Wolf Schwarz

Spatial Associations in Number-Related Tasks

Bimanual parity judgments of numerically small (large) digits are faster with the left (right) hand (the SNARC effect; Dehaene, Bossini, & Giraux, 1993). According to one explanation, this effect is culturally derived and reflects ontogenetic influences such as the direction of written language; it might therefore be limited to, or at least be larger with, pairs of lateralized effectors which are instrumental to the production and comprehension of written language. We report two experiments which test for SNARC effects with pedal responses, and compare these effects to manual results. Pedal responses yielded highly systematic SNARC effects; furthermore, these effects did not differ from manual SNARC effects. These results argue against accounts in which the SNARC effect is specific for effectors that are habitually associated with the production or comprehension of written language.

Is there an internal association of numbers to hands? The task set influences the nature of the SNARC effect

The SNARCeffect refers to the association of smaller numbers with the left and of larger numbers with the right side of extracorporal space (Dehaene, Bossini, & Giraux, 1993). We tested the assumption that, in addition to these associations, numbers are also related to participants’ hands. We report two experiments with vertically arranged buttons in which the nature of the SNARC effect depended on whether the task set was button or hand related: In the first case, a vertical location-related SNARC effect occurred, whereas in the second a handrelated SNARC effect was found. Our third experiment confirmed that space-related number representations dominate the SNARC effect when the buttons are arranged horizontally. We concluded that both effector- and space-related number representations can influence and modify the SNARC effect.

Response time models of delta plots with negative-going slopes

Delta plots (DPs) graphically compare reaction time (RT) quantiles obtained under two experimental conditions. In some research areas (e.g., Simon effects), decreasing delta plots (nDPs) have consistently been found, indicating that the experimental effect is largest at low quantiles and decreases for higher quantiles. nDPs are unusual and intriguing: They imply that RT in the faster condition is more variable, a pattern predicted by few standard RT models. We describe and analyze five classes of well-established latency mechanisms that are consistent with nDPs—exhaustive processing models, correlated stage models, mixture models, cascade models, and parallel channels models—and discuss the implications of our analyses for the interpretation of DPs. DPs generally do not imply any specific processing model; therefore, it is more fruitful to start from a specific quantitative model and to compare the DP it predicts with empirical data.

Aggregate and individual replication probability within an explicit model of the research process.

We study a model of the research process in which the true effect size, the replication jitter due to changes in experimental procedure, and the statistical error of effect size measurement are all normally distributed random variables. Within this model, we analyze the probability of successfully replicating an initial experimental result by obtaining either a statistically significant result in the same direction or any effect in that direction. We analyze both the probability of successfully replicating a particular experimental effect (i.e., the individual replication probability) and the average probability of successful replication across different studies within some research context (i.e., the aggregate replication probability), and we identify the conditions under which the latter can be approximated using the formulas of Killeen (2005a, 2007). We show how both of these probabilities depend on parameters of the research context that would rarely be known in practice. In addition, we show that the statistical uncertainty associated with the size of an initial observed effect would often prevent accurate estimation of the desired individual replication probability even if these research context parameters were known exactly. We conclude that accurate estimates of replication probability are generally unattainable.

Numerical distance effects in visual search

We present three experiments in which observers searched for a target digit among distractor digits in displays in which the mean numerical target–distractor distance was varied. Search speed and accuracy increased with numerical distance in both target-present and target-absent trials (Exp. 1A). In Experiment 1B, the target 5 was replaced with the letter S. The results suggest that the findings of Experiment 1A do not simply reflect the fact that digits that were numerically closer to the target coincidentally also shared more physical features with it. In Experiment 2, the numerical distance effect increased with set size in both target-present and target-absent trials. These findings are consistent with the view that increasing numerical target–distractor distance affords faster nontarget rejection and target identification times. Recent neurobiological findings (e.g., Nieder, 2011) on the neuronal coding of numerosity have reported a width of tuning curves of numerosity-selective neurons that suggests graded, distance-dependent coactivation of the representations of adjacent numbers, which in visual search would make it harder to reject numerically closer distractors as nontargets.

Brain signals do not demonstrate unconscious decision making: an interpretation based on graded conscious awareness.

Neuroscientific studies have shown that brain activity correlated with a decision to move can be observed before a person reports being consciously aware of having made that decision (e.g., Libet, Gleason, Wright, & Pearl, 1983; Soon, Brass, Heinze, & Haynes, 2008). Given that a later event (i.e., conscious awareness) cannot cause an earlier one (i.e., decision-related brain activity), such results have been interpreted as evidence that decisions are made unconsciously (e.g., Libet, 1985). We argue that this interpretation depends upon an all-or-none view of consciousness, and we offer an alternative interpretation of the early decision-related brain activity based on models in which conscious awareness of the decision to move develops gradually up to the level of a reporting criterion. Under this interpretation, the early brain activity reflects sub-criterion levels of awareness rather than complete absence of awareness and thus does not suggest that decisions are made unconsciously.

GSDT: An integrative model of visual search.

We present a new quantitative process model (GSDT) of visual search that seeks to integrate various processing mechanisms suggested by previous studies within a single, coherent conceptual frame. It incorporates and combines 4 distinct model components: guidance (G), a serial (S) item inspection process, diffusion (D) modeling of individual item inspections, and a strategic termination (T) rule. For this model, we derive explicit closed-form results for response probability and mean search time (reaction time [RT]) as a function of display size and target presence/absence. The fit of the model is compared in detail to data from 4 visual search experiments in which the effects of target/distractor discriminability and of target prevalence on performance (present/absent display size functions for mean RT and error rate) are studied. We describe how GSDT accounts for various detailed features of our results such as the probabilities of hits and correct rejections and their mean RTs; we also apply the model to explain further aspects of the data, such as RT variance and mean miss RT. (PsycINFO Database Record

Compensating tendencies in penalty kick decisions of referees in professional football: Evidence from the German Bundesliga 1963–2006

Abstract Using a large representative database (12,902 matches from the top professional football league in Germany), I show that the number (441) of two-penalty matches is larger than expected by chance, and that among these 441 matches there are considerably more matches in which each team is awarded one penalty than would be expected on the basis of independent penalty kick decisions (odds ratio = 11.2, relative risk = 6.34). Additional analyses based on the score in the match before a penalty is awarded and on the timing of penalties, suggest that awarding a first penalty to one team raises the referees penalty evidence criterion for the same team, and lowers the corresponding criterion for the other team.

Implications of Individual Differences in On-Average Null Effects.

Most psychological models are intended to describe processes that operate within each individual. In many research areas, however, models are tested by looking at results averaged across many individuals, despite the fact that such averaged results may give a misleading picture of what is true for each one. We consider this conundrum with respect to the interpretation of on-average null effects. Specifically, even though an experimental manipulation might have no effect on average across individuals, it might still have demonstrable effects—albeit in opposite directions—for many or all of the individuals tested. We discuss several examples of research questions for which it would be theoretically crucial to determine whether manipulations really have no effect at the individual level, and we present a method of testing for individual-level effects.

Exploring the origin of the number-size congruency effect: Sensitivity or response bias?

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.