@article{MillerUlrichSchwarz2009, author = {Miller, Jeff and Ulrich, Rolf and Schwarz, Wolfgang}, title = {Why jackknifing yields good latency estimates}, issn = {0048-5772}, doi = {10.1111/j.1469-8986.2008.00761.x}, year = {2009}, abstract = {We compared individual-participant and jackknife-based methods for scoring the onset latencies of event-related potential (ERP) components using a diffusion process as a model for an ERP. We studied "ramp-like" components in which the true ERP increases or decreases monotonically, except for noise. If the growth rates of such components vary across participants, the jackknife-based measure can easily have only 10\%-20\% as much error variance as the traditional method, and this advantage is magnified with more participants. We also studied boolean AND-shaped or "bump-like" components. Jackknifing generally yielded smaller error variances with these components too, especially when the component's peak amplitude varied across participants, but less so if the component's peak latency varied. These results help illuminate the reasons for the superiority of jackknife-based onset latency measures over traditional measures in recent simulations.}, language = {en} } @article{SchwarzEiselt2009, author = {Schwarz, Wolfgang and Eiselt, Anne-Kathrin}, title = {The perception of temporal order along the mental number line}, issn = {0096-1523}, doi = {10.1037/A0013703}, year = {2009}, abstract = {R. Sekuler, P. Tynan, and E. Levinson (1973) found that when 2 characters are presented side-by-side with a short onset asynchrony, subjectively, they often appear in a "first-left, then-right" order. The authors of this article conducted 6 experiments in which observers judged the temporal order (TOJs) in which 2 digits were presented. They found a consistent TOJ benefit (larger d') when the numerically smaller digit was presented first, even though this semantic information was irrelevant to the task and unrelated to the correct response. They concluded that digits located to the left of the mental number line are transmitted faster to a central comparison stage, which represents an "internal counterpart" to the Sekuler et al. (1973) finding regarding external locations. A corresponding benefit is found for letters pairs (e.g., A-Z) and also for mixed digit-letter pairs (e.g., I-Z).}, 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} } @article{Schwarz2003, author = {Schwarz, Wolfgang}, title = {Stovhastic cascade processes as a model of multi-stage concurrent information procesing}, year = {2003}, language = {en} } @article{SchwarzMueller2005, author = {Schwarz, Wolfgang and M{\"u}ller, Dana}, title = {Spatial associatiosn in number-related tasks : a comparison of manual and pedal responses}, year = {2005}, abstract = {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}, language = {en} } @article{KeusSchwarz2005, author = {Keus, I. M. and Schwarz, Wolfgang}, title = {Searching for the functional locus of the SNARC effect : evidence for a response-related origin}, year = {2005}, 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{DeBleserSchwarzBurchert2006, author = {De Bleser, Ria and Schwarz, Wolfgang and Burchert, Frank}, title = {Quantitative neurosyntactic analyses : the final word?}, issn = {0093-934X}, doi = {10.1016/j.bandl.2005.06.010}, year = {2006}, language = {en} } @article{KeusJenksSchwarz2005, author = {Keus, I. M. and Jenks, C. and Schwarz, Wolfgang}, title = {Psychophysiological evidence that the SNARC effect has a functional locus in a response selection stage}, year = {2005}, abstract = {When participants judge the parity of visually presented digits, left-hand responses are faster for numerically small numbers, whereas right-hand responses are faster for large numbers [SNARC effect; S. Dehaene, S. Bossini, P. Giraux, The mental representation of parity and number magnitude. J. Exp. Psychol. Gen., 122, (1993) 371-396]. The present study aimed to find more direct evidence for the functional locus of this effect by recording brain waves while participants performed speeded parity judgments giving manual responses. Our results show clear and robust SNARC effects in the response-locked event-related potentials (ERPs) compared to the stimulus-locked ERPs, confirming that the SNARC effect arises during response-related rather than stimulus-related processing stages. Further analyses of lateralized readiness potentials strongly suggest that the SNARC effect begins to emerge in a response-related stage prior to response preparation and execution, more specifically, in a response selection stage. (c) 2005 Elsevier B.V All rights reserved}, 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{SchwarzIschebeck2003, author = {Schwarz, Wolfgang and Ischebeck, A.}, title = {On the relative speed account of number-size interference effects in comparative judgments of numerals}, year = {2003}, language = {en} } @article{Schwarz2006, author = {Schwarz, Wolfgang}, title = {On the relationship between the redundant signals effect and temporal order judgments: Parametric data and a new model}, year = {2006}, abstract = {Paradigms used to study the time course of the redundant signals effect (RSE; J. O. Miller, 1986) and temporal order judgments (TOJs) share many important similarities and address related questions concerning the time course of sensory processing. The author of this article proposes and tests a new aggregate diffusion-based model to quantitatively explain both the RSE and TOJs and the relationship between them. Parametric data (13 stimulus onset asynchronies) from an experiment with pairs of visual stimuli (626-nm LEDs) confirm that, relative to central signals (3 degrees), peripheral signals (35 degrees) yield slower reaction times, more strongly modulated RSE time-course functions, and flatter TOJ psychometric functions. All of these qualitative features are well captured, even in quantitative detail, by the aggregate diffusion model}, language = {en} } @article{Schwarz2006, author = {Schwarz, Wolfgang}, title = {On the relation between the redundant signals effect and temporal order judgments: Parametric data and a new model}, year = {2006}, abstract = {Paradigms used to study the time course of the redundant signals effect (RSE; J. O. Miller, 1986) and temporal order judgments (TOJs) share many important similarities and address related questions concerning the time course of sensory processing. The author of this article proposes and tests a new aggregate diffusion-based model to quantitatively explain both the RSE and TOJs and the relationship between them. Parametric data (13 stimulus onset asynchronies) from an experiment with pairs of visual stimuli (626-nm LEDs) confirm that, relative to central signals (3 degrees), peripheral signals (35 degrees) yield slower reaction times, more strongly modulated RSE time-course functions, and flatter TOJ psychometric functions. All of these qualitative features are well captured, even in quantitative detail, by the aggregate diffusion model.}, language = {en} } @article{SchwarzEiselt2012, author = {Schwarz, Wolfgang and Eiselt, Anne-Kathrin}, title = {Numerical distance effects in visual search}, series = {Attention, perception, \& psychophysics : AP\&P ; a journal of the Psychonomic Society, Inc.}, volume = {74}, journal = {Attention, perception, \& psychophysics : AP\&P ; a journal of the Psychonomic Society, Inc.}, number = {6}, publisher = {Springer}, address = {New York}, issn = {1943-3921}, doi = {10.3758/s13414-012-0342-8}, pages = {1098 -- 1103}, year = {2012}, abstract = {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.}, language = {en} } @article{SchwarzKeus2004, author = {Schwarz, Wolfgang and Keus, I. M.}, title = {Moving the eyes along the mental number line : comparing SNARC effects with saccadic and manual responses}, year = {2004}, abstract = {Bimanual parity judgments about numerically small (large) digits are faster with the left (right) hand, even though parity is unrelated to numerical magnitude per se (the SNARC effect; Dehaene, Bossini, \& Giraux, 1993). According to one model, this effect reflects a space-related representation of numerical magnitudes (mental number line) with a genuine left-to-right orientation. Alternatively, it may simply reflect an overlearned motor association between numbers and manual responses-as, for example, on typewriters or computer keyboards-in which case it should be weaker or absent with effectors whose horizontal response component is less systematically associated with individual numbers. Two experiments involving comparisons of saccadic and manual parity judgment tasks clearly support the first view; they also establish a vertical SNARC effect, suggesting that our magnitude representation resembles a number map, rather than a number line}, language = {en} } @article{SchwarzMiller2010, author = {Schwarz, Wolfgang and Miller, Jeff O.}, title = {Locking the Wiener process to its level-crossing time}, issn = {0361-0926}, doi = {10.1080/03610920902755821}, year = {2010}, abstract = {We consider the specific transformation of a Wiener process {X(t), t >= 0} in the presence of an absorbing barrier a that results when this process is "time-locked" with respect to its first passage time T-a through a criterion level a, and the evolution of X(t) is considered backwards ( retrospectively) from T-a. Formally, we study the random variables defined by Y(t) = X(T-a - t) and derive explicit results for their density and mean, and also for their asymptotic forms. We discuss how our results can aid interpretations of time series "response-locked" to their times of crossing a criterion level.}, language = {en} } @article{MillerSchwarz2018, author = {Miller, Jeff and Schwarz, Wolfgang}, title = {Implications of Individual Differences in On-Average Null Effects}, series = {Journal of experimental psychology : General}, volume = {147}, journal = {Journal of experimental psychology : General}, number = {3}, publisher = {American Psychological Association}, address = {Washington}, issn = {0096-3445}, doi = {10.1037/xge0000367}, pages = {377 -- 397}, year = {2018}, abstract = {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.}, language = {en} } @article{SchwarzMiller2016, author = {Schwarz, Wolfgang and Miller, Jeff}, title = {GSDT: An Integrative Model of Visual Search}, series = {Journal of experimental psychology : Human perception and performance}, volume = {42}, journal = {Journal of experimental psychology : Human perception and performance}, publisher = {American Psychological Association}, address = {Washington}, issn = {0096-1523}, doi = {10.1037/xhp0000247}, pages = {1654 -- 1671}, year = {2016}, abstract = {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.}, 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} }