Temporal dynamics of numerosity judgments: Liberal responses across different reference numbers in early response onsets

Approximate numerosity judgments allow humans and animals to quickly estimate quantities without verbal processing. This ability obeys Weber's law, which states that the discrimination between two quantities depends on their ratio rather than their absolute difference, making numerosity perception robust across varying contexts. Despite its robustness, temporal constraints significantly influence numerosity judgments. Shorter response times have been associated with numerical overestimation; however, the consistency of this effect across different reference quantities and varying temporal demands remains unclear. To investigate this, an externally cued response-onset paradigm with cue delays ranging from 60 to 1000 ms was employed across four reference numbers (30, 40, 50, and 60), with comparison stimuli standardized by proportionally scaled numerical differences corresponding to one-fifth of each reference number. Comprehensive analysis methods were used, including relative bias and reaction time analysis, signal detection theory, and Bayesian psychometric modeling to examine changes in normalized perceived numerosity, sensitivity, and decision-making biases. The analyses revealed that shorter response onsets (60 and 120 ms) systematically yield an overestimation bias, which is associated with a liberal decision criterion and low sensory precision. Furthermore, posterior distributions indicated that as processing time increases, sensory evidence becomes more stable, thereby increasing the precision of numerical judgments. Crucially, this temporal effect was consistent across all reference conditions, suggesting that the approximate number system (ANS) operates in a ratio-sensitive manner independent of absolute quantity. These findings deepen the understanding of the ANS by showing how temporal constraints work with ratio-based processing, providing an extended behavioral approach for investigating the dynamic, scale-invariant nature of numerical cognition.