We next fit a linear mixed model to the data to examine the relationship between critical spacing estimates, perceived spacing (PSE), and position sensitivity (JND). Specifically, we fit a linear model with critical spacing as the dependent variable, with fixed effects of PSE and JND value, calculated from the perceived spacing task. To account for repeated measures between observers, each participant was entered as a random effect, with a separate intercept estimated for each participant.
The results of the linear model showed a significant, positive relationship between critical spacing and PSE values, with a fixed effect coefficient of 2.17 (standard error: 0.42), F(1,94) = 26.94, p < 0.001.
Figure 4A shows the unique relationship between PSE and critical spacing after taking into account the other variables in the model. These results support our hypothesis that, in areas where crowding is stronger (as indicated by larger critical spacing values), the point of subjective point of equality (PSE) was also larger. Larger PSE values correspond with smaller perceived spacing. Therefore, in locations where crowding is stronger, perceived spacing is smaller. The direction of this relationship was also consistent across participants (as shown in
Figure S2 in the
Supplemental materials; all participants had a positive slope when fitted individually). Furthermore, the results of the model showed a significant, positive relationship between the JND and critical spacing (
Figure 4B) with a fixed effect coefficient of 6.46 (standard error: 1.27), F(1,94) = 25.688, p < 0.001. These results demonstrate that position sensitivity is also positively associated with the strength of crowding. Areas with a higher JND (i.e., less sensitivity to changes in perceived spacing) were associated with larger critical spacing values, which means that participants experienced stronger crowding at those locations.
In addition, we verified that these effects were not dependent on our choice of parameters for fitting the psychometric functions. Rerunning the model with the addition of a lapse rate parameter to the psychometric fits revealed similar results, with a significant positive relationship between PSE and critical spacing, F(1,94) = 25.15,
p < .001, and a significant, positive relationship between JND and critical spacing, F(1,94) = 22.07,
p < .001 (see the
Supplemental Materials). We also verified that these results were consistent between sessions by analyzing the data separately for session 1 and session 2 (in contrast with the main analysis, which pooled responses from both sessions before fitting). In this analysis, we observed significant relationships between both the PSE and critical spacing and between the JND and critical spacing separately for each session, consistent with the finding that pairs of sessions within a given participant were well-correlated with one another (see the
Supplemental Materials).
One possibility is that there may be similarities in crowding and perceived spacing across participants that are seen at the group level. For example, on average, crowding is typically stronger in the upper visual field compared with the lower visual field (
He, Cavanagh, & Intriligator, 1996) and weaker along the horizontal meridian (
Liu, Jiang, Sun, & He, 2009). Given that critical spacing values and PSE values were similar across participants, we determined how much the effects from the main analysis are driven by participant idiosyncrasies compared with group-level effects. If PSE values show similar asymmetries, this process could produce a significant correlation that is driven by group-level effects. In other words, we separately analyzed the degree to which between-subject differences in perceived spacing are related to between-subject variability in crowding. To do this, we subtracted out the mean threshold estimates (the 80% and 50% estimates for critical spacing and PSE values, respectively) for each location (across participants) from every participant's threshold estimate at that location. These resulting values provide an estimate, for a given location, of the strength of crowding (or the size of the PSE values) for a participant relative to the group mean. We then proceeded to fit the same model to these values. The results indicate a significant positive relationship between the mean subtracted JND estimates and the critical spacing values,
F(1,94) = 9.28,
p = 0.003, with a coefficient of 3.82 (standard error, 1.25), indicating that the relationship between critical spacing and the JND was not driven exclusively by commonalities across participants and that individual differences in position sensitivity are associated with individual differences in crowding at different visual field locations. However, we did not observe the same positive relationship between the mean subtracted PSE values and the critical spacing estimates,
F(1,94) = 0.65,
p = 0.42. This finding suggests that, although stronger crowding is associated with smaller perceived separation, this factor is driven by group-level effects rather than individual idiosyncrasies.