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Article  |   October 2014
The effect of numerical magnitude on the perceptual processing speed of a digit
Author Affiliations
  • Shuang-Xia Li
    Department of Psychology and Behavioral Sciences, Zhejiang University, Hangzhou, China
    lsx440lsx@yeah.net
  • Yong-Chun Cai
    Department of Psychology and Behavioral Sciences, Zhejiang University, Hangzhou, China
    yccai@zju.edu.cn
Journal of Vision October 2014, Vol.14, 18. doi:10.1167/14.12.18
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      Shuang-Xia Li, Yong-Chun Cai; The effect of numerical magnitude on the perceptual processing speed of a digit. Journal of Vision 2014;14(12):18. doi: 10.1167/14.12.18.

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Abstract
Abstract
Abstract:

Abstract  In this study, we investigated whether the numerical information of a digit would affect perceptual processing speed for that digit. In Experiment 1, participants performed a temporal order judgment (TOJ) task in which they judged the order of two digits presented briefly to the left or right of the fixation point with a short asynchrony. The point of subjective simultaneity (PSS) was significantly shifted such that large numbers had to be presented before small numbers in order to be perceived as simultaneous, implying that small numbers are perceptually processed faster than large numbers. Given the susceptibility of a TOJ task to response bias, this result might have simply reflected the conceptual association between magnitude (e.g., small) and response selection (e.g., first). To exclude the potential influence of response bias, we adopted a simultaneity judgment (SJ) task in Experiment 2. Most participants in Experiment 2 had participated in Experiment 1. The participants judged whether the two digits were presented simultaneously or successively. The maximal possibility of simultaneous response was obtained when a large digit preceded a small digit by 5 ms, suggesting that small numbers were indeed perceived earlier than large numbers. Our findings indicated that small numbers were processed faster than large ones and that perceptual mechanisms contribute to this temporal advantage. In addition, although the TOJ and SJ task produced a similar processing speed advantage for small numbers, the PSSs of the two tasks were not correlated, which implied that different cognitive mechanisms were involved in the two tasks.

Introduction
Although temporal perception plays a critical role in our daily life as a basic cognitive ability, it is represented with limited precision and susceptible to many environmental and subjective factors. For example, there is increasing evidence that temporal perception can be modulated by numerical magnitude information, whereby either the perceived duration (Cappelletti, Freeman, & Cipolotti, 2009, 2011; Lu, Hodges, Zhang, & Zhang, 2009; Oliveri et al., 2008; Xuan, Zhang, He, & Chen, 2007) or the processing speed (Schwarz & Eiselt, 2009) of a number could be biased depending on the magnitude of the number, with smaller numbers perceived to last a shorter time and appear earlier than larger numbers. These findings have been interpreted as evidence supporting the hypothesis that representations of different magnitude dimensions (such as number, space, and time) share a common set of neural mechanisms (Bueti & Walsh, 2009; Walsh, 2003). 
The supporting evidence commonly comes from experiments using a comparison judgment paradigm, in which participants judge which of two stimuli has greater magnitude in one dimension (e.g., duration), and these judgments are usually modulated by the magnitude of another irrelevant dimension (e.g., numerical magnitude). However, this paradigm is prone to decisional biases; the interaction between different magnitude dimensions may reflect a genuine association between representations for different magnitude dimensions or simply be the result of a response bias (Schneider & Komlos, 2008). Many authors have cast doubts on the interpretation of previous findings related to magnitude interactions and argued that some results should be attributed to a response bias. For instance, Yates, Loetscher, and Nicholls (2012) examined the effect of numerical magnitude on duration perception by employing an “equality judgment” task, a task that requires participants to judge whether two stimuli are the same or different in terms of a specific dimension and is believed to be free from response biases. However, with this bias-free paradigm, they failed to confirm the previous finding that smaller numbers were perceived to appear for a shorter time. Therefore, the relationship between different magnitude dimensions needs to be further clarified through more sophisticated experimental paradigms. 
In this study, we attempted to examine whether the numerical information of a digit would affect perceptual processing speed. Schwarz and Eiselt (2009) have attempted to address this issue using a temporal order judgment (TOJ) task. In their experiments, two digits with different numerical magnitudes were presented asynchronously to the left and right side of a fixation point. Participants were required to judge the side (left or right) on which a digit appeared first. The results showed that participants made more correct responses for trials on which the small number (e.g., 1) preceded the large number (e.g., 9). The authors argued that this finding reflected different perceptual processing speed for digits of different magnitude, with smaller numbers transferring faster to a central onset comparison stage. However, because the TOJ task is a type of comparison judgment task, a response bias might have contributed to the TOJ effect. Indeed, after excluding the influence of response bias through the use of a simultaneity judgment (SJ) task (a type of equality task), Nicholls, Lew, Loetscher, and Yates (2011) found no difference between the perceived onset times for small and large numbers, suggesting that previous effects of numerical magnitude on temporal processing findings were mainly due to a response bias. 
Although Nicholls et al. (2011) did not find a difference in processing speed between small and large numbers, a number of studies have indicated that small numbers have an advantage over large ones in many tasks. For example, when required to produce numbers as randomly as possible (i.e., a random number generation [RNG] task), participants usually generate more small numbers than large ones (Boland & Hutchinson, 2000; Loetscher & Brugger, 2007; Rath, 1966), indicating a small number preference effect. Similarly, participants also show small number preference in a number bisection task: When participants are required to determine the numerical middle of a given digit pair, they generally respond with a number smaller than the true midpoint value (Göbel, Calabria, Farnè, & Rossetti, 2006; Loftus, Nicholls, Mattingley, Chapman, & Bradshaw, 2009; Longo & Lourenco, 2007). As the RNG and number interval bisection tasks mainly involve mental processing of number information at a high cognitive level, these results suggest that high-level neural processing may be modulated by small number preference effect. A recent study (Cai & Li, 2014) found that small numbers are more likely to attract visual attention than large numbers, implying that small number preference also exerts effects on early perceptual processes. Given the advantages of small numbers in various types of cognitive tasks, as well as the findings that a stimulus with attentional priority tends to be perceived as appearing earlier than an unattended stimulus (Nicholls & Yates, 2011; Schneider, & Bavelier, 2003; Shore, Spence, & Klein, 2001; Zampini, Shore, & Spence, 2005), small numbers should also have an advantage in perceptual processing speed. 
In the present study, we used a design similar to Nicholls et al. (2011) to further investigate how numerical magnitude affects processing speed of a digit. Two digits were presented to participants with a short asynchrony. In the experiments of Nicholls et al. (2011), the digit pair was displayed until the participants responded. This might lead to the two asynchronous digits lasting so long that the temporal difference (if any) in perception for the two digits could not be sensitively measured. Moreover, Cai and Li (2014) found that small number preference in capturing attention occurs only at short latencies after number onset, indicating that the magnitude effects are fast and transient. The temporal advantage for small numbers might also occur only in early phase poststimulus onset. To measure this possible early magnitude effect and to enhance the sensitivity of the measurement, a short stimulus presentation time (50 ms) was adopted in the present experiments. A TOJ task and an SJ task were used in the first and the second experiment, respectively. Most of the participants performed both of the two experiments, which allowed for a comparison of the correlations between the TOJ and SJ tasks. 
Method
Participants
Twenty-five right-handed students (16 females, mean age = 23 years, SD = 3.30) participated in this study. All participants had normal or corrected-to-normal eyesight and were naive to the purpose of the experiments. Sixteen of the participants performed both Experiments 1 and 2 and the other nine participants only performed Experiment 2. One participant was excluded from analysis in Exeriment 1 due to poor performance on the TOJ task and another one was excluded from Experiment 2 due to poor model fit. All procedures were approved by the Research Ethics Board of Zhejiang University. 
Apparatus and stimuli
Stimuli were generated using MATLAB and the Psychophysics Toolbox (Brainard, 1997; Pelli, 1997) and were displayed on a linearized 21-in. Dell UltraScan P1130 monitor (1024 × 768 resolution; 100-Hz refresh rate; Dell, Round Rock, TX). Participants viewed the monitor from a distance of 82 cm in an otherwise dark room. Participants' responses were collected using a parallel port response box. 
All stimuli appeared in black (0 cd/m2) against a gray background (35 cd/m2). A fixation point (0.33°) was always presented at the center of the screen. Number stimuli consisted of two Arabic digits (1.77° height, in Arial Black font), one with a small value and the other with a large value. The small digit was randomly selected from the set (1, 2) and the large digit from the set (8, 9), which resulted in four possible number pairs (1 and 8; 1 and 9; 2 and 8; and 2 and 9). Each digit was presented to either the left or right side of the fixation point at a distance of 3.5°. To equate the luminance of the two digits, the edges of each digit image were slightly adjusted such that the total pixels of each image were identical. 
Procedure
A schematic of a trial sequence is shown in Figure 1. Participants were seated in front of the monitor with their body midline and the response box aligned with the center of the screen. Each trial started with the presentation of the fixation point, and participants were asked to maintain fixation through the trial. After 500 ms, the two digits were briefly flashed for 50 ms, one to the left and the other to the right of the fixation point. The stimulus-onset asynchrony (SOA) between the two digits was 0, ±10, ±20, ±30, or ±50 ms. Positive SOAs indicated that the small digit was presented first, and negative SOAs indicated that the large digit was presented first. In Experiment 1, participants were required to judge the side on which the digit had first appeared (the TOJ task). In Experiment 2, the participants were required to indicate whether the digits were presented simultaneously or successively (the SJ task). Participants were informed that the numerical information was irrelevant to these tasks. Responses were registered by pressing one of two buttons on a response box with either the left or the right thumb, and no feedback was provided. Participants were encouraged to respond as accurately as possible within 5 s, although response speed was not stressed. 
Figure 1
 
A schematic of a trial sequence of Experiments 1 and 2. A numerically small and a numerically large digit were briefly presented to participants, one to the left and the other to the right of the fixation point. The SOA between the two digits was 0, ±10, ±20, ±30, or ±50 ms. Positive SOAs indicate that the small digit was presented first, and negative SOAs indicate that the large digit was presented first. In Experiment 1, participants judged whether the left or the right number was presented first (a TOJ task). In Experiment 2, participants judged whether the two digits were presented simultaneously (an SJ task). The time course of the two digits is shown at the bottom of the Figure.
Figure 1
 
A schematic of a trial sequence of Experiments 1 and 2. A numerically small and a numerically large digit were briefly presented to participants, one to the left and the other to the right of the fixation point. The SOA between the two digits was 0, ±10, ±20, ±30, or ±50 ms. Positive SOAs indicate that the small digit was presented first, and negative SOAs indicate that the large digit was presented first. In Experiment 1, participants judged whether the left or the right number was presented first (a TOJ task). In Experiment 2, participants judged whether the two digits were presented simultaneously (an SJ task). The time course of the two digits is shown at the bottom of the Figure.
In each experiment, each participant completed 432 trials, broken into four blocks. Each of the four digit pairs was randomly presented and repeated 12 times at each SOA, resulting in 48 trials for each SOA condition. For the 16 participants who performed both of the two experiments, they completed the two experiments in two sessions on separate days, and the order of the experiments was counterbalanced across participants. Prior to each experimental session, participants performed 60 practice trials corresponding to the subsequent experimental task. 
Results
Experiment 1: TOJ task
In Experiment 1, we aimed to replicate the temporal processing advantage for small numbers in a TOJ task, which was reported in previous studies (Nicholls et al., 2011; Schwarz & Eiselt, 2009). The main modification of the procedure, as compared with the original studies, was that the two digits were presented briefly (for 50 ms) instead of continuously appearing on the screen. 
The TOJ data was analyzed across all participants. The proportion of “small number first” responses was plotted as a function of SOA and fitted with a cumulative normal distribution function for each participant. Averaged data is shown in Figure 2. The R2 values for curve fitting ranged from 0.88 to 0.98, with a mean of 0.93, suggesting that the fit was good for each participant. The fitted functions were used to calculate the point of subjective simultaneity (PSS) for each participant by estimating the SOA value at which the fitted curve crossed 0.5 on the y-axis (i.e., the point where participants reported “small number first” and “large number first” equally often). Across all participants, the estimated PSS values ranged from −21.6 ms to 1.2 ms, with a mean of −7.1 ms (SE = 1.7). The negative mean PSS value suggests that the large number must be presented 7.1 ms prior to the small number for the numbers to be perceived as simultaneous. A one-sample t test using the point of subjective equality (PSE) values revealed that participants' PSS values significantly shifted from the true simultaneity point (i.e., 0 ms SOA) toward to the “large number first” condition, t(14) = −4.17, p = 0.001. Our results replicate previous findings that small numbers were judged to appear earlier in a TOJ task (Nicholls et al., 2011; Schwarz & Eiselt, 2009), which indicated a temporal processing advantage for small numbers over large numbers. 
Figure 2
 
Results of Experiment 1. The proportion of “small number first” responses plotted against the SOA. Negative SOAs reflect the presentation of numerically large numbers first, and positive SOAs refer to small numbers presented first. The solid line is the best-fitting cumulative normal distribution, averaged across participants. The arrow shows the mean PSS across participants. The shift towards negative SOA value indicates that numerically large numbers must precede small numbers for them to be perceived as simultaneous.
Figure 2
 
Results of Experiment 1. The proportion of “small number first” responses plotted against the SOA. Negative SOAs reflect the presentation of numerically large numbers first, and positive SOAs refer to small numbers presented first. The solid line is the best-fitting cumulative normal distribution, averaged across participants. The arrow shows the mean PSS across participants. The shift towards negative SOA value indicates that numerically large numbers must precede small numbers for them to be perceived as simultaneous.
There is extensive evidence for the spatial representation of numbers on a mental number line, with small numbers on the left side and large numbers on the right side (Hubbard, Piazza, Pine, & Dehaene, 2005; but see Gevers et al., 2010; Harvey, Klein, Petridou, & Dumoulin, 2013). Given the findings of an association between number and space, it is possible that the spatial layout of a number pair (e.g., small number to the left and large number to the right side of the fixation point, or the reverse) might influence the TOJ task. To examine this question, we compared the PSSs of spatially congruent (i.e., small number on the left and large number on the right) and spatially incongruent (i.e., large number on the left and small number on the right) conditions. The mean PSS for the congruent condition was significantly shifted toward the “large number first” SOAs, M = −19.5 ms, SE = 6.6, t(14) = −2.97, p = 0.01, while the mean PSS for the incongruent condition did not significantly deviate from the true simultaneity point, M = 2.4 ms, SE = 4.7, t(14) = 0.51, p = 0.62. There was a significant difference between the PSSs in the two conditions, t(14) = −2.19, p = 0.046. This result indicates that the space-number congruity could influence the numerical effect of the TOJ task, because small numbers were judged to appear earlier in the spatially congruent condition but not in the incongruent condition. This follows the same tendency found in the Nicholls et al. (2011) study, although the difference of the PSSs for the two congruity conditions did not reach significance in their study. However, an alternative explanation could also explain this effect of spatial layout of numbers. Sekuler, Tynan, and Levinson (1973) reported that in the TOJ task, participants tended to perceive the left stimulus as occurring first, when the two stimuli were actually presented together. This left-first benefit would lead to a more negative PSS when the small number was on the left (i.e., the congruent condition) and a more positive PSS when the small number was on the right (i.e., the incongruent condition). 
As documented in the abundant literature (Nicholls et al., 2011; Schneider & Komlos, 2008; Yates et al., 2012; Yates & Nicholls, 2011), the TOJ task (a comparison judgment task) is susceptible to the influence of decisional bias. Therefore, the leftward shift of PSS might either reflect a genuine temporal processing advantage for small numbers at the perceptual stage or response biases, in which the concept of early in temporal order was associated to the concept of small in numerical magnitude. To ascertain whether small numbers have the perceptual advantage in temporal processing, we conducted Experiment 2 using the SJ task, which is a type of equality task believed to be free from response biases (Nicholls et al., 2011; Schneider & Komlos, 2008; Yates et al., 2012; Yates & Nicholls, 2011). 
Experiment 2: SJ task
In Experiment 2, we further investigated the temporal advantage for small numbers by using the SJ task, a paradigm that eliminated the influence of response biases. This design is similar to that of experiment 2 in Nicholls et al. (2011), but in that experiment, no temporal effect of small number preference was found. Consistent with their procedure, we used a short stimulus presentation time (50 ms). As Cai and Li (2014) found that small numbers show an advantage in a perceptual task (i.e., visual attention task) only at the short SOA, the brief presentation of stimuli in the current study was expected to be helpful for the measurement of the temporal advantage for small numbers. 
The proportion of “simultaneous” responses was plotted as a function of SOA. Averaged data is shown in Figure 3. As seen from the figure, participants were prone to make “simultaneous” responses when large numbers were presented first than when small numbers were presented first. This was confirmed by a two-way ANOVA analysis (with the number order [large number first, small number first] and SOA [10, 20, 30, and 50 ms] as factors) showing a significant main effect of the number order, F(1, 23) = 31.62, p < 0.001. There was a significant interaction between the two factors, F(3, 69) = 4.59, p = 0.005, suggesting that this effect depended on SOA. Subsequent t tests demonstrated that the effect was significant or marginally significant at short SOAs (10 ms SOA: t[23] = 4.22, p < 0.001; 20 ms SOA: t[23] = 3.60, p = 0.002; 30 ms SOA: t[23] = 1.98, p = 0.06) but not at the long SOA (50 ms SOA: t[23] = −0.28, p = 0.79). 
Figure 3
 
Results of Experiment 2. The proportions of “simultaneous” responses plotted against the SOA. The solid line is the fitted psychometric function, averaged across all participants. The arrow shows the average PSS. The arrow shifts towards negative valves, indicating that numerically large numbers must precede small numbers in order for them to be perceived as simultaneous.
Figure 3
 
Results of Experiment 2. The proportions of “simultaneous” responses plotted against the SOA. The solid line is the fitted psychometric function, averaged across all participants. The arrow shows the average PSS. The arrow shifts towards negative valves, indicating that numerically large numbers must precede small numbers in order for them to be perceived as simultaneous.
To estimate PSS, the raw data was fitted with a smooth curve for each participant. As the function was not symmetric around the peak, a model-based psychometric function was used to fit the data (Alcalá-Quintana & García-Pérez, 2013; García-Pérez & Alcalá-Quintana, 2012). The index of the goodness-of-fit of the model for each participant was computed based on Pearson's chi-square test statistics χ2 and the log-likelihood ratio statistics G2. The χ2 and G2 values for curve fitting ranged from 0.78 to 6.10 and 0.78 to 6.09, respectively, with each p > 0.05, suggesting that the fit was good. The PSS value for each participant was derived from these fitted curves by estimating the SOA value at which the “simultaneous” response reached maximum value. Across all participants, the PSS value varied from −30.9 ms to 13.6 ms, with a mean value of −4.8 ms (SE = 1.6). The negative PSS value suggests that numerically large numbers must precede small numbers if they are to be perceived as simultaneous. A one-way t test revealed that the average PSS value was significantly different from the true simultaneity point (i.e., 0 ms SOA; t[23] = −2.95, p = 0.007. This result indicated that participants had a systematic tendency to perceive the numbers as simultaneous when numerically large numbers appeared first, rather than when the two numbers were actually presented simultaneously, suggesting that the temporal advantage for small numbers occurred even though the response bias was eliminated. 
Examination of number-space congruity revealed no significant difference between congruent and incongruent conditions, M = −3.3 ms, SE = 2.0 vs. M = −3.0 ms, SE = 2.6; paired-test, t(23) = −0.23, p = 0.91, which suggested that the spatial configuration did not exert an effect in the SJ task. This result was inconsistent with the result of the TOJ task, in which the PSS was modulated by the number-space congruity. This inconsistency implied that effects of number-space congruity depended on experimental tasks. This idea was supported by the finding of Turconi, Campbell, and Seron (2006) that the number-space congruity effect that the response to spatially ascending digit pairs (e.g., 2 5) was faster than to descending pairs (e.g., 5 2) occurred only in an order judgment task (e.g., 2 5; ascending or descending order?) but not in a magnitude comparison task (e.g., 2 5; which is larger?). Given the fact that the order-related processes were involved in the TOJ task but not in the SJ task, it is not surprising that the number-space congruity effect was observed only in the TOJ task. 
The present result is different from the findings of Nicholls et al. (2011); in that study, no effect of numerical magnitude on the perception of digit simultaneity in the SJ task was found. This discrepancy might be due to the use of different number pairs in the two studies: Nicholls et al. (2011) used 2 and 9 as stimuli, while we used 1 or 2 and 8 or 9 as stimuli. Because the digit 1 has a simple form that is dramatically different from other numbers, the present numerical effect might be attributed mainly to the contribution of number pairs including the digit 1. If this was the case, the present result could be due to the morphological differences in number pairs. To test this possibility, we separately analyzed data from number pairs either including 1 (i.e., number pairs 1, 8 and 1, 9) or not including 1 (i.e., number pairs 2, 8 and 2, 9). The temporal advantage for small numbers was found for both the number pairs, including 1: M = −3.9 ms, SE = 1.5, t(23) = −2.56, p = 0.018 and not including 1: M = −3.8 ms, SE = 1.2, t(23) = −3.23, p = 0.004, and there was no difference between them, t(23) = −0.07, p = 0.95. This result suggests that the temporal advantage for small numbers could not be explained by the morphological differences in the number stimuli. 
Discussion
A previous study reported that numerical information can influence judgments about the temporal order of two numbers, with the numerically smaller number perceived to occur earlier than the larger one (Schwarz & Eiselt, 2009). However, it is disputed whether this numerical effect on the time dimension reflected faster perceptual processing of small digits or a response bias, whereby participants were prone to associate small numbers with occurring first on a conceptual level. Schwarz and Eiselt (2009) found that both the advantage of processing speed for small numbers and the response bias contributed to the small-number-first benefit in a TOJ task. Nicholls and colleagues (2011), however, found no temporal processing advantage for small numbers using a response-bias free task (an SJ task). Therefore, Nicholls et al. suggested that previous findings might be mainly a result of response bias, and there were no actual differences in perceptual processing speed between numerically small and large numbers. 
In this study, we adopted the TOJ and SJ tasks to further investigate how numerical information affected the processing speed of a digit. In Experiment 1, we replicated the numerical effect found in previous research using the TOJ task (Schwarz & Eiselt, 2009; experiment 1 in Nicholls et al., 2011): Numerically larger numbers needed to precede smaller ones by approximately 7 ms to be perceived as presented simultaneously, even though the numerical information was irrelevant to the task. However, as discussed above, it was possible that this effect was simply due to a response bias. To eliminate the influence of the response bias, an SJ task was employed in Experiment 2, while the other design factors were held constant. Because responses of “simultaneous” and “successive” in the SJ task were orthogonal to numerical magnitudes, there would be no concept association that could cause a response bias. With this bias-free paradigm, the PSS occurred when large numbers preceded small numbers by approximately 5 ms. This finding indicated that numerically smaller numbers were perceived to occur earlier, even though the response biases were eliminated and suggested a genuine temporal processing advantage for small numbers. 
Our study was inspired by the work of Nicholls et al. (2011), in which no processing speed advantage was found for small numbers in the SJ task. The discrepancy between the current results and those of Nicholls et al. (2011) may be related to the major difference between the two studies: the digit presentation duration. In the Nicholls et al. study, the two digits were presented until a response was made. By contrast, in the present study, each number was presented only for 50 ms. The long stimuli presentation duration in the Nicholls et al (2011) study might mask potential subtle effects of numerical magnitude on temporal processing; therefore, the short presentation duration used in the current study might be more sensitive for the measurement of the numerical magnitude effect. Second, Cai and Li (2014) found that small numbers are more potent in attracting visual attention than larger numbers only at short latencies after number onset, suggesting that potential magnitude effects on perception are fast and transient. Therefore, the effects on temporal processing might only manifest when the numbers were presented briefly, as in the present study. 
In our study, the TOJ and SJ tasks were both used to measure temporal sensitivity. Majority of the participants (i.e., 16 of 25 participants) performed both of the two experiments, and a similar PSS was obtained for the two tasks (−7.1 ms in the TOJ task vs. −4.8 ms in the SJ task). It is intriguing to ask whether the similar PSS reflected common cognitive mechanisms underlying the two tasks. There is evidence that supports (Baron, 1969; Vroomen, Keetels, de Gelder, & Bertelson, 2004) and evidence that challenges (García-Pérez & Alcalá-Quintana, 2012; Van Eijk, Kohlrausch, Juola, & van de Par, 2008; Vatakis, Navarra, Soto-Faraco, & Spence, 2008) the idea that the TOJ and SJ tasks share common cognitive mechanisms. However, a correlation analysis for the present results indicated that the PSS estimates from the two tasks were not positively correlated (R = −0.39, p = 0.17), supporting the argument that the TOJ and SJ tasks measure different cognitive processes (Van Eijk et al., 2008; Vatakis et al., 2008). The TOJ task may involve more cognitive processing than the SJ task (Yates & Nicholls, 2011). For example, to successfully complete the TOJ task in the present study, participants had to associate the perceived temporal information (early vs. late) with the spatial information (left vs. right), whereas they only needed to process the temporal information (simultaneous or sequential) in the SJ task. In addition, as the TOJ task is susceptible to response biases, the PSS of the TOJ task might be contaminated by these biases. Moreover, most of the participants reported that the TOJ task was more difficult and required more effort to respond than in the SJ task. Therefore, the SJ task requires relatively less cognitive resources and appears more resistant to fatigue. We cannot determine which of the above factors explain the lack of correlation between the two tasks based on the current data. Nonetheless, we believe that SJ task is the optimal method for the measurement of perceptual synchrony because of its cognitive simplicity, bias-free characteristics, and ease of performance. 
It is worth noting that although the SJ task is an appropriate task for measuring the PSS, which was estimated according to the peak of simultaneous response curve, the height and the width of the curve may have been influenced by response criteria. For example, if a participant was biased to assume the numbers appeared together when he/she was not sure of the correct response, the simultaneous response option would be selected more often, leading to a shallow simultaneous response curve. This may explain why we obtained fewer simultaneous responses (with maximal proportion of 0.80) than did Nicholls et al. (2011) (with maximal proportion of 1). Our participants may have been more conservative in making simultaneous response. This could be a characteristic of the present participants or caused by the high difficulty of the task due to brief stimulus presentation. 
The present results indicated that there is an association between the magnitude of number and time, consistent with the theory that different magnitude dimensions (such as number, space, and time) are mediated by a common set of neural mechanisms (Walsh, 2003). One possible explanation for the faster processing speed associated with small numbers, suggested by Schwarz and Eiselt (2009), is that small numbers, located on the left side of the mental number line, are read out earlier than large numbers from iconic memory, in which visual information is usually read out in a left-to-right order. The earlier read-out small numbers would arrive at a central comparison stage earlier and thus be perceived as occurring first. However, it is worth noting that this explanation is inconsistent with the finding that there is no difference for the PSS between the number-space congruent and incongruent conditions in the SJ task. Another possibility is that visual attention may play a role in the temporal processing speed of small numbers. Small numbers have been found to be more likely to attract visual attention than large numbers (Cai & Li, 2014). Given the fact that the attended stimuli are processed faster than the unattended stimuli (Shore et al., 2001; Stelmach & Herdman, 1991; Yates & Nicholls, 2011), it is likely that the attentional preference for small numbers leads to the temporal processing advantage. These interpretations are not necessarily mutually exclusive, and the present study cannot determine the exact cognitive mechanism mediating the processing speed advantage for small numbers. 
Conclusions
The present study showed that numerical magnitude affects the perceptual processing speed of a digit. When two numbers with different numerical values were presented in rapid succession, the numerically larger number had to precede the smaller one in order to be perceived as simultaneous, indicating that small numbers are processed faster than large numbers. This processing speed advantage for small numbers was observed even when the response bias was eliminated, suggesting that the interaction between number and time occurs at a perceptual processing level. 
Acknowledgment
This work was supported by the Natural Science Foundations of China (31100731), Zhejiang Provincial Natural Science Foundation of China (LY13C090001), Zhejiang Province Education Department (Y201122208), and the Fundamental Research Funds for the Central Universities. 
Commercial relationships: none. 
Corresponding author: Yong-Chun Cai. 
Email: yccai@zju.edu.cn. 
Address: Department of Psychology and Behavioral Sciences, Xixi Campus, Zhejiang University, Hangzhou, China. 
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Figure 1
 
A schematic of a trial sequence of Experiments 1 and 2. A numerically small and a numerically large digit were briefly presented to participants, one to the left and the other to the right of the fixation point. The SOA between the two digits was 0, ±10, ±20, ±30, or ±50 ms. Positive SOAs indicate that the small digit was presented first, and negative SOAs indicate that the large digit was presented first. In Experiment 1, participants judged whether the left or the right number was presented first (a TOJ task). In Experiment 2, participants judged whether the two digits were presented simultaneously (an SJ task). The time course of the two digits is shown at the bottom of the Figure.
Figure 1
 
A schematic of a trial sequence of Experiments 1 and 2. A numerically small and a numerically large digit were briefly presented to participants, one to the left and the other to the right of the fixation point. The SOA between the two digits was 0, ±10, ±20, ±30, or ±50 ms. Positive SOAs indicate that the small digit was presented first, and negative SOAs indicate that the large digit was presented first. In Experiment 1, participants judged whether the left or the right number was presented first (a TOJ task). In Experiment 2, participants judged whether the two digits were presented simultaneously (an SJ task). The time course of the two digits is shown at the bottom of the Figure.
Figure 2
 
Results of Experiment 1. The proportion of “small number first” responses plotted against the SOA. Negative SOAs reflect the presentation of numerically large numbers first, and positive SOAs refer to small numbers presented first. The solid line is the best-fitting cumulative normal distribution, averaged across participants. The arrow shows the mean PSS across participants. The shift towards negative SOA value indicates that numerically large numbers must precede small numbers for them to be perceived as simultaneous.
Figure 2
 
Results of Experiment 1. The proportion of “small number first” responses plotted against the SOA. Negative SOAs reflect the presentation of numerically large numbers first, and positive SOAs refer to small numbers presented first. The solid line is the best-fitting cumulative normal distribution, averaged across participants. The arrow shows the mean PSS across participants. The shift towards negative SOA value indicates that numerically large numbers must precede small numbers for them to be perceived as simultaneous.
Figure 3
 
Results of Experiment 2. The proportions of “simultaneous” responses plotted against the SOA. The solid line is the fitted psychometric function, averaged across all participants. The arrow shows the average PSS. The arrow shifts towards negative valves, indicating that numerically large numbers must precede small numbers in order for them to be perceived as simultaneous.
Figure 3
 
Results of Experiment 2. The proportions of “simultaneous” responses plotted against the SOA. The solid line is the fitted psychometric function, averaged across all participants. The arrow shows the average PSS. The arrow shifts towards negative valves, indicating that numerically large numbers must precede small numbers in order for them to be perceived as simultaneous.
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