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ORIGINAL RESEARCH

published: 16 November 2017

doi: 10.3389/fpsyg.2017.02013

Frontiers in Psychology | www.frontiersin.org 1November 2017 | Volume 8 | Article 2013

Edited by:

Andriy Myachykov,

Northumbria University,

United Kingdom

Reviewed by:

Herbert Heuer,

Leibniz Research Centre for Working

Environment and Human Factors (LG),

Germany

Fuhong Li,

Jiangxi Normal University, China

*Correspondence:

Attila Krajcsi

krajcsi@gmail.com

†These authors have contributed

equally to this work.

Specialty section:

This article was submitted to

Cognition,

a section of the journal

Frontiers in Psychology

Received: 29 August 2017

Accepted: 03 November 2017

Published: 16 November 2017

Citation:

Krajcsi A and Kojouharova P (2017)

Symbolic Numerical Distance Effect

Does Not Reﬂect the Difference

between Numbers.

Front. Psychol. 8:2013.

doi: 10.3389/fpsyg.2017.02013

Symbolic Numerical Distance Effect

Does Not Reﬂect the Difference

between Numbers

Attila Krajcsi 1

*†and Petia Kojouharova 1, 2†

1Department of Cognitive Psychology, Institute of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary, 2Doctoral

School of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary

In a comparison task, the larger the distance between the two numbers to be compared,

the better the performance—a phenomenon termed as the numerical distance effect.

According to the dominant explanation, the distance effect is rooted in a noisy

representation, and performance is proportional to the size of the overlap between

the noisy representations of the two values. According to alternative explanations,

the distance effect may be rooted in the association between the numbers and the

small-large categories, and performance is better when the numbers show relatively high

differences in their strength of association with the small-large properties. In everyday

number use, the value of the numbers and the association between the numbers and

the small-large categories strongly correlate; thus, the two explanations have the same

predictions for the distance effect. To dissociate the two potential sources of the distance

effect, in the present study, participants learned new artiﬁcial number digits only for the

values between 1 and 3, and between 7 and 9, thus, leaving out the numbers between

4 and 6. It was found that the omitted number range (the distance between 3 and 7) was

considered in the distance effect as 1, and not as 4, suggesting that the distance effect

does not follow the values of the numbers predicted by the dominant explanation, but it

follows the small-large property association predicted by the alternative explanations.

Keywords: symbolic number processing, numerical distance effect, analog number system, discrete semantic

system

THE NUMERICAL DISTANCE EFFECT AND ITS EXPLANATIONS

In a symbolic number comparison task, performance is better (i.e., error rates are lower and

reaction times are shorter) when the numerical distance is relatively large, e.g., comparing 1 vs.

9 is easier than comparing 5 vs. 6 (Moyer and Landauer, 1967). There are several explanations for

this phenomenon termed the numerical distance eﬀect.

According to the dominant model, numbers are stored on a continuous (analog) and noisy

representation called the Analog Number System (ANS). The numbers are stored as noisy signals,

and the closer the two numbers on the ANS, the larger the overlap of the two respective signal

distributions is. As comparison performance is better when the overlap is relatively small, the

large distance number pairs are easier to process because of the smaller overlap between the

signals (Dehaene, 2007). More speciﬁcally, the ANS works according to Weber’s law; therefore,

the comparison performance depends on the ratio of the two numbers to be compared (Moyer and

Landauer, 1967). In fact, the distance eﬀect is the consequence of this ratio eﬀect because larger

Krajcsi and Kojouharova Symbolic Distance Effect as Association

distance also means higher ratio. The ratio eﬀect is also thought

to be the cause of the numerical size eﬀect: Comparison

performance is better for smaller numbers than for larger

numbers because smaller number pairs have larger ratio than

larger number pairs with the same distance (Moyer and

Landauer, 1967). The ANS is thought to be the essential base

of numerical understanding (Dehaene, 1992), and numerical

distance eﬀect is believed to be a diagnostic signal of the ANS

activation while solving a numerical task.

However, there could be another explanation for the cause of

the distance eﬀect. Recently, it has been proposed that symbolic

numerical eﬀects, such as the distance and size eﬀects, can be

explained by a representation similar to the mental lexicon or

conceptual networks, where nodes of the network represent the

digits, and connections between them are formed according

to their semantic and statistical relations (Krajcsi et al., 2016).

In this model, termed the Discrete Semantic System (DSS)

model, the numerical distance and size eﬀects are rooted in two

diﬀerent mechanisms, even if the combination of these eﬀects

looks similar to the formerly supposed ratio eﬀect. According

to the model, the size eﬀect might depend on the frequencies

of the numbers: Smaller numbers are more frequent than

larger numbers (Dehaene and Mehler, 1992); therefore, smaller

numbers are easier to process, producing the numerical size

eﬀect. A similar frequency-based explanation of the size eﬀect

could be found in the model of Verguts et al. (2005). At the same

time, numerical distance eﬀect could be based on the relations of

the numbers, for example, similar to the phenomenon in a picture

naming task, where priming eﬀect size depended on the semantic

distance between the prime and target pictures (Vigliocco et al.,

2002). There are several other alternative number processing

models with partly overlapping suppositions and predictions as

the DSS model (Nuerk et al., 2004; Verguts and Fias, 2004;

Verguts et al., 2005; Proctor and Cho, 2006; Leth-Steensen et al.,

2011; Pinhas and Tzelgov, 2012; Verguts and Van Opstal, 2014).

See the comparison of those models in Krajcsi et al. (2016) and in

Krajcsi et al. (in press). Supporting the alternative DSS model, it

has been found that the size eﬀect followed the frequency of the

digits in an artiﬁcial number notation comparison task (Krajcsi

et al., 2016). In addition, it has been shown in a correlational

study that in symbolic number comparison task, the distance

and the size eﬀects were independent (Krajcsi, 2017), reﬂecting

two independent mechanisms generating the two eﬀects. (See a

similar prediction for independent distance and size eﬀects in

Verguts et al., 2005; Verguts and Van Opstal, 2014).

Because of the DSS model and the empirical ﬁndings

demonstrating that the size eﬀect is a frequency eﬀect and that

the distance and size eﬀects are independent, it is essential to

reconsider how the distance eﬀect is generated. According to the

DSS model, diﬀerent explanations consistent with the supposed

network architecture are feasible. First, it is possible that based on

the values of the numbers, connections with diﬀerent strengths

between the numbers are formed—numbers with closer values

have stronger connections—and stronger connections create

interference in a comparison task, thereby resulting in a distance

eﬀect. This explanation is similar to the ANS model in a sense that

value-based semantic relations are responsible for the distance

eﬀect. As an alternative explanation, it is also possible that based

on previous experiences, numbers are associated with the “small”

and the “large” properties, e.g., large digits, such as 8 or 9, are

more strongly associated with “large,” and small digits, such

as 1 or 2, are more strongly associated with “small.” These

associations could inﬂuence the comparison decision, and the

number pairs with larger distance might be easier to process

because the associations of the two numbers with the small-large

properties diﬀer to a larger extent. A similar explanation has

been proposed earlier in a connectionist model, which model

predicted several numerical eﬀects successfully, and one key

component of this model was that the distance eﬀect relies on

the connection between the number layer and the “larger” nodes,

where relatively large numbers are associated with the “larger”

node more strongly than relatively small numbers (Verguts et al.,

2005).

Therefore, the explanations of the numerical distance eﬀect

suppose two diﬀerent sources for the eﬀect: According to the

ANS model and to the value-based DSS explanation, the eﬀect

is rooted in the values or the distance of the numbers, whereas in

the association-based DSS explanation and in the connectionist

model, the eﬀect is rooted in the strength of the associations

between the number and the small-large properties. The two

explanations are not exclusive; it is possible that both information

sources contribute to the distance eﬀect.

The two critical properties of the two explanations, i.e., the

values or distance of the numbers and the association between

the numbers and the small-large properties, strongly correlate in

the number symbols used in everyday numerical tasks. Therefore,

in those cases, one cannot specify their role in the distance

eﬀect. However, in a new artiﬁcial number notation, the two

factors (the distance of the values and the association) could be

manipulated independently. This is only possible if the distance

eﬀect is notation speciﬁc. Otherwise, the new symbols would get

the association strengths of the already known numbers, instead

of forming new association strengths between the new symbols

and the small-large properties. It is possible that the numerical

eﬀects are notation speciﬁc, as has been already demonstrated

in the case of the numerical size eﬀect: In an artiﬁcial number

notation comparison task, the size eﬀect followed the frequency

of the digits, which also means that the size eﬀect is notation

speciﬁc (Krajcsi et al., 2016).

THE AIM OF THE STUDY

The present study investigates whether in a new artiﬁcial number

notation, where the values of the digits and the small-large

associations do not necessarily correlate, the distance eﬀect is

inﬂuenced by the distance of the values or by the small-large

associations, or both. One way to dissociate the two properties is

to use a number sequence in which some of the values are omitted

(Figure 1). If the distance eﬀect is directed by the distance of the

values, then the measured distance eﬀect should be large around

the gap (in this example, the eﬀect should be measured as 4 units

large), whereas if the distance eﬀect is directed by the small-large

associations, then the measured distance eﬀect should be small

Frontiers in Psychology | www.frontiersin.org 2November 2017 | Volume 8 | Article 2013

Krajcsi and Kojouharova Symbolic Distance Effect as Association

FIGURE 1 | An example of the symbols and their meanings in the present

study. Arrows show the predicted distance effect size based on the

predictions of the two explanations.

around this gap, which is measured as an eﬀect with a single

unit distance, thereby supposing that the new digits were used

in a comparison task with equal probability. If both mechanisms

contribute to the distance eﬀect, then the distance eﬀect should

be measured somewhere between the single unit and the many

units (in this example 4 units) distance.

Why does the association explanation predict a distance

eﬀect of 1 distance around the gap? In a comparison task, the

association between a digit and the small-large properties may

depend on how many times the digit were judged as smaller or

larger. If the new digits are used with equal probability in the

comparisons (and if the distance eﬀect is notation speciﬁc), then

the probability of being smaller or larger than the other number

can be speciﬁed easily (see Table 1). In our example (Figure 1),

the number 1 is always smaller. Hence, the association frequency

is 100% with the small property and 0% with the large property.

The number 2 is smaller when compared with 3, 7, 8, and 9,

and larger when compared with 1. Therefore, the association

frequency is 80% small and 20% large. Continuing the example,

the association frequency is directly proportional to the order of

the symbols and not to their value. If the distance eﬀect depends

on the order, then the distance between 3 and 7 (i.e., the two digits

around the gap) is the same as any other neighboring digits (see

the speciﬁc values in Table 1).

The two explanations predict diﬀerent eﬀect sizes for the

distance eﬀect not only for the two numbers next to the gap

(e.g., for 3 vs. 7 on Figure 1 and Table 1) but also for any

number pairs in which the two numbers are on the opposing

side of the gap. The possible number pairs of the new symbols

seen on Figure 1 and their hypothetical distance eﬀect sizes

according to the two explanations can be seen on Figure 2

Columns and rows denote the two numbers to be compared,

and the cells show the distances of the value pairs (darker cells

mean smaller distance). In the value explanation (left side), the

predicted distance is the diﬀerence of the two numbers, whereas

in the association explanation (right side), the predicted distance

is computed based on the strength of the association with the

small-large properties when the numbers are presented with

equal probability, which is simply the order of those symbols in

that series. The comparison performance should be proportional

to the distance. Therefore, these ﬁgures show the performance

pattern predictions according to the two explanations. The results

will be displayed in a similar way as seen here because (a)

TABLE 1 | The chance of being smaller or larger in a comparison task when the

symbols are presented with equal probability.

Example symbols

Meaning of the symbols 1 2 3 7 8 9

Chance of being smaller in a comparison 100% 80% 60% 40% 20% 0%

Chance of being larger in a comparison 0% 20% 40% 60% 80% 100%

displaying the full stimulus space is more informative than

other indexes of distance eﬀects, as any systematic deviation

from the expected patterns could be observed, and (b) with the

relatively large number of cells, any systematic pattern could be a

convincing and critical information independent of the statistical

hypotheses tests.

In the present test, it is critical that the new symbols should

represent their intended values and not as a series that is

independent of the intended number meanings; otherwise, the

participants could consider the new symbols as numbers, e.g.,

from 1 to 6 because of their order in the new symbol series,

which in turn could generate the performance predicted by the

association explanation, even if the eﬀect would be based on their

values. One way to ensure that the new symbols are suﬃciently

associated to their intended values is to ensure that the priming

distance eﬀect works between the new and a well-known (for

example, Indo-Arabic) notation. In numerical comparison tasks,

the decision about the actual trial might be inﬂuenced by the

stimulus of the previous trial, and the size of the inﬂuence is

proportional to the numerical distance of the previous and actual

stimuli, which is termed as the priming distance eﬀect (PDE;

Koechlin et al., 1999; Reynvoet and Brysbaert, 1999). The PDE

is considered to be a sign of the relation between the symbols or

the overlap of their representations (Opstal et al., 2008). Earlier

experiments have shown that new artiﬁcial symbols can cause

PDE in Indo-Arabic numbers (Krajcsi et al., 2016), suggesting

that the new digits are not a series of symbols independent of

their intended values, but they can be considered as a notation for

the respective numbers. In the ANS framework, the PDE reﬂects

the representational overlap between the numbers; thus, the PDE

demonstrates that both notations appropriately activate the same

representation—the ANS.

To summarize, the present study investigates whether the

distance eﬀect follows the distance of the values of the numbers

(left of Figure 2) or the association of the small-large properties

(right of Figure 2) or both, in the case of a newly learned

notation (Figure 1), where some of the symbols are omitted.

If both explanations are true, then we expect a pattern in-

between the two ﬁgures, i.e., we should observe a break between

3 and 7 similar to the value explanation. However, the diﬀerence

between the two sides of the gap should not be as large as

in that explanation. All of these predictions only hold if the

distance eﬀect is notation speciﬁc; otherwise, the distance eﬀect

reﬂects the already well-known numbers, where the value and

the association strongly correlates, and the pattern seen on the

value model prediction can be expected. Consequently, only a

pattern seen on the right in Figure 2 can decide about the models,

because a pattern seen on the left can either mean a value-based

Frontiers in Psychology | www.frontiersin.org 3November 2017 | Volume 8 | Article 2013

Krajcsi and Kojouharova Symbolic Distance Effect as Association

FIGURE 2 | The expected distance effect pattern for the stimulus space used in the present study based on the value explanation (left side) and based on the

association explanation (right side). Speciﬁc values in the cells are the difference of the values (value model) or the difference of the order (association model) of the

numbers to be compared on an arbitrary scale. Darker color indicates worse performance.

distance eﬀect or it can mean that the distance eﬀect is notation

independent.

METHODS

In the present experiment, participants learned new symbols

(Figure 3), with the meaning of the numbers between 1 and 3,

and between 7 and 9 (Figure 1). Then a number comparison task

was performed with the new symbols (Figure 3).

Stimuli and Procedure

The new symbols were chosen from writing systems that were

mostly unknown to the participants (e.g., , , , ). The

characters had similar vertical and horizontal size, and similar

visual complexity, and the height of the symbols were ∼2 cm.

(As mostly the apparent size does not inﬂuence the eﬀects we

investigate here, the visual angle was not controlled strictly.)

Numbers were displayed in white on gray background. The

symbols were randomly assigned to values for all participants,

i.e., the same symbol could mean a diﬀerent value for diﬀerent

participants.

The participants ﬁrst learned new symbols for the numbers

between 1 and 3, and between 7 and 9 (Figure 3). To ensure

that the participants have learned them in the learning phase,

symbols were practiced until a threshold hit rate was reached.

In a trial, a new symbol and an Indo-Arabic digit were shown

simultaneously, and the participant decided whether the two

symbols denoted the same value by pressing the R or I key. The

stimuli were visible until response. After the response, auditory

feedback was given. In a block, all symbols were presented 10

times (60 trials in a block) in a randomized order. In half of the

trials, the symbols denoted the same values. The symbol learning

phase ended if the error rate in a completed block was smaller

than 5% or the participant could not reach that level in ﬁve

blocks.

In the following comparison task, the participants decided

which number is larger in a simultaneously presented new

symbol pair by pressing the R or I key (Figure 3). In a trial, two

numbers were shown until response, and the participants chose

the larger one. Numbers to be compared could be between 1 and

FIGURE 3 | Tasks in the new symbol experiment.

3, and between 7 and 9. After the response, auditory feedback was

given. All possible number pairs including the applied numbers,

excluding ties, were shown 15 times, thereby resulting in 450

trials.

Presentation of the stimuli and measurement of the responses

were managed by the PsychoPy software (Peirce, 2007).

Participants

Twenty-three university students participated in the experiment

for partial course credit. After excluding 4 participants showing

higher than 5% error rates (higher than the mean +the standard

deviation of the error rates in the original sample) in the

comparison task, the data of 19 participants was analyzed (16

females, mean age 22.2 years, standard deviation 4.6 years).

RESULTS

All participants successfully reached a lower than 5% error rate

within 3 blocks in the symbol learning task. Therefore, no

participants were excluded for not learning the symbols within

5 blocks.

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Krajcsi and Kojouharova Symbolic Distance Effect as Association

For all participants, the mean error rates and the mean

reaction times for correct responses were calculated for all

number pairs. Data of participants with higher than 5% mean

error rate were excluded (higher than the mean +the standard

deviation of the error rates in the original sample). The mean

error rates and reaction times of the group are displayed in

Figure 4 for the whole stimulus space. Visual inspection of

the error rate pattern suggests that partly the value model can

be observed, although the data are rather noisy, as reﬂected

in some outlier cells. In the case of reaction time, it is

more straightforward that the pattern is more in line with

the association model (see the two expected pure patterns in

Figure 2). In the reaction time data, one can also observe

the end eﬀect: number pairs including the largest number in

the range (i.e., 9) are faster to process (Scholz and Potts,

1974; Leth-Steensen and Marley, 2000). (There are diﬀerent

possibilities concerning what causes the end eﬀect. It is possible

that participants learn that 9 is the largest number in the actual

session; therefore, when 9 is displayed, no further consideration

is required in a comparison task. Alternatively, according to the

ANS model, it is possible that in the session, number 9 has

neighboring number only on one side, and the overlap between

the noisy signal distributions should be smaller, thereby leading

to a faster response; Balakrishnan and Ashby, 1991).

To test the results statistically, we ﬁrst ﬁt the two predictions of

the models (Figure 2) to the group average of the error rate and

the reaction time data (Figure 4) with a simple linear regression,

where one of the model predictions was the explanatory variable

and one of the behavioral performance measurements was the

dependent variable. Then the goodness of the ﬁt measured

as R2was calculated (R2columns given in Table 2), and the

correlations of two models were compared with the method

described by Steiger (1980) for every performance measurement

(diﬀerence of the group ﬁts column is given in Table 2). As an

alternative method, we calculated the R2values for every single

participant for both the value and association models, and the R2

of these model ﬁts, as ordinal variables, were compared pairwise

with Wilcoxon signed-rank test (Better model for the participants

column is given in Table 2).

To ﬁt the distance eﬀect appropriately, the end eﬀect should

also be considered, and its variance should be removed from

the data. Inspection of the descriptive data on Figure 4 suggests

that number pairs including the number 9 were involved in the

end eﬀect in the present study. One possibility to remove the

end eﬀect is to apply multiple linear regression, and beyond the

distance eﬀect regressor, an end eﬀect regressor (e.g., 1 if the

number pair includes 9, otherwise 0) also should be utilized.

The problem with this solution is that the end eﬀect not only

shortened the response latency for number pairs including 9 but

it also decreased the slope of the distance eﬀect in those cells

(see the less steep distance eﬀect in the row and column with 9

than in other rows and columns). As the end eﬀect is not added

linearly to the distance eﬀect, a multiple linear regression could

not describe this nonlinear aspect of the end eﬀect, which in turn

would distort the distance eﬀect results. As an alternative method,

to remove the end eﬀect, all cells with number pairs including

9 were removed from the analysis (i.e., the bottom row and the

right column on Figure 4) and only the distance eﬀect regressors

FIGURE 4 | Error rates (left) and reaction times (in ms, right) in the whole stimulus space.

TABLE 2 | Goodness of ﬁt of the models (measured as R2) and comparison of the correlations (Difference column) for the error rates, reaction times, and drift rates

patterns based on the group average data, and hypothesis tests for choosing the better model based on the participants’ data.

Linear model (Figure 2) Logarithm model

Value model

R2

Association

model R2

Difference of

the group ﬁts

Better model for

the participants

Value model

R2

Association

model R2

Difference of

the group ﬁts

Better model for

the participants

Error rate 0.709 0.708 Z=0.008,

p=0.993

T=73,

p=0.376

0.714 0.821 Z= −1.091,

p=0.275

T=92,

p=0.904

Reaction

time

0.543 0.790 Z= −2.294,

p=0.022

T=44,

p=0.040

0.457 0.817 Z= −3.646,

p<0.001

T=34,

p=0.014

Drift rate 0.526 0.861 Z= −3.647,

p<0.001

T=39,

p=0.024

0.425 0.874 Z= −5.748,

p<0.001

T=18,

p=0.002

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Krajcsi and Kojouharova Symbolic Distance Effect as Association

were used. Therefore, for all linear ﬁts (Table 2) in both the group

average and the participants level, the number pairs including 9

were removed.

Regarding the possible diﬀerence between the goodness of ﬁt

of the two models, we note that the diﬀerence is limited by the

fact that the two models correlate, e.g., the value model can be

considered as a modiﬁed association model with an additional

increase of the values in the top-right and bottom-left part of

the stimulus space seen in Figure 2. Therefore, if one model is

appropriate, then the other inappropriate model should show

some non-zero R2value too, although the R2should be smaller

than the R2of the appropriate model.

Results for the goodness of ﬁts (Table 2, linear model columns

on the left) show that in the error rates, the two models

are indistinguishable, and in the reaction time patterns, the

association model seems to describe the data better in line with

the visual inspection of the data.

Although error rate and reaction time data are highly

informative, the recently becoming more popular diﬀusion

model analysis could draw a more sensitive picture (Smith and

Ratcliﬀ, 2004; Ratcliﬀ and McKoon, 2008). In the diﬀusion

model, decision is based on a gradual accumulation of evidence

oﬀered by perceptual and other systems, and decision is made

when appropriate amount of evidence is accumulated. Reaction

time and error rates partly depend on the quality of the

information (termed the drift rate) upon which the evidence is

built. Drift rate is considered to be the most important parameter

that inﬂuences the number comparison performance and the

task diﬃculty (Dehaene, 2007). Importantly, observed reaction

time and error rate parameters can be used to recover the drift

rates (Ratcliﬀ and Tuerlinckx, 2002; Wagenmakers et al., 2007).

Drift rates can be more informative than the error rate or the

reaction time because drift rates reveal the sensitivity of the

background mechanisms more directly (Wagenmakers et al.,

2007). To recover the drift rates for all number pairs, the EZ

diﬀusion model was applied (Wagenmakers et al., 2007). The

EZ model supposes that some of the parameters do not play a

role in the response generation, and the model investigates and

recovers only the drift rate, the decision threshold, and the non-

decision time parameters. If one can suppose that only these three

parameters play a role in the responses, then the EZ model can be

utilized. Importantly, one essential advantage of this method is

that unlike most other diﬀusion parameter recovery methods, EZ

can be used when the number of trials per cells is relatively small.

For edge correction, we used the half trial solution, i.e., for error

rates of 0, 50, or 100%, the actual error rate was modiﬁed with

the percent value of 0.5 trial, e.g., in a cell with 15 trials and 0%

error rate, the corrected error rate was 0.5/15, which is 3.33% (see

the exact details about edge correction in Wagenmakers et al.,

2007). The scaling within-trials variability of drift rate was set to

0.1 in line with the tradition of the diﬀusion analysis literature.

Drift rates for all number pairs and participants were calculated.

The mean drift rates of the participants (Figure 5) show a similar

pattern observed above for the former descriptive data. Fitting

the two predictions of the models, the association model shows

again a better ﬁt (Table 2). In addition, (a) the largest diﬀerence

between the goodness of ﬁt of the two models can be observed

FIGURE 5 | Drift rate values in the whole stimulus space.

for the drift rates (compared to the error rate and the reaction

time data) and (b) the highest R2value is found for the drift rates,

thereby suggesting that the drift rate indeed captures the diﬃculty

of the comparison tasks more sensitively than the error rates or

the reaction times do.

The analysis above supposed that the distance eﬀect (either

coming from the value model or from the association model)

is linear. However, a logarithmic or a similar function with

decreasing change as the distance increases might be a better

option to describe the data. First, one cannot suppose a linear

distance eﬀect, because after a suﬃciently large distance, the

reaction time should be unreasonably short or even negative,

which would not make sense. Second, in a former artiﬁcial

symbol comparison task, where the missing size eﬀect did

not inﬂuence the distance eﬀect, the distance eﬀect was better

described with the logarithm function than with a linear function

(unpublished results in Krajcsi et al., 2016). For these reasons,

the analysis of goodness of ﬁt was repeated with logarithmic

distance eﬀect models, in which the regressors were the natural

logarithm of the values of the previously used linear models seen

in Figure 2. The results (Table 2, logarithm model columns on

the right) show that (a) for all three data types (error rate, reaction

time, and drift rate), the association model ﬁts better than it did

with the linear regressor models and (b) the diﬀerences of the

two models are larger than they were for the linear regressor

models. Overall, the largest diﬀerence between the value and the

association models can be seen in the logarithm model versions

for the drift rates.

While our present main interest is the nature of the distance

eﬀect, it is worth to note that no size eﬀect can be found in the

data: The regressor formed as the sum of the two numbers to

be compared (e.g., the regressor value for the 3 vs. 4 number

pairs is 7) does not ﬁt either the error rates (R2=0.001), or the

reaction time (R2=0.01), or the drift rate (see below) data (R2

=0.001). These data replicate the results of Krajcsi et al. (2016),

thereby conﬁrming that in new symbols with equal frequency of

numbers in a comparison task, the size eﬀect does not emerge

and also conﬁrm that the distance and size eﬀects may dissociate.

Relatedly, we note that the size eﬀect could not inﬂuence the ﬁt

of the distance eﬀect not only because the size eﬀect could not

be demonstrated in the present data but also because the size

eﬀect regressor (sum of the numbers to be compared) does not

correlate with distance eﬀect regressor (diﬀerence of the numbers

to be compared) at all.

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Krajcsi and Kojouharova Symbolic Distance Effect as Association

Reliability of the Results

To investigate the reliability of the present results, two additional

experiments are summarized here: (a) the whole experiment

was repeated with another sample and (b) the data of a follow-

up study was analyzed where the same paradigm was used

with Indo-Arabic numbers instead of new symbols to see if

the distance eﬀect can follow the associations of the numbers

and small-large responses in an already well-established notation

(Kojouharova and Krajcsi, Submitted). (a) In the replication

study, 41 university students participated. Four of them were

excluded, either because they did not reach the required

maximum 5% error rate after 5 blocks of symbol learning

or because they used wrong response keys. Five additional

participants were excluded, because they had higher than 6.5%

error rate (which was the mean +standard deviation error

rate in that sample) in the comparison task. As a result, the

data of 32 participants were analyzed (mean age was 21.0 years,

3 males). The error rate, reaction time, and drift rate means

for the whole stimulus space can be seen in Figure 6, and the

R2s of the models with the appropriate hypothesis tests are

displayed in Table 3. While the reaction time and drift rate means

replicate the results of the main study (although the diﬀerence

was signiﬁcant only with the comparison of the group ﬁts,

but not with the hypothesis test choosing the better ﬁt for the

participants), the error rates show the superiority of the value

model. (b) In the Indo-Arabic comparison task, 23 university

students participated. One participant was dyscalculic whose data

were excluded from further analysis, and 2 further participants

were excluded for having an error rate higher than 5%. Therefore,

the data of 20 participants were analyzed (mean age was 20.15

years, 4 males). The goodness of ﬁt of the logarithmic models

and their contrast can be seen in Table 4. The Indo-Arabic study

replicated the results of the main study, and also in the error rates,

the association model ﬁtted signiﬁcantly better than the value

model.

Looking strictly at the signiﬁcance of the results, the

replication shows a somewhat diﬀerent result pattern as the ﬁrst

measurement, because in error rate, the signiﬁcant diﬀerences

support the value model instead of the association model, and

in reaction time and drift rate, not all hypothesis tests are

signiﬁcant. Clearly, some non-signiﬁcant eﬀects might reﬂect not

only due to the lack of an eﬀect but also due to the lack of

statistical power, and signiﬁcant eﬀects can also be type-I errors

(there is especially a chance for this, when replication studies

ﬁnd opposing signiﬁcant eﬀects). To evaluate the accumulated

data, a mini meta-analysis was run on the three set of data

(Maner, 2014). Binary random-eﬀects with the DerSimonian-

Laird method (Viechtbauer, 2010; Wallace et al., 2012) was

performed on the logarithm model ﬁt data measuring the ratio

of participants where the association model was better than

the linear model. While the error rate does not show a clear

preference for any models (45.9% mean preference for the

association model with 95% CI of [17.5, 74.2%]), reaction time

and drift rate clearly prefer the association model (76.6% with

CI of [65.0, 88.2%] for reaction time and 72.9% with CI [58.2,

87.7%] for drift rate). Taken together, while the reaction time

and drift rate show the superiority of the association model,

the results of the error rates are ambiguous. It is important

to highlight that from the viewpoint of the present question,

reaction time and especially drift rates are more relevant. First,

reaction time data are usually considered to be more reliable and

sensitive than error rate, because error rate and reaction time data

measure two strongly correlating constructs. Error rate measures

it in a dichotomous scale, whereas reaction time is a continuous

scale. Therefore, the latter have more information about the

trial performance. Second, drift rate measures the diﬃculty of

FIGURE 6 | Error rates (top left), reaction times (in ms, top right), and drift rates (bottom) in the whole stimulus space in the replication study.

Frontiers in Psychology | www.frontiersin.org 7November 2017 | Volume 8 | Article 2013

Krajcsi and Kojouharova Symbolic Distance Effect as Association

TABLE 3 | Goodness of ﬁt of the models (measured as R2) and comparison of the correlations (Difference column) for the error rates, reaction times, and drift rates

patterns based on the group average data, and hypothesis tests for choosing the better model based on the participants’ data in the replication study.

Linear model (Figure 2) Logarithm model

Value model

R2

Association

model R2

Difference of the

group ﬁts

Better model for the

participants

Value model

R2

Association

model R2

Difference of

the group ﬁts

Better model for

the participants

Error rate 0.791 0.629 Z=2.041,

p=0.041

T=130,

p=0.012

0.862 0.724 Z=2.081,

p=0.037

T=150,

p=0.033

Reaction

time

0.610 0.719 Z= −1.258,

p=0.208

T=233,

p=0.562

0.517 0.713 Z= −2.236,

p=0.025

T=196,

p=0.204

Drift rate 0.768 0.914 Z= −2.727,

p=0.006

T=232,

p=0.550

0.695 0.929 Z= −4.284,

p<0.001

T=191,

p=0.172

TABLE 4 | Goodness of ﬁt of the models (measured as R2) and comparison of the

correlations (Difference column) for the error rates, reaction times, and drift rates

based on the group average data, and hypothesis tests for choosing the better

model based on the participants’ data in the Indo-Arabic study (Kojouharova and

Krajcsi, Submitted).

Logarithm model

Value

model R2

Association

model R2

Difference of

the group ﬁts

Better model for

the participants

Error rate 0.634 0.825 Z= −2.766,

p=0.006

T=17,

p=0.001

Reaction

time

0.749 0.917 Z= −3.737,

p<0.001

T=14,

p<0.001

Drift rate 0.681 0.864 Z= −3.080,

p=0.002

T=31,

p=0.006

the task more sensitively than error rates or reaction times in

themselves (Wagenmakers et al., 2007; this is also conﬁrmed by

the usually higher R2values for drift rates than for reaction times

or error rates). Therefore, we consider that reaction times and

drift rates reliably reﬂect the superiority of the association model

over the value model. At the same time, it might be a question of

future research whether heterogeneous error rates are the result

of random noise or whether there are aspects of performance that

partly reﬂects the functioning of the value model.

To summarize the results, it was found that (a) the association

model described the distance eﬀect better than the value model;

it measured with reaction time and drift rate, while error rate

displayed an inconsistent pattern, (b) drift rate draws more

straightforward picture than the reaction time or the error rate

data, (c) logarithmic type distance eﬀect describes the data more

precisely than the linear distance eﬀect, and ﬁnally, (d) size

eﬀect is absent in the present paradigm with uniform number

frequency distribution.

DISCUSSION

The present work investigated whether the numerical distance

eﬀect is rooted in the values of the numbers to be compared

or in the association between the numbers and the small-large

properties. In a new artiﬁcial number notation with omitted

numbers, the distance eﬀect measured with reaction time and

drift rate did not follow the values of the numbers, as it would

have been suggested in the mainstream ANS model (Moyer

and Landauer, 1967; Dehaene, 2007) or in the value-based

explanation of the DSS model. Instead, the eﬀect reﬂected the

association between the numbers and the small-large categories,

as proposed by the association-based explanation of the DSS

model or by the delta-rule connectionist model of numerical

eﬀects (Verguts et al., 2005). Measured with error rate, the results

were not conclusive, so it is the question of additional studies

whether the inconsistency in the error rate data is simply noise

or there are additional aspects of the distance eﬀect that should

be investigated with more sensitive methods.

Together with the present results, several ﬁndings converge to

the conclusion that the symbolic number comparison task cannot

be explained by the ANS. First, unlike the prediction of that

model suggesting that distance and size eﬀects are two ways to

measure the single ratio eﬀect, symbolic distance and size eﬀects

are independent (Krajcsi, 2017), and the distance eﬀect can be

present even when no size eﬀect can be observed (shown in the

present results and in Krajcsi et al., 2016). Second, the size eﬀect

follows the frequency of the numbers as demonstrated in Krajcsi

et al. (2016) and also in the present results, where the uniform

frequency of the digits induced no size eﬀect (i.e., the slope of

the size eﬀect is zero). Third, the present data demonstrated that

the distance eﬀect is not directed by the values of the digits as

predicted by the ANS model, but they are inﬂuenced by the

frequency of the association with the small and large categories

(see also the extension of the present ﬁndings for Indo-Arabic

numbers in Kojouharova and Krajcsi, Submitted).

The present and some previous results also characterize

the symbolic numerical comparison task; an alternative model

should take the following into consideration: (a) symbolic

distance and size eﬀects are independent (Krajcsi et al., 2016;

Krajcsi, 2017), (b) the eﬀects are notation independent (the

present results and Krajcsi et al., 2016), (c) the size eﬀect depends

on the frequency of the numbers (the present results and Krajcsi

et al., 2016), (d) the distance eﬀect depends on the association

between the numbers and the small-large categories (present

results), and (e) the distance eﬀect can be described with a

logarithm of the diﬀerence of the values (present results).

It is again highlighted that these results are not the

consequence of the possibility that the new symbols are not

related to their intended values and that the independent

Frontiers in Psychology | www.frontiersin.org 8November 2017 | Volume 8 | Article 2013

Krajcsi and Kojouharova Symbolic Distance Effect as Association

series of symbols would create a performance pattern similar

to the association model prediction, because it was already

shown that the new symbols prime the Indo-Arabic numbers,

thereby revealing that the new symbols denote their intended

values (Krajcsi et al., 2016). The present ﬁndings were also

replicated with Indo-Arabic numbers (Kojouharova and Krajcsi,

Submitted).

From a methodological point of view, it is worth to note that

in the present comparison task, the drift rate seemed to be the

most sensitive index to describe performance, which strengthens

the role of the diﬀusion model analysis, among others in cases

when sensitivity and statistical power are essential.

To summarize, the results revealed that in an artiﬁcial number

notation where some omitted numbers might create a gap,

the distance eﬀect followed the association with the small-

large properties and not the values of the numbers. This result

contradicts the Analog Number System model and the value-

based DSS explanation, which suggests that the distance eﬀect is

directed by the values or the ratio of the numbers. On the other

hand, the result is in line with the alternative association-based

DSS explanation and the delta-rule connectionist model, in

which the distance eﬀect is directed by the association between

the number nodes and the small-large nodes.

ETHICS STATEMENT

All studies reported here were carried out in accordance

with the recommendations of the Department of Cognitive

Psychology ethics committee with written informed consent

from all subjects. All subjects gave written informed consent in

accordance with the Declaration of Helsinki.

AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual

contribution to the work, and approved it for publication. Both

authors contributed equally to this work.

ACKNOWLEDGMENTS

We thank Ákos Laczkó and Gábor Lengyel for their comments

on an earlier version of the manuscript.

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Conﬂict of Interest Statement: The authors declare that the research was

conducted in the absence of any commercial or ﬁnancial relationships that could

be construed as a potential conﬂict of interest.

Copyright © 2017 Krajcsi and Kojouharova. This is an open-accessar ticle distributed

under the terms of the Creative Commons Attribution License (CC BY). The use,

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author(s) or licensor are credited and that the original publication in this journal

is cited, in accordance with accepted academic practice. No use, distribution or

reproduction is permitted which does not comply with these terms.

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