@phdthesis{Reike2017, author = {Reike, Dennis}, title = {A look behind perceptual performance in numerical cognition}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-407821}, school = {Universit{\"a}t Potsdam}, pages = {vi, 136}, year = {2017}, abstract = {Recognizing, understanding, and responding to quantities are considerable skills for human beings. We can easily communicate quantities, and we are extremely efficient in adapting our behavior to numerical related tasks. One usual task is to compare quantities. We also use symbols like digits in numerical-related tasks. To solve tasks including digits, we must to rely on our previously learned internal number representations. This thesis elaborates on the process of number comparison with the use of noisy mental representations of numbers, the interaction of number and size representations and how we use mental number representations strategically. For this, three studies were carried out. In the first study, participants had to decide which of two presented digits was numerically larger. They had to respond with a saccade in the direction of the anticipated answer. Using only a small set of meaningfully interpretable parameters, a variant of random walk models is described that accounts for response time, error rate, and variance of response time for the full matrix of 72 digit pairs. In addition, the used random walk model predicts a numerical distance effect even for error response times and this effect clearly occurs in the observed data. In relation to corresponding correct answers error responses were systematically faster. 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 be asymmetric to account for this asymmetry between correct and incorrect responses. Furthermore, the presented model provides a well-defined framework to investigate the nature and scale (e.g., linear vs. logarithmic) of the mapping of numerical magnitude onto its internal representation. In comparison of the fits of proposed models with linear and logarithmic mapping, the logarithmic mapping is suggested to be prioritized. Finally, we discuss how our findings can help interpret complex findings (e.g., conflicting speed vs. accuracy trends) in applied studies that use number comparison as a well-established diagnostic tool. Furthermore, a novel oculomotoric effect is reported, namely the saccadic overschoot effect. The participants responded by saccadic eye movements and the amplitude of these saccadic responses decreases with numerical distance. For the second study, an experimental design was developed that allows us to apply the signal detection theory to a task where participants had to decide whether a presented digit was physically smaller or larger. A remaining question is, whether the benefit in (numerical magnitude - physical size) congruent conditions is related to a better perception than in incongruent conditions. Alternatively, the number-size congruency effect is mediated by response biases due to numbers magnitude. The signal detection theory is a perfect tool to distinguish between these two alternatives. It describes two parameters, namely sensitivity and response bias. Changes in the sensitivity are related to the actual task performance due to real differences in perception processes whereas changes in the response bias simply reflect strategic implications as a stronger preparation (activation) of an anticipated answer. Our results clearly demonstrate that the number-size congruency effect cannot be reduced to mere response bias effects, and that genuine sensitivity gains for congruent number-size pairings contribute to the number-size congruency effect. Third, participants had to perform a SNARC task - deciding whether a presented digit was odd or even. Local transition probability of irrelevant attributes (magnitude) was varied while local transition probability of relevant attributes (parity) and global probability occurrence of each stimulus were kept constantly. Participants were quite sensitive in recognizing the underlying local transition probability of irrelevant attributes. A gain in performance was observed for actual repetitions of the irrelevant attribute in relation to changes of the irrelevant attribute in high repetition conditions compared to low repetition conditions. One interpretation of these findings is that information about the irrelevant attribute (magnitude) in the previous trial is used as an informative precue, so that participants can prepare early processing stages in the current trial, with the corresponding benefits and costs typical of standard cueing studies. Finally, the results reported in this thesis are discussed in relation to recent studies in numerical cognition.}, language = {en} } @article{ReikeSchwarz2017, author = {Reike, Dennis and Schwarz, Wolfgang}, title = {Exploring the origin of the number-size congruency effect}, series = {Attention, perception, \& psychophysics : AP\&P ; a journal of the Psychonomic Society, Inc.}, volume = {79}, journal = {Attention, perception, \& psychophysics : AP\&P ; a journal of the Psychonomic Society, Inc.}, publisher = {Springer}, address = {New York}, issn = {1943-3921}, doi = {10.3758/s13414-016-1267-4}, pages = {383 -- 388}, year = {2017}, abstract = {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.}, language = {en} } @article{ReikeSchwarz2019, author = {Reike, Dennis and Schwarz, Wolfgang}, title = {Aging effects on symbolic number comparison}, series = {Psychology and aging}, volume = {34}, journal = {Psychology and aging}, number = {1}, publisher = {American Psychological Association}, address = {Washington}, issn = {0882-7974}, doi = {10.1037/pag0000272}, pages = {4 -- 16}, year = {2019}, abstract = {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.}, language = {en} } @article{ReikeSchwarz2019, author = {Reike, Dennis and Schwarz, Wolfgang}, title = {Categorizing digits and the mental number line}, series = {Attention, perception, \& psychophysics : AP\&P ; a journal of the Psychonomic Society, Inc.}, volume = {81}, journal = {Attention, perception, \& psychophysics : AP\&P ; a journal of the Psychonomic Society, Inc.}, number = {3}, publisher = {Springer}, address = {New York}, issn = {1943-3921}, doi = {10.3758/s13414-019-01676-w}, pages = {614 -- 620}, year = {2019}, abstract = {Following the classical work of Moyer and Landauer (1967), experimental studies investigating the way in which humans process and compare symbolic numerical information regularly used one of two experimental designs. In selection tasks, two numbers are presented, and the task of the participant is to select (for example) the larger one. In classification tasks, a single number is presented, and the participant decides if it is smaller or larger than a predefined standard. Many findings obtained with these paradigms fit in well with the notion of a mental analog representation, or an Approximate Number System (ANS; e.g., Piazza 2010). The ANS is often conceptualized metaphorically as a mental number line, and data from both paradigms are well accounted for by diffusion models based on the stochastic accumulation of noisy partial numerical information over time. The present study investigated a categorization paradigm in which participants decided if a number presented falls into a numerically defined central category. We show that number categorization yields a highly regular, yet considerably more complex pattern of decision times and error rates as compared to the simple monotone relations obtained in traditional selection and classification tasks. We also show that (and how) standard diffusion models of number comparison can be adapted so as to account for mean and standard deviations of all RTs and for error rates in considerable quantitative detail. We conclude that just as traditional number comparison, the more complex process of categorizing numbers conforms well with basic notions of the ANS.}, language = {en} } @article{ReikeSchwarz2016, author = {Reike, Dennis and Schwarz, Wolfgang}, title = {One Model Fits All: Explaining Many Aspects of Number Comparison Within a Single Coherent Model-A Random Walk Account}, series = {Journal of experimental psychology : Learning, memory, and cognition}, volume = {42}, journal = {Journal of experimental psychology : Learning, memory, and cognition}, publisher = {American Psychological Association}, address = {Washington}, issn = {0278-7393}, doi = {10.1037/xlm0000287}, pages = {1957 -- 1971}, year = {2016}, 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.}, language = {en} } @article{SchwarzReike2017, author = {Schwarz, Wolfgang and Reike, Dennis}, title = {Regression away from the mean}, series = {British journal of mathematical and statistical psychology / British Psychological Society}, volume = {71}, journal = {British journal of mathematical and statistical psychology / British Psychological Society}, number = {1}, publisher = {Wiley}, address = {Hoboken}, issn = {0007-1102}, doi = {10.1111/bmsp.12106}, pages = {186 -- 203}, year = {2017}, abstract = {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.}, language = {en} } @article{SchwarzReike2018, author = {Schwarz, Wolfgang and Reike, Dennis}, title = {The number-weight illusion}, series = {Psychonomic bulletin \& review : a journal of the Psychonomic Society}, volume = {26}, journal = {Psychonomic bulletin \& review : a journal of the Psychonomic Society}, number = {1}, publisher = {Springer}, address = {New York}, issn = {1069-9384}, doi = {10.3758/s13423-018-1484-z}, pages = {332 -- 339}, year = {2018}, abstract = {When objects are manually lifted to compare their weight, then smaller objects are judged to be heavier than larger objects of the same physical weights: the classical size-weight illusion (Gregory, 2004). It is also well established that increasing numerical magnitude is strongly associated with increasing physical size: the number-size congruency effect e.g., (Besner \& Coltheart Neuropsychologia, 17, 467-472 1979); Henik \& Tzelgov Memory \& Cognition, 10, 389-395 1982). The present study investigates the question suggested by combining these two classical effects: if smaller numbers are associated with smaller size, and objects of smaller size appear heavier, then are numbered objects (balls) of equal weight and size also judged as heavier when they carry smaller numbers? We present two experiments testing this hypothesis for weight comparisons of numbered (1 to 9) balls of equal size and weight, and report results which largely conform to an interpretation in terms of a new number-weight illusion.}, language = {en} } @article{SchwarzReike2020, author = {Schwarz, Wolfgang and Reike, Dennis}, title = {The M{\"u}ller-Lyer line-length task interpreted as a conflict paradigm}, series = {Attention, perception, \& psychophysics : AP\&P ; a journal of the Psychonomic Society, Inc.}, volume = {82}, journal = {Attention, perception, \& psychophysics : AP\&P ; a journal of the Psychonomic Society, Inc.}, number = {8}, publisher = {Springer}, address = {New York}, issn = {1943-3921}, doi = {10.3758/s13414-020-02096-x}, pages = {4025 -- 4037}, year = {2020}, abstract = {We propose to interpret tasks evoking the classical M{\"u}ller-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 Muller-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).}, language = {en} }