A Thurstonian model for the dual pair (4IAX) discrimination method

In the dual pair method, the subject is presented with two stimuli in two pairs: One pair is composed of two samples of the same stimulus; the other pair is composed of two samples of different stimuli, one being the same as that in the identical pair. The task of the judge is to select the most different pair. The psychometric function for the dual pair method is derived and expressed in terms of a singly noncentral beta distribution. A table is provided that connects a measure of the degree of difference, d, to the probability of a correct response. This table assumes an unbiased observer and differencing decision rule. A table is provided to give an estimate of the variance of d , the experimental estimate of d. The power of the dual pair method is also investigated, and a formula to determine the sample size required to meet Type I and Type II error specifications is given. The dual pair method appears to be slightly less powerful than the duo-trio and the triangular methods. Experimental investigation is needed to explore the dual pair in applied research work.     

A markov model for discrimination learning

A theory for discrimination learning which incorporates the concept of an observing response is presented. The theory is developed in detail for experimental procedures in which two stimuli are employed and two responses are available to the subject. Applications of the model to cases involving probabilistic and nonprobabilistic schedules of reinforcement are considered; some predictions are derived and compared with experimental results.This research was supported by a grant from the National Science Foundation.     

A multivariate model for discrimination methods

We describe a multivariate model for a certain class of discrimination methods in this paper and discuss a multivariate Euclidean model for a particular method, the triangular method. The methods of interest involve the selection or grouping of stimuli drawn from two stimulus sets on the basis of attributes invoked by the subject. These methods are commonly used for estimation and hypothesis testing concerning possible differences between foods, beverages, odorants, tastants and visual stimuli.Mathematical formulation of the bivariate model for the triangular method is provided as well as extensive Monte Carlo results for up to 10-dimensional cases. The effect of correlation structure and variance inequality are discussed. Results from these methods (as probability of a correct response) are not monotonically related to the distance between the means of the stimulus sets from which the stimuli are drawn but depend in a particular way on dimensionality, correlation structure, and the relative orientation of the momentary sensory values in a multidimensional space. The importance of these results to the validity of these methods as currently employed is discussed and the possibility of developing a new approach to multidimensional scaling on the basis of this new theory is considered.     

A probabilistic model for the discrimination of visual number

This paper proposes a probabilistic model of how humans identify the number of dots within a briefly presented visual display. The model is an application of Thurstone’s law of comparative judgment, and it is assumed that the internal representation of numerosity consists of log-spaced random variables. The discrimination between any two different numerosities is consequently described as a function of max/rain, where max and rain are the larger and smaller numbers, respectively. The model was tested in two experiments in which the Weber fraction for numerosity, corresponding with the critical ratio of max and min, was found to have the value of 162. It was concluded that the classical span of subitizing numerosity is but a special case of the span of discrimination.     

A model for the time-order error in contrast discrimination

Trials in a temporal two-interval forced-choice discrimination experiment consist of two sequential intervals presenting stimuli that differ from one another as to magnitude along some continuum. The observer must report in which interval the stimulus had a larger magnitude. The standard difference model from signal detection theory analyses poses that order of presentation should not affect the results of the comparison, something known as the balance condition (J.-C. Falmagne, 1985, in Elements of Psychophysical Theory). But empirical data prove otherwise and consistently reveal what Fechner (1860/1966, in Elements of Psychophysics) called time-order errors, whereby the magnitude of the stimulus presented in one of the intervals is systematically underestimated relative to the other. Here we discuss sensory factors (temporary desensitization) and procedural glitches (short interstimulus or intertrial intervals and response bias) that might explain the time-order error, and we derive a formal model indicating how these factors make observed performance vary with presentation order despite a single underlying mechanism. Experimental results are also presented illustrating the conventional failure of the balance condition and testing the hypothesis that time-order errors result from contamination by the factors included in the model.     

A model for the time-order error in contrast discrimination

Trials in a temporal two-interval forced-choice discrimination experiment consist of two sequential intervals presenting stimuli that differ from one another as to magnitude along some continuum. The observer must report in which interval the stimulus had a larger magnitude. The standard difference model from signal detection theory analyses poses that order of presentation should not affect the results of the comparison, something known as the balance condition (J.-C. Falmagne, 1985, in Elements of Psychophysical Theory). But empirical data prove otherwise and consistently reveal what Fechner (1860/1966, in Elements of Psychophysics) called time-order errors, whereby the magnitude of the stimulus presented in one of the intervals is systematically underestimated relative to the other. Here we discuss sensory factors (temporary desensitization) and procedural glitches (short interstimulus or intertrial intervals and response bias) that might explain the time-order error, and we derive a formal model indicating how these factors make observed performance vary with presentation order despite a single underlying mechanism. Experimental results are also presented illustrating the conventional failure of the balance condition and testing the hypothesis that time-order errors result from contamination by the factors included in the model.     

Semiparametric Thurstonian Models for Recurrent Choices: A Bayesian Analysis

We develop semiparametric Bayesian Thurstonian models for analyzing repeated choice decisions involving multinomial, multivariate binary or multivariate ordinal data. Our modeling framework has multiple components that together yield considerable flexibility in modeling preference utilities, cross-sectional heterogeneity and parameter-driven dynamics. Each component of our model is specified semiparametrically using Dirichlet process (DP) priors. The utility (latent variable) component of our model allows the alternative-specific utility errors to semiparametrically deviate from a normal distribution. This generates a robust alternative to popular Thurstonian specifications that are based on underlying normally distributed latent variables. Our second component focuses on flexibly modeling cross-sectional heterogeneity. The semiparametric specification allows the heterogeneity distribution to mimic either a finite mixture distribution or a continuous distribution such as the normal, whichever is supported by the data. Thus, special features such as multimodality can be readily incorporated without the need to overtly search for the best heterogeneity specification across a series of models. Finally, we allow for parameter-driven dynamics using a semiparametric state-space approach. This specification adds to the literature on robust Kalman filters. The resulting framework is very general and integrates divergent strands of the literatures on flexible choice models, Bayesian nonparametrics and robust time series specifications. Given this generality, we show how several existing Thurstonian models can be obtained as special forms of our model. We describe Markov chain Monte Carlo methods for the inference of model parameters, report results from two simulation studies and apply the model to consumer choice data from a frequently purchased product category. The results from our simulations and application highlight the benefits of using our semiparametric approach.     

A diffusion model for discrimination of temporal numerosity

This paper presents a model for sequential discrimination between the temporal frequencies of two discrete events. The model results from adjoining a theory for continuous estimation of temporal numerosity to a mathematical representation of the response criterion. A simple set of assumptions is used to describe the time evolution of numerosity estimates by a stochastic diffusion process. The predictions of the model are extensively compared to a body of results presented in a previous report, with an emphasis on individual performances. It is shown that a number of response strategies, observed in actual experiments, can be predicted by generalizing the classical notion of a single-valued response threshold to a stochastic, time-varying process.     

A model of dual attitudes

When an attitude changes from A1 to A2, what happens to A1? Most theories assume, at least implicitly, that the new attitude replaces the former one. The authors argue that a new attitude can override, but not replace, the old one, resulting in dual attitudes. Dual attitudes are defined as different evaluations of the same attitude object: an automatic, implicit attitude and an explicit attitude. The attitude that people endorse depends on whether they have the cognitive capacity to retrieve the explicit attitude and whether this overrides their implicit attitude. A number of literatures consistent with these hypotheses are reviewed, and the implications of the dual-attitude model for attitude theory and measurement are discussed. For example, by including only explicit measures, previous studies may have exaggerated the ease with which people change their attitudes. Even if an explicit attitude changes, an implicit attitude can remain the same.     

A component model for stimulus variables in discrimination learning

A general function is derived describing the conditioning of a single stimulus component in a discriminative situation. This function, together with the combinatorial rules of statistical learning theory [5, 12], generates empirically testable formulas for learning of classical two-alternative discriminations, probabilistic discriminations, and discriminations based on the outcomes of preceding trials in partial reinforcement experiments.The researches on which this paper is based were facilitated by a grant from the National Science Foundation.     

Two-stage model for order discrimination

Ulrich (1987), exploring the shapes of psychometric functions obtained in the ternary-response paradigm, indicated several inconsistencies between observed and predicted relationships providing evidence against most popular models of temporal-order judgment. In this paper, a new model is presented. It assumes that there are two mechanisms involved in the order discrimination task: one is responsible for the recognition of whether or not the stimuli are successive, and the second is able to determine their order. The model was tested using Allan's (1975a, 1975b) and Ulrich's (1987) data; the model predictions were found to be consistent with the experimental results.     

Second guesses and the attention-switching model for successiveness discrimination

The present note demonstrates that Baron’s (1971) second-guess data are not inconsistent with the basic assumptions of Kristofferson’s attention-switching model.     

A discrete-trials/fixed-interval method of discrimination training

An efficient new method of discrimination training is described which has several advantages over older free-operant and discrete-trials procedures. Illustrative data on reversal learning in pigeons and goldfish are presented.     

A model for realism of confidence judgments: Implications for underconfidence in sensory discrimination

In a recent issue of this journal, Björkman, Juslin, and Winman (1993) presented a model of the calibration of subjective confidence judgments for sensory discrimination which they called “subjective distance theory.” They proposed that there was a robust underconfidence bias in such judgments, that the model predicted such a bias, and that two different models were needed for the calibration of subjective confidence for cognitive judgments and for sensory ones. This paper addresses issues they raised. It points out that they have not presented a new model, but rather a portion of a more general one, the “decision-variable partition model” originally proposed in Ferrell and McGoey (1980). This paper explores properties of the model and shows, contrary to Björkman, Juslin, and Winman’s hypotheses, that the model does not predict under-confidence, that the “hard-easy effect” can be observed with sensory discriminations, and that the model fits not only sensory, but also cognitive judgments.     

Likelihood ratio, optimal decision rules, and correct response probabilities in a signal detection theoretic, equal-variance Gaussian model of the observer in the 4IAX paradigm

This article provides a synthetic account of the likelihood ratio, optimal decision rules, and correct response probabilities in a signal detection theoretic model of the observer in the dual-pair comparison, or four-interval AX (4IAX), paradigm. The model assumes a static sampling process, resulting in four, equal-variance normally distributed (i.e., Gaussian) observations on each trial First, a likelihood ratio equation allowing for an arbitrary degree of correlation between observations is provided. Specific solutions for the cases of independent and highly correlated observations are then derived. It is shown that these solutions, and the associated decision rules, correspond to those provided independently in earlier publications. A modified 4IAX paradigm involving, as a standard, an additional stimulus (C) located medially between the A and the B stimuli is also considered. It is shown that the optimal (static, equal-variance, Gaussian) decision model for this paradigm is unaffected by correlation between observations and is equivalent to the standard 4IAX with highly correlated observations. Finally, we consider how, under the considered (static, equal-variance, Gaussian) model, the proportion of correct responses in the different versions of the 4IAX paradigm is related to d', and a solution for the case of independent observations is provided.