TY - THES A1 - Reike, Dennis T1 - A look behind perceptual performance in numerical cognition T1 - Ein Blick hinter die perzeptuellen Leistungen numerischer Kognition N2 - 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. N2 - Das Erkennen, Verstehen und Verwenden von Mengen sind beachtliche menschliche Fähigkeiten. Die Kommunikation numerischer Information fällt uns leicht, zudem beeinflussen numerische Informationen unser Handeln. Eine typische numerische Aufgabe ist der Mengenvergleich. Um solche Mengen zu beschreiben verwenden wir Ziffern als Symbole zur Bildung von Zahlen. Um Zahlen zu vergleichen, müssen wir auf die zuvor erlernte interne Zahlenrepräsentationen zurückgreifen. In dieser Dissertation werden drei Studien vorgestellt. Diese betrachten den Prozess des Zahlenvergleichs, die Interaktion numerischer und physikalischer Repräsentation und die strategische Nutzung numerischer Repräsentationen. In der ersten Studie sollten Versuchspersonen so schnell wie möglich die größere von zwei präsentierten Zahlen angeben. Sie sollten mit einer Sakkade in Richtung der größeren Zahl antworten. Eine Variante von Random Walk Modellen mit einem sparsamen Set an interpretierbaren Parameter wurde verwendet um die Reaktionszeit, die Fehlerrate und die Varianz der Reaktionszeit zu beschreiben. Auch für Fehlerzeiten sagt dieses Modell einen numerischen Distanzeffekt vorher, der sich in den Daten robust zeigt. Außerdem sind Fehlerzeiten schneller als korrespondierende Reaktionszeiten richtiger Antworten. Diese Asymmetrie lässt sich durch eine schiefe Schrittgrößenverteilung erklären, welche nicht zu den üblichen Standardannahmen von Random Walk Modellen gehört. Das vorgestellte Modell liefert einen definierten Rahmen um die Art und Skalierung (z.B. linear vs. logarithmisch) numerischer Repräsentationen zu untersuchen, wobei die Ergebnisse klar für eine logarithmische Skalierung sprechen. Abschließend wird ein Ausblick gegeben, wie dieses Modell helfen kann, komplexe Befunde (z.B. Geschwindigkeit vs. Genauigkeit) in Studien zu erklären, die Zahlenvergleiche als etabliertes Werkzeug verwenden. Außerdem beschreiben wir einen neuen okulomotorischen Effekt, den sakkadischen Overschoot Effekt. Für die zweite Studie wurde ein experimentelles Design entwickelt, das es ermöglicht die Signalentdeckungstheorie zu verwenden. Hierbei sollten Versuchspersonen die physikalische Größe von Ziffern beurteilen. Eine offene Frage ist, ob der Leistungsgewinn in (numerisch - physikalisch) kongruenten Bedingungen auf eine verbesserte Wahrnehmung oder auf einen numerisch induzierten Antwortbias zurückzuführen ist. Die Signalentdeckungstheorie ist das perfekte Werkzeug um zwischen diesen beiden Erklärungen zu unterscheiden. Dabei werden zwei Parameter beschrieben, die Sensitivität und der Antwortbias. Unsere Ergebnisse demonstrieren, dass der Zahlen-Größen Effekt nicht auf einen einfachen Antwortbias zurückzuführen ist. Vielmehr tragen wahre Sensitivitätsgewinne in kongruenten Bedingungen zur Entstehung des Zahlen-Größen Effekts bei. In der dritten Studie sollten die Versuchspersonen eine SNARC Aufgabe durchführen, wobei sie angeben sollten ob eine präsentierte Zahl gerade oder ungerade ist. Die lokale Wiederholungswahrscheinlichkeit des irrelevanten Attributes (Magnitude) wurde zwischen Versuchspersonen variiert. Die Versuchspersonen waren sensitiv für diese Wiederholungswahrscheinlichkeiten. Ein Leistungsgewinn zeigte sich bei tatsächlichen Wiederholungen des irrelevanten Attributes in der Bedingung mit hoher Wiederholungswahrscheinlichkeit des irrelevanten Attributes. Eine mögliche Interpretation ist, dass Informationen aus dem Vortrial als eine Art Hinweis betrachtet werden, so dass die Versuchspersonen im aktuellen Trial frühe Prozessstufen vorbereiten können, was zu entsprechenden Gewinnen und Kosten führt. Die in dieser Dissertation berichteten Ergebnisse werden abschließend diskutiert und in Relation zu aktuellen Studien im Bereich der numerischen Kognition gesetzt. KW - numerical cognition KW - mental number representation KW - numerische Kognition KW - mentale Zahlenrepräsentation Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-407821 ER - TY - JOUR A1 - Reike, Dennis A1 - Schwarz, Wolfgang T1 - Exploring the origin of the number-size congruency effect BT - sensitivity or response bias? JF - Attention, perception, & psychophysics : AP&P ; a journal of the Psychonomic Society, Inc. N2 - 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. KW - Numerical cognition KW - Number-size congruity effect KW - Signal detection theory Y1 - 2017 U6 - https://doi.org/10.3758/s13414-016-1267-4 SN - 1943-3921 SN - 1943-393X VL - 79 SP - 383 EP - 388 PB - Springer CY - New York ER - TY - JOUR A1 - Reike, Dennis A1 - Schwarz, Wolfgang T1 - Aging effects on symbolic number comparison BT - no deceleration of numerical information retrieval but more conservative decision-making JF - Psychology and aging N2 - 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. KW - numerical comparison KW - cognitive aging effects KW - numerical distance effect KW - random walk model KW - speed-accuracy trade-off Y1 - 2019 U6 - https://doi.org/10.1037/pag0000272 SN - 0882-7974 SN - 1939-1498 VL - 34 IS - 1 SP - 4 EP - 16 PB - American Psychological Association CY - Washington ER - TY - JOUR A1 - Reike, Dennis A1 - Schwarz, Wolfgang T1 - Categorizing digits and the mental number line JF - Attention, perception, & psychophysics : AP&P ; a journal of the Psychonomic Society, Inc. N2 - 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. KW - Categorization KW - Numerical distance effect KW - Mental number line KW - Diffusion models Y1 - 2019 U6 - https://doi.org/10.3758/s13414-019-01676-w SN - 1943-3921 SN - 1943-393X VL - 81 IS - 3 SP - 614 EP - 620 PB - Springer CY - New York ER - TY - JOUR A1 - Reike, Dennis A1 - Schwarz, Wolfgang T1 - One Model Fits All: Explaining Many Aspects of Number Comparison Within a Single Coherent Model-A Random Walk Account JF - Journal of experimental psychology : Learning, memory, and cognition N2 - 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. KW - numerical distance effect KW - random walk models KW - error latency KW - saccadic latency KW - saccadic amplitude Y1 - 2016 U6 - https://doi.org/10.1037/xlm0000287 SN - 0278-7393 SN - 1939-1285 VL - 42 SP - 1957 EP - 1971 PB - American Psychological Association CY - Washington ER - TY - JOUR A1 - Schwarz, Wolfgang A1 - Reike, Dennis T1 - Regression away from the mean BT - theory and examples JF - British journal of mathematical and statistical psychology / British Psychological Society N2 - 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. KW - bimodality KW - measurement error KW - non-normality KW - regression towards the mean KW - repeated measures KW - skewed distributions Y1 - 2017 U6 - https://doi.org/10.1111/bmsp.12106 SN - 0007-1102 SN - 2044-8317 VL - 71 IS - 1 SP - 186 EP - 203 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Schwarz, Wolfgang A1 - Reike, Dennis T1 - The number-weight illusion JF - Psychonomic bulletin & review : a journal of the Psychonomic Society N2 - 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. KW - Size-weight illusion KW - Number-size congruency effect KW - Numerical distance effect KW - Paired comparison KW - Reafference principle Y1 - 2018 U6 - https://doi.org/10.3758/s13423-018-1484-z SN - 1069-9384 SN - 1531-5320 VL - 26 IS - 1 SP - 332 EP - 339 PB - Springer CY - New York ER - TY - JOUR A1 - Schwarz, Wolfgang A1 - Reike, Dennis T1 - The Müller-Lyer line-length task interpreted as a conflict paradigm BT - A chronometric study and a diffusion account JF - Attention, perception, & psychophysics : AP&P ; a journal of the Psychonomic Society, Inc. N2 - 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 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). KW - Muller-Lyer illusion KW - Line perception KW - Conflict task KW - Diffusion model KW - Psychometric and chronometric function KW - Response bias KW - Sensitivity Y1 - 2020 U6 - https://doi.org/10.3758/s13414-020-02096-x SN - 1943-3921 SN - 1943-393X VL - 82 IS - 8 SP - 4025 EP - 4037 PB - Springer CY - New York ER -