One model fits all: Explaining many aspects of number comparison within a single coherent model—A random walk account.

Abstract

The time required to determine the larger of 2 digits decreases with their numerical distance, and, for a given distance, increases with their magnitude (Moyer & Landauer, 1967). One detailed quantitative framework to account for these effects is provided by random walk models. These chronometric models describe how number-related noisy partial evidence is accumulated over time; they assume that the drift rate of this stochastic process varies lawfully with the numerical magnitude of the digits presented. In a complete paired number comparison design we obtained saccadic choice responses of 43 participants, and analyzed mean saccadic latency, error rate, and the standard deviation of saccadic latency for each of the 72 digit pairs; we also obtained mean error latency for each numerical distance. Using only a small set of meaningfully interpretable parameters, we describe a variant of random walk models that accounts in considerable quantitative detail for many facets of our data, including previously untested aspects of latency standard deviation and error latencies. However, different from standard assumptions often made in random walk models, this account required that the distributions of step sizes of the induced random walks are asymmetric. We discuss how our findings can help in interpreting complex findings (e.g., conflicting speed vs. accuracy trends) in applied studies which use number comparison as a well-established diagnostic tool. Finally, we also describe a novel effect in number comparison, the decrease of saccadic response amplitude with numerical distance, and suggest an interpretation using the conceptual framework of random walk models. (PsycINFO Database Record

Categorizing digits and the mental number line

“…According to a prominent model of number processing (e.g., Dehaene & Brannon, 2011;Nieder, 2005;Reike & Schwarz, 2016Rugani & de Hevia, 2017;Walsh, 2003), humans automatically convert digits into analog representations much like sensory representations of physical attributes such as brightness or length. These representations are often thought to be approximate, preverbal, and highly over-learned, and to have a quasi-spatial format, often referred to metaphorically as a "mental number line".…”

“…For a decision involving a given digit i, response latencies and error rates depend on the response barriers and the retrieval rate (the drift μ i ) which represents the mean evidence increment per time unit contributed by that digit. Converging evidence suggests that diffusion models provide a coherent conceptual framework that accounts in considerable quantitative detail for the way in which humans compare numbers (e.g., Kamienkowski, Pashler, Dehaene, & Sigman, 2011;Reike & Schwarz, 2016;Schwarz & Ischebeck, 2003Sigman & Dehaene, 2005;Smith & Mewhort, 1998). More recently, the same model framework has been used to account, for example, for age effects in numerical cognition, both in young children (e.g., Thompson, Ratcliff, & McKoon, 2016) and in older adults (e.g., .…”

The number–weight illusion

“…Correspondingly, the way in which we learn, retrieve, process, and compare numerical magnitudes as symbolized by digits is among the best-investigated processes in cognitive psychology. According to a prominent model of number processing (e.g., Dehaene & Brannon, 2011;Maloney et al, 2010;Nieder, 2005;Reike & Schwarz, 2016;Rugani & de Hevia, 2017;Walsh, 2003), humans automatically convert digits into analog representations much like sensory representations of physical attributes such as brightness or size. These representations are often thought to be preverbal and highly over-learned, and to have a quasi-spatial format, a "mental number line" that preserves ordinal relations.…”

“…On this mental number line, smaller numbers are separated from their neighbors (e.g., 2-3) more distinctly than larger numbers (e.g., 7-8). This latter effect is reminiscent of Weber's law in judging physical attributes, and is typically modeled by assuming that the numbers 1 to 9 are internally represented on a mental number line that is logarithmically compressed (Dehaene, 2003;Gallistel & Gelman, 2005;Galton, 1880;Nieder, 2005;Reike & Schwarz, 2016). Convergent evidence supporting this view comes from experimental, comparative, physiological, and developmental psychology and has been extensively reviewed (Butterworth, 1999;Dechaene & Brannon, 2011;Whalen, Gallistel, & Gelman, 1999).…”

Exploring the origin of the number-size congruency effect: Sensitivity or response bias?

“…According to one influential model of number comparison (Moyer & Landauer, 1967; for recent formulations and review, see Ditz & Nieder, 2016;Ganor-Stern & Goldman, 2015;Gilmore, Attridge, & Inglis, 2011;Inglis & Gilmore, 2014; Maloney, Risko, Presto, Ansari, & Fugelsang, 2010;Nuerk, Moeller, Klein, Willmes, & Fischer, 2011;Reike & Schwarz, 2016;Sigman & Dehaene, 2005), digits are automatically converted into percept-like analog representations that are then in turn compared with each other, much like sensory representations of physical attributes such as brightness or orientation. One line of evidence consistent with this model derives from conflict paradigms in which the to-becompared digits are presented with varying physical (i.e., font) sizes.…”