Visual fixations and the computation and comparison of value in simple choice SUPPLEMENTARY MATERIALS

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1 Visual fixations and the computation and comparison of value in simple choice SUPPLEMENTARY MATERIALS Ian Krajbich 1 Carrie Armel 2 Antonio Rangel 1,3 1. Division of Humanities and Social Sciences, California Institute of Technology 2. Precourt Institute for Energy Efficiency, Stanford University 3. Computational and Neural Systems, California Institute of Technology 1

2 Table of Contents I. Model predictions for θ = 0 and θ = 1 In these figures we show the predictions of the null hypothesis models M1 (regular DDM) and M2 (full fixation bias) for each relevant figure from the text. Supplementary Item Title Page Figure S1 Figure S2 Figure S3 Figure S4 Figure S5 Figure S6 Figure S7 Figure S8 Figure S9 Figure S10 Null hypotheses for Fig. 2a Null hypotheses for Fig. 2b Null hypotheses for Fig. 2c Null hypotheses for Fig. 4b Null hypotheses for Fig. 4c Null hypotheses for Fig. 5a Null hypotheses for Fig. 5b Null hypotheses for Fig. 5c Null hypotheses for Fig. 5d Null hypotheses for Fig. 5e II. Individual level analyses In these figures we show histograms of individually obtained parameter estimates. Supplementary Item Title Page Figure S11 Figure S12 Individual histogram Fig. 5a bias histogram III. First fixation properties In these figures we show properties of the first fixations, analogous to Figure 3 of the text. Supplementary Item Title Page Figure S13 Figure S14 Figure S15 Fig. 3b for first fixations Fig. 3c for first fixations Fig. 3d for first fixations

3 IV. Fixation histograms In these figures we show histograms of first and middle fixation durations for all trials, as well as for trials with different levels of difficulty, along with their best log-normal fits. Supplementary Item Title Page Figure S16 Figure S17 Figure S18 Figure S19 Figure S20 Figure S21 Figure S22 Fixation histogram for all trials Fixation histogram with 0 difference Fixation histogram with 1 difference Fixation histogram with 2 difference Fixation histogram with 3 difference Fixation histogram with 4 difference Fixation histogram with 5 difference V. Alternative model predictions In these figures we show the predictions of the three alternative models from the text with different model parameter values to show that adjusting them does not help the models fit all the moments of the data. Supplementary Item Title Page Figure S23 Figure S24 Figure S25 Figure S26 Figure S27 Figure S28 Figure S29 Figure S30 Alternative model 1 Fig. 2b c Alternative model 1 Fig. 5b Alternative model 2 Alternative model 2 theta variants Alternative model 2 d variants Alternative model 2 sigma variants Alternative model 3 Alternative model 3 variants VI. Fixation distribution fitting In this table we show how well various distributions fit the empirical distribution of fixation durations. Supplementary Item Title Page Table S1 Fixation-distribution fitting 34 3

4 p(left chosen) left rating - right rating p(left chosen) left rating - right rating Figure S1: Null hypotheses for Fig. 2a. Reproduction of Fig. 2a from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The χ 2 goodness-of-fit statistics for θ = 0 and θ =1 were 39.6 (p=10-5 ) and 192 (p=10-16 ) respectively. 4

5 rt [ms] p=2.6e best rating - worst rating rt [ms] p=2.6e best rating - worst rating Figure S2: Null hypotheses for Fig. 2b. Reproduction of Fig. 2b from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The goodness-of-fit statistics for θ = 0 and θ =1 were p = 0.01 and p = respectively. 5

6 # of fixations p=3.9e-06 # of fixations p=3.9e best rating - worst rating best rating - worst rating Figure S3: Null hypotheses for Fig. 2c. Reproduction of Fig. 2c from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The goodness-of-fit statistics for θ = 0 and θ =1 were p = 10-5 and p = respectively. 6

7 p(last fixation to chosen) last seen item rating - other item rating p(last fixation to chosen) last seen item rating - other item rating Figure S4: Null hypotheses for Fig. 4b. Reproduction of Fig. 4b from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The χ 2 goodness-of-fit statistics for θ = 0 and θ =1 were 22.9 (p=0.01) and 622 (p=10-16 ) respectively. 7

8 time advantage [ms] last fixation duration [ms] time advantage [ms] last fixation duration [ms] Figure S5: Null hypotheses for Fig. 4c. Reproduction of Fig. 4c from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The goodness-of-fit statistics for θ = 0 and θ =1 were p = 0.96 and p = 0.04 respectively. 8

9 p(left chosen) last fix left last fix right p(left chosen) last fix left last fix right left rating - right rating left rating - right rating Figure S6: Null hypotheses for Fig. 5a. Reproduction of Fig. 5a from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The χ 2 goodness-of-fit statistics for θ = 0 and θ =1 were 5.76 (p=0.83) and 495 (p=10-16 ) respectively for last fixation left, and 13.6 (p=0.19) and 34.7 (p=0.0001) respectively for last fixation right. 9

10 p(left chosen) p= p(left chosen) p= final time advantage left [ms] final time advantage left [ms] Figure S7: Null hypotheses for Fig. 5b. Reproduction of Fig. 5b from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The χ 2 goodness-of-fit statistics for θ = 0 and θ =1 were 25.8 (p=0.0002) and 74 (p=10-13 ) respectively. 10

11 corrected p(left chosen) p= final time advantage left [ms] corrected p(left chosen) p= final time advantage left [ms] Figure S8: Null hypotheses for Fig. 5c. Reproduction of Fig. 5c from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The goodness-of-fit statistics for θ = 0 and θ =1 were p = and p = respectively. 11

12 p(first seen chosen) p=1e-06 p(first seen chosen) p=1e first fixation duration [ms] first fixation duration [ms] Figure S9: Null hypotheses for Fig. 5d. Reproduction of Fig. 5d from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The χ 2 goodness-of-fit statistics for θ = 0 and θ =1 were 1.16 (p=0.76) and 16.5 (p=0.0009) respectively. 12

13 corrected p(first seen chosen) p=6.3e-06 corrected p(first seen chosen) p=6.3e first fixation duration [ms] first fixation duration [ms] Figure S10: Null hypotheses for Fig. 5e. Reproduction of Fig. 5e from the text but with the simulation data from θ = 0 (left) and θ =1 (right). The goodness-of-fit statistics for θ = 0 and θ =1 were p = 0.1 and p = 10-9 respectively. 13

14 Figure S11: Individual histogram. Histogram of the best-fitting θ parameters based on a subject-by-subject MLE analysis where d and σ were fixed at their values ( and 0.02) from the group-level analysis, and we searched for θ from 0 to 1, in increments of

15 Histogram of biases Frequency biases Figure S12: Fig. 5a bias histogram. Histogram of the left-choice bias between last-fixation-left trials and last-fixation-right trials, subject by subject. This bias measure takes the average difference between the two curves in Fig. 5a. With d and σ fixed at their values ( and 0.02) from the group-level analysis, a subject with θ =1 would show a bias of 0 (first bin with 1 subject), a subject with θ = 0.3 would show a bias of 0.47 (fifth bin with 9 subjects), and a subject with θ = 0 would show a bias of 0.58 (sixth bin with 5 subjects). 15

16 first fixation duration [ms] fixated item rating Figure S13: Fig. 3b for first fixations. First fixation duration as a function of the fixated item s liking rating. A mixed-effects regression for fixation duration on liking rating yielded a coefficient of 9.4 ms/rating (p=0.004). 16

17 first fixation duration [ms] fixated item rating - unfixated item rating Figure S14: Fig. 3c for first fixations. First fixation duration as a function of the difference in liking ratings between the fixated item and the unfixated item. A mixed-effects regression for fixation duration on the difference in liking ratings yielded a coefficient of 4.3 ms/rating (p=0.1) suggesting no significant effect of relative value on first-fixation duration. 17

18 first fixation duration [ms] p= best rating - worst rating Figure S15: Fig. 3d for first fixations. First fixation duration as a function of the difference in liking ratings of the best and worst rated items. A mixed-effects regression for fixation duration on the absolute difference in liking ratings yielded a coefficient of ms/rating (p=0.98) indicating no significant effect of absolute rating-difference on first-fixation duration. 18

19 Density fixation duration Density fixation duration Figure S16: Fixation histogram for all trials. Fixation duration histograms across all trials with best-fitting log-normal distributions superimposed (solid line) for first fixations (left) and middle fixations (right). 19

20 Density fixation duration Density fixation duration Figure S17: Fixation histogram with 0 difference. Fixation duration histograms for trials with an absolute difference in liking ratings of 0, with best-fitting log-normal distributions superimposed (solid line) for first fixations (left) and middle fixations (right). 20

21 Density fixation duration Density fixation duration Figure S18: Fixation histogram with 1 difference. Fixation duration histograms for trials with an absolute difference in liking ratings of 1, with best-fitting log-normal distributions superimposed (solid line) for first fixations (left) and middle fixations (right). 21

22 Density fixation duration Density fixation duration Figure S19: Fixation histogram with 2 difference. Fixation duration histograms for trials with an absolute difference in liking ratings of 2, with best-fitting log-normal distributions superimposed (solid line) for first fixations (left) and middle fixations (right). 22

23 Density fixation duration Density fixation duration Figure S20: Fixation histogram with 3 difference. Fixation duration histograms for trials with an absolute difference in liking ratings of 3, with best-fitting log-normal distributions superimposed (solid line) for first fixations (left) and middle fixations (right). 23

24 Density fixation duration Density fixation duration Figure S21: Fixation histogram with 4 difference. Fixation duration histograms for trials with an absolute difference in liking ratings of 4, with best-fitting log-normal distributions superimposed (solid line) for first fixations (left) and middle fixations (right). 24

25 Density fixation duration Density fixation duration Figure S22: Fixation histogram with 5 difference. Fixation duration histograms for trials with an absolute difference in liking ratings of 5, with best-fitting log-normal distributions superimposed (solid line) for first fixations (left) and middle fixations (right). 25

26 rt [ms] p=2.6e-05 # of fixations p=3.9e best rating - worst rating best rating - worst rating rt [ms] p=2.6e-05 # of fixations p=3.9e best rating - worst rating best rating - worst rating Figure S23: Alternative model 1 Fig. 2b c. Replication of Fig. 2b (left column) and 2c (right column) for alternative model 1 with d increased to ms -1 (top row) and with σ increased to 0.06 (bottom row). 26

27 p(left chosen) p= final time advantage left [ms] p(left chosen) p= final time advantage left [ms] Figure S24: Alternative model 1 Fig. 5b. Replication of Fig. 5b for alternative model 1 with the numerator of the probability function decreased to (left) and (right). 27

28 p(left chosen) left rating - right rating rt [ms] p=2.6e best rating - worst rating # of fixations p=3.9e best rating - worst rating p(left chosen) p= final time advantage left [ms] Figure S25: Alternative model 2. Replication of Fig. 2a, 2b, 2c, and 5b with the alternative model 2 that has no barriers and exogenous reaction times. 28

29 time advantage [ms] last fixation duration [ms] p(left chosen) last fix left last fix right left rating - right rating time advantage [ms] p(left chosen) last fixation duration [ms] left rating - right rating last fix left last fix right time advantage [ms] p(left chosen) last fixation duration [ms] left rating - right rating last fix left last fix right Figure S26: Alternative model 2 theta variants. Replication of Fig. 4c (left column) and 5a (right column) for alternative model 2 with θ decreased to 0 (top row) and increased to 0.6 (middle row) and 0.9 (bottom row). 29

30 time advantage [ms] p(left chosen) last fixation duration [ms] left rating - right rating last fix left last fix right time advantage [ms] p(left chosen) last fixation duration [ms] left rating - right rating last fix left last fix right Figure S27: Alternative model 2 d variants. Replication of Fig. 4c (left column) and 5a (right column) for alternative model 2 with d decreased to ms -1 (top row) and increased to ms -1 (bottom row). 30

31 time advantage [ms] p(left chosen) last fixation duration [ms] left rating - right rating last fix left last fix right time advantage [ms] p(left chosen) last fixation duration [ms] left rating - right rating last fix left last fix right Figure S28: Alternative model 2 sigma variants. Replication of Fig. 4c (left column) and 5a (right column) for alternative model 2 with σ decreased to 0.01 (top row) and increased to 0.03 (bottom row). 31

32 p=2.6e-05 p(left chosen) left rating - right rating rt [ms] best rating - worst rating p(last fixation to chosen) last seen item rating - other item rating time advantage [ms] last fixation duration [ms] p(left chosen) last fix left last fix right p(first seen chosen) p=1e left rating - right rating first fixation duration [ms] Figure S29: Alternative model 3. Replication of Fig. 2a, 2b, 4b, 4c, 5a and 5d for alternative model 3 with barrier heights of 0.8 for fixated items. 32

33 p(left chosen) p= p(first seen chosen) p=1e final time advantage left [ms] first fixation duration [ms] p(left chosen) p= p(first seen chosen) p=1e final time advantage left [ms] first fixation duration [ms] Figure S30: Alternative model 3 variants. Replication of Fig. 5b (left column) and 5c (right column) for alternative model 3 with barrier heights of 0.5 (top row) and 0.2 (bottom row) for fixated items. 33

34 log normal distribution log likelihoods for other distributions meanlog sdlog log likelihood gamma cauchy negative binomial normal middle fixations all trials diff = NaN diff = diff = NaN diff = diff = diff = first fixations all trials diff = diff = diff = diff = diff = diff = Table S1. Fixation-distribution fitting. Best fitting parameters for various statistical distributions that were fit to the fixation data. The log-normal distributions consistently provided the best fit, as indicated by gray shading. Each row indicates the fit for either all the trials, or for only those trials with a particular absolute difference in liking ratings. In a couple cases the gamma distribution would not fit the data, resulting in a value of NaN. 34