Phantom array effect visibility is based on the correct response rate. Saccadic eye movement speed was measured using an eye tracker, the ViewPoint EyeTracker® System (Arrington Research, USA). We filtered out many potential artifacts of the tracking data to determine the peak speed of eye movement in addition to the temporal information of the moment the experiment light turns on and off.
Phantom array visibility at different viewing distances and directions
The participants’ correct response rates of the phantom array effect for the distance- and direction-based experiments are described in Table 1 and Fig. 1. The frequency and correct response rate were inversely proportional, which means that phantom array visibility decreased as the modulation frequency increased. In both experiments, the visibility of the phantom array effect was evaluated based on the correct response rate of the participant. If the participant selected the correct answer in all trials, the correct response rate would be 100%, and if the response was randomly selected regardless of the visibility of the phantom array, it would be 50%. Therefore, we set the average correct response rate to 75%, which is half-way, as the visibility threshold frequency of the phantom array effect.
Table 1
Mean (and standard deviation) of the correct response rates of the phantom array effect of all experiments. Frequency values above 75% correct response rate are marked in bold with gray background.
Experiment type | Session | Frequency |
1kHz | 2kHz | 3kHz | 4kHz | 5kHz | 6kHz | 7kHz | 8kHz | 9kHz | 10kHz | 11kHz |
Distance- based | Narrow (15°) | 89 (17) | 90 (16) | 80 (22) | 76 (22) | 73 (22) | 67 (22) | 66 (20) | 67 (17) | 70 (20) | 57 (23) | 54 (21) |
Wide (30°) | 94 (11) | 92 (11) | 87 (17) | 86 (17) | 78 (24) | 73 (21) | 66 (21) | 69 (25) | 63 (22) | 61 (20) | 60 (20) |
Direction-based | Right upward | 86 (19) | 87 (16) | 80 (20) | 72 (18) | 68 (20) | 63 (19) | 62 (18) | 57 (19) | 60 (17) | 50 (17) | 51 (16) |
Rightward | 94 (9) | 87 (17) | 86 (22) | 80 (26) | 80 (24) | 80 (19) | 70 (22) | 70 (26) | 65 (23) | 65 (20) | 59 (18) |
Right downward | 86 (22) | 79 (25) | 73 (25) | 65 (20) | 59 (23) | 58 (22) | 51 (20) | 61 (12) | 50 (15) | 50 (13) | 43 (16) |
Downward | 83 (24) | 81 (18) | 74 (21) | 63 (21) | 60 (25) | 66 (19) | 52 (26) | 56 (19) | 55 (20) | 51 (17) | 44 (16) |
The correct response rate at the threshold frequency of wide saccadic distance was higher than that of narrow saccadic distance (paired t-test, t(15)=-3.058, p = .003). Regarding saccadic direction, the correct response rate at the threshold frequency was high in the following order: rightward > right upward > right downward ≅ downward (ANOVA, F(3,42) = 9.951, p = .000).
Eye Movement Speed Analysis For Different Viewing Distances And Viewing Directions
The participants’ average speed based on peak saccadic eye movement when observing the phantom array effect is shown in Fig. 2. The average speeds of saccadic eye movement for the narrow and wide viewing distances are as follows: 349.31 ± 97.90°/s and 448.95 ± 143.94°/s, respectively. The speed of saccadic eye movement at a wide viewing distance was significantly faster than that of the narrow one (Fig. 2a, paired t-test, t(15)=-4.279, p = .001). Average speeds of saccadic eye movement in the right upward, rightward, right downward, and downward viewing directions are 393.64 ± 93.29°/s, 414.27 ± 98.99°/s, 388.79 ± 88.94°/s, and 358.40 ± 80.85°/s, respectively. Repeated measures ANOVA showed that the difference in saccadic peak velocity according to viewing direction was statistically significant (Fig. 2b, ANOVA, F(3,42) = 37.624, p = .000). The speed of saccadic eye movement in the rightward direction was significantly faster than in the other directions.
Subgroup differences in the speed of saccadic eye movement and visibility of the phantom array effect
To analyze the individual differences in saccadic eye movements and phantom array visibility, participants were further grouped through k-means according to the speed of saccadic eye movements in each exercise condition. The K value was selected based on the elbow method17, and in the case of the direction-based experiment, k-means was performed after reducing it to two dimensions through multidimensional scaling (MDS)18. Figure 3 shows the two-dimensional embedding and clustering of the distance-based (Fig. 3a) and direction-based experiments (Fig. 3b). Scatterplot examples of individual saccadic speeds in different direction pairs are also shown with the identification of high-, middle, and low subgroups (Fig. 3c).
Table 2 shows the subgroup mean and standard deviation of the saccadic eye movement speed (°/s) and peak threshold frequency (kHz) for different viewing distances and viewing directions.
Figure 4 shows the saccadic peak velocity and peak threshold frequency according to the subgroups classified based on saccadic peak velocity. To determine statistical significance, independent t-tests were performed for distance-based experiments and one-way ANOVA for direction-based experiments.
Regarding the distance-based experiment, the difference in saccadic peak velocity between subgroups was significant in narrow and wide distances (independent t-test, t(14)=-4.939, p = .000; t(14)=-7.128, p = .000), and the peak threshold frequency was statistically higher in the high-speed group than in the low-speed group in both narrow and wide distances (independent t-test, t(14)=-2.954, p = .010; t(14)=-3.818, p = .002).
For the direction-based experiment, the saccadic velocity between the three groups was statistically significant in the right upward, rightward, right downward, and downward directions (ANOVA, F(2,14) = 17.712, p = 0.000; F(2,14) = 17.810, p = 0.000; F(2,14) = 22.767, p = 0.000; F(2,14) = 17.103, p = 0.000). The difference in the peak threshold frequency was significant in the rightward and right downward directions (ANOVA, F(2,14) = 15.160, p = 0.001; F(2,14) = 7.235, p = 0.009).
Table 2
Subgroup mean (and standard deviation) of saccadic eye movement speed and peak threshold frequency at different viewing distances and viewing directions (°/sec).
Experiment type | Session | Saccade peak velocity group (n = number of participants) | Saccade peak velocity (°/s) | Peak threshold frequency (kHz) |
Distance-based | Narrow | High (n = 8) | 321.38 (65.25) | 8.75 (1.98) |
Low (n = 8) | 271.38 (50.72) | 4.88 (3.14) |
Wide | High (n = 8) | 576.38 (77.35) | 9.38 (1.77) |
Low (n = 8) | 427.38 (73.61) | 5.25 (2.49) |
Direction-based | Right upward | High (n = 6) | 478.35 (28.93) | 5.83 (2.23) |
Mid (n = 6) | 374.99 (64.44) | 4.83 (3.43) |
Low (n = 3) | 261.52 (63.59) | 4.00 (2.00) |
Rightward | High (n = 6) | 524.23 (27.35) | 10.00 (1.26) |
Mid (n = 6) | 387.34 (65.44) | 9.50 (1.52) |
Low (n = 3) | 296.39 (84.66) | 4.33 (2.08) |
Right downward | High (n = 6) | 483.67 (33.05) | 6.83 (1.47) |
Mid (n = 6) | 378.54 (54.77) | 5.83 (1.94) |
Low (n = 3) | 263.56 (54.91) | 2.33 (1.53) |
Downward | High (n = 6) | 437.15 (17.16) | 6.67 (3.83) |
Mid (n = 6) | 353.40 (56.10) | 6.33 (2.42) |
Low (n = 3) | 245.22 (67.56) | 4.00 (3.61) |