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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

This article was submitted to Cognition, a section of the journal Frontiers in Psychology

†These authors have contributed equally to this work.

This is an open-access article 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.

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.

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,

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,

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.,

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.,

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

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.,

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

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.

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

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% |

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

The expected distance effect pattern for the stimulus space used in the present study based on the value explanation

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.,

To summarize, the present study investigates whether the distance effect follows the distance of the values of the numbers (left of Figure

In the present experiment, participants learned new symbols (Figure

Tasks in the new symbol experiment.

The new symbols were chosen from writing systems that were mostly unknown to the participants (e.g.,

The participants first learned new symbols for the numbers between 1 and 3, and between 7 and 9 (Figure

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

Presentation of the stimuli and measurement of the responses were managed by the PsychoPy software (Peirce,

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).

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.

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

Error rates

To test the results statistically, we first fit the two predictions of the models (Figure ^{2} was calculated (^{2} columns given in Table ^{2} values for every single participant for both the value and association models, and the ^{2} 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

Goodness of fit of the models (measured as ^{2}) 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.

^{2} |
^{2} |
^{2} |
^{2} |
|||||
---|---|---|---|---|---|---|---|---|

Error rate | 0.709 | 0.708 | 0.714 | 0.821 | ||||

Reaction time | 0.543 | 0.790 | 0.457 | 0.817 | ||||

Drift rate | 0.526 | 0.861 | 0.425 | 0.874 |

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

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} value too, although the ^{2} should be smaller than the ^{2} of the appropriate model.

Results for the goodness of fits (Table

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, ^{2} value 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.

Drift rate values in the whole stimulus space.

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.,

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 (^{2} = 0.001), or the reaction time (^{2} = 0.01), or the drift rate (see below) data (^{2} = 0.001). These data replicate the results of Krajcsi et al. (

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 ^{2}s of the models with the appropriate hypothesis tests are displayed in Table

Error rates

Goodness of fit of the models (measured as ^{2}) 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.

^{2} |
^{2} |
^{2} |
^{2} |
|||||
---|---|---|---|---|---|---|---|---|

Error rate | 0.791 | 0.629 | 0.862 | 0.724 | ||||

Reaction time | 0.610 | 0.719 | 0.517 | 0.713 | ||||

Drift rate | 0.768 | 0.914 | 0.695 | 0.929 |

Goodness of fit of the models (measured as ^{2}) 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).

^{2} |
^{2} |
|||
---|---|---|---|---|

Error rate | 0.634 | 0.825 | ||

Reaction time | 0.749 | 0.917 | ||

Drift rate | 0.681 | 0.864 |

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, ^{2} values 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.

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,

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,

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.,

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 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.,

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.

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.

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.

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.

We thank Ákos Laczkó and Gábor Lengyel for their comments on an earlier version of the manuscript.