Modeling Response times for Two-choice Decisions

The diffusion model for two-choice real-time decisions is applied to four psychophysical tasks. The model reveals how stimulus information guides decisions and shows how the information is processed through time to yield sometimes correct and sometimes incorrect decisions. Rapid two-choice decisions yield multiple empirical measures: response times for correct and error responses, the probabilities of correct and error responses, and a variety of interactions between accuracy and response time that depend on instructions and task difficulty. The diffusion model can explain all these aspects of the data for the four experiments we present. The model correctly accounts for error response times, something previous models have failed to do. Variability within the decision process explains how errors are made, and variability across trials correctly predicts when errors are faster than correct responses and when they are slower. Making decisions is a ubiquitous part of everyday life. In psychology, besides being an object of study in its own right, decision making plays a central role in the tasks used to study basic cognitive functions such as memory, perception, and language comprehension. Frequently, the decisions required in these tasks are rapid two-choice decisions, decisions that are based on information that can be described as varying along a single dimension. Two key features of these decisions are that they occur over time—decisions are never reached instantaneously—and that they are error prone. In this article, we present a model to explain this class of decision processes. The goal is to understand what information drives the decision and how the decision process evolves over time to reach correct and incorrect decisions. The problem is difficult because potential models are constrained to explain multiple empirical measures that interact in complex ways. The measures include mean response times for correct and error responses, the shapes of the distributions of the response times, and the probabilities of correct and error responses. The relation between response time and accuracy is not fixed; it varies according to whether speed or accuracy of performance is emphasized and according to whether one or the other of the responses is more probable or weighted more heavily. In addition, the relation between probability of an error and error response time is not fixed but varies across levels of overall accuracy. Because of these complexities, no previous model has been completely successful. Often, models have dealt with only one measure—accuracy but not response time, or response time but not accuracy. Models that have dealt with response time have usually tried to explain only mean response times for correct responses, not the shapes of response time distributions or response times for errors. Modeling speed–accuracy relationships has usually not been attempted. In this article, we show how the diffusion model (Ratcliff, 1978, 1981, 1985, 1988; Ratcliff, Van Zandt, & McKoon, 1998) can explain all of these aspects of the data for two-choice perceptual decisions. For the first time, the model provides an integrated account of both the information that drives decisions and how that information is processed over time to produce correct and error responses. The main domain of application is to tasks on which response time is typically under a second. The model may apply when response time is greater than 1 s, but at much longer times, decisions are probably based on multiple decision attempts in which the first decision attempted was sometimes not made or made with too little confidence for the response to be based on that decision (with perhaps different information or different response criteria used in each successive attempt). A major weakness of all of the models for reaction time is the failure to account for error reaction times. Luce (1986) reviewed data and theory for error reaction times and concluded that there are few systematic studies of error reaction times that can be used to produce comprehensive empirical generalizations, nor is there a comprehensive theoretical account of error reaction times. Empirically, the relationship between correct and error reaction times varies: Sometimes errors are faster than correct responses (mainly when the task is easy and speed is emphasized); sometimes errors are slower than correct responses (mainly when the task is hard and accuracy is emphasized; see Luce, 1986; Swensson, 1972). Ratcliff et al. (1998, see also Smith & Vickers, 1988) presented data showing individual subjects had a crossover, with error responses faster than correct responses at high accuracy, and error responses slower than correct responses at low accuracy. This pattern is very difficult for models to produce; models predict slow errors or fast errors (e.g., Link & Heath, 1975), but most cannot predict both or predict crossovers. In this article, we show that the diffusion model can explain the relationship between correct and error responses across a range of experimental paradigms while at the same time fitting all the other response time and response probability aspects of the data. The key to the model’s success is variability in the decision process: We show this in experiments with perceptual stimuli, but the model is more general than this application; it can potentially have equal success for the twochoice cognitive tasks to which it has been applied previously. These tasks include shortand long-term recognition memory tasks, same/different letter-string matching, lexical decision tasks, numerosity judgments, and visual-scanning tasks (Ratcliff, 1978, 1981; Ratcliff et al., 1998; Strayer & Kramer, 1994).