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.     

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.     

Individual differences on speeded cognitive tasks: Comment on Chen,Hale, and Myerson (2007)

Chen, Hale, and Myerson (2007) recently reported a test of the difference engine model (Myerson, Hale, Zheng, Jenkins, & Widaman, 2003). This test evaluated whether the standard deviation (SD) is proportional to the amount of processing—that is, mean reaction time (RT)—in a speeded cognitive task. We show that this evaluation is not a test of the model because its finding is a consequence of relationships in the data. We argue any model structure that produces increasing values of RT as a function of difficulty, with different slopes for different individuals, necessarily produces a correlation between SD and mean RT. We illustrate this with a different model structure—that is, the diffusion model proposed by Ratcliff (1978)—showing that it produces a fan out between fast- and slow-group means and produces the correlation between SD and mean RT that matches the empirical result.     

Conflict tasks and the diffusion framework: Insight in model constraints based on psychological laws

Formal models of decision-making have traditionally focused on simple, two-choice perceptual decisions. To date, one of the most influential account of this process is Ratcliff’s drift diffusion model (DDM). However, the extension of the model to more complex decisions is not straightforward. In particular, conflicting situations, such as the Eriksen, Stroop, or Simon tasks, require control mechanisms that shield the cognitive system against distracting information. We adopted a novel strategy to constrain response time (RT) models by concurrently investigating two well-known empirical laws in conflict tasks, both at experimental and modeling levels. The two laws, predicted by the DDM, describe the relationship between mean RT and (i) target intensity (Piéron’s law), (ii) standard deviation of RT (Wagenmakers–Brown’s law). Pioneering work has shown that Piéron’s law holds in the Stroop task, and has highlighted an additive relationship between target intensity and compatibility. We found similar results in both Eriksen and Simon tasks. Compatibility also violated Wagenmakers–Brown’s law in a very similar and particular fashion in the two tasks, suggesting a common model framework. To investigate the nature of this commonality, predictions of two recent extensions of the DDM that incorporate selective attention mechanisms were simulated and compared to the experimental results. Both models predict Piéron’s law and the violation of Wagenmakers–Brown’s law by compatibility. Fits of the models to the RT distributions and accuracy data allowed us to further reveal their relative strengths and deficiencies. Combining experimental and computational results, this study sets the groundwork for a unified model of decision-making in conflicting environments.     

A fast numerical algorithm for the estimation of diffusion model parameters

In this paper, we describe a new algorithmic approach for parameter estimation in Ratcliff's [(1978). A theory of memory retrieval. Psychological Review, 85 (2), 59-108] diffusion model. This problem, especially if inter-trial variabilities of parameters are included in the model, is computationally very expensive; the parameter estimation procedure often takes a long time even with today's high-speed computers. The algorithm described here makes the calculation of the cumulative distribution functions for predicted process durations computationally much less expensive. This improvement is achieved by solving the Kolmogorov backward equation numerically instead of employing the previously used closed form solution. Additionally, the algorithm can determine the optimum fit for one of the model parameters (the starting point z) directly, thereby reducing the dimension of the parameter search space by one. The resulting method is shown to be notably faster than the standard (closed-form solution) method for parameter estimation.     

Integrating Impairments in Reaction Time and Executive Function Using a Diffusion Model Framework

Using Ratcliff’s diffusion model and ex-Gaussian decomposition, we directly evaluate the role individual differences in reaction time (RT) distribution components play in the prediction of inhibitory control and working memory (WM) capacity in children with and without ADHD. Children with (n?=?91, $ \overline{\mathrm{x}} $ age?=?10.2 years, 67 % male) and without ADHD (n?=?62, $ \overline{\mathrm{x}} $ age?=?10.6 years, 46 % male) completed four tasks of WM and a stop signal reaction time (SSRT) task. Children with ADHD had smaller WM capacities and less efficient inhibitory control. Diffusion model analyses revealed that children with ADHD had slower drift rates (v) and faster non-decision times (Ter), but there were no group differences in boundary separations (a). Similarly, using an ex-Gaussian approach, children with ADHD had larger τ values than non-ADHD controls, but did not differ in μ or σ distribution components. Drift rate mediated the association between ADHD status and performance on both inhibitory control and WM capacity. τ also mediated the ADHD-executive function impairment associations; however, models were a poorer fit to the data. Impaired performance on RT and executive functioning tasks has long been associated with childhood ADHD. Both are believed to be important cognitive mechanisms to the disorder. We demonstrate here that drift rate, or the speed at which information accumulates towards a decision, is able to explain both.     

The EZ diffusion method: Too EZ?

The diffusion model (Ratcliff, 1978) for fast two-choice decisions has been successful in a number of domains. Wagenmakers, van der Maas, and Grasman (2007) proposed a new method for fitting the model to data (“EZ”) that is simpler than the standard chisquare method (Ratcliff & Tuerlinckx, 2002). For an experimental condition, EZ can estimate parameter values for the main components of processing using only correct response times (RTs), their variance, and accuracy, not error RTs or the shapes of RT distributions. Wagenmakers et al. suggested that EZ produces accurate parameter estimates in cases in which the chi-square method would fail-specifically, experimental conditions with small numbers of observations or with accuracy near ceiling. In this article, I counter these claims and discuss EZ’s limitations. Unlike the chi-square method, EZ is extremely sensitive to outlier RTs and is usually less efficient in recovering parameter values, and it can lead to errors in interpretation when the data do not meet its assumptions, when the number of observations in an experimental condition is small, or when accuracy in an experimental condition is high. The conclusion is that EZ can be useful in the exploration of parameter spaces, but it should not be used for meaningful estimates of parameter values or for assessing whether or not a model fits data.     

A test of the diffusion model explanation for the worst performance rule using preregistration and blinding

People with higher IQ scores also tend to perform better on elementary cognitive-perceptual tasks, such as deciding quickly whether an arrow points to the left or the right Jensen (2006). The worst performance rule (WPR) finesses this relation by stating that the association between IQ and elementary-task performance is most pronounced when this performance is summarized by people’s slowest responses. Previous research has shown that the WPR can be accounted for in the Ratcliff diffusion model by assuming that the same ability parameter—drift rate—mediates performance in both elementary tasks and higher-level cognitive tasks. Here we aim to test four qualitative predictions concerning the WPR and its diffusion model explanation in terms of drift rate. In the first stage, the diffusion model was fit to data from 916 participants completing a perceptual two-choice task; crucially, the fitting happened after randomly shuffling the key variable, i.e., each participant’s score on a working memory capacity test. In the second stage, after all modeling decisions were made, the key variable was unshuffled and the adequacy of the predictions was evaluated by means of confirmatory Bayesian hypothesis tests. By temporarily withholding the mapping of the key predictor, we retain flexibility for proper modeling of the data (e.g., outlier exclusion) while preventing biases from unduly influencing the results. Our results provide evidence against the WPR and suggest that it may be less robust and less ubiquitous than is commonly believed.     

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.     

Interpreting the parameters of the diffusion model: an empirical validation

The diffusion model (Ratcliff, 1978) allows for the statistical separation of different components of a speeded binary decision process (decision threshold, bias, information uptake, and motor response). These components are represented by different parameters of the model. Two experiments were conducted to test the interpretational validity of the parameters. Using a color discrimination task, we investigated whether experimental manipulations of specific aspects of the decision process had specific effects on the corresponding parameters in a diffusion model data analysis (see Ratcliff, 2002; Ratcliff & Rouder, 1998; Ratcliff, Thapar, & McKoon, 2001, 2003). In support of the model, we found that (1) decision thresholds were higher when we induced accuracy motivation, (2) drift rates (i.e., information uptake) were lower when stimuli were harder to discriminate, (3) the motor components were increased when a more difficult form of response was required, and (4) the process was biased toward rewarded responses.     

Response time distributional evidence for distinct varieties of number attraction

Speakers are known to make subject–verb agreement errors both when a number-mismatching noun intervenes between the head of the subject phrase and the verb (e.g., ¹The key to the cabinets are on the table) and in configurations in which there is a number-mismatching noun that does not intervene (e.g., ¹The cabinets that the key open are on the second floor). Using a two-choice response time (RT) paradigm, Staub (2009) found that correct agreement decisions were also slowed in both cases. The present article reports a new experiment designed to explore whether these two RT effects are qualitatively similar or different. Fitting of the ex-Gaussian distribution (Ratcliff, 1979) to individual subjects’ RT data, in each condition, demonstrated that the effect of an intervening number attractor on correct RT is due to both a shifting of the distribution to the right and to increased skewing, while the effect of a non-intervening attractor is almost entirely a skewing effect. A non-parametric vincentizing procedure supported these conclusions. These findings are taken to support the view that these two types of number attraction involve distinct processing mechanisms.     

A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models

Group-level variance estimates of zero often arise when fitting multilevel or hierarchical linear models, especially when the number of groups is small. For situations where zero variances are implausible a priori, we propose a maximum penalized likelihood approach to avoid such boundary estimates. This approach is equivalent to estimating variance parameters by their posterior mode, given a weakly informative prior distribution. By choosing the penalty from the log-gamma family with shape parameter greater than 1, we ensure that the estimated variance will be positive. We suggest a default log-gamma(2,λ) penalty with λ→0, which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero, thus remaining consistent with the data while being nondegenerate. We also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior. Our default method provides better estimates of model parameters and standard errors than the maximum likelihood or the restricted maximum likelihood estimators. The log-gamma family can also be used to convey substantive prior information. In either case—pure penalization or prior information—our recommended procedure gives nondegenerate estimates and in the limit coincides with maximum likelihood as the number of groups increases.     

Predictions from a model of global psychophysics about differences between perceptual and physical matches

A well-known phenomenon is that ??matched?? successive signals do not result in physical identity. This phenomenon has mostly been studied in terms of how much the second of two signals varies from the first, which is called the time-order error (TOE). Here, theoretical predictions led us to study the more general question of how much the matching signal differs from the standard signal, independent of the position of the matching signal as the first or second in a presentation. This we call non-equal matches (NEM). Using Luce??s (Psychological Review, 109, 520?C532, 2002, Psychological Review, 111, 446?C454, 2004, Psychological Review, 115, 601, 2008, Psychological Review, 119, 373?C387, 2012) global psychophysical theory, we predicted NEM when an intensity z is perceived to be ??1 times a standard signal x.?? The theory predicts two different types of individual behaviors for the NEM, and these predictions were evaluated and confirmed in an experiment. We showed that the traditional definition of TOE precludes the observation, and thus the study, of the NEM phenomenon, and that the NEM effect is substantial enough to alter conclusions based on data that it affects. Furthermore, we demonstrated that the custom of averaging data over individuals clearly leads to quite misleading results. An important parameter in this modeling is a reference point that plays a central role in creating variability in the data, so that the key to obtaining regular data from respondents is to stabilize the reference point.     

Adult age differences in interference from a prospective-memory task: a diffusion model analysis

People often slow down their ongoing activities when they must remember an intended action, known as the cost or interference effect of prospective memory (PM). Only a few studies have examined adult age differences in PM interference, and the specific reasons underlying such differences are not well understood. The authors used a model-based approach to reveal processes underlying PM interference and age differences in these processes. Older and younger adults first performed a block of an ongoing lexical decision task alone. An embedded event-based PM task was added in a second block. Simultaneously accounting for the changes in response time distributions and error rates induced by the PM task, Ratcliff’s (Psychological Review 85:59–108, 1978) diffusion model was used to decompose the nonlinear combination of speed and accuracy into psychologically meaningful components. Remembering an intention not only reduced processing efficiency in both age groups, but also prolonged peripheral nondecision times and induced response cautiousness. Overall, the findings suggest that there are multiple, but qualitatively similar factors underlying PM task interference in both age groups.     

The efficient computation of the cumulative distribution and probability density functions in the diffusion model

An algorithm is described to efficiently compute the cumulative distribution and probability density functions of the diffusion process (Ratcliff, 1978) with trial-to-trial variability in mean drift rate, starting point, and residual reaction time. Some, but not all, of the integrals appearing in the model’s equations have closed-form solutions, and thus we can avoid computationally expensive numerical approximations. Depending on the number of quadrature nodes used for the remaining numerical integrations, the final algorithm is at least 10 times faster than a classical algorithm using only numerical integration, and the accuracy is slightly higher. Next, we discuss some special cases with an alternative distribution for the residual reaction time or with fewer than three parameters exhibiting trialto-trial variability.     

A reexamination of mental arithmetic models and data: A reply to Ashcraft

In a review of the chronometric literature, M. Ashcraft (1982, Developmental Review, 2, 213–236) concludes that adults store each basic arithmetic fact in a table-like retrieval network. In my commentary (1983, Developmental Review, 3, 225–230), I argued that procedural knowledge (stored rules, principles, or heuristics) might be a cognitively more economical basis for generating many number combinations. In this paper, I draw an analogy between this alternative model of number fact representation and how computers efficiently reconstruct arithmetic combinations, note that the research findings do not clearly support any one model of mental arithmetic, and attempt to address Ashcraft's (1983, Developmental Review, 3, 231–235) criticisms of my model.     

Characterizing belief bias in syllogistic reasoning: A hierarchical Bayesian meta-analysis of ROC data

The belief-bias effect is one of the most-studied biases in reasoning. A recent study of the phenomenon using the signal detection theory (SDT) model called into question all theoretical accounts of belief bias by demonstrating that belief-based differences in the ability to discriminate between valid and invalid syllogisms may be an artifact stemming from the use of inappropriate linear measurement models such as analysis of variance (Dube et al., Psychological Review, 117(3), 831–863, 2010). The discrepancy between Dube et al.’s, Psychological Review, 117(3), 831–863 (2010) results and the previous three decades of work, together with former’s methodological criticisms suggests the need to revisit earlier results, this time collecting confidence-rating responses. Using a hierarchical Bayesian meta-analysis, we reanalyzed a corpus of 22 confidence-rating studies (N =?993). The results indicated that extensive replications using confidence-rating data are unnecessary as the observed receiver operating characteristic functions are not systematically asymmetric. These results were subsequently corroborated by a novel experimental design based on SDT’s generalized area theorem. Although the meta-analysis confirms that believability does not influence discriminability unconditionally, it also confirmed previous results that factors such as individual differences mediate the effect. The main point is that data from previous and future studies can be safely analyzed using appropriate hierarchical methods that do not require confidence ratings. More generally, our results set a new standard for analyzing data and evaluating theories in reasoning. Important methodological and theoretical considerations for future work on belief bias and related domains are discussed.     

Effects of implicit fear of failure on cognitive processing: A diffusion model analysis

Whereas previous studies suggest that individuals with high implicit fear of failure (FF) perform worse on various indicators of general performance, the underlying mechanisms of this effect have not yet been understood. In our experimental study, 280 participants worked on a binary color discrimination task. Half of the participants were frustrated by means of negative performance feedback, while the control group received mainly positive feedback. We employed a diffusion model analysis (Ratcliff in Psychol Rev 85(2):59–108, 1978) to disentangle the different components involved in the execution of the task. Results revealed that participants in the frustration condition adopted more conservative decision settings (threshold separation parameter of the diffusion model). Besides, high implicit FF was related to slow information accumulation (drift), and this relation was stronger in the frustration condition. Participants with higher FF further showed reduced learning rates during the task. Task related intrusive thoughts are discussed as mechanism for reduced performance of high FF individuals. We conclude that diffusion model analyses can contribute to a better understanding of the mechanisms underlying the effects of psychological motives.