We analyzed global preference (defined as the proportion of global responses in a similarity judgment task with Navon-like hierarchical stimuli) as a function of four main factors: one characteristic of the participants (their age) and three characteristics of the stimuli (the sensory modality in which they were delivered, visual or haptic, the size of their local components, and the size of the global shape—as set by the density of the local elements). We used a generalized linear-mixed model to establish which of these factors had a significant impact.
Table 2 shows the results of the complete model including all factors and their interactions.
We then proceeded to simplify the model by removing each of the factors and testing whether this affected the goodness of the fit, as gauged from the likelihood ratio “LR” for the models with and without each factor. We observed that the only factor that could be removed at no cost is the size of the global shape (or the density of the local elements: LR = 10.53; delta DF = 12;
p value = 0.57). This suggests that global size (density) failed to affect global preference, while both local element size and sensory modality had significant effects.
Figure 2 supports this conclusion by showing the average global preference across all observers (irrespectively of their age), which was clearly affected by the size of local elements (different colors: higher global preference for smaller elements) and by stimulus modality (empty/filled symbols: higher global preference for vision, at least for larger elements), but not global size (circles and diamonds, nearly overlapping in all cases).
Table 3 shows the model (after removing the factor global size), which revealed a significant three-way interaction (F(1, 9281) = 3.93;
p = 0.019) between factors modality, size and age, indicating that size differentially affected vision and haptics over development. Moreover, the simplified model (i.e., without global size) was preferable also according to the Akaike Information Criterion (AIC = 8997.8) as compared with the more complex one (i.e., including global size, AIC = 9011.2). Because the two models did not significantly differ, the simplest model was chosen (
Agresti, 2003).
Inspection of
Figure 3 allows for interpreting this interaction. For large stimuli, local-global preference was largely constant across age, and it was markedly different across modalities, with global preference emerging only for visual, not haptic stimuli (
Figure 3C). However, for medium and small stimuli (
Figures 3A,
3B), results were similar for the two modalities, both showing a gradual increase of global preference with age, with a trace of increased global preference for our stimuli composed of medium-sized elements.
In other words, the size of the local elements modulated the concordance across modalities: local-global preferences in haptics and vision covaried across age groups for stimuli made of the smallest local elements, whereas large cross-modal differences emerged for stimuli made of the largest local elements. This is also supported by the correlation analysis in
Figure 3D, plotting the proportion of global responses in haptics (y-axis) versus vision (x-axis) across participants; although all correlations are highly significant, there is a trend for a steeper slope for stimuli made of smaller local elements.
Figure 3D also highlights the scatter of individual participants’ results, with some individuals expressing diametrically opposite judgments across modalities (e.g., 100% global responses for vision and <50% global responses for haptics or vice versa). To test whether these idiosyncratic cross-modal differences are related with age, we defined an index of visuo-haptic concordance as the absolute value of the difference in global responses across the two modalities (
Figure 4).
For stimuli made of the largest local elements, cross-modal differences were large, as expected since the average global preference for visual and haptic stimuli were different (
Figures 2 and
3; red symbols and lines). The results for stimuli made of the smallest local elements are more interesting. Although, on average, global preference was similar for visual and haptic stimuli, some participants showed large cross-modal differences (of opposite signs, which make them disappear in the averages of
Figures 2 and
3, blue symbols and lines, and only emerge in
Figure 4 where absolute values are used). These differences were particularly large in young children, smaller in adults. We verified that these trends are statistically significant with a new linear mixed model entered with the absolute value of the cross-modal difference for each stimulus configuration, studied as function of local element size (categorical variable with three levels: large, medium, small) and the participants’ age (continuous variable, log-transformed), plus a random intercept to account for inter-individual variability. This revealed a significant local-size by age interaction (F(1,2324) = 11.54;
p = 6.94 · 10
−4) and no other significant main effects. Importantly, we excluded that the large cross-modal differences seen in young children could be explained by the random variability of their judgments. We measured the split-half reliability by correlating the proportion of global responses on even and odd trials (within modality and local-element size). The results show that even young participants (from six years old to 35.37, median 11.6 years) had very high internal consistency, for both visual and haptic stimuli (split-half reliability in vision: r(58) = 0.96;
p= 8 · 10
−33, logBF = 29.63 and haptics: r(58) = 0.87;
p = 6 · 10
−19, logBF = 16.03).
As a final step of our analyses, we checked whether our results could be better accounted for by a covariate of the local element size, local element numerosity. One could imagine that the reduced global preference for stimuli made of largest (and fewer) local elements is driven by the degradation of the global information related to the low numerosity of components (
Kimchi & Palmer, 1982;
Martin, 1979).
We had the opportunity to test this hypothesis by comparing global responses for (target) stimuli of different global shapes: triangles and squares, that were necessarily composed of a different number of local elements (see
Table 1). If the number of composing elements affected the quality of global information, one would predict higher global preference when the stimuli are composed of more numerous elements.
Figure 5 shows that our results do not conform to this prediction – if anything, there is a trace of an opposite tendency. For example, triangles made of three large local items showed higher (not lower) global preference than squares made of four local items. Moreover, small local element stimuli, having the highest numerosity variation (global forms were composed of 10, 15 or 16 local items), show the smallest (not the largest) change in global preference across global shapes. Given the latter observation, we suggest that the tendency for slightly stronger global preference for triangle versus square global shapes is not a consequence of the numerosity of local elements. Rather, it may be an effect of shape complexity, where simpler shapes (defined by less vertices) are easier to segment or more salient.