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Hang Zhang, Laurence Maloney; Testing the sampling model of frequency and probability distortion in a visual numerosity task. Journal of Vision 2013;13(9):1041. doi: https://doi.org/10.1167/13.9.1041.
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© ARVO (1962-2015); The Authors (2016-present)
People distort probability information in many tasks. The pattern of distortion can be captured by a linear transformation of log odds: lo(w(p))=γ*lo(p)+b, where lo(p)=log(p/(1-p)). The slope coefficient γ is roughly .6-.7 in decision making (Kahneman & Tversky, 1979, Econometrica) but typically greater than 1 in equivalent motor tasks (Wu, Delgado, Maloney, 2008, PNAS). What determines γ? In Zhang & Maloney (2012, Frontiers in Neuroscience), subjects estimated the relative frequency of black or white dots in a visual display. We found that their γ decreased with the total number of dots, from 0.88 for 200 dots to 0.73 for 600 dots. We developed a sampling model that could quantitatively account for the numerosity effect. The model assumes one single slope γ0 for the same individual and attributes the observed different γ’s to the binomial variation in sampling from the display. The model predicts γ to be greater than one when sample size (associated with numerosity) is small enough. It further predicts a specific relationship between the maximum and minimum γ of the same individual. Here we reported a new experimental test of the model.
Methods: On each trial subjects judged the average relative frequency of black (white) dots in the two displays. Each display, presented for 0.7 second, consisted of randomly scattered black and white dots. Each display was a binomial random sample from a population, where the numerosity (8,12,18,27) and the population relative frequency (0.1,0.2,…,0.9) were varied across trials. Each numerosity condition had 162 trials, randomly mixed. Twelve participated.
Results: The predictions of the model were confirmed. (1) The γ decreased with display numerosity, F(3,33)=20.7,p<.001. (2) The maximum γ (numerosity 8) was 1.15 (10/12 subjects >1). (3) The observed minimum γ, varying across subjects from 0.66 to 1.23, was indistinguishable from that predicted from the maximum γ, t(11)=-0.84,p=.42.
Meeting abstract presented at VSS 2013
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