Trisha Van Zandt

How to fit a response time distribution

Among the most valuable tools in behavioral science is statistically fitting mathematical models of cognition to data—response time distributions, in particular. However, techniques for fitting distributions vary widely, and little is known about the efficacy of different techniques. In this article, we assess several fitting techniques by simulating six widely cited models of response time and using the fitting procedures to recover model parameters. The techniques include the maximization of likelihood and least squares fits of the theoretical distributions to different empirical estimates of the simulated distributions. A running example is used to illustrate the different estimation and fitting procedures. The simulation studies reveal that empirical density estimates are biased even for very large sample sizes. Some fitting techniques yield more accurate and less variable parameter estimates than do others. Methods that involve least squares fits to density estimates generally yield very poor parameter estimates.

Redefine Statistical Significance

We propose to change the default P-value threshold for statistical significance from 0.05 to 0.005 for claims of new discoveries.

A comparison of two response time models applied to perceptual matching

Two models, a Poisson race model and a diffusion model, are fit to data from a perceptual matching task. In each model, information about the similarity or the difference between two stimuli accumulates toward thresholds for either response. Stimulus variables are assumed to influence the rate at which information accumulates, and response variables are assumed to influence the level of the response thresholds. Three experiments were conducted to assess the performance of each model. In Experiment 1, observers performed under different response deadlines; in Experiment 2, response bias was manipulated by changing the relative frequency ofsame anddifferent stimuli. In Experiment 3, stimulus pairs were presented at three eccentricities: foveal, parafoveal, and peripheral. We examined whether the race and diffusion models could fit the response time and accuracy data through changes only in response parameters (for Experiments 1 and 2) or stimulus parameters (for Experiment 3). Comparisons between the two models suggest that the race model, which has not been studied extensively, can account for perceptual matching data at least as well as the diffusion model. Furthermore, without the constraints on the parameters provided by the experimental conditions, the diffusion and the race models are indistinguishable. This finding emphasizes the importance of fitting models across several conditions and imposing logical psychological constraints on the parameters of models.

On the ability to inhibit thought and action: General and special theories of an act of control

Response inhibition is an important act of control in many domains of psychology and neuroscience. It is often studied in a stop-signal task that requires subjects to inhibit an ongoing action in response to a stop signal. Performance in the stop-signal task is understood as a race between a go process that underlies the action and a stop process that inhibits the action. Responses are inhibited if the stop process finishes before the go process. The finishing time of the stop process is not directly observable; a mathematical model is required to estimate its duration. Logan and Cowan (1984) developed an independent race model that is widely used for this purpose. We present a general race model that extends the independent race model to account for the role of choice in go and stop processes, and a special race model that assumes each runner is a stochastic accumulator governed by a diffusion process. We apply the models to 2 data sets to test assumptions about selective influence of capacity limitations on drift rates and strategies on thresholds, which are largely confirmed. The model provides estimates of distributions of stop-signal response times, which previous models could not estimate. We discuss implications of viewing cognitive control as the result of a repertoire of acts of control tailored to different tasks and situations.

STATISTICAL MIMICKING OF REACTION TIME DATA : SINGLE-PROCESS MODELS, PARAMETER VARIABILITY, AND MIXTURES

Statistical mimicking issues involving reaction time measures are introduced and discussed in this article. Often, discussions of mimicking have concerned the question of the serial versus parallel processing of inputs to the cognitive system. We will demonstrate that there are several alternative structures that mimic various existing models in the literature. In particular, single-process models have been neglected in this area. When parameter variability is incorporated into single-process models, resulting in discrete or continuous mixtures of reaction time distributions, the observed reaction time distribution alone is no longer as useful in allowing inferences to be made about the architecture of the process that produced it. Many of the issues are raised explicitly in examination of four different case studies of mimicking. Rather than casting a shadow over the use of quantitative methods in testing models of cognitive processes, these examples emphasize the importance of examining reaction time data armed with the tools of quantitative analysis, the importance of collecting data from the context of specific process models, and the importance of expanding the database to include other dependent measures.

Time-dependent Poisson counter models of response latency in simple judgment

An important class of sequential-sampling models for response time (RT) assumes that evidence for competing response alternatives accrues in parallel and that a response is made when the evidence total for a particular response exceeds a criterion. One member of this class of models is the Poisson counter model, in which evidence accrues in unit increments and the waiting time between increments is exponentially distributed. This paper generalizes the counter model to allow the Poisson event rate to vary with time. General expressions are obtained for the RT distributions for the two- and the m-alternative cases. Closed-form expressions are obtained for response probabilities under a proportional-rates assumption and for mean RT under conditions in which the integrated event rate increases as an arbitrary power of time. An application in the area of early vision is described, in which the Poisson event rates are proportional to the outputs of sustained and transient channels.

An application of the poisson race model to confidence calibration.

In tasks as diverse as stock market predictions and jury deliberations, a persons feelings of confidence in the appropriateness of different choices often impact that persons final choice. The current study examines the mathematical modeling of confidence calibration in a simple dual-choice task. Experiments are motivated by an accumulator model, which proposes that information supporting each alternative accrues on separate counters. The observer responds in favor of whichever alternatives counter first hits a designated threshold. Confidence can then be scaled from the difference between the counters at the time that the observer makes a response. The authors examine the overconfidence result in general and present new findings dealing with the effect of response bias on confidence calibration.

A dynamic stimulus-driven model of signal detection

Signal detection theory forms the core of many current models of cognition, including memory, choice, and categorization. However, the classic signal detection model presumes the a priori existence of fixed stimulus representations--usually Gaussian distributions--even when the observer has no experience with the task. Furthermore, the classic signal detection model requires the observer to place a response criterion along the axis of stimulus strength, and without theoretical elaboration, this criterion is fixed and independent of the observers experience. We present a dynamic, adaptive model that addresses these 2 long-standing issues. Our model describes how the stimulus representation can develop from a rough subjective prior and thereby explains changes in signal detection performance over time. The model structure also provides a basis for the signal detection decision that does not require the placement of a criterion along the axis of stimulus strength. We present simulations of the model to examine its behavior and several experiments that provide data to test the model. We also fit the model to recognition memory data and discuss the role that feedback plays in establishing stimulus representations.

An Integrated Model of Discrete Choice and Response Time

Computer and web-based interviewing tools have made response times ubiquitous in marketing research. These data are used as an indicator of data quality by practitioners, and of latent processes related to memory, attributes and decision making by academics. We investigate a Poisson race model with choice and response times as dependent variables. The model facilitates inference about respondent preference for choice alternatives, their diligence in providing responses, and the accessibility of attitudes/the speed of thinking. Thus, the model distinguishes respondents who are quick to think versus those who react quickly but without much thought. Empirically, we find support for the endogenous nature of response times and demonstrate that models that treat response times as exogenous variables may result in misleading inferences.

Was it a car or a cat I saw? An Analysis of Response Times for Word Recognition

We model the response times for word recognition collected in experimental trials conducted on four subjects. Because of the sequential nature of the experiment and the fact that several replications of similar trials were conducted on each subject, the assumption of i.i.d. response times within experimental conditions (often encountered in the psychology literature) is untenable. We consider Bayesian hierarchical models in which the response times are described as conditionally independent Weibull random variables given the parameters of the Weibull distribution. The sequential dependencies, as well as the effects of response accuracy, word characteristics, and subject specific learning processes are incorporated via a linear regression model for the logarithm of the scale parameter of the Weibull distribution. We compare the inferences from our analysis with those obtained by means of instruments that are commonly used in the cognitive psychology arena. We pay close attention to the quality of the fits, the adequacy of the assumptions, and their impact on the inferential conclusions.