Douglas Vickers

Evidence for an Accumulator Model of Psychophysical Discrimination

Recent theoretical approaches to the problem of psychophysical discrimination have produced what may be classified as ‘ statistical decision ’ or ‘ data accumulation ’ models. While the former have received much attention their application to judgment and choice meets with some difficulties. Among the latter, the two types which have received most attention are a ‘ runs ’ and a ‘ recruitment ’ model, but neither seems able to account for all of the relevant data. It is suggested instead that an ‘ accumulator ’ model, in which sampled events may vary in magnitude as well as probability, can be developed to give a good account of much of the available data on psychophysical discrimination. Two experiments are reported, in which the subject presses one of two keys as soon as he has decided whether the longer of two simultaneously presented lines is on the left or right. Results are found to be inconsistent with a runs or recruitment process, but to accord well with predictions from the accumulator model. Oth...

Perceptual Indices of Performance: The Measurement of ‘Inspection Time’ and ‘Noise’ in the Visual System

Indices of human performance are less clear-cut than analogous measures in the physical sciences for several reasons. Variables which can alter performance include the noise upon which sensory signals are imposed, the bias shown by an observer towards one or another alternative, the use of an optional-stopping strategy for examining sensory input, and the accumulation of statistical information over time concerning the source of the input. The development of a theoretical appreciation of these variables is traced and, on this basis, a measure of the noise, which sets a limit to discriminative capacity, is suggested. The proposed index is simply the standard deviation of the best fitting normal ogive, calculated for the psychometric function obtained in a forced-choice discrimination task by the method of constant stimuli, the discriminanda being presented for 100 ms, in random order, and followed by appropriate backward masking. The index is thus closely related to traditional psychophysical measures, but is distinguished by the detailed specification of conditions under which it should be obtained. Analysis of data from previous experiments gives some indication of the order of magnitude that might be expected from a carefully controlled determination of the measure. In addition, three experiments were carried out to evaluate this suggestion, and to test its underlying rationale. In the first two, observers were required to discriminate, by pressing one of two keys, between two lines of markedly different length, exposed in random order for ten different durations. In the third, stimulus exposure was held constant, while ten different stimulus differences were presented in random order. Results from the first two experiments yielded estimates of minimum inspection time close to 100 ms, and were inconsistent with the view that observers can abandon an optional-stopping procedure of processing sensory information in favour of responding by a deadline. Measures of noise calculated in the third experiment were of the order expected on the proposed rationale, while response latencies were again inconsistent with the notion of deadline responding. Further analysis of the results of the three experiments suggests that measures of inspection time and noise, together with a third index related to the degree of caution exercised by an observer, appear to be stable and consistent descriptors of performance. The wider implications of the successful use of this kind of perceptual index of performance are discussed with reference to the measurement of visual acuity, as a means for detecting effects of environmental stress, and as a conceptual framework for the understanding of individual differences.

The accumulator model of two-choice discrimination

Abstract The accumulator model of two-choice discrimination conceives the decision process as a race between competing evidence totals in discrete time and continuous state space. General expressions for the terminating probabilities for the model are derived, and a tractable version considered in which the increment distributions are exponential. Expressions for response probabilities, first passage time distributions, and a “balance of evidence” theory of response confidence are presented, and the resulting model compared to that obtained under the more usual assumption of truncated normal increments. The exponential model is fitted to data from an experiment exhibiting both faster and slower mean error times across a range of discriminability levels, and is shown to satisfactorily account for these when augmented with the assumption of negatively correlated within-condition variation in discriminability and decision criterion. Cross-paradigmatic support for the fitted model is provided by an estimate of sampling time of the order of 100 msec, in agreement with estimates obtained from a technique employing brief stimulus exposures.

The rationale for the inspection time index

Abstract The inspection time (IT) index proposed and tested by Vickers, Nettelbeck and Willson [Perception1, 263–295 (1972)] imposed a number of experimental constraints in order to measure the time required by the S to make a single observation of the sensory input, the constraints in question being determined by certain general assumptions regarding the decision process mediating the Ss responses. These constraints are re-examined in the light of recent experimental evidence which necessitates some modifications to the underlying conceptualization, particularly with reference to the assumptions concerning the sensory classification of observations and the factors influencing performance change. Implications for the measurement and interpretation of the IT index are then considered.

Effects of alternating set for speed or accuracy on response time, accuracy and confidence in a unidimensional discrimination task ☆

Abstract Three approaches to a theoretical analysis of confidence judgments are considered: one linking confidence to the number of sensory observations, one based on a distinction between ‘state’ and ‘process’ factors, and a ‘balance of evidence’ hypothesis developed from an accumulator model of discrimination. An experiment is described in which observers were asked to decide which of two parallel lines was the longer, and to rate their confidence in each judgment. Each observers set for speed or accuracy was manipulated over successive blocks of trials, and effects on response time, accuracy, and confidence were examined. Under an accuracy set, observers produced a higher percentage of correct responses, had longer response times, and made more confident judgments than under a set for speed. Within both speed and accuracy blocks, however, confidence ratings were inversely related to response time. The data on response accuracy, time, and confidence indicate certain deficiencies in either of the first two approaches, but were well accounted for by the third.

Dynamic Models of Simple Judgments: I. Properties of a Self-Regulating Accumulator Module

This is the first of two papers comparing connectionist and traditional stochastic latency mechanisms with respect to their ability to account for simple judgments. In this paper, we show how the need to account for additional features of judgment has led to the formulation of progressively more sophisticated models. One of these, a self-regulating, generalized accumulator process, is treated in detail, and its simulated performance across a sample of tasks is described. Since an adaptive decision module of this kind possesses all the ingredients of intelligent behavior, it is eminently suited as a basic computing element in more complex networks.

Discriminating between the frequency of occurrence of two alternative events

Abstract An experiment was carried out in which subjects were asked to decide which of two regularly flashing lamps, one red and one blue, was set to come on more frequently. The results were compared with the predictions of a random walk, a runs, and a recruitment model of the decision process, and significant differences found in all three cases. Three alternative models are examined: a random walk with diminishing criterion; a ‘variable recruitment’ model; and an ‘accumulator’ process, in which not only the probability but also the magnitude of the alternative events is supposed to vary. It is concluded that the accumulator process shows the greatest potential for describing the data.

Human Performance on Visually Presented Traveling Salesperson Problems with Varying Numbers of Nodes.

We investigated the properties of the distribution of human solution times for Traveling Salesperson Problems (TSPs) with increasing numbers of nodes. New experimental data are presented that measure solution times for carefully chosen representative problems with 10, 20, . . . 120 nodes. We compared the solution times predicted by the convex hull procedure proposed by MacGregor and Ormerod (1996), the hierarchical approach of Graham, Joshi, and Pizlo (2000), and by five algorithms drawn from the artificial intelligence and operations research literature. The most likely polynomial model for describing the relationship between mean solution time and the size of a TSP is linear or near-linear over the range of problem sizes tested, supporting the earlier finding of Graham et al. (2000). We argue the properties of the solution time distributions place strong constraints on the development of detailed models of human performance for TSPs, and provide some evaluation of previously proposed models in light of our findings.

The roles of the convex hull and the number of potential intersections in performance on visually presented traveling salesperson problems

The planar Euclidean version of the traveling salesperson problem requires finding the shortest tour through a two-dimensional array of points. MacGregor and Ormerod (1996) have suggested that people solve such problems by using a global-to-local perceptual organizing process based on the convex hull of the array. We review evidence for and against this idea, before considering an alternative, local-to-global perceptual process, based on the rapid automatic identification of nearest neighbors. We compare these approaches in an experiment in which the effects of number of convex hull points and number of potential intersections on solution performance are measured. Performance worsened with more points on the convex hull and with fewer potential intersections. A measure of response uncertainty was unaffected by the number of convex hull points but increased with fewer potential intersections. We discuss a possible interpretation of these results in terms of a hierarchical solution process based on linking nearest neighbor clusters.

Experimental paradigms emphasising state or process limitations. I: Effects on speed-accuracy tradeoffs

The distinction between E (experimenter-controlled) and S (subject-controlled) experimental paradigms is related to the macro- and the micro-tradeoff between speed andaccuracy, and to the distinction between state and process (or resource and data) limitations. A discrimination task is described, involving E and S conditions, which respectively emphaise state or process limitations on the quantity or quality of information utilised by the subject. Results from both conditions show a clear difference between macro- and micro-tradeoffs, are inconsistent with a number of fixed- and variable-sample decision models, but can be explained in terms of an accumalator process. On this basis, a taxonomy of performance functions is proposed, and it is concluded that differences between macro- and macro-tradeoffs and distinctions between state and process limitations, can be understood only in terms of a detailed model of how information is processed by the subject.