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

Physical size modulates the efficiency of digit comparison, depending on whether the relation of numerical magnitude and physical size is congruent or incongruent (Besner & Coltheart, Neuropsychologia, 17, 467-472, 1979), the number-size congruency effect (NSCE). In addition, Henik and Tzelgov (Memory & Cognition, 10, 389-395, 1982) first reported an NSCE for the reverse task of comparing the physical size of digits such that the numerical magnitude of digits modulated the time required to compare their physical sizes. Does the NSCE in physical comparisons simply reflect a number-mediated bias mechanism related to making decisions and selecting responses about the digit's sizes? Alternatively, or in addition, the NSCE might indicate a true increase in the ability to discriminate small and large font sizes when these sizes are congruent with the digit's symbolic numerical meaning, over and above response bias effects. We present a new research design that permits us to apply signal detection theory to a task that required observers to judge the physical size of digits. Our results clearly demonstrate that the NSCE cannot be reduced to mere response bias effects, and that genuine sensitivity gains for congruent number-size pairings contribute to the NSCE. Numbers and numerical information play a central role in how humans represent, communicate, and respond to quantity-related aspects of their environments. Correspondingly, a considerable amount of research has addressed many cognitive, neuronal, developmental, and clinical aspects of how we process quantitative information (for general review, see Dehaene, 2011;Nieder, 2005).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. According to the analog representation model, the digit's (task-irrelevant) physical size should modulate the efficiency of the comparison process, depending on whether the relation of numerical magnitude and physical size is congruent (as in 8 -2) or incongruent (e.g., 8 -2). This number-size congruency effect (NSCE) has indeed first been reported by Besner and Coltheart (1979), and has since then often been confirmed and extended (e.g., Algom,

Using a standard repeated measures model with arbitrary true score distribution and normal error variables, we present some fundamental closed-form results which explicitly indicate the conditions under which regression effects towards (RTM) and away from the mean are expected. Specifically, we show that for skewed and bimodal distributions many or even most cases will show a regression effect that is in expectation away from the mean, or that is not just towards but actually beyond the mean. We illustrate our results in quantitative detail with typical examples from experimental and biometric applications, which exhibit a clear regression away from the mean ('egression from the mean') signature. We aim not to repeal cautionary advice against potential RTM effects, but to present a balanced view of regression effects, based on a clear identification of the conditions governing the form that regression effects take in repeated measures designs.

We propose to interpret tasks evoking the classical Müller-Lyer illusion as one form of a conflict paradigm involving relevant (line length) and irrelevant (arrow orientation) stimulus attributes. Eight practiced observers compared the lengths of two line-arrow combinations; the length of the lines and the orientation of their arrows was varied unpredictably across trials so as to obtain psychometric and chronometric functions for congruent and incongruent line-arrow combinations. To account for decision speed and accuracy in this parametric data set, we present a diffusion model based on two assumptions: inward (outward)-pointing arrows added to a line (i) add (subtract) a separate, task-irrelevant drift component, and (ii) they reduce (increase) the distance to the barrier associated with the response identifying this line as being longer. The model was fitted to the data of each observer separately, and accounted in considerable quantitative detail for many aspects of the data obtained, including the fact that arrow-congruent responses were most prominent in the earliest RT quartile-bin. Our model gives a specific, process-related meaning to traditional static interpretations of the Müller-Lyer illusion, and combines within a single coherent framework structural and strategic mechanisms contributing to the illusion. Its central assumptions correspond to the general interpretation of geometrical-optical illusions as a manifestation of the resolution of a perceptual conflict (Day & Smith, 1989;Westheimer, 2008). Keywords Müller-Lyer illusion • Line perception • Conflict task • Diffusion model • Psychometric and chronometric function • Response bias • SensitivityThe classical Müller-Lyer (1889) illusion is one of the most striking and robust demonstrations in visual perception: a line of given length appears subjectively longer (shorter) when inward (outward)-pointing arrows are attached to its ends 1 . Many aspects of this illusion have been thoroughly investigated, including geometrical and physical parameters defining the stimuli (DeLucia, 1993;Dragoi & Lockhead, 1999), neurophysiological correlates of the illusion 1 Note that some authors call inward-pointing arrows (> <) "outward arrows" (Day, 1972), "outgoing wings" (Coren & Girgus, 1972), or "tail fins" (Wang et al., 1998).

Whereas many cognitive tasks show pronounced aging effects, even in healthy older adults, other tasks seem more resilient to aging. A small number of recent studies suggests that number comparison is possibly one of the abilities that remain unaltered across the life span. We investigated the ability to compare single-digit numbers in young (19-39 years; n = 39) and healthy older (65-79 years; n = 39) adults in considerable detail, analyzing accuracy as well as mean and variance of their response time, together with several other well-established hallmarks of numerical comparison. Using a recent comprehensive process model that parsimoniously accounts quantitatively for many aspects of number comparison (Reike & Schwarz, 2016), we address two fundamental problems in the comparison of older to young adults in numerical comparison tasks: (a) to adequately correct speed measures for different levels of accuracy (older participants were significantly more accurate than young participants), and (b) to distinguish between general sensory and motor slowing on the one hand, as opposed to a specific age-related decline in the efficiency to retrieve and compare numerical magnitude representations. Our results represent strong evidence that healthy older adults compare magnitudes as efficiently as young adults, when the measure of efficiency is uncontaminated by strategic speed-accuracy trade-offs and by sensory and motor stages that are not related to numerical comparison per se. At the same time, older adults aim at a significantly higher accuracy level (risk aversion), which necessarily prolongs processing time, and they also show the well-documented general decline in sensory and/or motor functions. (PsycINFO Database Record