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Sabine Raphael, Barbara Dillenburger, Michael Morgan; Numerosity Estimation Is Not Derived Only from Density and Size Judgments. Journal of Vision 2011;11(15):59. doi: 10.1167/11.15.59.
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A previous study (Morgan et al., 2011) has shown that when observers estimate the numerosity of a large number of dots (>60) drawn in a notional circle they could be combining separate estimations of size and density. We predicted from this that if estimates of size could be made noisier, errors in dot numerosity estimation would rise. We made size estimation more difficult by drawing the dots in irregularly shaped notional polygons with random numbers of vertices. We used a 2AFC task with a 64 dot standard and a test that either changed in size (covarying with numerosity) or in a separate session in density (co-varying with numerosity). The test and standard had different pseudo-randomly generated shapes that also varied from trial to trial. The subjects were asked to chose which stimulus was larger or denser depending on the condition. In a third condition we mixed size and density varying trials and asked subjects to estimate which stimulus had the larger number of dots. The arrangement of dots in notional polygons did indeed increase the Weber fraction for size judgment substantially relative to the previous experiment with circles. However contrary to our prediction this had little effects on thresholds for numerosity. Numerosity, size and density Weber fractions were not significantly different. A model with a single noise source for numerosity limiting performance in all three tasks is a better fit to the data than the Morgan et al. (2011) model with different noise sources for size and density and no numerosity mechanism. Our results therefore are consistent with a pure numerosity mechanism (Ross & Burr, 2010) or with a mechanism that encodes numerosity as the ratio of responses from a pair of filters tuned to low and high spatial frequencies as suggested by Dakin et al. (2011).
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