Embodied Magnitude Processing: On the Relation Between the SNARC Effect and Perceived Reachability

The SNARC Effect

A common representation for numbers and space and its link with action is also substantiated by the horizontal spatial numerical association of response codes (SNARC) effect: In tasks where participants have to judge the magnitude (e.g., smaller or greater than five) or the parity (e.g., odd or even) of a number the responses to small numbers are faster with the left response key (compared to the right response key), and vice versa (Dehaene et al., 1993). The SNARC effect has been shown for various number notations (e.g., Arabic digits or number words: Dehaene et al., 1993; Lindemann et al., 2007; Nuerk et al., 2005) and visual, auditory, and haptic stimuli (Dehaene et al., 1993; Lindemann et al., 2007; Nuerk et al., 2005). The SNARC effect is stable across modalities and number notations and, hence, is well-suited to investigate the sensorimotor grounding of numerical cognition in action.

Such a grounding would suggest that the SNARC effect is influenced by different action-relevant aspects, especially those related to the effectors (e.g., the hands). Surprisingly, no major influence of handedness on the SNARC effect (Cipora et al., 2019; Dehaene et al., 1993; Müller & Schwarz, 2007), and contradictory results for finger counting habits were found (Fabbri, 2013; Fischer, 2008; Tschentscher et al., 2012; see also Shaki & Fischer, 2021, they show that finger counting habits influence the association between space and magnitude).

However, previous studies have shown that the spatial arrangement of the effectors influences the SNARC effect. For example, response times are modulated when the response hands are crossed (Müller & Schwarz, 2007; Viarouge et al., 2014; Wood et al., 2006). Even though all three studies use different instructions and find different modulations of the SNARC effect by crossed hands, all authors indicate that the SNARC effect is influenced by spatial representational (a spatial representation serves as a reference frame to encode spatial information, i.e., a coordinate system) as well as hand-based (hands serve as a reference, i.e., the position of the left hand represents the left side of space) frames of reference. In most studies investigating the horizontal SNARC effect, the representational and hand-based frames of references are aligned, as responses on the left side are conducted with the left hand, and responses on the right side are conducted with the right hand. Hence, the horizontal (left and right side of the space) and the hand-based (left and right hand) SNARC effects are not separable in those settings.

Further studies have shown that the response layout can influence the spatial dimension along which numbers are mentally represented (Mourad & Leth-Steensen, 2017; Sixtus et al., 2019). SNARC effects were found not only on the horizontal (left-to-right) axis but also on the vertical (bottom-to-top) and sagittal (near-to-far) axes. Winter et al. (2015) pointed out that the vertical SNARC effects have been studied far less than horizontal/hand-based SNARC effects. However, several studies documented an association between small magnitudes and the bottom as well as large magnitudes and the top when responses are arranged vertically (Aleotti et al., 2020; Hartmann et al., 2014). Similarly, studies investigating sagittal SNARC effects indicate an association between small/large magnitudes and near/far space, respectively, when sagittal responses were required (Marghetis & Youngstrom, 2014; Santens & Gevers, 2008). Some studies even investigated SNARC effects on two or more axes at the same time (Aleotti et al., 2020; Aleotti et al., 2023; Hesse & Bremmer, 2017; Mourad & Leth-Steensen, 2017; see Winter et al., 2015 for a review) and indeed found co-existing SNARC effects on different axes.

Nevertheless, the SNARC effect does not always occur on the dimension along which the response buttons are arranged. For example, Wiemers et al. (2017) arranged response buttons on the vertical axis (one button was placed below and one on a tabletop). They found no vertical SNARC effect when the hand-to-button mapping changed only once, no matter whether the participants were standing or sitting. However, when they changed the hand-to-button mapping after every 16 trials, they found a vertical SNARC effect.

In typical fronto-parallel display setups, the sagittal and vertical axes can easily be confused. This happened, for example, in two studies that claimed to investigate spatial numerical associations on the vertical axis but presented stimuli in a sagittal arrangement on a table (Göbel, 2015) or mixed a vertical stimulus arrangement with a sagittal response arrangement (Li et al., 2017). In our opinion, it is essential to carefully distinguish vertical and sagittal arrangements when vertical and sagittal SNARC effects are investigated. One good possibility is to use virtual reality arrangements, such as the one used by Lohmann et al. (2018).

In this study, digits were presented at different sagittal distances in a virtual environment. Participants were instructed to judge whether the presented digits were smaller or larger than five (magnitude comparison) by closing either their right or left hand. Lohmann et al. (2018) found no sagittal SNARC effect but a modulation of the horizontal/hand-based SNARC effect by distance, with the largest SNARC effect for stimuli presented close to the virtual hands on the sagittal axis. This shows that both the effectors' spatial layout and the stimuli's spatial location influence the SNARC effect. It seems that the SNARC effect can be realized within body-centric (e.g., near and far from the body) as well as effector-centered (e.g., near and far from the hands) frames of reference (Wood et al., 2006). The absence of a sagittal SNARC effect might be explained by the horizontal response arrangement, which primed a horizontal number representation (for similar results with a vertical SNARC effect, see Gevers, Lammertyn, et al., 2006). In the horizontal response setting, the representational and hand-based frames of reference are aligned. The pattern found in the horizontal/hand-based SNARC effect implies enhanced processing of stimuli presented close to the hands. While responses to stimuli close to the hand were not generally faster or slower, the SNARC effect became more extreme regarding response time differences. More precisely, (congruent) responses with the left/right hand to small/large numbers were faster, while (incongruent) responses with the left/right hand to large/small numbers were slowed down when stimuli were presented close to the hands. This might be due to a higher salience for the space close to the hands (near hand effect; Reed et al., 2006) or the modulation of spatial attention for objects presented close to the hands (e.g., Tseng et al., 2012).

There are several additional factors influencing the SNARC effect. Most importantly, the SNARC effect is more pronounced in slower responses (Cipora et al., 2019; Wood et al., 2008). Further, the task itself influences the SNARC effect. In parity or magnitude comparison tasks, numbers are processed differently; hence, the SNARC effect differs (Georges et al., 2017; van Dijck et al., 2009). For example, the SNARC effect for parity judgment tasks can be described by a linear decrease in the response time superiority of left-side responses with magnitude. In contrast, the SNARC effect in magnitude comparison tasks can be described more accurately by a categorical predictor¹, which groups the numbers into those smaller and larger than the critical number (Gevers et al., 2010; see Weis et al., 2018 for comparisons of parity judgment and magnitude comparison tasks and Wood et al., 2008 for a meta-analysis). Moreover, in a parity judgment task, both magnitude and its relation to space are automatically activated (although irrelevant). In contrast, only the relation to space has to be automatically activated in a magnitude comparison task (Weis et al., 2018).

Also, personal factors such as gender (Bull et al., 2013; but see Weis et al., 2018), age (D. Hoffmann et al., 2014; Wood et al., 2008), and mathematical skills (D. Hoffmann et al., 2014) were found to influence the strength of the SNARC effect. Furthermore, participants who read from right to left show an attenuated or even reversed SNARC effect (Shaki & Fischer, 2014; Shaki et al., 2009; Zebian, 2005).

In conclusion, the SNARC effect allows for examining the grounding of number processing in actions. Although the SNARC effect is stable across modalities and number notations and hence seems to be rather amodal, it can be influenced by modal information, such as the effector placement and the distance of the presented number to the body or the effector. Presumably, the properties of the spatial representation of the space close to the effectors – the peripersonal space – can modulate the SNARC effect.

Peri- and Extrapersonal Space

There are several different definitions of peripersonal space. We follow the definition of Bufacchi and Iannetti (2018), who define peripersonal space as a set of continuous fields, which describe physiological reactions evoked by a stimulus' behavioral relevance. This relevance reflects whether contact with the stimulus is desirable (leading to a reaching movement, for instance) or not (resulting in a defensive movement). Bufacchi and Iannetti (2018) postulate graded response fields for the peripersonal space instead of in-or-out zones to account for the observations that indicators measuring the peripersonal space change gradually with distance to different body parts in all three dimensions. Such behavioral findings were corroborated on a neuronal level: Neurons were found that become increasingly active with stimuli presented at nearer distances (Canzoneri et al., 2012; Sambo & Iannetti, 2013). The activation of those fields depends on the intended interactions and the axes that are the focus of attention (Bufacchi & Iannetti, 2018). The space outside the peripersonal space, which cannot be directly acted upon with the body, is called the extrapersonal space (Cléry et al., 2015).

Lohmann et al. (2018) found that the SNARC effect is largest when digits are presented close to the hands and second largest in extrapersonal space. This suggests that the SNARC effect is not modulated by sagittal distance alone but specifically by the distance between the digit and the effector. The described results imply a close coupling between spatial numerical associations and bodily representations in general and peripersonal space in particular. If this is indeed the case, it would support the assumption of ATOM that the general magnitude system is rooted in or closely linked to sensorimotor experience and is not as amodal or abstract as sometimes suggested.

The Current Study

We investigated the hand-based and sagittal SNARC effects when stimuli are presented close to and far from the hands to further examine the relationship between spatial numerical associations and continuous spatial body representations. More precisely, we investigated the hand-based and sagittal SNARC effects modulations by the sagittal distance between the participant's hands, body, and stimuli. Such modulations would support the assumption that the SNARC effect is at least partially grounded in sensorimotor experience and would provide an example of modal influences on the SNARC effect.

In a previous study by Lohmann et al. (2018), the largest SNARC effect was found for stimuli presented close to the hands, suggesting a stronger dependence of the SNARC effect (in peripersonal space) on the sagittal distance between the digit and the hands compared to the sagittal distance between the digit and the body. However, those distances were confounded because the digits were presented at different sagittal distances while the positions of the hands and the body were fixed. To avoid confounding variables, the sagittal response arrangement in the current study distinguished between the influences of those distances.

To realize an immersive setup and a realistic depth impression, we used a similar virtual reality (VR) setup as Lohmann et al. (2018). Participants performed a magnitude comparison task (indicating whether a number is smaller or larger than five) with digits 1 to 4 and 6 to 9 presented at different distances on the sagittal axis. The magnitude comparison task was used for two reasons: First, magnitude judgment tasks seem related to spatial working memory instead of verbal working memory as in parity judgment (van Dijck et al., 2009). Second, only the relation to space has to be automatically activated rather than the magnitude and its relation to space, as in a parity judgment task (Weis et al., 2018). During the experiment, hand movements were tracked using markerless optical motion capture to allow participants to respond directly by closing their hand².

In contrast to Lohmann et al. (2018), who arranged the hands on the horizontal axis, in the current study, a sagittal response arrangement was realized where either the left or the right hand was close to the participants, while the other hand was more distant to the body. Hence, compared to Lohmann et al. (2018), the SNARC effect in the current study is not influenced by representational frames of reference as the responses are sagitally aligned. Participants were instructed to respond to digits with small or large magnitude with either the close or the far hand. So, the sagittal axis was used for response and stimulus arrangement, and hence, this axis was primed. Accordingly, we expected to find a sagittal SNARC effect (see also Gronau et al., 2017). Meanwhile, we expected an SNARC effect along the horizontal axis, as it was shown to be robust even if the effectors were arranged on another axis (Gevers, Verguts, et al., 2006; Wiemers et al., 2017). Given the background described above, we developed the following hypotheses:

We expected to find a sagittal SNARC effect: Responses to digits with a small magnitude are faster with the hand close to the body (compared to the hand distant from the body), and responses to digits with a large magnitude are faster with the hand distant from the body (compared to the hand close to the body). We predicted this effect to be influenced by the sagittal distance between the presented digits and the hands in the way that the greater the distance, the smaller the SNARC effect.

We expected to find a hand-based SNARC effect: Responses to digits with a small magnitude are faster with the left hand (compared to the right hand), and responses to digits with a large magnitude are faster with the right hand (compared to the left hand). We also expected the sagittal distance between the presented digit and the hands to influence this hand-based SNARC effect in the way that the greater the sagittal distance, the smaller the hand-based SNARC effect.

Results following the hypotheses would suggest that the SNARC effect and, therefore, the mental magnitude representation can be grounded in sensorimotor experience and influenced by the distance between stimuli and effectors. This would further support ATOM's assumptions that the general magnitude system is rooted in – or closely linked to – sensorimotor experience and is not as amodal or abstract as sometimes suggested.

We evaluated these hypotheses on the mean response times level as in standard SNARC analyses. Additionally, continuous motion tracking allowed us to obtain the maximum hand closure (MHC) of the incorrect hand for each trial and the according time (MHCT). These kinematic measures enabled us to evaluate response conflicts in the different experimental conditions by registering the strength of incorrect hand closure and its time, even when these conflicts do not lead to a wrong response.

Participants

An a priori power analysis revealed that for the hand-based SNARC effect – in terms of a two-way interaction between magnitude and response hand – a power of at least .90 and a partial eta-squared of ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .51 can be reached by analyzing the data of 22 participants. For the power analysis, a Monte Carlo Simulation was conducted based on the data of Lohmann et al. (2018). The respective R script and the data can be found in Supplementary Material S1 and S2. The results of Lohmann et al. (2018) were chosen as reference data because the experiment was also conducted in VR and within a similar setting. The study was preregistered (see Supplementary Materials).

Participants were recruited through internal university communication channels primarily. Consequently, the participants were probably mostly students. Only participants with normal or corrected to normal vision were recruited for the study. Participation was voluntary, and participants received monetary compensation or course credit in exchange for their participation. All participants provided written informed consent. The ethics committee for psychological research at the University of Tuebingen³ has approved the study protocol.

From 15.03.2022 to 30.06.2022, we collected the data from 31 participants to reach 22 participants with correct responses in at least 75% of the trials of any factor combinations. The included participants' ages ranged from 18 to 31 years (M = 23.32, SD = 3.18). Five participants reported to be male, 17 to be female, 20 reported to be right-handed, and two to be left-handed, and all participants read from left to right. All participants stated German as their mother tongue, except one who reported Spanish as their mother tongue. Three participants had already completed their master’s, and all the others had a high-school diploma. Their grades in math reached from 3 to 15 (M = 10.43, SD = 3.14; one participant did not report any math grade). Details of the assessments can be found in the Procedure section below.

Material and Apparatus

Design

We used a 4 (spatial displacements) × 2 (digit categories) × 2 (response hands, left and right) × 2 (response distances, close and distant hand position) factorial, within-subject design (see Figure 2 for an overview of the conditions). Therefore, we presented the digits 1 to 4 (small magnitudes) and 6 to 9 (large magnitudes) on the four distances 10 cm, 30 cm, 50 cm, and 70 cm in each block. We chose those distances instead of the ones used by Lohmann et al. (2018) for ergonomic reasons. We varied the response hands and the response distances across blocks. Participants responded with the left hand to digits smaller than five and the right hand to digits larger than five, or vice versa. Each of these response mappings was combined with two different hand positions. Either the right hand was held at a distance of 40 cm and the left at 20 cm, or vice versa. This resulted in different instructions for each of the four blocks. The order of the trials in a block was randomized, and the order of the blocks was balanced. Together, these factors resulted in 32 combinations repeated 24 times each. Thus, every participant performed 768 trials in four blocks with 192 trials each.

The influence of these factors on the response time, the MHC of the wrong hand, and the respective time of the MHCT were measured. The measure of MHC is meant to roughly reflect the degree of an involuntary response preparation (incorrect responses).

Procedure

The experiment took about 90 minutes. A video of the procedure can be found in Supplementary Materials S8. First, we obtained demographic information because there are some indications that these factors might influence the SNARC effect. We asked for age (self-report: open text field), gender (self-report: male/female/diverse), handedness (self-report: left/right), reading direction (self-report: from left to right/from right to left/both), education level (self-report: Hauptschulabschluss [secondary school leaving certificate]/Realschulabschluss [comprehensive school leaving certificate]/Abitur [high-school diploma]/apprenticeship/completed bachelor/completed master/Ph.D./other), last grade in mathematics (self-report: 0 – very bad to 15 – very good, like in the German high school exam grading pattern), finger counting habits (described below), and mother tongue (self-report: open text field). Finger counting habits were assessed using the instruction: "Please clap into your hands, then count with your fingers from 1 to 10 and indicate the order in which you counted with your fingers." followed by a picture of two hands similar to Wasner et al. (2014).

Verbal instruction regarding the VR equipment followed. Then, the HMD was put on, and the scene was calibrated. Therefore, the participants put their hands in the starting position, and the experimenter adjusted the position of the virtual hand such that the participants were in a comfortable seating and hand position. Afterward, the participants were asked to extend their right arm and touch a cardboard box with the tip of their middle finger. Participants were instructed that this position is the most distant reachable point. The position of the cardboard box corresponded to the outer limit of the radial tracking range and the sixth horizontal white line in the VR, which reflected the border of reachable space (see Figure 1). This provided a haptic indicator for the bounds of the task space.

After that, the sensorimotor exploration task and the magnitude comparison task were conducted. Both are described in detail below. Ultimately, participants were asked to complete a presence questionnaire (IPQ; Schubert et al., 2001) and a questionnaire to assess finger-counting habits (Fischer, 2008). The IPQ questionnaire⁵ is used to test for a sufficient level of immersion and spatial perception in VR, which is necessary to ensure a similar influence of the space and the body on number processing as in reality. We assessed finger counting habits with two different assessments to be able to analyze influences of situated cognition because Wasner et al. (2014) showed that the proportion of left- and right-starters differs considerably, depending on how you ask participants. In their (between-participant) study, 72% started with their right hand when they counted spontaneously. However, 62% reported starting with their left hand when they were given a finger-counting questionnaire and had their pen in their hand. Because of such considerable differences, we used both assessments.

Analysis

Response Times

The results of the repeated measures ANOVA on the response times are shown in Table 1. The ANOVA revealed significant interaction effects of response distance and digit category (sagittal SNARC effect; F(1, 21) = 4.48 p = .046, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .18) and response distance and spatial displacement (spatial congruency effect; F(3, 63) = 64.18, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .75).

The interaction between response distance and digit category indicates the expected sagittal SNARC effect, as responses to small numbers were faster with the near hand (M_near = 624.81 ms) compared to the far hand (M_far = 656.37 ms; t(21) = -4.28, p < .001, Cohen's d = 0.91), while responses to large numbers did not differ (M_near = 645.20 ms, M_far = 648.61 ms; t(21) = -0.37, p = .718, Cohen's d = 0.08). The influence of the spatial displacement of the numbers did not yield a significant interaction with the sagittal SNARC effect. The effect size of the sagittal SNARC effect was largest when numbers were presented in extrapersonal space (Figure 6; very near: t(21) = 1.45, p = .322, Cohen's d = 0.31; near: t(21) = 1.68, p = .322, Cohen's d = 0.36; far: t(21) = 1.59, p = .322, Cohen's d = 0.34; very far: t(21) = 2.37, p = .111, Cohen's d = 0.50).

The interaction between response distance and spatial displacement showed a significant response time advantage when numbers were presented in the very near space, and the response was conducted with the near hand (M_near = 626.57 ms, M_far = 713.33 ms; t(21) = -10.55, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 2.25). For all other spatial displacements, no difference was found (near: M_near = 633.31 ms, M_far = 647.23 ms; t(21) = 1.34, p = .390, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.42; far: M_near = 643.06 ms, M_far = 625.92 ms; t(21) = 2.53, p = .059 ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.54; very far: M_near = 636.88 ms, M_far = 626.45 ms; t(21) = 1.58, p = .389, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.34).

Further, main effects of response distance (M_near = 634.87 ms, M_far = 652.34 ms; F(1, 21) = 12.27, p = .002, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.37), number magnitude (M_small = 640.39 ms, M_large = 646.63 ms; F(1, 21) = 4.96, p = .037, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.19), and spatial displacement (M_verynear = 668.46 ms, M_near = 639.99 ms, M_far = 634.15 ms, M_veryfar = 631.54 ms; F(3, 63) = 24.67, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.54) were found. Further t-tests showed that the response times to numbers presented in the very near space were significantly larger compared to the response times to stimuli presented further away from the body (very near vs. near: t(21) = 5.85, p < .001, Cohen's d = 1.25; very near vs. far: t(21) = 6.89, p < .001, Cohen's d = 1.47; very near vs. very far: t(21) = 7.00, p < .001, Cohen's d = 1.49; all other tests: p > .117).

Linear Regression With Number Magnitude and Digit Category

Including number magnitude (1, 2, 3, 4, 6, 7, 8, or 9) in addition to the magnitude (small or large) in a linear model predicting response time revealed similar results as the ANOVA on the response times with two main differences: significant interactions between the digit category and target magnitude (F(1, 21) = 236.60, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.09) and response hand and response distance (F(1, 21) = 35.02, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.35). Further, marginally significant interactions between digit category, target magnitude, response hand, and response distance (interaction between horizontal and sagittal SNARC-like effects; F(1, 21) = 3.90, p = .053, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.06) and between digit category, target magnitude, response hand, and response distance (sagittal SNARC-like effects; F(1, 64) = 2.94, p = .091, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.04) were found. No significant interaction between the response hand and digit category on response time was found (horizontal SNARC-like effect; F(1, 21) = 0.00, p = .961, ${\mathrm{\eta }}_{\mathrm{p}}^2$ < 0.01). An overview of all results can be found in Table 2.

The analysis of the MHC did not show any significant predictions (p > .058). The analysis of MHCT only revealed a significant interaction effect between response distance and spatial displacement (spatial congruency effect; F(3, 63) = 12.54, p < .001, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.37). The respective main effects of spatial displacement (F(3, 63) = 5.43, p = .002, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.21), and response distance (F(1, 19) = 4.86, p = .039, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = 0.19) also got significant (all other tests p > .107)⁸. Hence, these analyses did not provide further information.

Explorative Results

Although not significant, the response times descriptively showed a hand-based SNARC pattern: Responses to small numbers were faster with the left compared to the right hand, and the response pattern to large numbers was reversed. So, while we do not wish to interpret non-significant results as significant, we want to notice that the non-significance of the results could also arise due to the low effect size and relatively low sample size. A further investigation of the hand-based SNARC for the different spatial displacements showed a marginally significant hand-based SNARC effect when the numbers were presented between the hands (t(21) = 2.43, p = .097, Cohen's d = 0.52; left bottom graphic in Figure 7).

When the numbers were presented very near and far from the body, the hand-based SNARC effect did not reach significance (very near: t(21) = 1.20, p = .535, Cohen's d = 0.26; far: t(21) = 1.34, p = .535, Cohen's d = 0.30; see Figure 7 left panels). When the numbers were presented very far from the body (extrapersonal space), the effect size was very low and far from any significance (t(21) = 0.28, p = .780, Cohen's d = 0.06).

Additionally, we calculated the mean response times for each participant per combination of digit category spatial displacement, response distance, and response hand. We calculated dRTs to further analyze the hand-based (dRT: response time of the right minus response time of the left hand) and sagittal (dRT: response time of the far minus response time of the near hand) SNARC effect. Based on both the sagittal and hand-based dRTs, we conducted a linear regression for each person for each spatial displacement, predicting the dRTs for each digit category. The differences in the resulting beta-coefficients of the sagittal and hand-based dRTs between different spatial displacements were tested using two separate ANOVAs. The beta-coefficients of the horizontal hand-based and sagittal dRTs were compared. Note that both analyses are based on the same data but with different dRT tailored to different questions. The mean beta coefficients are shown in Table 3. We further analyzed the quadratic contrasts for the ANOVAs.

Both the ANOVA of the hand-based (F(3, 63) = 2.07, p = .113, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .09) as well as the ANOVA of the sagittal (F(3, 63) = 0.30, p = .825, ${\mathrm{\eta }}_{\mathrm{p}}^2$ = .01) dRT found no difference between the beta-coefficients for different spatial displacements (see Table 3). The analysis of the quadratic contrasts of the hand-based dRT revealed a marginally significant result (t(63) = 1.71, p = .092), indicating that the hand-based SNARC was largest near the effectors and then decreased with increasing distance from the effectors (corroborating Lohmann et al., 2018, who found similar results). The analysis of the quadratic contrasts of the sagittal dRT was non-significant (t(63) = -0.77, p = .446). The beta-coefficients of the hand-based and sagittal dRTs did not differ (t(87) = 0.88, p = .379).

Implications

In contrast to Lohmann et al. (2018), we found a sagittal and no (or at best marginally) significant hand-based SNARC effect, with the only main change between these experiments being the response layout. These results support the assumption of a general magnitude system that is rooted in, or closely linked to, sensorimotor experience, as proposed by ATOM (Bueti & Walsh, 2009; Walsh, 2003), as well as a bidirectional mapping between motor commands and sensory changes, as described in common codes (Hommel et al., 2001). They even align with more general theories of task processing and the resource-oriented pre-activation of task sets for particular events (Butz, 2022; Frings et al., 2020; Heald et al., 2021; Heald et al., 2023). Such task sets may include numbers expected to be processed, which would be in line with the working memory account of the SNARC effect (van Dijck & Fias, 2011; van Dijck et al., 2009). Moreover, trial preparations may pre-activate possible stimulus locations in space, inducing a sagittal stimulus axis, the possible sagittal response locations, and horizontal left versus right response decisions.

Response-influencing interactions unfold on modal sensory- and motor-respective axes as well as on somewhat amodal, magnitude-respective axial arrangements. Thereby, a multitude of axial interactions applies, including purely spatial compatibility effects, which side-step the amodal magnitude component, and magnitude-dependent interactions, which reveal flexible bidirectional modal-to-amodal associations.

Conclusion

In this preregistered report, we intended to examine the proposed grounding of magnitude processing in the sensorimotor system. We initially expected to find a hand-based and a sagittal SNARC effect. Moreover, we expected both SNARC effects to decrease with a larger distance between digit and hands because we expected that reachability would influence the strength of the axial activities. In line with our expectations, we found a sagittal SNARC effect, but unexpectedly, the hand-based SNARC effect did not reach significance. These results further indicate the influence of the response layout on the mental orientation of the number representation. Meanwhile, purely spatial compatibility effects play a significant role, which we had not considered sufficiently when proposing the study.

Moreover, several of the results should be scrutinized further. For example, it should be investigated whether spatial compatibility effects systematically differ from magnitude-mediated effects. Do they yield respective response biases simply in a weighted, linear manner? Do the sagittal and the hand-based SNARC effect both occur with larger sample sizes? Do they interact? We believe that further experimental studies will help to answer these questions. However, probably only an actual process model implementation will be able to fully explain the observed effects on David Marr's computational and algorithmic levels (Marr, 1982), aiming at generating predictions about sequential trial dynamics (Butz, 2022; Heald et al., 2021; Heald et al., 2023; Steyvers et al., 2019; Zénon et al., 2019) and about the performance in related but different stimulus and response arrangements.