Two connectionist frameworks, GRAIN (J. L. McClelland, 1993) and brain-state-in-a-box (J. A. Anderson, 1991), and R. Ratcliff's (1978) diffusion model were evaluated using data from a signal detection task. Dependent variables included response probabilities, reaction times for correct and error responses, and shapes of reaction-time distributions. The diffusion model accounted for all aspects of the data, including error reaction times that had previously been a problem for all response-time models. The connectionist models accounted for many aspects of the data adequately, but each failed to a greater or lesser degree in important ways except for one model that was similar to the diffusion model. The findings advance the development of the diffusion model and show that the long tradition of reaction-time research and theory is a fertile domain for development and testing of connectionist assumptions about how decisions are generated over time.

Connectionist and diffusion models of reaction time.

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.

How to fit a response time distribution.

Response times are one of the most commonly collected dependent measures in empirical investigations of human cognition. Issues of response time analysis are becoming more important as models are developed that make explicit predictions about the shape of the response time distribution. This chapter addresses the general problem of response time analysis. First, I present the concept that response time data form a random sample from some distribution. I discuss how random variables are characterized, defining the density, distribution, survivor and hazard functions. I pay special attention to problems of estimation, including familiar issues involving central tendency (mean) and dispersion (variance) and the effects of outliers on these statistics. I also address parameter estimation in the context of specific models of the response time distribution. Methods for obtaining empirical estimates of the distribution, density and hazard functions are presented. Finally, in the last section of the chapter, I demonstrate and discuss the use of distributional analysis in testing processing models of cognition.


Keywords:

distribution;
estimation;
model testing;
modeling;
statistical procedures

Analysis of Response Time Distributions

A theory is presented that establishes a dominance hierarchy of potential distinctions (order relations) between two distributions. It is proposed that it is worthwhile for researchers to ascertain the strongest possible distinction, because all weaker distinctions are logically implied. Implications of the theory for hypothesis testing, theory construction, and scales of measurement are considered. Open problems for future research are outlined. There are many occasions in psychological research when experimenters are concerned with two or more treatments and their effects on experimental groups. In most instances, the measurements assessing the treatments are assumed to be on at least an ordinal scale. The measurements typically form a sample probability distribution. The investigator may then ask the question as to whether the two distributions differ from one another in some way. The way chosen is often through the testing of summary statistics, particularly sample means, but there are techniques that test entire distributions against one another. In this article I develop a theory that shows that some differences between the distributions are stronger than others, in that the stronger ones imply the weaker ones but not vice versa. In fact, a dominance hierarchy of distinctions between two arbitrary distributions is developed: A distinction may imply another, be implied by another, be equivalent to another, or none of these. The dominance hierarchy is put into the form of an implication graph. The power of the results is that they are, except for special cases to be specified later, general across distributions. That is, most of the relations do not depend on a particular type of distribution such as the normal distribution. Furthermore, as long as the measurements are on at least an ordinal scale, again with the exception of the special cases, they

/pdf/truth-and-consequences-of-ordinal-differences-in-statistical-1yqk5aru1p.pdf

Truth and consequences of ordinal differences in statistical distributions: toward a theory of hierarchical inference.

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.

/pdf/statistical-mimicking-of-reaction-time-data-single-process-3o4ci92f7f.pdf

Statistical mimicking of reaction time data: Single-process models, parameter variability, and mixtures.

A constrained quadratic spline is proposed as an estimator of the hazard function of a random variable. A maximum penalized likelihood procedure is used to fit the estimator to a sample of psychological response times. The results of a small simulation study suggest that, with a sample size of 500, the procedure may provide a reasonably precise estimate of the shape of a hazard function.

Bruce Bloxom

Papers

A Constrained Spline Estimator of a Hazard Function.