To investigate the factors related to visual search time, we conducted regression analyses based on the data of study B. The analysis was not performed for study A because of the small number of subjects.
Directness and gaze speed appeared to be highly associated with search time, as shown in
Figure 9 , which plots the search times versus the product of directness and gaze speeds. Each data point represents an eccentricity (15°, 22°, or 29°) under a cue condition (without cues and with auditory or contour cues). We proposed the following model to describe the relationship. Natural logarithms were applied to convert multiplication to addition so that simple linear regression analysis could be performed.
\[\mathrm{ln}(t)\ {=}\ k_{0}\ {+}\ k_{1}\ {\times}\ \mathrm{ln}(dir)\ {+}\ k_{2}\ {\times}\ \mathrm{ln}(spd)\]
where
t is the measured search time under a cue condition for an eccentricity,
dir is the measured directness,
spd is the angular gaze speed in deg/s, and
k 0 , k 1, and
k 2 are coefficients. Regression results were
k 0 = –0.77,
k 1 = –0.92, and
k 2 = –1.36. This model could explain 90% of the variance in the observed search time (
R 2 = 0.90;
df = 78). Search time was significantly related to directness (
P < 0.001) and gaze speed (
P < 0.001).
As mentioned, directness increased with VF and decreased with eccentricity (except for auditory-cue search; see “Directness of Search Path”). To examine the relationship between search time and VF, we tested a model similar to
equation 1describing the relationship between search time and eccentricity, VF, and gaze speed
\[\mathrm{ln}(t)\ {=}\ b_{0}\ {+}\ b_{1}\ {\times}\ \mathrm{ln}(ecc)\ {+}\ b_{2}\ {\times}\ \mathrm{ln}(vf)\ {+}\ b_{3}\ {\times}\ \mathrm{ln}(spd)\]
where
t and
spd have the same definitions as in
equation 1 ,
vf is the binocular horizontal VF size in degrees,
ecc is the target eccentricity, and
b 0 , b 1 , b 2, and
b 3 are coefficients to be derived. Because neither eccentricity nor VF had any significant correlation with gaze speed, as reported, eccentricity, VF, and gaze speed can be considered orthogonal variables.
Because the effect of eccentricity on directness varied between cue conditions
(Fig. 8b) , we conducted three separate multiple regression analyses.
Table 1lists the results of regression analyses based on the model described in
equation 2 . The model explained 63%, 67%, and 79% of the variance in the search time for without-cue, auditory-cue, and contour-cue searches, respectively. Eccentricity, VF, and gaze speed were all significantly correlated with search time (
P ≤ 0.006) except for the eccentricity factor in auditory-cue search (
P = 0.143), which was consistent with the result from the previous ANOVA.
Based on the model, larger eccentricity required longer search time for both without-cue and contour-cue searches. However, the strength of the eccentricity effect was stronger for contour-cue searches (
b 1 = 1.52 vs. 0.75). This suggests that there may be an eccentricity at which the search times for contour-cue searches and without-cue searches will be the same. This crossover point can be thought of as the eccentricity threshold, below which contour-cue searches take shorter time than without-cue searches. We derived the eccentricity threshold using the following equation:
\[b_{0\mathrm{w}}\ {+}\ b_{1\mathrm{w}}\ {\times}\ \mathrm{ln}(ecc)\ {+}\ b_{2\mathrm{w}}\ {\times}\ \mathrm{ln}(vf)\ {+}\ b_{3\mathrm{w}}\ {\times}\ \mathrm{ln}(spd_{\mathrm{w}})\ {>}\ b_{0\mathrm{c}}\ {+}\ b_{1\mathrm{c}}\ {\times}\ \mathrm{ln}(ecc)\ {+}\ b_{2\mathrm{c}}\ {\times}\ \mathrm{ln}(vf)\ {+}\ b_{3\mathrm{c}}\ {\times}\ \mathrm{ln}(spd_{\mathrm{c}})\]
and
\[(b_{1\mathrm{w}}\ {-}\ b_{1\mathrm{c}})\mathrm{ln}(ecc)\ {>}\ b_{0\mathrm{c}}\ {-}\ b_{0\mathrm{w}}\ {+}\ (b_{2\mathrm{c}}\ {-}\ b_{2\mathrm{w}})\ {\times}\ \mathrm{ln}(vf)\ {+}\ b_{3\mathrm{c}}\ {\times}\ \mathrm{ln}(spd_{\mathrm{w}})\ {-}\ b_{3\mathrm{w}}\ {\times}\ \mathrm{ln}(spd_{\mathrm{w}})\]
where the notations have the same definition as in
equation 2 , and subscripts
w and
c denote the without-cue search and contour-cue search, respectively. After substituting the coefficients listed in
Table 1into
equation 3 , it becomes
\[{-}0.77\ {\times}\ \mathrm{ln}(ecc)\ {>}\ {-}0.5{-}0.67\ {\times}\ \mathrm{ln}(vf){-}1.39\ {\times}\ \mathrm{ln}(spd_{\mathrm{c}})\ {+}\ 1.11\ {\times}\ \mathrm{ln}(spd_{\mathrm{w}})\]
and
\[\mathrm{ln}(ecc)\ {<}\ 0.65\ {+}\ 0.87\ {\times}\ \mathrm{ln}(vf)\ {+}\ 1.81\ {\times}\ \mathrm{ln}(spd_{\mathrm{c}})\ {-}\ 1.44\ {\times}\ \mathrm{ln}(spd_{\mathrm{w}})\]
Note that the greater-than sign becomes a less-than sign because the coefficient of ln(
ecc) is negative in the derivation of
equation 4 .
Figure 10plots the eccentricity threshold and VF size, assuming the gaze speed of the without-cue search is always 63 deg/s—the actual average gaze speed without cues in study B. In the figure, the solid line represents that gaze speed with contour cues is 33 deg/s—the measured average gaze speed with contour cues. The dashed line represents (for an assumption) that gaze speed with contour cues could be increased to 38 deg/s. As shown, the dashed line is above the solid line. It means that as gaze speed with the device increases, the eccentricity threshold becomes larger. In other words, the device would then be helpful for patients to search for targets within a larger eccentricity, such as from point A to point B. Similarly, it also suggests that when the gaze speed increases, the VF required to gain benefit from the device would become smaller. In other words, the device could become helpful for patients with smaller VFs, such as from point C to point D. We believe that gaze speed can be improved through practice or training with the device.
Based on the predicted eccentricity threshold, we further examined the ratio of eccentricity threshold to VF radius (because eccentricity is also a radius measure), which we define as the expansion ratio (ER). This ratio represents, relative to the VF, the area within which patients would find targets faster when using the contour cues than without.ER as a function of VF for study B is also plotted in
Figure 10with × signs and triangles for the two gaze speeds. It can be seen that patients with smaller VFs have larger ERs than those with larger VFs. This result does not contradict our finding that patients with larger VFs benefited from the device while the patients with smaller VFs did not. Because ER is a relative measure, the same tested eccentricities required higher ERs for patients with smaller VFs than patients with larger VFs. For instance, the regression model predicts that at a gaze speed of 33 deg/s, a patient with a VF of 7° would achieve an ER of 4.2, which is larger than that of a patient with a VF of 12° (ER of 3.9). However, the patient with the 7° VF could benefit from the device for targets within an eccentricity of 14.7° (4.2 × 7/2), which is smaller than that of the patient with the 12° VF, within an eccentricity of 23.4° (3.9 × 12/2). Therefore, when eccentricity of 15° is tested, we would likely observe that the patient with the 7° VF cannot benefit from the device but that the patient with the 12° VF can.