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.     

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.     

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.     

Vocational choice: A decision making perspective

We propose a model of vocational choice that can be used for analyzing and guiding the decision processes underlying career and job choices. Our model is based on research in behavioral decision making (BDM), in particular the choice goals framework developed by Bettman, Luce, and Payne (1998). The basic model involves two major processes. First, the selection of a decision strategy according to four choice goals: maximizing decision accuracy, minimizing cognitive effort, minimizing negative emotion, and maximizing justifiability of the decision. Second, the construction of situation-specific preferences, which can reflect irrelevant task and context factors such as the evaluation mode. This basic model is extended to account for social influences and the long decision time typical of most career and job decisions. We review research on vocational choice in light of this model, discuss normative implications for counseling, and outline a research agenda for studying vocational choice from a behavioral decision making perspective.     

The response dynamics of preferential choice

The ubiquity of psychological process models requires an increased degree of sophistication in the methods and metrics that we use to evaluate them. We contribute to this venture by capitalizing on recent work in cognitive science analyzing response dynamics, which shows that the bearing information processing dynamics have on intended action is also revealed in the motor system. This decidedly “embodied” view suggests that researchers are missing out on potential dependent variables with which to evaluate their models—those associated with the motor response that produces a choice. The current work develops a method for collecting and analyzing such data in the domain of decision making. We first validate this method using widely normed stimuli from the International Affective Picture System (Experiment 1), and demonstrate that curvature in response trajectories provides a metric of the competition between choice options. We next extend the method to risky decision making (Experiment 2) and develop predictions for three popular classes of process model. The data provided by response dynamics demonstrate that choices contrary to the maxim of risk seeking in losses and risk aversion in gains may be the product of at least one “online” preference reversal, and can thus begin to discriminate amongst the candidate models. Finally, we incorporate attentional data collected via eye-tracking (Experiment 3) to develop a formal computational model of joint information sampling and preference accumulation. In sum, we validate response dynamics for use in preferential choice tasks and demonstrate the unique conclusions afforded by response dynamics over and above traditional methods.     

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.     

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.     

A comparison of bounded diffusion models for choice in time controlled tasks

The Wiener diffusion model (WDM) for 2-alternative tasks assumes that sensory information is integrated over time. Recent neurophysiological studies have found neural correlates of this integration process in certain neuronal populations. This paper analyses the properties of the WDM with two different boundary conditions in decision making tasks in which the time of response is indicated by a cue. A dual reflecting boundary mechanism is proposed and its performance is compared with a well-established absorbing boundary in the cases of the WDM, the WDM with extensions, and the WDM with prior probability. The two types of boundary influence the dynamics of the model and introduce differential weighting of evidence. Comparisons with Ornstein-Uhlenbeck models are also done, and it is shown that the WDM with both types of boundary achieves similar performance and produces similar fits to existing behavioural data. Further studies are proposed to distinguish which boundary mechanism is more consistent with experimental data.     

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.     

The dynamics of deferred decision

Decision makers are often unable to choose between the options that they are offered. In these settings they typically defer their decision, that is, delay the decision to a later point in time or avoid the decision altogether. In this paper, we outline eight behavioral findings regarding the causes and consequences of choice deferral that cognitive theories of decision making should be able to capture. We show that these findings can be accounted for by a deferral-based time limit applied to existing sequential sampling models of preferential choice. Our approach to modeling deferral as a time limit in a sequential sampling model also makes a number of novel predictions regarding the interactions between choice probabilities, deferral probabilities, and decision times, and we confirm these predictions in an experiment. Choice deferral is a key feature of everyday decision making, and our paper illustrates how established theoretical approaches can be used to understand the cognitive underpinnings of this important behavioral phenomenon.     

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.     

The point of no return in choice reaction time: controlled and ballistic stages of response preparation

A countermanding procedure and race model are used to assess separately the effects of experimental factors before and after the "point of no return" in response preparation. The results reveal details about processes that so closely precede the initiation of movement that they cannot be inhibited. These processes appear to be affected by the repetition of stimulus-response pairs, but not by the physical or semantic properties of the stimuli. A model of response preparation is supported in which response inhibition depends upon the outcome of a race between independent excitatory and inhibitory processes, and reaction time is the sum of the durations of at least two stages, separated by the point of no return.     

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.     

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.     

The precise time course of lexical activation: MEG measurements of the effects of frequency, probability, and density in lexical decision

Visually presented letter strings consistently yield three MEG response components: the M170, associated with letter-string processing (Tarkiainen, Helenius, Hansen, Cornelissen, & Salmelin, 1999); the M250, affected by phonotactic probability, (Pylkk?nen, Stringfellow, & Marantz, 2002); and the M350, responsive to lexical frequency (Embick, Hackl, Schaeffer, Kelepir, & Marantz, 2001). Pylkk?nen et al. found evidence that the M350 reflects lexical activation prior to competition among phonologically similar words. We investigate the effects of lexical and sublexical frequency and neighborhood density on the M250 and M350 through orthogonal manipulation of phonotactic probability, density, and frequency. The results confirm that probability but not density affects the latency of the M250 and M350; however, an interaction between probability and density on M350 latencies suggests an earlier influence of neighborhoods than previously reported.     

Decision by sampling: The role of the decision environment in risky choice

Decision by sampling (DbS) is a theory about how our environment shapes the decisions that we make. Here, I review the application of DbS to risky decision making. According to classical theories of risky decision making, people make stable transformations between outcomes and probabilities and their subjective counterparts using fixed psychoeconomic functions. DbS offers a quite different account. In DbS, the subjective value of an outcome or probability is derived from a series of binary, ordinal comparisons with a sample of other outcomes or probabilities from the decision environment. In this way, the distribution of attribute values in the environment determines the subjective valuations of outcomes and probabilities. I show how DbS interacts with the real-world distributions of gains, losses, and probabilities to produce the classical psychoeconomic functions. I extend DbS to account for preferences in benchmark data sets. Finally, in a challenge to the classical notion of stable subjective valuations, I review evidence that manipulating the distribution of attribute values in the environment changes our subjective valuations just as DbS predicts.     

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.     

On the dissociation between reaction time and response duration as a function of lexical and sublexical reading: An examination of phonetic decoding and computational models

Neurobiological models of reading account for two ways in which orthography is converted to phonology: (1) familiar words, particularly those with exceptional spelling-sound mappings (e.g., shoe) access their whole-word lexical representations in the ventral visual stream, and (2) orthographically unfamiliar words, particularly those with regular spelling-sound mappings (i.e., pseudohomophones [PHs], which are orthographically novel but sound like real words; e.g., shue) are phonetically decoded via sublexical processing in the dorsal visual stream. The present study used a naming task in order to compare naming reaction time (RT) and response duration (RD) of exception and regular words to their PH counterparts. We replicated our earlier findings with words, and extended them to PH phonetic decoding by showing a similar effect on RT and RD of matched PHs. Given that the shorter RDs for exception words can be attributed to the benefit of whole-word processing in the orthographic word system, and the longer RTs for exception words to the conflict with phonetic decoding, our PH results demonstrate that phonetic decoding also involves top-down feedback from phonological lexical representations (e.g., activated by shue) to the orthographic representations of the corresponding correct word (e.g., shoe). Two computational models were tested for their ability to account for these effects: the DRC and the CDP+. The CDP+ fared best as it was capable of simulating both the regularity and stimulus type effect on RT for both word and PH identification, although not their over-additive interaction. Our results demonstrate that both lexical reading and phonetic decoding elicit a regularity dissociation between RT and RD that provides important constraints to all models of reading, and that phonetic decoding results in top-down feedback that bolsters the orthographic lexical reading process.     

Using diffusion models to understand clinical disorders

Sequential sampling models provide an alternative to traditional analyses of reaction times and accuracy in two-choice tasks. These models are reviewed with particular focus on the diffusion model (Ratcliff, 1978) and how its application can aid research on clinical disorders. The advantages of a diffusion model analysis over traditional comparisons are shown through simulations and a simple lexical decision experiment. Application of the diffusion model to a clinically relevant topic is demonstrated through an analysis of data from nonclinical participants with high- and low-trait anxiety in a recognition memory task. The model showed that after committing an error, participants with high-trait anxiety responded more cautiously by increasing their boundary separation, whereas participants with low-trait anxiety did not. The article concludes with suggestions for ways to improve and broaden the application of these models to studies of clinical disorders.     

The role of epistemic motivation in individuals’ response to decision complexity

Integrating findings on the effects of more alternatives with findings on the effects of more attributes, we offer a motivational decision-making model, suggesting that epistemic motivation moderates individuals’ responses to complex information. Study 1 empirically investigated the shared essence of four conceptualizations of epistemic motivation, further distinguishing it from the maximizing/satisficing motivation. A series of experiments indicate that epistemic motivation moderates the effect of complex information on one’s discomfort with a decision (Studies 2–4) and on the tendency to implement one’s choice in action (Study 3). Taken together, our findings indicate that individuals with low epistemic motivation experience more discomfort and are less likely to implement their decision when faced with complex information whereas those high on epistemic motivation portray a weaker or even an opposite effect. The consistent findings across conceptualizations (dispositional Need-for-Cognitive-Closure and manipulated Openness vs. Conservation values) indicate the robustness of the findings and the important role of epistemic motivation in complex decisions.