On the mean and variance of response times under the diffusion model with an application to parameter estimation

We give closed form expressions for the mean and variance of RTs for Ratcliff’s diffusion model [Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59-108] under the simplifying assumption that there is no variability across trials in the parameters. These expressions are more general than those currently available. As an application, we demonstrate their use in a method-of-moments estimation procedure that addresses some of the weaknesses of the EZ method [Wagenmakers, E.-J., van der Maas, H. L. J., & Grasman, R. P. P. P. (2007). An EZ-diffusion model for response time and accuracy. Psychonomic Bulletin & Review, 14, 3-22], and illustrate this with lexical decision data. We discuss further possible applications.     

Strategic flexibility in response preparation: Effects of cue validity on reaction time and pupil dilation

This study examined the ability of participants to strategically adapt their level of response preparation to the predictive value of preparatory cues. Participants performed the finger-precuing task under three levels of cue validity: 100, 75 and 50% valid. Response preparation was indexed by means of reaction time (RT) and pupil dilation, the latter providing a psychophysiological index of invested effort. Results showed a systematic increase in RT benefits (generated by valid cues) and RT costs (generated by invalid cues) with increments in the predictive value of cues. Converging with these behavioural effects, pupil dilation also increased systematically with greater cue validity during the cue-stimulus interval, suggesting more effortful response preparation with increases in cue validity. Together, these findings confirm the hypothesis that response preparation is flexible and that it can be strategically allocated in proportion to the relative frequency of valid/invalid preparatory cues.     

Response-level probability effects on reaction time: Now you see them,now you don't

Many reaction time (RT) experiments have tested for response-level probability effects. Their results have been mixed, which is surprising because psychophysiological studies provide clear evidence of motor-level changes associated with an anticipated response. A survey of the designs used in the RT studies reveals many potential problems that could conceal the effects of response probability. We report five new RT experiments testing for response-level probability effects with the most promising of the previous designs—that of Blackman (1972 Blackman, A. R. 1972. Influence of stimulus and response probability on decision and movement latency in a discrete choice reaction task. Journal of Experimental Psychology, 92: 128–133. doi:10.1037/h0032169[Crossref], [PubMed] , [Google Scholar])—and with new designs. Some of these experiments yield evidence of response-level probability effects, but others do not. It appears that response-level probability effects are present primarily in simple tasks with a strong emphasis on response preparation, possibly because participants only expend effort on response preparation in these tasks.     

The simplest complete model of choice response time: linear ballistic accumulation

We propose a linear ballistic accumulator (LBA) model of decision making and reaction time. The LBA is simpler than other models of choice response time, with independent accumulators that race towards a common response threshold. Activity in the accumulators increases in a linear and deterministic manner. The simplicity of the model allows complete analytic solutions for choices between any number of alternatives. These solutions (and freely-available computer code) make the model easy to apply to both binary and multiple choice situations. Using data from five previously published experiments, we demonstrate that the LBA model successfully accommodates empirical phenomena from binary and multiple choice tasks that have proven difficult for other theoretical accounts. Our results are encouraging in a field beset by the tradeoff between complexity and completeness.     

From Poisson shot noise to the integrated Ornstein-Uhlenbeck process: Neurally principled models of information accumulation in decision-making and response time

In the diffusion model of decision-making, evidence is accumulated by a Wiener diffusion process. A neurally motivated account of diffusive evidence accumulation is given, in which diffusive accumulation arises from an interaction between neural integration processes operating on short and long time scales. The short time scale process is modeled as a Poisson shot noise process with exponential decay. Stimulus information is coded by excitatory-inhibitory shot noise pairs. The long time scale process is modeled as algebraic integration, possibly implemented as a first-order autoregressive process realized by recurrent connections within a population of neurons. At high intensities, an excitatory-inhibitory shot noise pair converges weakly to an Ornstein-Uhlenbeck (OU) velocity process. The integrated OU process, or OU displacement process, obtained by integrating the velocity process over time, is indistinguishable at long times from the Wiener process. Diffusive information accumulation may therefore be characterized as an integrated OU process whose properties mimic those of the Wiener process.     

Reaction time and intelligence: Comparing associations based on two response modes

People who score highly on intelligence tests also tend to have faster and less variable reaction times. Effect size estimates for the reaction time–intelligence association are larger in samples that are more representative of the population. However, such samples have often been tested on a reaction time device that requires reading a number and processing its association with a specific response location (Cox, Huppert, & Whichelow, 1993). Here, we use this device and another reaction time device (Dykiert et al., 2010) that is similar, except that the responses require less processing; subjects simply press a button that is adjacent to the stimulus light. We focus on the possibility that lights as stimuli require less higher-order cognitive engagement than numbers, and then test whether parameters from these two tasks are highly correlated and similarly associated with age and higher cognitive abilities. Both tasks measured simple and choice reaction times and their intra-individual variation across trials. The parameters of the two tasks were very highly correlated and parameters from both tasks were similarly associated with age, social factors, and differences in higher cognitive abilities. The respective choice reaction time parameters from either task accounted for much of the age- and higher cognitive ability-associations of the other task's parameters. These findings are important in establishing that the effect sizes of higher cognitive ability associations with processing speed measures may be found when the processing demands are minimal.     

Drawing conclusions from choice response time models: A tutorial using the linear ballistic accumulator

Cognitive models of choice and response times can lead to deeper insights into the processes underlying decisions than standard analyses of accuracy and response time data. The application of these models, however, has historically been reserved for the authors of the models, and their associates. Recently, choice response time models have become more accessible through the release of user-friendly software for estimating their parameters. The aim of this tutorial is to provide guidance about the process of using these parameter estimates and associated model fits to make conclusions about experimental data. We use an application of one response time model, the linear ballistic accumulator, as an example to demonstrate the steps required to select an appropriate parametric characterization of a data set. We also discuss how to evaluate the quality of the agreement between model and data, including guidelines for presenting model predictions for group-level data.     

Fast and accurate calculations for first-passage times in Wiener diffusion models

We propose a new method for quickly calculating the probability density function for first-passage times in simple Wiener diffusion models, extending an earlier method used by [Van Zandt, T., Colonius, H., & Proctor, R. W. (2000). A comparison of two response-time models applied to perceptual matching. Psychonomic Bulletin & Review, 7, 208-256]. The method relies on the observation that there are two distinct infinite series expansions of this probability density, one of which converges quickly for small time values, while the other converges quickly at large time values. By deriving error bounds associated with finite truncation of either expansion, we are able to determine analytically which of the two versions should be applied in any particular context. The bounds indicate that, even for extremely stringent error tolerances, no more than 8 terms are required to calculate the probability density. By making the calculation of this distribution tractable, the goal is to allow more complex extensions of Wiener diffusion models to be developed.     

An integrated perspective on the relation between response speed and intelligence

Research in the field of mental chronometry and individual differences has revealed several robust regularities (Jensen, 2006). These include right-skewed response time (RT) distributions, the worst performance rule, correlations with general intelligence (g) that are more pronounced for RT standard deviations (RTSD) than they are for RT means (RTm), an almost perfect linear relation between individual differences in RTSD and RTm, linear Brinley plots, and stronger correlations between g and inspection time (IT) than between g and RTm. Here we show how all these regularities are manifestations of a single underlying relationship, when viewed through the lens of Ratcliff’s diffusion model ( [Ratcliff, 1978] and [Ratcliff et al., 2008] ). The single underlying relationship is between individual differences in general intelligence and individual differences in “drift rate”, which is just the speed of information processing in Ratcliff’s model. We also test and confirm a strong prediction of the diffusion model, namely that the worst performance rule generalizes to phenomena outside of the field of intelligence. Our approach provides an integrative perspective on intelligence findings.     

Crossed-uncrossed difference (CUD) on reaction time and movement time in three two-choice experiments

The present study addresses the robustness and the reliability of the crossed-uncrossed difference (CUD) on a reaction time (RT) and a movement time (MT) component of a prescribed unimanual response to lateralized stimuli. Experiment 1 demonstrated positive CUDs both when a visual warning signal (WS) and an auditory reaction signal (RS) appeared on corresponding and non-corresponding sides of the body. Experiment 2 showed effect of handedness on CUD calculated among right-handers and left-handers. Experiment 3 investigated CUDs through five successive days, indicating that CUDs became steady and reliable although practice affected both RT and MT. All experiments indicated CUD on RT and MT. In addition, Experiments 1 and 2 demonstrated hemispheric asymmetries in favour of an anatomical theory while Experiment 3 did not show any asymmetries and supports an S-R compatibility theory.     

Placing inspection time, reaction time, and perceptual speed in the broader context of cognitive ability: The VPR model in the Lothian Birth Cohort 1936

The idea that information processing speed is related to cognitive ability has a long history. Much evidence has been amassed in its support, with respect to both individual differences in general intelligence and developmental trajectories. Two so-called elementary cognitive tasks, reaction time and inspection time, have been used to compile this evidence, but most studies have used either one or the other. Relations between speed and fluid intelligence have tended to be stronger than those between speed and crystallized intelligence, but studies testing this have confounded verbal abilities with crystallized intelligence and spatial/perceptual abilities with fluid intelligence. Questions have also been raised regarding whether speed contributes directly to general intelligence or to more specific cognitive abilities to which general intelligence also contributes. We used 18 ability and speed measures in the Lothian Birth Cohort 1936, assessed at approximately age 70, to construct alternative versions of the Verbal-Perceptual-Image Rotation (Johnson & Bouchard, 2005a) model of cognitive ability to test different hypotheses regarding these issues. Though differences in the extents to which our models fit the data were relatively small, they suggested that reaction and inspection time tasks were comparable indicators of information processing speed with respect to general intelligence, that verbal and spatial abilities were similarly related to information processing speed, and that spatial, verbal, and perceptual speed abilities were more directly related to information processing speed than was general intelligence. We discuss the theoretical implications of these results.     

Children with developmental coordination disorder (DCD) and their ability to disengage ongoing attentional focus: more on inhibitory function

Children with developmental coordination disorder (DCD), along with their Control counterparts, completed two endogenous, spatial precue tasks. When the precue arrow was informative (.80) with respect to target location, the spatial precue effect results demonstrated that children with DCD take significantly longer than Control individuals to volitionally disengage (inhibit) attention from an endogenously cued location (i.e., a disengagement inhibition deficit). When the precue was uninformative (.25), we found, contrary to a common assumption, that the precue arrow automatically moved attention in the direction of the arrow, and, in addition, that DCD children may also be less able to inhibit the precued-induced urge to move attention (i.e., an initiation inhibition deficit). This type of inhibitory difficulty was also indicated for manual response inclinations produced on catch trials. Overall, DCD children appeared to have an elevated difficulty suppressing the initiation of incorrect, stimulus-provoked movement urges, be they manual or attention in nature.     

Interhemispheric transmission time in an auditory two-choice reaction task

An unimanual auditory choice reaction task was performed by 16 right-handed male subjects. Upon release of a central ready button subjects pressed a target button on their right or left side depending on the ear in which they heard a tone. A significant 'Ear' x 'Hand' interaction effect appeared on both reaction time (RT) and the first component of movement (MT₁), showing that uncrossed conditions give rise to faster responses than crossed conditions. Interhemispheric transmission time (IHTT) was estimated from the difference of response latency in the two types of conditions. No asymmetry in the speed of information transfer between the hemispheres was found. IHTT was 16 ms in the case of RT, which supported previous research with visual reaction tasks, and IHTT for MT₁ was 13 ms, which has not been reported before.     

Intra-individual reaction time variability in schizophrenia, depression and borderline personality disorder

Intra-individual reaction time variability (IIV) in neuropsychological task performance reflects short term fluctuations in performance. Increased IIV has been reported in patients with schizophrenia and could be related to a deficient neural timing mechanism, but the role of IIV in adult patients with other psychiatric disorders has not been established. Therefore, we compared IIV measures obtained in a Go/Nogo task from patients with schizophrenia, major depression and borderline personality disorder. IIV was increased for patients with schizophrenia. When correcting for differences in mean reaction time, depressive and borderline patients also showed increased IIV. Importantly, all groups showed a strong association between IIV and accuracy of task performance. This suggests that increased IIV might be a sensitive marker for the efficiency of top-down attentional control in all diagnostic groups. Aside from these similarities, the complete results including measures of IIV, mean reaction time and accuracy show differential patterns for patients with schizophrenia compared to those with borderline personality disorder or depression. These results are discussed with respect to common versus disorder-specific neural mechanisms underlying increased IIV.     

Testing the hypothesized effect of dysgenic fertility on intelligence with existing reaction time data: A comment on Woodley,te Nijenhuis,and Murphy (2013)

Dysgenic fertility has supposedly resulted in a decline in general intelligence (g) over time. In light of evidence that simple visual reaction time (RT) is inversely related to IQ, Woodley et al. (2013) tested the hypothesized dysgenic effect by subjecting to a meta-regression simple visual RT data collected over 100 years in 15 studies. This analysis found that RT had significantly increased according to a linear function over this time period. Woodley et al. then used this result to estimate the rate at which g had declined over the same period. The present comment points out that there are large gaps in the distribution of RTs analyzed by Woodley et al. with respect to year tested, and that RT in males did not vary as a function of year in the 13 studies published from 1941 on. It is concluded that although existing data are consistent with the idea that g has been adversely affected by dysgenic fertility, it cannot be determined at what rate g has fallen over time.     

Response preparation with static versus moving hands

This research tested the response inhibition account of the hand-advantage found in the finger pre-cuing task. According to this account, the advantage of preparing two fingers on one hand (represented in one hemisphere) as opposed to preparing two fingers on two hands (represented in two hemispheres) is due, in part, to a response inhibition process that operates more efficiently within than between hemispheres. In this view, supplying extra activation to both hemispheres by moving the hands should decrease the within-hemisphere inhibition advantage. Twelve participants performed the finger pre-cuing task with static and moving hands. As predicted by the response inhibition account, the hand-advantage, present with the hands at rest, decreased with the hands moving.     

How to use the diffusion model: Parameter recovery of three methods: EZ, fast-dm, and DMAT

Parameter recovery of three different implementations of the Ratcliff diffusion model was investigated: the EZ model (Wagenmakers, van der Maas, & Grasman, 2007), fast-dm (Voss & Voss, 2007), and DMAT (Vandekerckhove & Tuerlinckx, 2007). Their capacity to recover both the mean structure and individual differences in parameter values was explored. The three methods were applied to simulated data generated by the diffusion model, by the leaky, competing accumulator (LCA) model (Usher & McClelland, 2001) and by the linear ballistic accumulator (LBA) model (Brown & Heathcote, 2008). Results show that EZ and DMAT are better capable than fast-dm in recovering experimental effects on parameters. EZ was best in recovering individual differences in parameter values. When data were generated by the LCA model, the diffusion model estimates obtained with all three methods correlated well with corresponding LCA model parameters. No such one-on-one correspondence could be established between parameters of the LBA model and the diffusion model.     

Processing time predictions of current models of perception in the classic additive factors paradigm

This article explores the consequences for factorial additivity in a Sternberg [(1969). The discovery of processing stages: Extensions of donders method In: W.G. Koster (Ed.), Attention and performance II, Acta Psychologica, 30, 276-315] additive-factors paradigm of the assumptions adopted by models of perception that relate the representation of a stimulus to decision time. Three example models, signal detection theory with the latency-distance hypothesis, stochastic general recognition theory, and a random walk model of exemplar classification, are interrogated to determine what type of interaction they predict factors will yield in a hypothetical factorial (choice) reaction time experiment in which the ‘empirical’ factors’ effects are manifest as parameter changes. All frameworks make the critical assumption that decision time depends on the perceptual representation of the stimulus as well as the architecture. As a consequence, nonadditivity of factors thought to affect different “stages” in the classical approach emerges within the current modeling approach. The nature of this influence is revealed through analytic investigations and simulation. Earlier empirical findings of failures of selective influence that have defied adequate explanation are reinterpreted in light of the present findings.     

Attention,spatial integration,and the tail of response time distributions in Stroop task performance

A few studies have examined selective attention in Stroop task performance through ex-Gaussian analyses of response time (RT) distributions. It has remained unclear whether the tail of the RT distribution in vocal responding reflects spatial integration of relevant and irrelevant attributes, as suggested by Spieler, Balota, and Faust (2000 Spieler, D. H., Balota, D. A. and Faust, M. E. 2000. Levels of selective attention revealed through analyses of reaction time distributions. Journal of Experimental Psychology: Human Perception and Performance, 26: 506–526. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Here, two colour–word Stroop experiments with vocal responding are reported in which the spatial relation between colour and word was manipulated. Participants named colours (e.g., green; say “green”) while trying to ignore distractors that were incongruent or congruent words (e.g., red or green), or neutral series of Xs. The vocal RT was measured. Colour words in colour, white words superimposed onto colour rectangles (Experiment 1), and colour rectangles combined with auditory words (Experiment 2) yielded Stroop effects in both the leading edge and the tail of the RT distributions. These results indicate that spatial integration is not necessary for effects in the tail to occur in vocal responding. It is argued that the findings are compatible with an association of the tail effects with task conflict.