Relative numerosity discrimination has been studied experimentally in adults (Burr & Ross, 2008; Durgin, 1995, 2008; Ross & Burr, 2010) infants (Xu & Spelke, 2000), and non-human 4 mu (Brannon et al., 2001; Gallistel, 1989; Leslie, Gelman, & Gallistel, 2008), using psychophysics (Barlow, 1978), fMRI (Harvey et al., 2013; Piazza et al., 2007), and single unit physiology (Nieder, 2005). It has been suggested that there is a ‘visual sense of number’ (Burr & Ross, 2008) and that ‘Vision senses number directly’ (Ross & Burr, 2010) for large numbers of tokens. Here we attempt to discover whether there is indeed a mechanism for numerosity separate from density and size of textures. A common-used strategy for measuring relative numerosity thresholds is to scatter the tokens within a confined area, such as a circle (Burr & Ross, 2008; Durgin, 1995; Raphael, Dillenburger, & Morgan, 2013). In these circumstances, changing the number of tokens must change either the area of the pattern or the density of items. Weber fractions for numerosity are lower when the numerosity change is accompanied by a change in area (Raphael, Dillenburger, & Morgan, 2013), in agreement with other studies showing that a high-precision, one-dimensional mechanism is responsible for area discrimination of circles (Morgan, 2005; Nachmias, 2011). Therefore, experiments with circular textures may overestimate the accuracy of true numerosity discrimination. Randomly interleaving size-varying and density-varying trials (Burr & Ross, 2008; Raphael, Dillenburger, & Morgan, 2013) does not solve this problem, since observers may use whichever of the two independently noisy signals, size or density, is larger on a particular trial (Raphael, Dillenburger, & Morgan, 2013). For these reasons, we thought it desirable to repeat the experiment of Burr and Ross (2008) using stimuli with non-circular polygonal outlines (Fig. 1). We compared four conditions: (1) density-varying trials alone (2) area varying trials alone (3) interleaved area-density trials where the observers made a numerosity discrimination and (4) which is the same as condition 3, but in addition observers had to decide whether the difference was in area or in density. We expected area thresholds for random polygons to be higher than those for circles, and the first question was whether this would also raise thresholds for numerosity. In additional conditions subjects discriminated changes in density or changes in size when numerosity was constant.
Thresholds (Weber Fractions) are shown separately for the different conditions, Trial Types and observers in Fig. 2 and Table 2. The observer’s thresholds (σ) are shown by circles with error bars indicating 95% confidence limits. The colored bars show predictions of a 2-channel size-density model, to be described below. The horizontal lines depict predictions of a 3-channel size-density-numerosity model.
These findings are confirmed by the statistical pairwise comparisons in Table 2.
Fig. 3 and Table 3 compare the Weber fractions of the polygon experiment against those of the previously published circle experiment of the four subjects that took part in both experiments (Raphael, Dillenburger, & Morgan, 2013). It is shown that thresholds on size-varying trials are indeed lower with circular texture than for polygons in all conditions.
In Fig. 4 which shows the Weber Fractions of size varying trials against density varying trials for each observer and condition Lysogen can be seen that the thresholds for size-varying and density-varying trials, in size, density and number conditions are indeed similar. In trials of the Extended Conditions when numerosity is kept constant and density and size vary reciprocally, discrimination is significantly impaired (significant in 9 out of 10 comparisons, 5 observers×2 Trial Types). Though, comparing the Single Trial Type conditions for size and density with the same Trial Type of the extended conditions of size and density reveals no increase in threshold for size judgment. Hence, observers are not worse in density and size judgments when stimuli are interleaved with trials that offer a less reliable cue.