@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{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} }