On the relation between the mean and the variance of a diffusion model response time distribution

Almost every empirical psychological study finds that the variance of a response time (RT) distribution increases with the mean. Here we present a theoretical analysis of the nature of the relationship between RT mean and RT variance, based on the assumption that a diffusion model (e.g., Ratcliff (1978) Psychological Review, 85, 59-108; Ratcliff (2002). Psychonomic Bulletin & Review, 9, 278-291), adequately captures the shape of empirical RT distributions. We first derive closed-form analytic solutions for the mean and variance of a diffusion model RT distribution. Next, we study how systematic differences in two important diffusion model parameters simultaneously affect the mean and the variance of the diffusion model RT distribution. Within the range of plausible values for the drift rate parameter, the relation between RT mean and RT standard deviation is approximately linear. Manipulation of the boundary separation parameter also leads to an approximately linear relation between RT mean and RT standard deviation, but only for low values of the drift rate parameter.     

Modeling response signal and response time data

The diffusion model (Ratcliff, 1978) and the leaky competing accumulator model (LCA, Usher & McClelland, 2001) were tested against two-choice data collected from the same subjects with the standard response time procedure and the response signal procedure. In the response signal procedure, a stimulus is presented and then, at one of a number of experimenter-determined times, a signal to respond is presented. The models were fit to the data from the two procedures simultaneously under the assumption that responses in the response signal procedure were based on a mixture of decision processes that had already terminated at response boundaries before the signal and decision processes that had not yet terminated. In the latter case, decisions were based on partial information in one variant of each model or on guessing in a second variant. Both variants of the diffusion model fit the data well and both fit better than either variant of the LCA model, although the differences in numerical goodness-of-fit measures were not large enough to allow decisive selection between the models.     

A diffusion model account of response time and accuracy in a brightness discrimination task: fitting real data and failing to fit fake but plausible data

A brightness discrimination experiment was performed to examine how subjects decide whether a patch of pixels is “bright” or “dark,” and stimulus duration, brightness, and speed versus accuracy instructions were manipulated. The diffusion model (Ratcliff, 1978) was fit to the data, and it accounted for all the dependent variables: mean correct and error response times, the shapes of response time distributions for correct and error responses, and accuracy values. Speed-accuracy manipulations affected only boundary separation (response criteria settings) in the model. Drift rate (the rate of accumulation of evidence) in the diffusion model, which represents stimulus quality, increased as a function of stimulus duration and stimulus brightness but asymptoted as stimulus duration increased from 100 to 150 msec. To address the argument that the diffusion model can fit any pattern of data, simulated patterns of plausible data are presented that the model cannot fit.     

Single- versus dual-process models of lexical decision performance: insights from response time distributional analysis

This article evaluates 2 competing models that address the decision-making processes mediating word recognition and lexical decision performance: a hybrid 2-stage model of lexical decision performance and a random-walk model. In 2 experiments, nonword type and word frequency were manipulated across 2 contrasts (pseudohomophone-legal nonword and legal-illegal nonword). When nonwords became more wordlike (i.e., BRNTA vs. BRANT vs. BRANE), response latencies to nonwords were slowed and the word frequency effect increased. More important, distributional analyses revealed that the Nonword Type = Word Frequency interaction was modulated by different components of the response time distribution, depending on the specific nonword contrast. A single-process random-walk model was able to account for this particular set of findings more successfully than the hybrid 2-stage model.     

A direct test of the unequal-variance signal detection model of recognition memory

Analyses of the receiver operating characteristic (ROC) almost invariably suggest that, on a recognition memory test, the standard deviation of memory strengths associated with the lures (σ_lure) is smaller than that of the targets (σ_target). Often, σ_lure/σ_target ≈ 0.80. However, that conclusion is based on a model that assumes that the memory strength distributions are Gaussian in form. In two experiments, we investigated this issue in a more direct way by asking subjects to simply rate the memory strengths of targets and lures using a 20-point or a 99-point strength scale. The results showed that the standard deviation of the ratings made to the targets (s _target) was, indeed, larger than the standard deviation of the ratings made to the lures (s _lure). Moreover, across subjects, the ratios _lure/s _target correlated highly with the estimate of σ_lure/σtarget obtained from ROC analysis, and both estimates were, on average, approximately equal to 0.80.     

A dual-process model of reactions to perceived stigma

The authors propose a theoretical model of individual psychological reactions to perceived stigma. This model suggests that 2 psychological systems may be involved in reactions to stigma across a variety of social contexts. One system is primarily reflexive, or associative, whereas the other is rule based, or reflective. This model assumes a temporal pattern of reactions to the stigmatized, such that initial reactions are governed by the reflexive system, whereas subsequent reactions or "adjustments" are governed by the rule-based system. Support for this model was found in 2 studies. Both studies examined participants' moment-by-moment approach-avoidance reactions to the stigmatized. The 1st involved participants' reactions to persons with HIV/AIDS, and the 2nd, participants' reactions to 15 different stigmatizing conditions.     

Fitting the ratcliff diffusion model to experimental data

Many experiments in psychology yield both reaction time and accuracy data. However, no off-the-shelf methods yet exist for the statistical analysis of such data. One particularly successful model has been the diffusion process, but using it is difficult in practice because of numerical, statistical, and software problems. We present a general method for performing diffusion model analyses on experimental data. By implementing design matrices, a wide range of across-condition restrictions can be imposed on model parameters, in a flexible way. It becomes possible to fit models with parameters regressed onto predictors. Moreover, data analytical tools are discussed that can be used to handle various types of outliers and contaminants. We briefly present an easy-touse software tool that helps perform diffusion model analyses.     

Nonword reading: Comparing dual-route cascaded and connectionist dual-process models with human data

Two prominent dual-route computational models of reading aloud are the dual-route cascaded (DRC) model, and the connectionist dual-process plus (CDP+) model. While sharing similarly designed lexical routes, the two models differ greatly in their respective nonlexical route architecture, such that they often differ on nonword pronunciation. Neither model has been appropriately tested for nonword reading pronunciation accuracy to date. We argue that empirical data on the nonword reading pronunciation of people is the ideal benchmark for testing. Data were gathered from 45 Australian-English-speaking psychology undergraduates reading aloud 412 nonwords. To provide contrast between the models, the nonwords were chosen specifically because DRC and CDP+ disagree on their pronunciation. Both models failed to accurately match the experiment data, and both have deficiencies in nonword reading performance. However, the CDP+ model performed significantly worse than the DRC model. CDP++, the recent successor to CDP+, had improved performance over CDP+, but was also significantly worse than DRC. In addition to highlighting performance shortcomings in each model, the variety of nonword responses given by participants points to a need for models that can account for this variety. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

A mixed Rasch model of dual-process conditional reasoning

A fine-grained dual-process approach to conditional reasoning is advocated: Responses to conditional syllogisms are reached through the operation of either one of two systems, each of which can rely on two different mechanisms. System1 relies either on pragmatic implicatures or on the retrieval of information from semantic memory; System2 operates first through inhibition of System1, then (but not always) through activation of analytical processes. It follows that reasoners will fall into one of four groups of increasing reasoning ability, each group being uniquely characterized by (a) the modal pattern of individual answers to blocks of affirming the consequent (AC), denying the antecedent (DA), and modus tollens (MT) syllogisms featuring the same conditional; and (b) the average rate of determinate answers to AC, DA, and MT. This account receives indirect support from the extant literature and direct support from a mixed Rasch model of responses given to 18 syllogisms by 486 adult reasoners.     

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.     

Cognitive and motivational processes underlying coping flexibility: a dual-process model

Discriminative facility was proposed as a cognitive process and need for closure was proposed as a motivational process underlying coping flexibility. The dual-process model posits that need for closure influences discriminative facility, which in turn modifies coping flexibility and psychological adjustment. In Study 1, results of structural equation modeling provided support for the dual-process model. This model was further examined using experimental methods (Study 2) and a prospective design (Study 3). Consistent with the dual-process model, results from all 3 studies showed that participants who were more motivated to seek alternative coping strategies tended to encode stressful situations in a more differentiated way. These individuals used a greater variety of strategies to fit different situational demands and were better adjusted.     

Simple matrix methods for analyzing diffusion models of choice probability, choice response time, and simple response time

Diffusion processes (e.g., Wiener process, Ornstein-Uhlenbeck process) are powerful approaches to model human information processes in a variety of psychological tasks. Lack of mathematical tractability, however, has prevented broad applications of these models to empirical data. This tutorial explains step by step, using a matrix approach, how to construct these models, how to implement them on a computer, and how to calculate the predictions made by these models. In particular, we present models for binaries choices for unidimensional and multiattribute choice alternatives; for simple reaction time tasks; and for three alternatives choice problems.     

A dual-process motivational model of attitudes towards vegetarians and vegans

Vegetarians and vegans comprise a minority of most western populations. However, relatively little research has investigated the psychological foundations of attitudes towards this minority group. The following study employs a dual process model of intergroup attitudes to explore the motivational basis of non-vegetarians' attitudes towards vegetarians and vegans. Participants were 1,326 individuals recruited through advertisements in a national newspaper in New Zealand. Non-vegetarian participants first completed measures of ideological attitudes and social worldviews, and then were randomly assigned to complete a measure of outgroup attitudes towards either vegetarians or vegans. Although non-vegetarians' attitudes towards both vegetarians and vegans were generally positive, attitudes towards vegans were significantly less positive than attitudes towards vegetarians, and male participants expressed significantly less positive attitudes towards both outgroups than female participants. The structural equation model predicting attitudes towards vegetarians and vegans fit the data well and explained a significant amount of the variance in attitudes.     

A ballistic model of choice response time

Almost all models of response time (RT) use a stochastic accumulation process. To account for the benchmark RT phenomena, researchers have found it necessary to include between-trial variability in the starting point and/or the rate of accumulation, both in linear (R. Ratcliff & J. N. Rouder, 1998) and nonlinear (M. Usher & J. L. McClelland, 2001) models. The authors show that a ballistic (deterministic within-trial) model using a simplified version of M. Usher and J. L. McClelland's (2001) nonlinear accumulation process with between-trial variability in accumulation rate and starting point is capable of accounting for the benchmark behavioral phenomena. The authors successfully fit their model to R. Ratcliff and J. N. Rouder's (1998) data, which exhibit many of the benchmark phenomena.     

An EZ-diffusion model for response time and accuracy

The EZ-diffusion model for two-choice response time tasks takes mean response time, the variance of response time, and response accuracy as inputs. The model transforms these data via three simple equations to produce unique values for the quality of information, response conservativeness, and nondecision time. This transformation of observed data in terms of unobserved variables addresses the speed—accuracy trade-off and allows an unambiguous quantification of performance differences in two-choice response time tasks. The EZ-diffusion model can be applied to data-sparse situations to facilitate individual subject analysis. We studied the performance of the EZ-diffusion model in terms of parameter recovery and robustness against misspecification by using Monte Carlo simulations. The EZ model was also applied to a real-world data set.     

A diffusion model explanation of the worst performance rule for reaction time and IQ

The worst performance rule for cognitive tasks [Coyle, T.R. (2003). IQ, the worst performance rule, and Spearman's law: A reanalysis and extension. Intelligence, 31, 567-587] in which reaction time is measured is the result that IQ scores correlate better with longer (i.e., 0.7 and 0.9 quantile) reaction times than shorter (i.e., 0.1 and 0.3 quantile) reaction times. We show that this pattern of correlations can be predicted by the diffusion model [Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59-108], in two ways: either assuming that the rate of accumulation of information toward a decision is higher for higher IQ subjects or assuming that the criterial amounts of information they require before a decision are lower. Importantly, the model explains both reaction times and accuracy, so the two possibilities can be distinguished.     

A dual-process model of affective and instrumental attitudes in predicting physical activity

IntroductionA novel dual-process model based upon the affective-reflective theory which distinguishes between affective attitude and instrumental attitude at an explicit and implicit level was used to predict physical activity.MethodUndergraduate students (N = 114) completed a lab-based study at two time-points, spaced two weeks apart. Participants completed self-report measures of the theory of planned behavior constructs, including explicit affective attitude and explicit instrumental attitude. Implicit affective attitude and implicit instrumental attitude were measured using single category implicit association tests.ResultsExplicit affective attitude and perceived behavioral control indirectly predicted physical activity through intention, and explicit affective attitude, implicit affective attitude, and intention predicted physical activity directly with a modest effect size (R² = 0.24).DiscussionFindings suggest explicitly and implicitly held evaluative information are conceptually distinct, and affective evaluations are key in guiding physical activity behavior regardless of whether such evaluations are implicit or consciously accessible.