Numerical Perception Biased by Saliency

Numerical Perception Biased by Saliency Naama Katzin1,2, Avishai Henik1,2 and Moti Salti2 1 Department of Psychology, Ben-Gurion University of the Negev, Beer-Sheva, Israel 2 Zlotowski Center for Neuroscience, Ben-Gurion University of the Negev, Beer-Sheva, Israel How do people process numerosity? Do they rely on general magnitude processing (e.g. area, density, etc.)1,2,3? Alternatively, do they depend on a designated module underlying numerosity judgements4,5? In a 2016 paper, Cicchini and colleagues6 show results that strongly support the latter. They demonstrated that humans automatically perceive and spontaneously use numerosity rather than other physical magnitudes (i.e. area of convex hull or density) when asked to make comparison judgments. Here we present an alternative account for their findings. We suggest that saliency of the different attributes of the stimuli (i.e. numerosity, area of convex hull and density) can bias participant’s strategy. We show that in Cicchini et al.’s design, indeed numerosity was more salient than the other stimuli dimensions. We demonstrate that when saliency of another property is increased, participants tend to rely on it instead of numerosity. Taken together, this challenges Cicchini et al.’s conclusion that numerosity is processed automatically. Cicchini et al. used an elegant design in an attempt to find the automatic strategy used in numerical decisions. They used two different discrimination tasks. In the first task, subjects were presented with three patches of dots and were instructed to pick the ‘odd one out’. Two of the three patches were similar in all dimensions: numerosity, density and area of convex hull. The odd patch differed in two dimensions. The subjects were not given information regarding which dimensions are the odd ones out. Their results showed that subjects tended to rely on numerosity even when it was not the odd dimension. Moreover, subjects were more sensitive to differences in numerosity. The strength of this task is in its non-biased design. The naïve subjects are not guided or primed to use a specific strategy, and therefore, the authors assumed subjects would choose the most intuitive and efficient strategy. The authors suggested that the fact that subjects relied on numerosity even when it was not beneficial suggests that numerical processing is primary and does not need to rely on general magnitudes. In the second task, subjects were presented with two patches of dots and were instructed, in different blocks, to choose the more numerous, denser, or larger (in the area of convex hull) array. Their results showed that even when instructed to make a decision based on density or area, subjects relied on numerosity. The authors concluded that numerical cognition relies on a designated organ for number and is not supported by a general magnitude processing system. Despite the innovativeness and ecological characteristics of the experimental design, it lacks crucial control for an alternative account. In a world that is rich with stimuli, people have a hard time separating the wheat from the chaff. This is often resolved by relying on salient information even when it is not the most relevant to the task. This is very apparent in everyday life, for example, when people overestimate frequencies of salient events (i.e. the availability heuristic7). It also takes effect in experimental situations. The Stroop effect, which is one of the most robust effects in cognitive psychology, could be eliminated or even reversed when saliency of the stimulus dimensions is manipulated8. As acknowledged by Cicchini and colleagues6, the dot array stimuli convey several dimensions other than numerosity. The authors mention area of convex hull and density but there are of course many more (total surface area of the dots, average size of the dots, dot circumference, etc.). If numerosity was more salient than the other dimensions in the aforementioned study then one could not conclude that “humans extract number information, directly and spontaneously, via dedicated mechanisms”6. Saliency obeys Weber’s law. Accordingly, noticing the difference between two stimuli depends on their ratio is (i.e. smaller value divided by larger value). The lower the ratio, the more salient the difference is. Table 1 shows the area of convex hull and density ratios that were used in Cicchini et al.’s first experiment. The numerical ratio can be derived from these ratios (by multiplying area ratio by density ratio). The table demonstrates that the numerical ratio was always smaller than the ratios of area of the convex hull and/or density, or equal to one of them. Consequently, the numerical dimension was more salient than the other two dimensions, and therefore more likely to be used by the subjects. In the second experiment, the ratios of the numerosities varied from 0.09 to 0.75. The ratios of density and area varied from 0.59 to 1. Again, this means that discriminating numerosities would have been easier than discriminating area or density. Table 1| Numerical Ratios of the Stimuli Convex Hull Density Ratio Area Ratio 0.35 0.5 0.7 1 0.35 0.1225 0.175 0.245 0.35 0.5 0.175 0.25 0.35 0.5 0.7 0.245 0.35 0.49 0.7 1 0.35 0.5 0.7 1 Note. The data is adapted from Cicchini et al.’s (2016) paper. Density (D) and area of convex hull (H) ratios were transformed from octaves to ratios. Since all dots had the same size, the numerical ratio was equal to the ratio of the total surface area of the dots. Total dot surface (𝑆 = 𝜋 ∑ 𝑟 2) ratio is a product of CH ratio and density ratio: 𝐷∗𝐻 = 𝑆 𝐷1 𝐻1 𝑆1 𝑁1 ∗ = = 𝐷2 𝐻2 𝑆2 𝑁2 We hypothesized that participants would not rely on numerosity automatically, but instead, rely on the most salient dimension. Accordingly, we ran an experiment similar to Cicchini’s Exp 1 in which area of convex hull was the most salient dimension. To this end, we used the same two-dimensional space representing density, area of convex hull and numerosity used in Cicchini et al.’s study and a similar ‘odd one out’ paradigm. We created an alternative set of stimuli in which the ratio of the area of convex hull is the product of the numerical and density ratio, making it most salient. The results show that indeed participants tended to rely more on area of convex hull. Results and Discussion As in Cicchini et al.’s first experiment, the general procedure was to measure discrimination thresholds for stimuli that varied in ratio over two dimensions: numerosity and density. The ratio of the area of the convex hull is the product of numerosity and density ratio. We then measure discrimination thresholds in all dimensions: numerosity, density and convex hull area. Discrimination thresholds in the number-density space The task was an ‘odd one out’ task, in which participants are asked to choose which patch of dots seemed different to them, without knowing the dimensions that differed. From the three stimuli displayed two were identical standards (24 or 36), and the odd stimulus that differed by at least two dimensions (see Figure 1 A and B). Figure 1 plots proportion of error rates (pooled across six subjects) for standard 24 (Figure 1 C) and 36 (Figure 1 D). We followed Cicchini et al.’s analysis. The data was fitted by 2D Gaussian functions, described in ellipses. According to Cicchini and colleagues, the orientation of the short radius of the ellipse is informative to the dimension that was used by the observers to perform the task (0° - numerosity, 45° - area of convex hull, 90° - density). The ratio between the radii reflects the magnitude of this effect. As we expected, participants relied on area of convex-hull. The angle of the short radius was 45 for standard 24 and 41.62 for standard 36. In other words, change in ratio of convex hull area caused the greatest change in error rates, compared to numerosity and density. In addition, we measured the aspect ratio of the radii of the ellipse. The aspect ratio for standard 24 was 1.82, and for standard 36 it was 1.69. There is a qualitative difference between our results and Cicchini et al.’s results. The orientation of the ellipse demonstrated that participants were most sensitive to area of convex hull in the current study, while Cicchini et al. participants were most sensitive to numerosity. Therefore we conclude that subjects tend to rely on the most salient dimension. Indeed, we see some difference in the effects’ magnitudes as reflected in the ellipses aspect ratios. However, these are two separate experiments with a different sample of participants and stimuli. Figure 1| Stimuli and discrimination boundaries. (A+B) Example of stimulus for a standard of 24 (A) and 36 (B) dots. As can be seen, the area of the convex hull of the odd patch (smaller in A and bigger in B) is the most salient. (C+D) 2D psychometric function of thresholds in the numerosity/density space for a standard of 24 dots (C) and standard of 36 (D). Interpolated per cent correct responses are plotted as a function of log numerosity and log density, described by the ellipse. Cicchini et al. present an experimental design that, in principle, does not constrain the variety of strategies a participant can use. This approach has the potential to unveil basic mechanisms underlying number perception. However, in their design other latent factors biased the participants. Specifically, the low numerical ratio compared to the other dimensions biased participants to rely on it. Our results show that this strategy could be changed if different dimension is more salient. In order to truly reveal the observer’s spontaneous strategy one must equate the saliency of dimensions. In two recent studies, we showed that when the saliency of numerosity and convex hull area was equated in a dot comparison task, convex hull area was processed more automatically than numerosity9,10. To conclude, our results show that the conviction that numerosity is perceived spontaneously is not supported. Instead, spontaneous perception is guided by saliency. Methods: Participants: 6 participants (5 women, mean age 25.84 (SD=3.65)) with normal or corrected-to-normal vision participated in the experiment. The experiment was approved by the departmental ethics committee. Stimuli: Stimuli were created with Psychtoolbox in Matlab11. Each stimulus consisted of 3 patches of dots for 2 numerical standards: 24 and 36. In each triplet 2 patches, the standards, were identical in numerosity, density and convex hull area. Density and convex hull area differed at most by 5% (i.e., 𝐷𝑒𝑛𝑠𝑖𝑡𝑦\𝐶𝑜𝑛𝑣𝑒𝑥 ℎ𝑢𝑙𝑙 𝑎𝑟𝑒𝑎 1 𝐷𝑒𝑛𝑠𝑖𝑡𝑦\𝐶𝑜𝑛𝑣𝑒𝑥 ℎ𝑢𝑙𝑙 𝑎𝑟𝑒𝑎 2 ≤ 0.05). Numerosity and density of the odd patch was determined by the numerical and density ratio (ratios 0.5-1 in jumps of 0.1, and their inverse ratios). Actual density ratio was accurate up to 4% (for example – ratio 0.5 was 0.48-0.52). Convex hull area ratio was the product of the numerical and density ratio. The patches were located in the middle of the screen around a fixation mark (see Figure 1A and 1B). Location of the odd cloud was random. Experimental procedure: Each trial began with a fixation presented in the middle of the screen for 500 ms, then the stimulus was presented for 250ms. After the participant chose the odd-patch with the arrow keys (right, left, up), a black screen was presented for 500 ms, and then the next trial began. The experiment began with 10 practice trials followed by 6 blocks, 242 trials each, 3 consecutive blocks for each standard: 24 and 36, in total 726 trials for each standard. Order of standards was counterbalanced across participants. Data analysis: Data was analyzed by plotting percent correct responses as a function of the density and numerosity ratio of the stimulus. This yielded the maps in Figure 1C and 1D. The axes are logarithmic, hence the forward diagonal represents convex hull area (numerosity times density). Data were fitted with 2D elliptical Gaussian functions. The elliptical Gaussian has five free parameters: orientation of short-radius, widths of the short and long radii and the position of the center in the number and density space. The orientation of the short radius is the axis of maximal sensitivity as subjects discriminate best when the odd stimulus varied in that direction. The ratio between long and short radii is an index of selectivity of the sensitivity. References 1. Gebuis, T. & Reynvoet, B. The interplay between nonsymbolic number and its continuous visual properties. J. Exp. Psychol. Gen. 141, 642–648 (2012). 2. Leibovich, T. & Henik, A. Magnitude processing in non-symbolic stimuli. Front. Psychol. 4, 375 (2013). 3. Salti, M., Katzin, N., Katzin, D., Leibovich, T. & Henik, A. One tamed at a time : a new approach for controlling continuous magnitudes in numerical comparison tasks. Behav. Res. Methods. 1-8 (2016). 4. Dehaene, S. The Number Sense: How The Mind Creates Mathematics. Oxford University Press (1997). 5. 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