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Symbolic Numerical Distance Effect Does Not Reflect the Difference between Numbers

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Symbolic Numerical Distance Effect Does Not Reflect the Difference between Numbers

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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.
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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,
distribution or reproduction in other forums is permitted, provided the original
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.
Frontiers in Psychology | www.frontiersin.org 10 November 2017 | Volume 8 | Article 2013
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Learning mathematics requires fluency with symbols that convey numerical magnitude. Algebra and higher-level mathematics involve literal symbols, such as "x", that often represent numerical magnitude. Compared to other symbols, such as Arabic numerals, literal symbols may require more complex processing because they have strong pre-existing associations in literacy. The present study tested this notion using same-different tasks that produce less efficient judgments for different magnitudes that are closer together compared to farther apart (i.e., same-different distance effects). Twenty-four adolescents completed three same-different tasks using Arabic numerals, literal symbols, and artificial symbols. All three symbolic formats produced same-different distance effects, showing literal and artificial symbol processing of numerical magnitude. Importantly, judgments took longer for literal symbols than artificial symbols on average, suggesting a cost specific to literal symbol processing. Taken together, results suggest that literal symbol processing differs from processing of other symbols that represent numerical magnitude.
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In elementary symbolic number processing, the comparison distance effect (in a comparison task, the task is more difficult with smaller numerical distance between the values) and the priming distance effect (in a number processing task, actual number is easier to process with a numerically close previous number) are two essential phenomena. While a dominant model, the approximate number system model, assumes that the two effects rely on the same mechanism, some other models, such as the discrete semantic system model, assume that the two effects are rooted in different generators. In a correlational study, here we investigate the relation of the two effects. Critically, the reliability of the effects is considered; therefore, a possible null result cannot be attributed to the attenuation of low reliability. The results showed no strong correlation between the two effects, even though appropriate reliabilities were provided. These results confirm the models of elementary number processing that assume distinct mechanisms behind number comparison and number priming.
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Humans have the subjective impression of a rich perceptual experience, but this perception is riddled with errors that might be produced by top-down expectancies or failures in feature integration. The role of attention in feature integration is still unclear. Some studies support the importance of attention in feature integration (Paul & Schyns, 2003), whereas others suggest that feature integration does not require attention (Humphreys, 2016). Understanding attention as a heterogeneous system, in this study we explored the role of divided (as opposed to focused – Experiment 1) attention, and endogenous-exogenous spatial orienting (Experiments 2 and 3) in feature integration. We also explored the role of feature expectancy, by presenting stimulus features that were completely unexpected to the participants. Results demonstrated that both endogenous and exogenous orienting improved feature integration while divided attention did not. Moreover, a strong and consistent feature expectancy effect was observed, demonstrating perceptual completion when an unexpected perceptual feature was presented in the scene. These results support the feature confirmation account (Humphreys, 2016), which proposes that attention is important for top-down matching of stable representations.
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In the numerical Stroop task, participants are presented with two digits that differ in their numerical and physical size and are requested to respond to which digit is numerically larger. Commonly, slower responses are observed when the numerical distance between the digits is small (the distance effect) and when the numerical and physical size are incongruent (the size-congruency effect). The current study will use proportion manipulation, which consists of two experimental lists with high versus low frequency of trials belonging to different conditions, as a tool to reduce these effects. Specifically, it will be used to examine how these two interference effects depend on each other, and how a reduction of one effect will affect the other. In Experiment 1, the size-congruency proportions were manipulated; in Experiment 2, the distance proportions were manipulated. The results show that manipulating size-congruency proportions modulates the size-congruency effect but not the distance effect, while manipulating the distance proportions modulates the distance effect but not the size-congruency effect. These results demonstrate for the first time that the distance effect can be modulated by the distance proportions. Furthermore, these results indicate that proportion manipulation is specific and only modulates the variable being manipulated. Together, these results shed new light on the specificity of proportion manipulation in the context of numerical information processing. These results are further discussed in the context of various numerical models that suggest a different relationship between these effects and demonstrate how proportion manipulation can aid to investigate numerical processes.
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Lay abstract: Many families seeking early evaluations for autism spectrum disorder face long waitlists, must often travel to centers with appropriate expertise, and are frequently told by providers to "wait and see." This results in significant stress for families and delayed supports to infants and their caregivers who could benefit. This study evaluated whether telehealth could be used to identify and evaluate infants with early autism spectrum disorder characteristics in the first year of life. In this study, we evaluated 41 infants via telehealth using a standard set of probes and scored behavior related to social communication, play, imitation, and other developmental domains. We found the majority of infants demonstrated elevated likelihood of autism spectrum disorder on both parent-reported questionnaires and examiner-rated behavior. Caregiver ratings of the overall utility of the protocol used in this study were high. Overall, this study demonstrates the feasibility for telehealth-based approaches to evaluate infants' with elevated likelihood of autism spectrum disorder in the first year of life, which could help to improve families' access to care and to expand our capacity to conduct studies evaluating possible intervention supports.
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Human number understanding is thought to rely on the analog number system (ANS), working according to Weber’s law. We propose an alternative account, suggesting that symbolic mathematical knowledge is based on a discrete semantic system (DSS), a representation that stores values in a semantic network, similar to the mental lexicon or to a conceptual network. Here, focusing on the phenomena of numerical distance and size effects in comparison tasks, first we discuss how a DSS model could explain these numerical effects. Second, we demonstrate that the DSS model can give quantitatively as appropriate a description of the effects as the ANS model. Finally, we show that symbolic numerical size effect is mainly influenced by the frequency of the symbols, and not by the ratios of their values. This last result suggests that numerical distance and size effects cannot be caused by the ANS, while the DSS model might be the alternative approach that can explain the frequency-based size effect.
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The R environment provides a natural platform for developing new statistical methods due to the mathematical expressiveness of the language, the large number of existing libraries, and the active developer community. One drawback to R, however, is the learning curve; programming is a deterrent to non-technical users, who typically prefer graphical user interfaces (GUIs) to command line environments. Thus, while statisticians develop new methods in R, practitioners are often behind in terms of the statistical techniques they use as they rely on GUI applications. Meta-analysis is an instructive example; cutting-edge meta-analysis methods are often ignored by the overwhelming majority of practitioners, in part because they have no easy way of applying them. This paper proposes a strategy to close the gap between the statistical state-of-the-science and what is applied in practice. We present open-source meta-analysis software that uses R as the underlying statistical engine, and Python for the GUI. We present a framework that allows methodologists to implement new methods in R that are then automatically integrated into the GUI for use by end-users, so long as the programmer conforms to our interface. Such an approach allows an intuitive interface for non-technical users while leveraging the latest advanced statistical methods implemented by methodologists.
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Numerical and non-numerical order processing share empirical characteristics (distance effect and semantic congruity), but there are also important differences (in size effect and end effect). At the same time, models and theories of numerical and non-numerical order processing developed largely separately. Currently, we combine insights from 2 earlier models to integrate them in a common framework. We argue that the same learning principle underlies numerical and non-numerical orders, but that environmental features determine the empirical differences. Implications for current theories on order processing are pointed out. (PsycINFO Database Record (c) 2014 APA, all rights reserved).
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In Experiment 1, participants classified a 1st number ( prime) as smaller or larger than 5 and then performed the same task again on a 2nd number ( target). In Experiment 2, participants classified a target number as smaller or larger than 5, while unknown to them a masked number was displayed for 66 ms prior to the target. Primes and targets appeared in Arabic notation, in verbal notation, or as random dot patterns. Two forms of priming were analyzed: quantity priming (a decrease in response times with the numerical distance between prime and target) and response priming (faster responses when the prime and target were on the same side of 5 than when they were not). Response priming transferred across notations, whereas quantity priming generally did not. Under conditions of speeded processing, the internal representation of numerical quantities seems to dissociate into multiple notation-specific subsystems. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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Interference between number magnitude and other properties can be explained by either an analogue magnitude system interfering with a continuous representation of the other properties or by discrete, categorical representations in which the corresponding number and property categories interfere. In this study, we investigated whether parity, a discrete property which supposedly cannot be stored on an analogue representation, could interfere with number magnitude. We found that in a parity decision task the magnitude interfered with the parity, highlighting the role of discrete representations in numerical interference. Additionally, some participants associated evenness with large values, while others associated evenness with small values, therefore, a new interference index, the dual index was introduced to detect this heterogeneous interference. The dual index can be used to reveal any heterogeneous interference that were missed in previous studies. Finally, the magnitude-parity interference did not correlate with the magnitude-response side interference (Spatial-Numerical Association of Response Codes [SNARC] effect) or with the parity-response side interference (Markedness Association of Response Codes [MARC] effect), suggesting that at least some of the interference effects are not the result of the stimulus property markedness.
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Numerical distance and size effects (easier number comparison with large distance or small size) are mostly supposed to reflect a single effect, the ratio effect, which is the consequence of the analogue number system (ANS) activation, working according to Weber’s law. In an alternative model, symbolic numbers can be processed by a discrete semantic system (DSS), in which the distance and the size effects could originate in two independent factors: the distance effect depends on the semantic distance of the units, and the size effect depends on the frequency of the symbols. While in the classic view both symbolic and nonsymbolic numbers are processed by the ANS, in the alternative view only nonsymbolic numbers are processed by the ANS, but symbolic numbers are handled by the DSS. The current work contrasts the two views, investigating whether the size of the distance and the size effects correlate in nonsymbolic dot comparison and in symbolic Indo-Arabic comparison tasks. If a comparison is backed by the ANS, the distance and the size effects should correlate, because the two effects are merely two ways to measure the same ratio effect, however, if a comparison is supported by other system, for example the DSS, the two effects might dissociate. In the current measurements the distance and the size effects correlated very strongly in the dot comparison task, but they did not correlate in the Indo-Arabic comparison task. Additionally, the effects did not correlate between the Indo-Arabic and the dot comparison tasks. These results suggest that symbolic number comparison is not handled by the ANS, but by an alternative representation, such as the DSS. Find the postprint at the project's website at https://sites.google.com/site/mathematicalcognition/home/discrete-semantic-system
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A number of scholars recently have argued for fundamental changes in the way psychological scientists conduct and report research. The behavior of researchers is influenced partially by incentive structures built into the manuscript evaluation system, and change in researcher practices will necessitate a change in the way journal reviewers evaluate manuscripts. This article outlines specific recommendations for reviewers that are designed to facilitate open data reporting and to encourage researchers to disseminate the most generative and replicable studies. These recommendations include changing the way reviewers respond to imperfections in empirical data, focusing less on individual tests of statistical significance and more on meta-analyses, being more open to null findings and failures to replicate previous research, and attending carefully to the theoretical contribution of a manuscript in addition to its methodological rigor. The article also calls for greater training and guidance for reviewers so that they can evaluate research in a manner that encourages open reporting and ultimately strengthens our science. © The Author(s) 2014.
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In this chapter, I put together the first elements of a mathematical theory relating neuro- biological observations to psychological laws in the domain of numerical cognition. The starting point is the postulate of a neuronal code whereby numerosity—the cardinal of a set of objects—is represented approximately by the firing of a population of numerosity detectors. Each of these neurons fires to a certain preferred numerosity, with a tuning curve which is a Gaussian function of the logarithm of numerosity. From this log- Gaussian code, decisions are taken using Bayesian mechanisms of log-likelihood compu- tation and accumulation. The resulting equations for response times and errors in classical tasks of number comparison and same-different judgments are shown to tightly fit behavioral and neural data. Two more speculative issues are discussed. First, new chronometric evidence is presented supporting the hypothesis that the acquisition of number symbols changes the mental number line, both by increasing its precision and by changing its coding scheme from logarithmic to linear. Second, I examine how symbolic and nonsymbolic representations of numbers affect performance in arithmetic compu- tations such as addition and subtraction.
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Attempted to discriminate between 2 theories concerning the strategies Ss use when trying to remember a linear ordering. Rating-scale theory argues that Ss represent the items to be learned as points along an imaginary spatial continuum. End-term anchoring theory suggests that serial orderings are learned from the end points inward. A scaling analysis of proportions correct in an experiment with 40 undergraduates suggests that neither theory alone is sufficient. The Ss apparently represented ordering information along 2 dimensions, the 1st representing the distance between items on the imaginary continuum and the 2nd representing the distance of the items from the center of the ordering. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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In psychological research, it is desirable to be able to make statistical comparisons between correlation coefficients measured on the same individuals. For example, an experimenter (E) may wish to assess whether 2 predictors correlate equally with a criterion variable. In another situation, the E may wish to test the hypothesis that an entire matrix of correlations has remained stable over time. The present article reviews the literature on such tests, points out some statistics that should be avoided, and presents a variety of techniques that can be used safely with medium to large samples. Several numerical examples are provided. (18 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)