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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).

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.

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.

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.

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. 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.