slower to stimuli for
which the numerical and the physical dimension activated
a different decision.
8. Experiment 4
So far, in our behavioral experiments and simulations,
we investigated the interaction between numerical size
as the relevant dimension and physical size as the irrelevant dimension. In the shared decisions model, the numerical and the physical size route are formally equivalent and
therefore interchangeable. The model thus predicts a similar congruity effect when the relevant dimension is physical size and the irrelevant dimension is numerical size. In
experiment 4, we investigate whether the novel findings
with the close/far task in Experiment 3 can be replicated
when the relevant and irrelevant dimensions are
interchanged.
108
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
Fig. 7. Experiment 3: observed (panels A and C) and simulated (panels B and D) RT. Panels A and B: congruity defined according to the shared
representation account. Panels C and D: congruity defined based on the specific decisions for this task. (Error bars for the model data were too small to
depict. To aid visual inspection of the effect sizes, the model data of all simulations are presented on the same scale and so are the behavioral data of all
experiments. The range of values on the Y axis is the same for all simulations, whereas it changes between experiments for the behavioral data.)
8.1. Method
8.1.1. Participants
Twenty students from Ghent University were paid to
participate in an experiment of 30 min. 14 participants
were female, 17 were right-handed. Their mean age was
20 years (range: 18–24 years). All participants had normal
or corrected to normal vision. They signed an informed
consent prior to the experiment.
8.1.2. Stimuli and apparatus
The experimental setting was the same as in Experiments 1–3. The stimuli were exactly the same as in Experiments 2 and 3.
8.1.3. Experimental procedure
The experimental procedure was very similar to that of
Experiment 3. The most important difference was that the
physical dimension was now the relevant dimension and
the numerical dimension was the irrelevant dimension.
Hence, participants had to press the left or the right button
according to the physical distance (close/far) between the
physical size of the target stimulus and the mean physical
size. All combinations of numerical and physical sizes were
presented. In this way, 16 different stimuli could be constructed. Each stimulus was repeated four times on each
of the 10 blocks in the experiment (640 trials). In one half
of the experiment, participants were instructed to press
the left key for a stimulus with physical size that was close
to the mean physical size (b or g) and the right key for a
physical size that was far from the mean physical size (a
or h). In the other five blocks, this response mapping was
reversed. The order of the response mappings was counterbalanced between subjects. At the beginning of the experiment, all physical sizes, including the mean physical size,
were displayed on screen. To further familiarize the participants with the different physical sizes, 25 practice trials
were presented. No details about the numerical sizes were
given. Participants were instructed to ignore the differences in numerical size.
8.2. Results and discussion
The RT analyses were based on the mean RTs per condition. Like in Experiment 3, in the first ANOVA, congruity
was defined according to the shared representation account (i.e. in the same way as in the magnitude judgment
task). For this analysis, the main effect of congruity was not
significant [F(1, 19) < 1; g2p < :01] (see Fig. 8A). In the second ANOVA, congruity was defined according to the shared
decisions account. For this analysis, the main effect of congruity was significant [F(1, 19) = 6.24; p < .05; g2p ¼ :25]
(see Fig. 8C). To directly test possible differences between
Experiment 3 and 4, we analyzed the data from these
two experiments with an ANOVA with a 2 (congruity defined according to the shared decisions account: congruent
or incongruent) ! 2 (relevant dimension: numerical or
physical) design in which congruity was treated as a within
subjects factor and relevant dimension as a between subjects factor. Evidently, there was a main effect of congruity
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
109
Fig. 8. Experiment 4: observed (panels A and C) and simulated (panels B and D) RT. Panels A and B: congruity defined according to the shared
representation account. Panels C and D: congruity defined based on the specific decisions for this task. (Error bars for the model data were too small to
depict. To aid visual inspection of the effect sizes, the model data of all simulations are presented on the same scale and so are the behavioral data of all
experiments. The range of values on the Y axis is the same for all simulations, whereas it changes between experiments for the behavioral data.).
[F(1, 35) = 10.97; p < .01; g2p ¼ :24]. There was also a main
effect of relevant dimension [F(1, 35) = 27.66; p < .001;
g2p ¼ :44]. Participants responded faster when the numerical dimension was relevant than when the physical dimension. Importantly, there was no interaction between
congruity and relevant dimension F(1, 35) < 1; g2p < :01].
On average, participants did not respond in time or
made an error on 8.2% of all trials. The same ANOVAs as
for the RT analyses were run for the arcsine transformed
error rates in. The ANOVA in which congruity was defined
according to the shared representation account did not
show a significant effect of congruity [F(1, 19) = 1.51;
p = .23; g2p ¼ :07]. There was a main effect of congruity
when it was defined according to the shared decisions account [F(1, 19) = 9.44; p < .01; g2p = .33].
9. Simulation 4
9.1. Method
The model simulation was exactly the same as in Simulation 3, except that now the activity from the numerical
dimension was attenuated relative to the activity from
the physical dimension.
9.2. Results and discussion
The model gave the correct response to all stimuli.
Simulated RT is plotted in Fig. 8. The results from the model are in accordance with the behavioral results and very
similar to the results of Simulation 3. In particular, simulated RT was sensitive to congruity as defined by the
shared decision account rather than the shared representation account.
10. General discussion
In this study, we compared the shared representation
and the shared decisions accounts of the size congruity effect. Both accounts were hitherto underspecified. In addition, we have argued that the shared representation
account suffers from a conceptual indeterminacy problem.
The shared decisions account, on the other hand, can be
considered as belonging to a broader theoretical framework, explaining congruity effects in terms of dual route
processing. We specified this account in a dual route
model, based on the number comparison mechanism in
the model of Verguts et al. (2005). We simulated the
congruity effect and its interactions with distance on the
relevant and the irrelevant dimension (Experiment 1, Simulation 1). In Experiment 2, using two different tasks but
identical stimuli across tasks, the size congruity effect
was shown to be dependent on the task instructions in a
way predicted by the shared decisions account. In Experiment 3, it was shown that the task not only modulates
the presence of a size congruity effect, but that it also
predicts specific congruity effects different from those in
a number comparison task. In Experiment 4, we showed
that interchanging the relevant and the irrelevant dimensions does not necessarily have an influence on the size
110
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
congruity effect. Indeed, whereas there was an overall difference in RTs between Experiments 3 and 4, there was no
interaction with size congruity. Our dual route model, as
an instantiation of the shared decisions account, was able
to simulate all of these effects. In contrast, it is difficult
to see how the results from Experiments 2–4 would be explained by an account in which the size congruity effect
originates from a level where numerical and physical size
would be jointly represented in an analog format (shared
representation account).
Our computational model can be related to the coalescence model of Schwarz and Ischebeck (2003). Interestingly, this diffusion model approach accurately describes
RT and accuracy in a standard size congruity paradigm,
and has the potential to also investigate more detailed aspects of RT and accuracy distributions. However, its current formulation does not distinguish between the
different accounts on how numerical and physical size
interact. As the authors state themselves ‘‘. . . there is nothing in the formalism of the coalescence model per se that
could distinguish between these accounts in terms of an
early versus late interaction’’ (p. 520). The computational
model presented here has a complementary advantage.
Whereas it incorporates a simpler way of simulating RTs
and errors, it explicitly models how physical and numerical
size interact. In doing so, it makes only the necessary
assumptions (see Appendix B). We believe that joining
the individual qualities of these two approaches may provide a more realistic model of how quantities are compared; we leave that project for a follow-up study.
An open question is to what extent the size congruity
effect can be observed when numerical size is paired with
quantitative dimensions other than physical size. Size congruity effects have already been observed between numerical size and quantities such as luminance (Cohen Kadosh
& Henik, 2006; Cohen Kadosh et al., 2008; Pinel et al.,
2004), line length (Dormal & Pesenti, 2007, 2009), duration
(Dormal, Andres, & Pesenti, 2008) and dot size (Gebuis,
Cohen Kadosh, de Haan, & Henik, 2009). In each case, the
dual route model presented here predicts that the effects
will originate from the decision level and not from the
representational level. However, this does not imply that
each pair of dimensions should exhibit equal size congruity
effects. Some dimensions may be more discriminable
(Pansky & Algom, 1999), and some pairs of dimensions
might have more similar decisions. For example, in a comparison task, numerical and physical size may have more
similar decisions (smaller and larger) than numerical size
and brightness (smaller versus less bright and larger versus
more bright). In general, we predict that each quantitative
dimension can interfere with any other to the extent that
the irrelevant dimension is effective in automatically activating the decision units that are used for the task at hand.
We have shown that the shared decisions model presented in this study can account for both well-known
and novel congruity effects. However, this does not
exclude the possibility that a shared representation for
different quantities exists. Strong evidence in favor of a
shared representation would be single neurons that show
selectivity for quantity in different dimensions. Several
authors have put forward the hypothesis that the neurons
that are initially recruited to represent space and time are
used later in development to represent number as well
(Bueti & Walsh, 2009; Cohen Kadosh & Walsh, 2008; Cohen
Kadosh et al., 2008). In search of such neurons, Tudusciuc
and Nieder (2007) presented dot patterns and line lengths
to rhesus monkeys while recording neural activity from
the intraparietal sulcus (IPS), a site where neurons selective to number have been observed (Nieder & Miller,
2004). Only about 3–4% of the 400 neurons in IPS they recorded from had both a preferred numerosity and a preferred line length. These cells would thus be good
candidates to represent quantity independent of the specific dimension. However, if these neurons represent a
common metric, then their preferred numerosity should
correlate with their preferred line length. This was not
the case. Hence, not even these few neurons can be said
to have a common metric for number and length. It was,
however, the case that number-selective and length-selective neurons were spatially intermixed in the IPS. One possibility is that (a subset of) neurons in IPS are naturally
‘‘biased’’ for coding quantitative dimensions, and that different neurons become recruited for different dimensions
depending on which dimensions are important for the
organism. This account remains however to be tested.
Appendix A. Calibration of the stimuli for Experiment 1
On the first day of Experiment 1, participants performed
a calibration procedure. In this procedure, the physical
sizes of the stimuli were adjusted in order to minimize
the response time (RT) differences between comparing
numerical sizes 1 and 2 and physical sizes a and b, between
comparing numerical sizes 2 and 7 and physical sizes b and
g and between comparing numerical sizes 7 and 8 and
physical sizes g and h. The values for a, b, g, and h were defined as factors with which the height of the digits in the
first block was multiplied. The experiment started with a
numerical comparison task, in which this factor was fixed
at 1 for all stimuli. This corresponded to a height of 7.4 visual degrees. 20 repetitions of the stimulus pairs 1–2, 2–7,
and 7–8 (i.e., 60 trials) were presented. The order of the
digits in a pair was randomized. At the end of these 60 trials, median RTs for each pair were calculated and participants were informed that now the physical comparison
task had to be performed. Another 60 trials were presented, but now consisting of 20 repetitions of each of
the physical size pairs a–b, b–g, and g–h. For the first 60 trials of physical comparison, size a was set to 0.3, b to 0.55, g
to 1.45 and h to 1.70 times the height of the stimuli in the
numerical comparison task. The arbitrary symbols ‘‘#’’,
‘‘@’’, ‘‘&’’ and ‘‘}’’ were randomly assigned to these physical
heights. As in the first 60 trials, participants had to press on
the side of the smaller or larger stimulus, but now based on
its physical size. After a block of 60 + 60 = 120 trials, the
mean differences in RT between pair 1–2 and pair a–b
(DRT1), between pair 2–7 and pair b–g (DRT2) and between
pair 7–8 and pair g–h (DRT3) were calculated. If this difference was positive (i.e., a higher RT for the numerical pair
than for the physical pair), then the distance between the
two physical values was decreased to make them less
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
discriminable. If the difference was negative, then the distance between the physical values was increased. More
RT 2
specifically, D1000
was first added to a and b and subtracted
RT 1
RT 3
from g and h. Then, D500
was added to a and D500
was subtracted from h. Ten such blocks of 120 trials were presented in the calibration phase. In this way, the values
for a, b, g and h were determined for each participant separately and used in the size congruity task on the second
day of the experiment.
Appendix B. Neural network model implementation
B.1. Simulation 1
The model in Simulation 1 implements a size congruity
task according to the shared decisions account. It is based
on the Verguts et al. (2005) number comparison model.
For simplicity, we did not implement input or output layers in the model, only the representation and decision layers are included (for a similar approach, see Van Opstal
et al., 2008). The physical dimension is implemented by
adding two extra ‘‘representation layers’’ (one for the right
stimulus and one for the left; see Fig. 2) with a similar
structure as the number line layers in the original model
of Verguts et al. (2005). The numerical size representation
layers and the physical size representation layers are connected to the same decision units. These units are labeled
with the decisions left larger and right larger. Each representation layer has 9 units. Numerical size is represented
by an activation pattern that corresponds to a Gaussianlike function (Verguts & Fias, 2004). Physical size is represented identically; it is assumed that there are 9 sizes (a–i)
and the distance between two successive sizes is equal to
the distance between two successive numerical sizes. The
net input activation to the decision units was transferred
using a log-sigmoid function. The simulated RT of the network was calculated as
1
ðjo1 $ o2 j þ cÞ
with o1 and o2 being the activation of the decision units
and c being a small constant, equal to 1 for all simulations.
Using the delta rule, the network was trained separately to
give the correct response for the numerical comparison
task and the physical comparison task, respectively. When
trained on one dimension, the units of the representation
layer of the other dimension were set to zero. Training
was performed using a gradient descent with momentum
learning function. The learning rate was set to 5. During
training, weights were adjusted to minimize the average
squared error (mse) between the values of the output units
and the target values for these units (i.e. [0 1] or [1 0]). The
model was trained until this mse was smaller than 0.001.
Behavioral performance in Experiment 1 was simulated
by testing the trained model with the numerical sizes 1,
2, 7 and 8, and the physical sizes a, b, g and h. During testing, the activation of the representation layer units on the
irrelevant dimension was multiplied by a parameter H
with the value 0.15; this is a proxy to biasing the relevant
dimension via task demand units implemented in earlier
111
models (e.g., Botvinick, Braver, Barch, Carter, & Cohen,
2001; Cohen et al., 1990). It is also similar to the H parameter in the coalescence model by Schwarz and Ischebeck
(2003). In their diffusion model approach, this parameter
reflects ‘‘. . .the degree to which the drift rate component
of the irrelevant attribute enters into the overall drift rate.’’
Training and testing of the model was repeated as many
times as there were participants in the corresponding
behavioral experiment. The means of the simulated RTs
over these replications are presented in the results section.
The size congruity effect originates in the model from
the fact that, although numerical and physical sizes are
represented separately, the decision units are shared for
the two dimensions. The H parameter biases the relevant
dimension over the irrelevant with respect to the activity
that reaches the decision units. The distance effects on
the relevant and the irrelevant dimension are caused by
the pattern of weights between the representations of
numerical and physical size and the decision units, just like
in the model of Verguts et al. (2005; also see Van Opstal
et al., 2008). The specific interaction pattern between distance on the relevant and irrelevant dimension is caused
by the non-linear function transforming the activation of
the decision units into response time.
We tested the effect of varying parameter values on the
performance of the model. The gradient descent with
momentum learning function and the learning rate were
chosen for computational efficiency only. Increasing the
learning to e.g. 50 and choosing an even more efficient
learning function (gradient descent with momentum and
adaptive learning rate) resulted in very fast learning (the
performance goal was reached in less than 100 iterations),
but caused only minor quantitative and no qualitative
changes in the simulated RT that the model produced.
The same was true when learning was made very slow
by setting the learning rate to 5 and choosing a less efficient learning function (gradient descent without momentum nor adaptive learning rate). Decreasing the
performance goal (e.g. to a value as low as 0.0001) only
had a minor influence on the behavior of the model, but
naturally increased the training time. Increasing or
decreasing the width of the activation spread in the
numerical and physical size representation layers also
had only a minor influence and did not change the results
qualitatively. This is because the comparison mechanism
in our model is based on the pattern of weights between
the representation and the decision units and not on the
representation itself (for a detailed explanation, see Van
Opstal et al., 2008; Verguts et al., 2005). Finally, we tested
the influence of varying the value of the H parameter.
Decreasing this value (e.g. to 0.05) led to a diminished size
congruity effect. Increasing its value led to a bad performance of the model. For values of 0.30 or higher, the model
made many errors, especially in the conditions that were
the slowest in the original simulation.
B.2. Simulation 2
Only minor adjustments were made to the model of
Simulation 1 (see Fig. 5A). There was now only one representation layer for the numerical dimension and one for
112
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
the physical dimension, as only one stimulus was presented in each trial. The decision units were now labeled
smaller (than the mean size) and larger (than the mean size).
For the parity judgment task, the model architecture was
similar, but the decision units were labeled odd and even
(see Fig. 5B). The models were trained separately for
numerical size input and physical size input. Because physical sizes cannot be categorized as odd or even, the model
was trained to have zero activation for both decision units
when physical sizes were presented in the parity task.
B.3. Simulation 3
The model used in Simulation 3 is identical to that in Simulation 2 (see Fig. 5), except for the decisions used to perform
the task. The decision units were now labeled close (to the
mean size) and far (from the mean size). The model was
trained separately on numerical size and physical size input.
Similarly to the behavioral experiment, the model was
trained to answer close when numerical sizes 4 or 6 were
presented and far when numerical sizes 1 or 9 were presented. For physical size input, the model was trained to answer close for physical sizes b and g and far for sizes a and h.
B.4. Simulation 4
The only difference with Simulation 3 is that here, the
physical dimension was treated as the relevant dimension
and the numerical dimension as the irrelevant.
References
Ansari, D., Fugelsang, J. A., Dhital, B., & Venkatraman, V. (2006).
Dissociating response conflict from numerical magnitude processing
in the brain: An event-related fMRI study. NeuroImage, 32, 799–805.
Algom, D., Dekel, A., & Pansky, A. (1996). The perception of number from
the separability of the stimulus: The Stroop effect revisited. Memory &
Cognition, 24, 557–572.
Besner, D., & Coltheart, M. (1979). Ideographic and alphabetic processing
in skilled reading of English. Neuropsychologia, 17, 467–472.
Botvinick, M. M., Braver, T. S., Barch, D. M., Carter, C. S., & Cohen, J. D.
(2001). Conflict monitoring and cognitive control. Psychological
Review, 108, 624–652.
Bueti, D., & Walsh, V. (2009). The parietal cortex and the representation of
time, space, number and other magnitudes. Philosophical Transactions
of the Royal Society B – Biological Sciences, 364, 1831–1840.
Cantlon, J. F., Platt, M. L., & Brannon, E. M. (2009). Beyond the number
domain. Trends in Cognitive Sciences, 13, 83–91.
Chiou, R. Y.-C., Chang, E. C., Tzeng, O. J.-L., & Wu, D. H. (2009). The
common magnitude code underlying numerical and size processing
for action but not for perception. Experimental Brain Research, 194,
553–562.
Cohen, J. D., Dunbar, K., & McClelland, J. L. (1990). On the control of
automatic processes – A parallel distributed processing account of the
Stroop effect. Psychological Review, 97, 332–361.
Cohen, J. D., Servan-Schreiber, D., & McClelland, J. L. (1992). A parallel
distributed processing approach to automaticity. The American Journal
of Psychology, 105, 239–269.
Cohen Kadosh, R., Cohen Kadosh, K., & Henik, A. (2008). When brightness
counts: The neuronal correlate of numerical-luminance interference.
Cerebral Cortex, 18, 337–343.
Cohen Kadosh, R., Cohen Kadosh, K., Linden, D. E. J., Gevers, W., Berger, A.,
& Henik, A. (2007). The brain locus of interaction between number
and size: A combined functional magnetic resonance imaging and
event-related potential study. Journal of Cognitive Neuroscience, 19,
957–970.
Cohen Kadosh, R., & Henik, A. (2006). A common representation for
semantic and physical properties. Experimental Psychology, 53, 87–94.
Cohen Kadosh, R., Lammertyn, J., & Izard, V. (2008). Are numbers special?
An overview of chronometric, neuroimaging, developmental and
comparative studies of magnitude representation. Progress in
Neurobiology, 84, 132–147.
Cohen Kadosh, R., & Walsh, V. (2008). From magnitude to natural
numbers: A developmental neurocognitive perspective. Behavioral
and Brain Sciences, 31, 647–648.
Coles, M. G. H. (1989). Modern mind-brain reading: Psychophysiology,
physiology, and cognition. Pscyhophysiology, 26, 251–269.
Dormal, V., Andres, M., & Pesenti, M. (2008). Dissociation of numerosity
and duration processing in the left intraparietal sulcus: A transcranial
magnetic stimulation study. Cortex, 44, 462–469.
Dormal, V., & Pesenti, M. (2007). Numerosity-length interference – A
Stroop experiment. Experimental Psychology, 54, 289–297.
Dormal, V., & Pesenti, M. (2009). Common and specific contributions of
the intraparietal sulci to numerosity and length processing. Human
Brain Mapping, 30, 2466–2476.
Gebuis, T., Cohen Kadosh, R., de Haan, E., & Henik, A. (2009). Automatic
quantity processing in 5-year olds and adults. Cognitive Processing, 10,
133–142.
Gevers, W., Ratinckx, E., De Baene, W., & Fias, W. (2006a). Further
evidence that the SNARC effect is processed along a dual-route
architecture – Evidence from the lateralized readiness potential.
Experimental Psychology, 53, 58–68.
Gevers, W., Verguts, T., Reynvoet, B., Caessens, B., & Fias, W. (2006b).
Numbers and space: A computational model of the SNARC effect.
Journal of Experimental Psychology – Human Perception and
Performance, 32, 32–44.
Hackley, S. A., & Miller, J. (1995). Response complexity and precue interval
effects on the lateralized readiness potential. Psychophysiology, 32,
230–241.
Henik, A., & Tzelgov, J. (1982). Is 3 greater than 5 – The relation between
physical and semantic size in comparison tasks. Memory & Cognition,
10, 389–395.
Kaufmann, L., Koppelstaetter, F., Delazer, M., Siedentopf, C., Rhomberg, P.,
Golaszewski, S., et al. (2005). Neural correlates of distance and
congruity effects in a numerical Stroop task: An event-related fMRI
study. NeuroImage, 25, 888–898.
Masaki, H., Wild-Wall, N., Sangals, J., & Sommer, W. (2004). The functional
locus of the lateralized readiness potential. Psychophysiology, 41,
220–230.
Meck, W. H., & Church, R. M. (1983). A mode control model of counting
and timing processes. Journal of Experimental Psychology: Animal
Behavior Processes, 9, 320–334.
Nieder, A., & Miller, E. K. (2004). A parieto-frontal network for visual
numerical information in the monkey. Proceedings of the
National Academy of Sciences of the United States of America, 101,
7457–7462.
Loftus, G. R., & Masson, M. E. J. (1994). Using confidence-intervals in
within-subject designs. Psychonomic Bulletin & Review, 1, 476–490.
Pansky, A., & Algom, D. (1999). Stroop and Garner effects in comparative
judgment of numerals: The role of attention. Journal of Experimental
Psychology – Human Perception and Performance, 25, 39–58.
Pinel, P., Piazza, M., Le Bihan, D., & Dehaene, S. (2004). Distributed and
overlapping cerebral representations of number, size, and luminance
during comparative judgments. Neuron, 41, 983–993.
Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85,
59–108.
Santens, S., & Gevers, W. (2008). The SNARC effect does not imply a
mental number line. Cognition, 108, 263–270.
Schwarz, W., & Heinze, H. J. (1998). On the interaction of numerical and
size information in digit comparison: A behavioral and event-related
potential study. Neuropsychologia, 36, 1167–1179.
Schwarz, W., & Ischebeck, A. (2003). On the relative speed account of
number-size interference in comparative judgment of numerals.
Journal of Experimental Psychology – Human Perception and
Performance, 29, 507–522.
Stevens, M., Lammertyn, J., Verbruggen, F., & Vandierendonck, A. (2006).
Tscope: A C library for programming cognitive experiments on the MS
Windows platform. Behavior Research Methods, 38, 280–286.
Stroop, J. R. (1935). Studies of interference in serial verbal reactions.
Journal of Experimental Psychology, 18, 643–662.
Szücs, D., & Soltész, F. (2008). The interaction of task-relevant and taskirrelevant stimulus features in the number/size congruency
paradigm: An ERP study. Brain Research, 1190, 143–158.
Tudusciuc, O., & Nieder, A. (2007). Neural population coding of
continuous and discrete quantity in the primate posterior parietal
cortex. Proceedings of the National Academy of Sciences of the United
States of America, 104, 14513–14518.
S. Santens, T. Verguts / Cognition 118 (2011) 97–113
Usher, M., & McClelland, J. L. (2001). The time course of perceptual choice:
The leaky, competing accumulator model. Psychological Review, 108,
550–592.
Van Opstal, F., Gevers, W., De Moor, W., & Verguts, T. (2008). Dissecting
the symbolic distance effect: Comparison and priming effects in
numerical and nonnumerical orders. Psychonomic Bulletin & Review,
15, 419–425.
Verguts, T., & Fias, W. (2004). Representation of number in animals and
humans: A neural model. Journal of Cognitive Neuroscience, 16,
1493–1504.
Verguts, T., & Fias, W. (2008). Symbolic and nonsymbolic pathways of
number processing. Philosophical Psychology, 21, 539–554.
113
Verguts, T., Fias, W., & Stevens, M. (2005). A model of exact small-number
representation. Psychonomic Bulletin & Review, 12, 66–80.
Walsh, V. (2003). A theory of magnitude: common cortical metrics
of time, space and quantity. Trends in Cognitive Sciences, 7,
483–488.
Xuan, B., Zhang, D., He, S., & Chen, X. (2007). Larger stimuli are judged to
last longer. Journal of Vision, 7, 1–5.
Yeung, N., Botvinick, M. M., & Cohen, J. D. (2004). The neural basis of error
detection: Conflict monitoring and the error-related negativity.
Psychological Review, 111, 931–959.
Zorzi, M., & Umiltà, C. (1995). A computational model of the Simon effect.
Psychological Research – Psychologische Forschung, 58, 193–205.