Weber's law is a fundamental psychophysical principle. It states that the just noticeable difference (JND) between stimuli increases with stimulus magnitude; consequently, larger stimuli should be estimated with larger variability. However, visually guided grasping seems to violate this expectation: When repeatedly grasping large objects, the variability is similar to that when grasping small objects. Based on this result, it was often concluded that grasping violated Weber's law. This astonishing finding generated a flurry of research, with contradictory results and potentially far-reaching implications for theorizing about the functional architecture of the brain. We show that previous studies ignored nonlinearities in the scaling of the grasping response. These nonlinearities result from, for example, the finger span being limited such that the opening of the fingers reaches a ceiling for large objects. We describe how to mathematically take these nonlinearities into account and apply this approach to our own data, as well as to the data of three influential studies on this topic. In all four datasets, we found that—when appropriately estimated—JNDs increase with object size, as expected by Weber's law. We conclude that grasping obeys Weber's law, as do essentially all sensory dimensions.

*SD*

_{MGA}) as a proxy for the corresponding JND and found that

*SD*

_{MGA}does not increase with object size. From this they concluded that Weber's law is violated in grasping.

^{1}Again, Ganel et al. (2008) calculated the within-subjects standard deviations of each of these responses (

*SD*

_{Response}) as a proxy for the corresponding JND and found for both perceptual tasks that

*SD*

_{Response}did increase with object size. From this, they concluded that Weber's law holds for perceptual tasks—in accordance with the well-known ubiquity of Weber's law in most sensory dimensions (Teghtsoonian, 1971).

*SD*

_{Response}) as a proxy for the corresponding JND, as did subsequent studies on Weber's law in grasping. However, we will show that this approach is only valid if there is a perfectly linear relationship between stimulus and response. Any small nonlinearity will make this approach problematic and can lead to erroneous conclusions. For ease of exposition, we will focus on grasping, where the typically measured response is MGA, such that the within-subjects standard deviation of the response is

*SD*

_{MGA}, but all our arguments apply equally well to other tasks and responses. For the sake of generality, we use the term

*SD*

_{Response}to subsume multiple possible responses but refer to

*SD*

_{MGA}in concrete cases of grasping. Before describing why it is problematic to use

*SD*

_{MGA}as a proxy for JND, we first need to describe a more general problem.

*identical*to the object size. For example, it is well known that the MGA is always larger than the to-be-grasped object, such that there is a safety margin (cf. Uccelli, Pisu, & Bruno, 2021) that prevents the fingers from colliding with the object (for a laborious measurement of this response function, see Figure 6a of Hesse & Franz, 2009; for a comprehensive review, see Figure 6a of Smeets & Brenner, 1999).

*erroneously equating the response with the stimulus*, because they would be confusing the response (here, MGA) with the stimulus (here, physical object size that prompted the motor system to prepare the grasp) by implicitly assuming that those two were identical. Instead, what the researchers should do is to

*invert*the response function. Whereas they first measured MGA as a function of object size, now they needed to calculate object size as a function of MGA in order to

*find the stimulus that elicited the response*. We will see that a similar problem exists when

*SD*

_{MGA}is used as proxy for JND.

*SD*

_{MGA}used as proxy for JND?

*compares*two objects of different sizes and decides whether they are of equal or different sizes. The JND is then the difference in physical sizes between the two objects at which the participant responds 50% of the time “different” and 50% “equal.” Weber's law states that the JND is (roughly) proportional to the absolute size of the object. However, such a comparison is not possible in grasping and even less so in “natural grasping” (target-oriented grasping; see Goodale, Jakobson, & Keillor, 1994) as is prescribed by the perception–action model if one wants to measure the dorsal stream (see General discussion). These problems arise because grasping is typically targeted at a single object and does not easily allow for a comparison of two objects.

*SD*

_{MGA}for grasping) as a proxy for JND. Following their lead, subsequent studies also based their investigations of Weber's law in grasping on

*SD*

_{MGA}or on similar measures (see General discussion). However, we will show that this choice again constitutes an erroneous equalization of response and stimulus because

*SD*

_{MGA}is related to the variability in the

*response*, whereas JND is related to the variability of the

*stimulus*(it gives us the amount we would have to change the physical object until this change is detected).

*s*

_{1}and responds with a certain MGA

_{1}and

*SD*

_{MGA1}(Figure 1). Now, we increase the stimulus until this change can be detected in the response. For this detection, we have to set a certain threshold, but this is not critical and standard practice in signal detection models. Therefore, let us set the threshold to ±1

*SD*

_{MGA1}, such that we consider the physical size change as being detected when it creates a change in MGA that corresponds to 1

*SD*

_{MGA1}(our results would not change had we used a factor other than 1 here). Now, all we have to do is determine how much we have to change the stimulus to create this 1-

*SD*

_{MGA1}change. To determine this, we have to invert the response function and map

*SD*

_{MGA1}back to the stimuli. Let us call the change in stimulus size that is needed to elicit a 1-

*SD*

_{MGA1}change the \({\widehat {JND}_1}\) (for “estimated JND”). This \({\widehat {JND}_1}\) tells us how much we would need to change the stimulus

*s*

_{1}to detect a 1-

*SD*

_{MGA1}change in the response. Appendix A gives the corresponding math: All one has to do is to calculate at every object size the local slope (the first derivative) of the response function that relates object size to MGA and then to divide the

*SD*

_{MGA1}by this local slope to obtain \({\widehat {JND}_1}\) (the math is similar to the famous error propagation formula in statistics). We then can use \({\widehat {JND}_1}\) as a proxy for the JND for this stimulus.

*SD*

_{MGA}directly as a proxy for JND. Thereby, they equated the MGA with the physical stimulus size, erroneously equating response and stimulus. In a nutshell: Their intuition to use an

*SD*as a proxy for JND is acceptable, but they used the wrong

*SD*—at the level of the response, rather than the stimulus.

*SD*

_{MGA}instead of \(\widehat {JND}\) would not have dramatic consequences if the response function that relates object size to MGA were perfectly linear. In that case, all local slopes are equal (i.e., for each object size the response function has the same slope) and the transformation from

*SD*

_{MGA}to \(\widehat {JND}\) is always by the same constant factor (because we always divide by the same slope). Therefore,

*SD*

_{MGA}could still be used as a proxy for JND when one wanted to assess Weber's law. However, in grasping the response function is not linear. This slight nonlinearity has relatively large effects when trying to use

*SD*

_{MGA}as a proxy for JND instead of \(\widehat {JND}\).

*s*

_{2}in Figure 1. This stimulus has exactly the same

*SD*

_{MGA}as the stimulus with size

*s*

_{1}, but, because the response function is slightly bent, we have to change the physical size of

*s*

_{2}much more than that of

*s*

_{1}to achieve the same effect on the MGA. Although

*SD*

_{MGA2}and

*SD*

_{MGA1}are identical (which would be interpreted by Ganel et al., 2008 as a violation of Weber's law), \({\widehat {JND}_2}\) is much larger than \({\widehat {JND}_1}\)—just as expected by Weber's law!

*SD*

_{MGA}is even smaller for large than for small stimuli (Bruno, Uccelli, Viviani, & Sperati, 2016; Löwenkamp et al., 2015; Utz et al., 2015), and nevertheless Weber's law could still hold (i.e., \(\widehat {JND}\) could still increase as predicted by Weber's law). All this can only be tested and decided when the appropriate proxy for JND is used.

*SD*

_{MGA}is used as a proxy for JND. To avoid this pitfall, researchers need to first calculate \(\widehat {JND}\) and use this as a proxy for JND. Only then does it make sense to draw inferences about Weber's law. In the following, we will apply this approach to grasping and manual estimation using four different datasets: Experiment 1 consists of newly collected data using a design that was specifically optimized for this purpose. Then, we present three reanalyses of two previously published studies (Heath et al., 2011; Löwenkamp et al., 2015) and the pioneer study on this subject (Ganel et al., 2008).

*k*is between 0.02 and 0.06, as we would expect from classic studies) (cf. McKee & Welch, 1992; Teghtsoonian, 1971).

*N*= 20 participants. We calculated \(\widehat {JND}\) as described above and in Appendix A. In a nutshell, at each object size and for each participant, we divided the within-subjects standard deviation of the responses (

*SD*

_{Response}) by the local slope of the response function (Figure 1), resulting in \(\widehat {JND}\). Weber's law holds when \(\widehat {JND}\) increases linearly with object size.

*g*(

*s*) =

*a*+

*bs*+

*cs*

^{2}) were fitted for each participant (Appendix A). To allow for a meaningful interpretation of the linear term

*b*of the quadratic regression, the predictor (i.e., size) was centered on its mean, such that the linear term

*b*of the quadratic regression describes the slope at the mean object size (i.e., 35 mm) and equals the slope

*b*of a simple linear regression. \(\widehat {JND}\) was calculated by dividing the within-subjects standard deviation of the response by the local slope of the participant's individual quadratic regression function at each size for each participant (Appendix A). Linear regressions relating \(\widehat {JND}\) to object size were then fitted for each participant in order to assess Weber's law. The linear regression allowed for a non-zero intercept. Strictly speaking, Weber's law does not include an intercept. However, it is known that the generalized form of Weber's law (which includes an intercept) is a better descriptor of behavior, and it is standard practice to model Weber's law with a non-zero intercept (Baird & Noma, 1978; Brown, Galanter, Hess, & Mandler, 1962; Miller, 1947). We discuss this issue further below.

*p*values of 0.001 or less are depicted as

*p*< 0.001. Between-participant means and corresponding standard errors are depicted as mean ± 1

*SEM.*We also report 95% confidence intervals (CIs) for the slopes of the linear regression of

*SD*s and \(\widehat {JND}\)s on object size (for the studies where we have the full data).

*b*

_{MGA}= 1.04 ± 0.013, and the quadratic coefficient was

*c*

_{MGA}= −0.0037 ± 0.0005 mm

^{−1}. The residuals of the linear and quadratic fits are depicted in Appendix B. The residuals of the linear fit indicate a systematic relationship, which disappears in the residuals of the quadratic fit. The sign of the quadratic coefficient was negative, indicating a concave relationship between MGA and object size: The responsiveness of MGA decreased with increasing object size.

*SD*

_{MGA}did not scale with object size,

*b*= 0.002 ± 0.010,

*t*(19) = 0.18,

*p*= 0.859, 95% CI, −0.018 to 0.022 (Figure 3c). Thus, we replicated the finding of previous studies that the uncertainty of the response in grasping does not increase with object size. Based on such a result, it would often be concluded that grasping violated Weber's law (Ganel et al., 2008). However—as we have shown above—this conclusion would be premature. We first have to calculate \({\widehat {JND}_{{\rm{MGA}}}}\) before we can assess Weber's law. To do this, we need to divide the SD

_{MGA}by the local slope of the response function (computed from quadratic regression on individual participants) at that object size. These slopes are shown in Figure 3d.

*k*

_{MGA}= 0.040 ± 0.013,

*t*(19) = 3.18,

*p*= 0.005, 95% CI, 0.014–0.066 (Figure 3e). This value of Weber's constant fits nicely within the expected range from the literature for size perception (0.02–0.06) (McKee & Welch, 1992; Teghtsoonian, 1971). Thus, when using an appropriate proxy for the JNDs, the uncertainty of the grasping response does increase with object size and thus follows Weber's law.

*b*

_{ME}= 1.05 ± 0.021, and the quadratic coefficient,

*c*

_{ME}= 0.0033 ± 0.001 mm

^{−1}, was different from zero. The residuals (Appendix B) show a systematic relationship, which disappears for the quadratic fit.

*SD*

_{ME}increased with object size,

*b*= 0.056 ± 0.008,

*t*(19) = 6.74,

*p*< 0.001, 95% CI, 0.038–0.073 (Figure 3c). This finding is consistent with previous studies that also found such a scaling for ME. Based on such a result, it would often be concluded that manual estimation follows Weber's law (Ganel et al., 2008). But again, this conclusion would be premature. We first have to calculate \(\widehat {JND}\) before we can assess Weber's law. Again, we divided the

*SD*

_{ME}by the local slope (see Figure 3d) to calculate the \(\widehat {JND}\).

*k*

_{ME}= 0.023 ± 0.014,

*t*(19) = 1.65,

*p*= 0.116, 95% CI, −0.006 to 0.051 (Figure 3e). We cannot claim that this result differs from zero, but crucially,

*k*approached the typical magnitude expected for size perception (0.02 - 0.06; McKee & Welch, 1992; Teghtsoonian, 1971). Given that we found clearly significant Weber's constant in manual estimation in the other studies we analyzed, and since researchers do not question that manual estimation adheres to Weber's law, we think this single non-significant result is no reason to question Weber's law in manual estimation.

*SD*

_{Response}as a proxy for JND. Based on this approach, manual estimation seems to follow Weber's law and grasping seems to violate it (Ganel et al., 2008). However, this conclusion would be premature because

*SD*

_{Response}is not a good proxy for JND when the response function is nonlinear. Therefore, we first have to calculate \(\widehat {JND}\) before we can assess Weber's law. When we do this, then grasping and manual estimation both seem to follow Weber's law, and the corresponding Weber constants

*k*are in the range we would expect from the literature for size perception. This raises the question of how general our results are. To assess this, we reanalyzed three studies from the literature and calculated the Weber constants

*k*based on \(\widehat {JND}\). For better comparison, we depicted all those Weber constants in Figure 4.

*b*of the regression function for MGA was

*b*

_{MGA}= 0.84 ± 0.009 and for ME was

*b*

_{ME}= 1.02 ± 0.02. The quadratic term

*c*was negative for MGA (

*c*

_{MGA}= −0.0042 ± 0.0003 mm

^{−1}), and positive for ME (

*c*

_{ME}= 0.0021 ± 0.0004 mm

^{−1}). As in Experiment 1, there was a concave relationship between MGA and object size (sign of

*c*was negative); that is, the responsiveness of MGA decreased with increasing object size, whereas the responsiveness of ME changed much less.

*SD*

_{MGA}did not scale significantly with object size,

*b*= −0.019 ± 0.01,

*t*(14) = −1.88,

*p*= 0.08, 95% CI, −0.041 to 0.003 (Figure 5c), but

*SD*

_{ME}increased significantly with object size,

*b*= 0.062 ± 0.007,

*t*(14) = 8.56,

*p*< 0.001, 95% CI, 0.046–0.078. Based on such a pattern of results, it would often be concluded that grasping violated Weber's law, whereas manual estimation followed Weber's law (Ganel et al., 2008). However, we again have to calculate \(\widehat {JND}\) first, before we can assess Weber's law.

*k*

_{MGA}= 0.054 ± 0.014,

*t*(14) = 3.91,

*p*= 0.002, 95% CI, 0.024–0.083 (Figure 5e). Similarly, in manual estimation, \({\widehat {JND}_{{\rm{ME}}}}{\rm{\;}}\;\)also increased significantly, with a Weber constant of

*k*

_{ME}= 0.035 ± 0.011,

*t*(14) = 3.26,

*p*= 0.006, 95% CI, 0.012–0.058. In short, grasping and manual estimation show Weber constants that are perfectly in the range we expect for size perception: 0.02 to 0.06 (McKee & Welch, 1992; Teghtsoonian, 1971) (Figure 4).

*SEM*and 95% confidence intervals.

*b*for MGA was

*b*

_{MGA}= 0.76 and for ME was

*b*

_{ME}= 0.88. The sign of the quadratic term

*c*was negative for MGA (

*c*

_{MGA}= −0.0038 mm

^{−1}) and positive for ME (

*c*

_{ME}= 0.0044 mm

^{−1}).

*SD*

_{MGA}(

*b*= 0.001 ± 0.004; 95% CI, −0.012 to 0.014) did not change with object size (Figure 6c), whereas

*SD*

_{ME}increased with object size (

*b*= 0.062 ± 0.003; 95% CI, 0.052–0.072), confirming the typically obtained result in such studies. Based on this pattern of results, it would often be concluded that manual estimation follows Weber's law but grasping does not (Ganel et al., 2008; Heath et al., 2011). But again, this conclusion would be premature. We first have to calculate \(\widehat {JND}\) before we can assess Weber's law.

*k*

_{MGA}= 0.078 ± 0.011 (95% CI, 0.043–0.113). In manual estimation,\(\;{\widehat {JND}_{{\rm{ME}}}}\;\)also increased with object size, resulting in a Weber constant of

*k*

_{ME}= 0.027 ± 0.003 (95% CI, 0.017–0.036) (Figure 6e). These values again fit nicely with the expected range from the literature for size perception: 0.02 to 0.06 (McKee & Welch, 1992; Teghtsoonian, 1971) (Figure 4). In short, even for data from another laboratory (Heath et al., 2011), we find indications that grasping follows Weber's law if an appropriate proxy for JND is used. This raises the question of whether that is also true for the first, most influential landmark study on this topic (Ganel et al., 2008).

*n*= 13), manual estimation (

*n*= 11), and perceptual adjustment (i.e., adjustment of a comparison line on a monitor, the response to which we refer to as ADJ;

*n*= 6). All tasks were performed in full-vision conditions. We obtained the data at the participant level from the authors. For further details on the experimental procedure, see the original publication.

*b*of the regression function for MGA was

*b*

_{MGA}= 0.66 ± 0.015; for ME, it was

*b*

_{ME}= 0.63 ± 0.16; and for ADJ it was

*b*

_{ADJ}= 1.07 ± 0.02. The quadratic term

*c*for MGA was

*c*

_{MGA}= −0.0011 ± 0.0002 mm

^{−1}; for ME, it was

*c*

_{ME}= −0.00001 ± 0.0003 mm

^{−1}; and for ADJ, it was

*c*

_{ADJ}= −0.0003 ± 0.0003 mm

^{−1}(see also Appendix C). It is interesting to note that the values of

*c*for ME and ADJ are very close to zero, because the response function is almost perfectly linear (we will further discuss this below).

*SD*

_{MGA}, linear term

*b*= 0.0003 ± 0.004,

*t*(12) = 0.078,

*p*= 0.939, 95% CI, −0.008 to 0.009, did not scale with object size, whereas

*SD*

_{ME},

*b*= 0.035 ± 0.005,

*t*(10) = 7.23,

*p*< 0.001, 95% CI, 0.025–0.046, and

*SD*

_{ADJ},

*b*= 0.054 ± 0.01,

*t*(5) = 5.46,

*p*= 0.003, 95% CI, 0.029–0.079, scaled with object size (Figure 7c). This was interpreted as an absence of Weber's law in grasping by Ganel et al. (2008). However, as we have shown, conclusions about Weber's law should only be made after calculating the \(\widehat {JND}{\rm{s}}.\)

*k*for MGA,

*k*

_{MGA}= 0.019 ± 0.009,

*t*(12) = 2.01,

*p*= 0.068, 95% CI, −0.002 to 0.039, ME,

*k*

_{ME}= 0.062 ± 0.010,

*t*(10) = 6.05,

*p*< 0.001, 95% CI, 0.039–0.084, and ADJ,

*k*

_{ADJ}= 0.055 ± 0.013,

*t*(5) = 4.07,

*p*= 0.01, 95% CI, 0.020–0.089, were again consistent with the expected values from the literature on size perception: 0.02 to 0.06 (McKee & Welch, 1992; Teghtsoonian, 1971) (Figure 4). The Weber's constant in grasping did not reach significance; however, the value was in the expected range. Also, the 95% CI overlapped with the expected range of

*k.*Further, in the three other studies we analyzed, the Weber's constant in grasping was highly significant. Ganel et al. (2008) had only 13 participants, which may have been too few, leading to low power. Overall and based on the results from all studies, we conclude that there is evidence for grasping following Weber's law.

*SD*

_{MGA}) as a proxy for JND—starting with Ganel et al. (2008). We showed that this were only acceptable if the response function were linear. If, however, the response function is nonlinear—as is the case in grasping—then we first have to transform the response variability back to the corresponding stimulus variability (Figure 1). That is, we have to transform the hitherto used

*SD*

_{MGA}to corresponding \(\widehat {JND}{\rm{s}}\) (Appendix A). Only then does it make sense to assess Weber's law.

*SD*

_{MGA}does not increase with object size. Traditionally, this result would have been interpreted as a violation of Weber's law (Ganel et al., 2008); however, the response function of MGA in grasping is nonlinear. It is concave, such that responsiveness decreases for larger object sizes (one reason could be because the finger span is limited and grasping needs to trade-off a safety margin with the ability to still enclose the target object). Therefore, it is not appropriate to use

*SD*

_{MGA}as a proxy for JND. Instead, we need to transform

*SD*

_{MGA}back to the corresponding variability at the stimulus level. When we do this and calculate \({\widehat {JND}_{{\rm{MGA}}}}\), then we find that \({\widehat {JND}_{{\rm{MGA}}}}\) does increase with object size. The corresponding Weber constant

*k*(i.e., the slope of the linear function relating \({\widehat {JND}_{{\rm{MGA}}}}\) to object size) is perfectly in the range we would expect from the literature for size perception (Figure 4).

*SD*

_{MGA}does not increase with object size; (b) the response function of MGA in grasping is nonlinear; and (c) \(\widehat {JND}\;\)seems to increase with object size and the corresponding Weber constant is well in the expected range (Figure 4). We conclude that there is evidence for Weber's law in grasping.

*IQR*

_{Response}) or similar measures of dispersion to assess Weber's law instead of the most often used within-subject standard deviation (

*SD*

_{Response}). The main motivation was the well-established fact that these measures are more robust against extreme values and outliers than the

*SD*

_{Response}. However, our arguments apply equally to these alternative measures because they still quantify the variability at the level of the response, not the stimulus. Due to the nonlinear response function in grasping, it is therefore still possible for the

*IQR*

_{Response}to not scale with object size, whereas the corresponding \(\widehat {JND}\) does (the arguments are analogous to those we present for

*SD*

_{Response}in Figure 1). Therefore, even when

*IQR*

_{Response}is used, it should be divided by the local slope at every object size to obtain the corresponding \(\widehat {JND}\).

*SD*

_{Response}because this is the measure that was used by studies inferring a strong violation of Weber's law in grasping. Studies based on

*IQR*

_{Response}or other alternative measures typically favored the biomechanical constraints approach which is consistent with our main conclusions and which we discuss below.

*SD*as a proxy for JND. Are they all wrong?

*SD*

_{Response}for this assessment (for a list, see Ganel, Freud, & Meiran, 2014). The question now arises whether all those studies need to be reanalyzed by calculating the \(\widehat {JND}\)s instead. This would require dividing at each stimulus magnitude the

*SD*

_{Response}by the local slope of the response function (i.e., the function relating stimulus magnitude to response).

*SD*

_{Response}would at each stimulus magnitude be divided by the same constant value, such that \(\widehat {JND} = \;\frac{{S{D_{Response}}}}{{Constant}}\). This constant scaling of

*SD*

_{Response}will not change the assessment of whether the \(\widehat {JND}\)s scale with object size, such that the answer to the question of whether the sensory domain adheres to Weber's law will not change. In many cases, the slopes will even be close to 1, such that

*SD*

_{Response}≈ \(\widehat {JND}\) so that both measures will even give the same numerical answer (e.g., the perceptual adjustment task of Ganel et al., 2008) (see Figure 7d). However, in situations where the response function clearly deviates from linearity, as is the case for grasping, the

*SD*

_{Response}is not appropriate for assessing Weber's law, and \({\widehat {JND}_{{\rm{Response}}}}\) must be calculated.

*generalized version*of Weber's law (Miller, 1947; Ono, 1967). This is also the approach we used, and our results are nicely consistent with the literature (Figure 4). However, the

*strict version*of Weber's law predicts a proportional relationship between stimulus magnitude and JND—that is, a linear function with a zero-intercept (e.g., equation 3.1 of Baird & Noma, 1978). Therefore, the question arises whether it would be better to use this strict version of Weber's law on the current data.

*SD*

_{Response}as a proxy for JND, because this method also finds non-zero intercepts for grasping and manual estimation (Ganel et al., 2008; Heath et al., 2011) (see also Appendix C).

*SD*at each object size for each participant should be divided by the local slope (see each panel d in Figures 3 and 5–7) of the response function (for that participant) at that object size. Essentially, the \(\widehat {JND\;}\) is a ratio with the measured slope in the denominator. If the measured slope for any participant is a small value close to zero, this can lead to inflated values (and variability) in the final \(\widehat {JND\;}\). This problem of ratios is well known from statistical calibration (Buonaccorsi, 2001), and there also exist methods to ameliorate this problem (Franz, 2007; von Luxburg & Franz, 2009). One reason why measured slopes may be close to zero is not having enough trials (approximately <20 trials per object size). Therefore, researchers applying this method to their own data and other data need to be careful that there were sufficient trials per object size. In Experiment 1, we used 50 trials per object size, and we reanalyzed only those studies that had 20 or more trials per object size.

*SD*

_{MGA}as a proxy for JND; therefore, it is difficult to judge the results. For a full assessment, we would need to transfer

*SD*

_{MGA}to \(\widehat {JND}\) first. This is all the more important as it is plausible that those manipulations change not only the variability of the response but also the response function, in one or the other way (larger or smaller safety margin, depending on the specific manipulation). For example, it is well known that the safety margin of MGA is smaller in 2D and pantomimed grasping, conditions where no object is physically grasped. Weber's law should then be assessed by calculating the \(\widehat {JND}s\).

*SD*

_{MGA}does not increase with object size.

*SD*

_{MGA}does not increase with object size. The difference to the present study is that we now have a toolkit that allows us to quantitatively assess whether Weber's law does or does not hold,

*independent*of the exact mechanisms that cause the response function to be bent. It is important to note here that our results do not rest upon biomechanical constraints being the reason for nonlinearity in the grasping response function—it is just one plausible mechanism that has been discussed by others in the literature.

*b*

_{skew}= −0.0085 ± 0.003; 95% CI, −0.015 to −0.002), but not in manual estimation (

*b*

_{skew}= −0.0037 ± 0.004; 95% CI, −0.012 to 0.005). Note that we found this result even within a range of medium objects (i.e., 20–50 mm in Experiment 1; see also Löwenkamp et al., 2015), which have been termed “functionally graspable” and for which it was sometimes assumed that the influence of biomechanical constraints can be excluded (Ayala et al., 2018; Heath et al., 2017). Nevertheless, it seems plausible that optimization processes in the generation of the MGA, such as the generation of comfortable or efficient grip apertures, may cause these effects even at small, graspable object sizes. A recent study (Uccelli et al., 2021) assessed Weber's law in small-to-medium object sizes (5–40 mm) and reported that the skewness of the MGA increased with object size up to 40 mm. In a previous study by the same group (Bruno et al., 2016), where stimuli larger than 40 mm were also used, small effects for a negative scaling of skewness at objects larger than 40 mm were found. We agree with those authors’ conclusion that a study investigating the skewness of the MGA in the full range of object sizes up to the limit of the handspan would be useful to make definite claims (Uccelli et al., 2021). It is possible that the skewness follows an inverse-U–shaped function of object size.

*b*

_{skew}= 0.011 ± 0.005; 95% CI, 0.000–0.021, with MAS values for 19 out of 20 participants in grasping). This scaling of skewness with hand size was negligible at our smallest object size (20 mm).

*decreasing*variability in the MGA with object size, or an apparent

*inversion*of Weber's law in grasping. This result was especially confusing, because none of the theories proposed to explain the absence of Weber's law in grasping (perception–action model or double-pointing hypothesis) could accommodate this finding. Utz et al. (2015) reasoned that biomechanical constraints on the finger aperture could cause ceiling effects in grasping large objects. These constraints combined with a nonlinear grasping response function can readily explain this strange result of apparently inverted Weber's law. When grasping a large object, the finger aperture will be larger than the to-be-grasped object (including the safety margin), and this is capped by the maximum possible opening of the hand. As object sizes increase after the point where object size + safety margin is close to the maximum possible hand opening, the safety margin will decrease in compensation, thus leading to decreasing variability in the response. When we reanalyzed Löwenkamp et al. (2015), the apparent inversion of Weber's law disappeared and the \(\widehat {JND}\) scaled positively with object size, as expected by Weber's law, and, notably, with values of Weber's constant

*k*(slope) consistent with the literature (Figure 4). Therefore, our approach can readily explain these inconsistencies in the large literature on Weber's law in grasping.

*SD*

_{MGA}as proxy for JND and therefore forsook the stimulus level. We showed that this is problematic when the response function is nonlinear (as in grasping) and that instead \(\widehat {JND}\) must be calculated, which brings us back to the stimulus level. If we do this, then grasping follows Weber's law—in our own data, as well as in previous studies that claimed a violation of Weber's law for grasping. Our method is general and can also be used in other tasks and sensory domains whenever a direct assessment of JND is not possible.

**Open materials and data:**The associated materials and data are available at https://osf.io/ent2y/.

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*SD*

_{MGA}

*SD*

_{MGA}is not an appropriate proxy for the corresponding JND and that we have to use \(\widehat {JND}\) instead. Here, we describe how to calculate \(\widehat {JND}\). For ease of exposition, we will derive our formulae for the special case of grasping, albeit all formulae can directly be generalized to other responses (for example, manual estimation or size adjustment). Our scenario is shown in Figure A1 (which is similar to Figure 1 of the main text but contains more technical details). A participant is presented with objects of different sizes

*s*. When the participant grasps those objects, their sizes are transferred to MGAs by the response function

*g*(

*s*). For simplicity of exposition, we focus on the stimulus with physical size

*s*

_{1}.

*s*

_{1}, resulting in an MGA of

*g*(

*s*

_{1}). We now increase the size of the object by a small amount (which we denote as ∆

*s*

_{1}), such that the second object has the size

*s*

_{1}+ Δ

*s*

_{1}, resulting in an MGA of

*g*(

*s*

_{1}+ ∆

*s*

_{1}). The effect of increasing the size of the object by ∆

*s*

_{1}on MGA in grasping is then

*g*(

*s*) at the position

*s*

_{1}, as \({f_1}{:{=}}\;\frac{{dg}}{{ds}}{\big|_{{s_1}}}\), such that we obtain

*s*

_{1}, then MGA is increased by this amount times the slope

*f*

_{1}(the first derivative) of the response function

*g*(

*s*) that relates object size to MGA.

*SD*s, such that

*g*

_{1}and want to know to which change in object size ∆

*s*

_{1}this corresponds. Given our local linear approximation, this is now easy to do. We only need to invert Equation 5 such that we obtain

*SD*

_{MGA}and want to know to which standard deviation this corresponds at the level of physical object size, we just need to divide

*SD*

_{MGA}by the slope

*f*

_{1}(the first derivative) of the response function

*g*(

*s*) that relates object size to MGA.

*k*, which indicates to which degree \(\widehat {JND}\) increases with object size, as predicted by Weber's law. This parameter is traditionally referred to as the Weber constant or Weber fraction. Numerical values for all these fitted coefficients are also given in Appendix C.

*RSE*), where full participant data were available.

*a*being the constant source of variability and

*k*being interpreted as Weber's constant. The equation can be reformulated to

*a*

^{2}corresponding to the intercept and

*k*

^{2}corresponding to the slope (for further details, see Smeets & Brenner, 2008). A small glitch can occur when the regression results in negative values for

*a*

^{2}and

*k*

^{2}, such that

*a*and

*k*are undefined. In these cases, reduced models with the corresponding parameter set to zero were fitted (e.g., if

*a*

^{2}would be negative in the full model, then a model with zero intercept is fitted).

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