Grasping follows Weber's law: How to use response variability as a proxy for JND

Abstract

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.

Figure 1.

Illustration of the apparent violation of Weber's law in grasping due to the nonlinear response function g(s). First, suppose we already knew that Weber's law holds in grasping, such that uncertainty about object size is smaller for small objects than for large objects (compare ${\widehat{JND}}_1$ with ${\widehat{JND}}_2$, respectively). This larger uncertainty is, however, not necessarily reflected in the variability of the response because the response function in grasping becomes shallower for large objects (compare SD_MGA1 and SD_MGA2, respectively). This illustrates how the nonlinear response function can mask an underlying adherence to Weber's law in grasping. Now, consider what needs to be done if a researcher only knows SD_MGA and the response function but not $\widehat{JND}$. The researcher would need to invert the response function and map SD_MGA back to the corresponding uncertainty at the level of object size. Only then would the researcher arrive at the correct estimates for $\widehat{JND}$. For mathematical details, see Appendix A.

Figure 2.

(a) Setup of Experiment 1 for grasping and manual estimation. (b) Infrared light-emitting diodes (IREDs) were fixed to the index finger and thumb to record the trajectories of the movements.

Figure 3.

Results of Experiment 1 for grasping and manual estimation. (a) Mean responses as a function of object size. Quadratic regressions are shown as solid color curves; linear regressions as black dashed lines. (b) Difference between response (and quadratic fit) and physical stimulus size. The nonlinearity in the responses can be seen in the curvature of these differences. (c) SD_Response as a function of object size. (d) The local slope at every object size in the grasping and manual estimation. This is the value that the SD is divided by to calculate $\widehat{JND}.$ (e) $\widehat{JND}$ as a function of object size. Error bars depict ±1 SEM (between subjects).

Figure 4.

Weber constant k based on $\widehat{JND}$for all of our analyses. From the literature, we expect k to be between 0.02 and 0.06 for visual size perception, as indicated by the shaded area. The k values we obtained are very close to this expected range. Values are shown as mean ± SEM. The values for H11 are dashed because we only had aggregate data for this study. Exp 1, Experiment 1; L15, Löwenkamp et al. (2015); H11, Heath et al. (2011); G08, Ganel et al. (2008).

Figure 5.

Reanalysis of Löwenkamp et al. (2015). (a) Mean responses as a function of object size. Quadratic regressions are shown as solid color curves; linear regressions as black dashed lines. (b) Difference between response (and quadratic fit) and physical stimulus size. The nonlinearity in the responses can be seen in the curvature of these differences. (c) SD_Response as a function of object size. (d) The local slope at every object size in the grasping and manual estimation. This is the value that the SD is divided by to calculate $\widehat{JND}.$ (e) $\widehat{JND}$ as a function of object size. Error bars depict ±1 SEM (between subjects).

Figure 6.

Reanalysis of Heath et al. (2011). (a) Mean responses as a function of object size. Quadratic regressions are shown as solid color curves; linear regressions as black dashed lines. (b) Difference between response (and quadratic fit) and physical stimulus size. The nonlinearity in the responses can be seen in the curvature of these differences. (c) SD_Response as a function of object size. (d) The local slope at every object size in grasping and manual estimation. This is the value that the SD is divided by to calculate $\widehat{JND}.$ (e) $\widehat{JND}$ as a function of object size. Error bars are absent because individual participant data were not available.

Figure 7.

Reanalysis of Ganel et al. (2008). (a) Mean responses as a function of object size. Quadratic regressions are shown as solid color curves; linear regressions as black dashed lines. (b) Difference between response (and quadratic fit) and physical stimulus size. The nonlinearity in the responses can be seen in the curvature of these differences. (c) SD_Response as a function of object size. (d) The local slope at every object size in grasping, manual estimation, and adjustment. This is the value that the SD is divided by to calculate $\widehat{JND}.$ (e) $\widehat{JND}$ as a function of object size. Error bars depict ±1 SEM (between subjects).

Figure 8.

Skewness as a function of hand size. The skewness in the response of every participant at every object size is plotted as a function of the hand size or the maximum aperture separation (MAS) of that participant. Also depicted are the regression lines for skewness as a function of MAS.

Figure A1.

Concave (“bent”) grasping response function g(s).

Figure B1.

Residuals of the mean in the linear and quadratic regression functions for each dataset. The error bars depict ±1 RSE, where full participant data were available. RSE = residual standard errors.