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.     

Individual differences in the use of simplification strategies in a complex decision-making task

Decision time results were used to assess the strategies that 90 college undergraduates used in a complex decision-making task. Trend analyses revealed that the functions relating choice time to the number of choice alternatives in a set and the number of attributes comprising those alternatives contained linear (increasing) components. In addition, for a portion of the subjects, there was a quadratic effect of the number of attributes available to the decision maker on choice time, suggesting that these subjects adopted simplification strategies at high levels of task complexity. Reliable individual differences in these trend components were observed, consistent with individual differences in motivation and/or processing capacities. These individual differences were included in an information-processing model of decision behavior that captured the choice time data observed in this study. Subjects' ratings of apartments were used as a basis to assess the extent to which the use of simplification strategies resulted in preference reversals. Contrary to expectation, subjects whose choice times contained quadratic components demonstrated fewer preference reversals at high levels of information load.     

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.     

Context effects in multi-alternative decision making: empirical data and a Bayesian model

For decisions between many alternatives, the benchmark result is Hick's Law: that response time increases log-linearly with the number of choice alternatives. Even when Hick's Law is observed for response times, divergent results have been observed for error rates-sometimes error rates increase with the number of choice alternatives, and sometimes they are constant. We provide evidence from two experiments that error rates are mostly independent of the number of choice alternatives, unless context effects induce participants to trade speed for accuracy across conditions. Error rate data have previously been used to discriminate between competing theoretical accounts of Hick's Law, and our results question the validity of those conclusions. We show that a previously dismissed optimal observer model might provide a parsimonious account of both response time and error rate data. The model suggests that people approximate Bayesian inference in multi-alternative choice, except for some perceptual limitations.     

Visual memory and stimulus repetition effects

Recent investigations of memory for randomly configured patterns indicate that visual memory can involve distinct short-term and long-term components. The appearance of a visual recency effect that is confined to the last-presented item is believed to result from the active visualization of this item during the retention interval. Studies of the retention of familiar visual information have also suggested that the short-term effects observed are a result of active visualization. In a review of these studies, however, we argue that the effects obtained with familiar visual information are not necessarily a result of active visualization and, indeed, may not involve anything other than long-term visual memory. For example, Rabbitt and Vyas (1979) observed a visual recency effect in a serial choice reaction time task involving familiar information. That this recency effect was confined to the final item accords with the results obtained with unfamiliar visual information. However, this choice reaction time task did not require subjects to remember previous stimuli, so it is unlikely that they actively visualized them. With the case for a distinct short-term visual memory currently resting on the recency effect interpreted as reflecting a process of active visualization, this result is especially important. In the second part of the present paper, we report a series of experiments that provides an understanding of the visual recency effect in the serial choice reaction time task. We conclude from these studies that this effect is not due to visualization or to a visual trace either decaying or being overwritten by a succeeding stimulus.     

Intensity and number of alternatives in hue identification: Piéron’s law and choice reaction time

Two experiments investigated the effects of intensity and the number of response alternatives in hue identification. Three questions were addressed. First, what is the relationship between these effects? If intensity and number of alternatives affect different processes in series, these factors would have additive effects on reaction time (RT). Other models predict a multiplicative interaction. These models assume that when a stimulus is presented, evidence accrues over time for each response alternative, and that when the evidence for a particular alternative exceeds its criterion, the response is made. A natural auxiliary assumption is that intensity affects the rate of evidence accrual, while the number of alternatives affects the criterion. The second question addressed is, how does choice RT change as a function of intensity? Piéron used a power function to describe the effect of intensity on simple RT. This paper considers three candidates for intensity’s effect on choice RT: the power, logistic, and Michaelis functions. Each candidate function was tested in an additive model and a multiplicative model. The best account was given by the power function in an additive model. The last question is, does each process produce an output that can be classified as correct or incorrect? Schweickert (1985) proposed that factors that selectively influenced such processes would have additive effects on log percent correct. Instead, an interaction was found. Perhaps the assumption that the process affected by intensity produces a single output is wrong.     

On the reduction of choice reaction times with practice

Information theory concepts have been discussed in relation to data from choice reaction time experiments. Specifically, it has been stated that choice reaction time is proportional to the logarithm of the number of randomly appearing alternative stimuli. It is suggested that the reported increase in choice reaction times with an increased number of alternatives is the result of insufficient practice. Data are provided to show that, with sufficient practice, this increase does not occur between two and four choices.     

A crossed random effects diffusion model for speeded semantic categorization decisions

Choice reaction times (RTs) are often used as a proxy measure of typicality in semantic categorization studies. However, other item properties have been linked to choice RTs as well. We apply a tailored process model of choice RT to a speeded semantic categorization task in order to deconfound different sources of variability in RT. Our model is based on a diffusion model of choice RT, extended to include crossed random effects (of items and participants). This model retains the interesting process interpretation of the diffusion model’s parameters, but it can be applied to choice RTs even in the case where there are few or no repeated measurements of each participant-item combination. Different aspects of the response process are then linked to different types of item properties. A typicality measure turns out to predict the rate of information uptake, while a lexicographic measure predicts the stimulus encoding time. Accessibility measures cannot reliably predict any component of the decision process.     

选项数量和属性重要性对决策中信息加工的影响

本研究采用决策过程研究方法中的信息板技术,以现实生活中的职业选择作为决策任务,考察了选项数量和选项属性的重要性对决策过程中信息加工的影响方式。结果表明:选项数量对决策过程的信息搜索深度、搜索模式和策略的补偿性产生了影响;选项属性的重要性并不影响决策者对属性的查看时间,但影响了决策者对属性的点击次数。     

Choice reaction times for skilled responses

Two kinds of choice reaction time experiments are reported, both of which make use of a highly overlearned sensori-motor response. When a response is required for each stimulus presented, no increase in reaction time occurs as a function of the number of alternative stimuli available. It is proposed that the increase in choice reaction times commonly thought to accompany an increase in the number of alternative choices provided reflects the unpractised state of the responder. When a response is required for only one out of n possible stimuli, a slight but consistent increase in reaction time takes place with an increase in the number of alternatives. An analogy is drawn between the second experiment and a vigilance task and an expectancy hypothesis is invoked to explain the results.     

Precueing response factors in choice reaction time: a word of caution

This note discusses the assumption researchers make when movement attributes are precued in a choice reaction time (RT) paradigm-i.e., that the precue only reduces the number of response alternatives, so that changes in RT reflect the programming time for the unprecued attributes. However, it is argued that since the precue reduces the number of stimulus alternatives, changes in RT reflect changes in both stimulus processing and response programming. Therefore, the interpretation made by Klapp (1977), using the precue paradigm, that response selection and response timing are not independent sequential stages in the programming sequence, is not necessarily correct.     

Precueing Response Factors in Choice Reaction Time

This note discusses the assumption researchers make when movement attributes are precued in a choice reaction time (RT) paradigm—i.e., that the precue only reduces the number of response alternatives, so that changes in RT reflect the programming time for the unprecued attributes. However, it is argued that since the precue reduces the number of stimulus alternatives, changes in RT reflect changes in both stimulus processing and response programming. Therefore, the interpretation made by Klapp (1977), using the precue paradigm, that response selection and response timing are not independent sequential stages in the programming sequence, is not necessarily correct.     

Individual differences in components of reaction time distributions and their relations to working memory and intelligence

The authors bring together approaches from cognitive and individual differences psychology to model characteristics of reaction time distributions beyond measures of central tendency. Ex-Gaussian distributions and a diffusion model approach are used to describe individuals' reaction time data. The authors identified common latent factors for each of the 3 ex-Gaussian parameters and for 3 parameters central to the diffusion model using structural equation modeling for a battery of choice reaction tasks. These factors had differential relations to criterion constructs. Parameters reflecting the tail of the distribution (i.e., tau in the ex-Gaussian and drift rate in the diffusion model) were the strongest unique predictors of working memory, reasoning, and psychometric speed. Theories of controlled attention and binding are discussed as potential theoretical explanations.     

Structural equation modeling of approval voting data

The method of approval voting is a commonly used voting procedure in which each judge selects a subset of the alternatives. By postulating that the random utilities associated with the choice options in approval voting elections follow a multivariate normal distribution under the Thurstonian framework, Regenwetter, Ho, and Tsetlin (2007) attempted to integrate the normative theories and individual variabilities in modeling social behavior. However, their approach is limited to only three alternatives, due to computational intractability as the number of alternatives increases. In this article, we reparameterize extensions of their models under the structural equation modeling framework and propose the use of limited information methods for estimating model parameters. As a result, we are able to extend their previous approach to the analysis of approval voting data with any number of alternatives. Two applications are presented to illustrate the usefulness of such an approach.     

Visual binding of English and Chinese word parts is limited to low temporal frequencies

Some perceptual mechanisms manifest high temporal precision, allowing reports of visual information even when that information is restricted to windows smaller than 50 ms. Other visual judgments are limited to much coarser time scales. What about visual information extracted at late processing stages, for which we nonetheless have perceptual expertise, such as words? Here, the temporal limits on binding together visual word parts were investigated. In one trial, either the word 'ball' was alternated with 'deck', or 'dell' was alternated with 'back', with all stimuli presented at fixation. These stimuli restrict the time scale of the rod identities because the two sets of alternating words form the same image at high alternation frequencies. Observers made a forced choice between the two alternatives. Resulting 75% thresholds are restricted to 5 Hz or less for words and nonword letter strings. A similar result was obtained in an analogous experiment with Chinese participants viewing alternating Chinese characters. These results support the theory that explicit perceptual access to visual information extracted at late stages is limited to coarse time scales.     

The optimal number of alternatives at a choice point with travel time considered

The problem of optimal design is considered for tasks that can be represented as n choice points with a_i alternatives per choice point. Examples are mazes or multiple choice tests. Some fixed time t′ elapses between choice points, say traveling from one to the other or reading a question, and additional time t is spent considering each alternative at the choice point. For fixed total time, the goal is to find the optimum number of choice points and the optimum number of alternatives at each choice point. In general, the longer the time spent between choice points, the greater the optimum number of alternatives at each point.     

Limits to decision-making ability in direct multiattribute alternative evaluation

In recent years there has been considerable interest in how decision strategies in choice problems depend on such characteristics as the number of available criteria and the number of available alternatives. Along with multicriteria choice problems, classification tasks are treated where alternatives are classified into several decision (evaluative) classes. We believe that in classification problems one's ability in information processing is limited and depends on task complexity. In this paper we describe two experiments involving categorization of a large number of multiattribute alternatives into two or four evaluative classes. The results demonstrate that the subjects use different decision strategies and that there is a limit in human ability to implement these two tasks.     

On adaptation, maximization, and reinforcement learning among cognitive strategies

Analysis of binary choice behavior in iterated tasks with immediate feedback reveals robust deviations from maximization that can be described as indications of 3 effects: (a) a payoff variability effect, in which high payoff variability seems to move choice behavior toward random choice; (b) underweighting of rare events, in which alternatives that yield the best payoffs most of the time are attractive even when they are associated with a lower expected return; and (c) loss aversion, in which alternatives that minimize the probability of losses can be more attractive than those that maximize expected payoffs. The results are closer to probability matching than to maximization. Best approximation is provided with a model of reinforcement learning among cognitive strategies (RELACS). This model captures the 3 deviations, the learning curves, and the effect of information on uncertainty avoidance. It outperforms other models in fitting the data and in predicting behavior in other experiments.     

The attentional dynamics of masked detection

A dichoptic masking procedure was used to test whether the mask-dependent cuing effects found in luminance detection by P. L. Smith (2000a) were due to integration masking or interruption masking. Attentional cuing enhanced detection sensitivity (d') when stimuli were backwardly masked with either dichoptic or monoptic masks, whereas no cuing effect was found with unmasked stimuli, implying the mask dependencies were due to interruption of stimulus processing in visual cortex by the mask. The effect is predicted by a gated diffusion process model in which masks interrupt stimulus processing and attention controls the flow of information to a sequential-sampling decision mechanism. The model correctly predicts different patterns of performance for detection and discrimination and cuing effects in simple reaction time.