Using Hierarchical Linear Models to Examine Approximate Number System Acuity: The Role of Trial-Level and Participant-Level Characteristics

Abstract

The ability to intuitively and quickly compare the number of items in collections without counting is thought to rely on the Approximate Number System (ANS). To assess individual differences in the precision of peoples’ ANS representations, researchers often use non-symbolic number comparison tasks in which participants quickly choose the numerically larger of two arrays of dots. However, some researchers debate whether this task actually measures the ability to discriminate approximate numbers or instead measures the ability to discriminate other continuous magnitude dimensions that are often confounded with number (e.g., the total surface area of the dots or the convex hull of the dot arrays). In this study, we used hierarchical linear models (HLMs) to predict 132 adults’ accuracy on each trial of a non-symbolic number comparison task from a comprehensive set of trial-level characteristics (including numerosity ratio, surface area, convex hull, and temporal and spatial variations in presentation format) and participant-level controls (including cognitive abilities such as visual-short term memory, working memory, and math ability) in order to gain a more nuanced understanding of how individuals complete this task. Our results indicate that certain trial-level characteristics of the dot arrays contribute to our ability to compare numerosities, yet numerosity ratio, the critical marker of the ANS, remains a highly significant predictor of accuracy above and beyond trial-level characteristics and across individuals with varying levels of math ability and domain-general cognitive abilities.

To what extent is dot comparison an appropriate measure of approximate number system?

“…There was a ratio effect on accuracy for fully incongruent trials, but not for congruent ones. A similar interaction between ratio and congruency was found in recent studies with adults (e.g., Braham et al, 2018;Reynvoet et al, 2021). As in Reynvoet et al 's study, the present finding may be explained by a ceiling effect in fully congruent trials and the low accuracy in incongruent trials, especially in fully incongruent ones.…”

“…High accuracy in fully congruent trials has been reported previously in other studies (e.g., Gilmore et al, 2013 ; Wilkey et al, 2021 ). Braham et al (2018) suggested that when trials are difficult due to the ratio being high (small differences between the two quantities), children may use other cues to solve the comparison. In the fully incongruent trials, this strategy decreased accuracy.…”

“…Secondly, given that differences in performance across trial types depend on the visual properties of the stimuli ( Gebuis and Reynvoet, 2012a , b ), we expect children to be more accurate on trials that are congruent with the visual properties of the stimuli than on incongruent ones (e.g., Clayton et al, 2015 ) and particularly inaccurate when all visual cues are negatively correlated with numerosity (e.g., Clayton et al, 2019 ). Thirdly, provided that recent studies with adults suggest that the numerosity ratio effect is also influenced by the continuous visual properties of the stimuli ( Braham et al, 2018 ; Reynvoet et al, 2021 ), we expect the numerosity ratio effect to be larger in incongruent trials than in congruent ones. As stated earlier, the congruency effect has been interpreted as the effort to suppress the response based on visual features to respond according to the numerosity of the arrays ( Szucs et al, 2013 ), so we also predict that inhibitory control skills will influence performance in incongruent trials but not in congruent ones.…”

Food Quantity Discrimination in Angelfish (Pterophyllum scalare): The Role of Number, Density, Size and Area Occupied by the Food Items

“…Next (Experiment 2), we examine the influence of the number of items and density when these features are in conflict while keeping the size of the food items equal. Last, another non-numerical variable that has been shown to influence non-symbolic numerical discrimination is the convex hull, the perimeter that encompasses all of the items in a set (e.g., DeWind and Brannon, 2016;Gilmore et al, 2016;Braham et al, 2018). This variable usually covaries with other continuous as well as numerical attributes of contrasted sets in food quantity discrimination studies with fish.…”

Modeling Magnitude Discrimination: Effects of Internal Precision and Attentional Weighting of Feature Dimensions

“…Participants were shown a stimulus set containing segregated groups of blue and yellow dots, and were instructed to respond, in separate blocks, which of the two groups was larger with respect to each feature. We chose total surface area and convex hull because they are some of the most‐frequently investigated dimensions in the number literature (e.g., area: Brannon et al., 2006; DeWind, Bonner, & Brannon, 2020; Odic et al., 2013; Tomlinson et al., 2020; convex hull: Braham et al., 2018; Clayton et al., 2015; Clayton & Gilmore, 2015; Gilmore et al., 2016; Norris et al., 2019) and additionally, these dimensions can be varied separately from one another, which allowed us to quantify each of their individual contributions. Importantly, the stimuli were identical across the three tasks, which enables us to directly compare responses to a selfsame image.…”