Daniel M. Ennis

A multidimensional stochastic theory of similiarity

A multidimensional theory of similarity in which the mental representations of stimulus objects are assumed to be drawn from multivariate normal distributions is described. A distance-based similarity function is defined and the expected value of similarity is derived. This theory is the basis for a possible explanation of paradoxical results with highly similar stimuli regarding the form of the similarity function and the distance metric. A stochastic approach to multidimensional scaling based on samedifferent judgments is demonstrated using artificial and real data sets. The theory of similarity presented is used as a basis for a Thurstonian extension of Shepard’s model of identification performance.

A multivariate model for discrimination methods

Abstract 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 SIMPLE MULTIVARIATE PROBABILISTIC MODEL FOR PREFERENTIAL AND TRIADIC CHOICES

Multidimensional probabilistic models of behavior following similarity and choice judgements have proven to be useful in representing multidimensional percepts in Euclidean and non-Euclidean spaces. With few exceptions, these models are generally computationally intense because they often require numerical work with multiple integrals. This paper focuses attention on a particularly general triad and preferential choice model previously requiring the numerical evaluation of a 2n-fold integral, wheren is the number of elements in the vectors representing the psychological magnitudes. Transforming this model to an indefinite quadratic form leads to a single integral. The significance of this form to multidimensional scaling and computational efficiency is discussed.

Mathematical Formulation of Multivariate Euclidean Models for Discrimination Methods.

Multivariate models for the triangular and duo-trio methods are described in this paper. In both cases, the mathematical formulation of Euclidean models for these methods is derived and evaluated for the bivariate case using numerical quadrature. Theoretical results are compared with those obtained using Monte Carlo simulation which is validated by comparison with previously published theoretical results for univariate models of these methods. This work is discussed in light of its importance to the development of a new theory for multidimensional scaling in which the traditional assumption can be eliminated that proximity measures and perceptual distances are monotonically related.

Replicated difference and preference tests: how to account for inter-trial variation☆

Abstract Binomial tests are often used in sensory difference and preference testing. Two assumptions underlie this use: (1) responses are independent and (2) choice probabilities do not vary from trial to trial. In many applications, the latter assumption is violated. In this paper we account for variation in inter-trial choice probabilities using the beta distribution. The result of combining the binomial with the beta distribution is a compound distribution known as the beta-binomial. We show how to use the beta-binomial model for replicated difference and preference tests such as those used to support product claims.

Probabilistic psychophysics with noisy stimuli

Abstract Mental representations of physical objects may vary from moment to moment. This occurs because of stimulus variation prior to the initiation of a neural signal and also because of neural variation in the neural mechanism. A psychophysical transformation function formalizes the connection between stimulus measures and mental representations, and this function will be assumed to be a one-to-one function (monotonic in one dimension). A very general equation is derived for the probability density function (pdf) of the momentary psychological magnitudes given any particular positive stimulus pdf, any one-to-one psychophysical transformation function and any neural pdf. Special cases of this general model are applied to the method of paired comparisons. These models for paired comparisons are derived on the basis of assumed lognormally distributed stimulus values on a stimulus continuum, a log or power psychophysical transformation and added neural normally distributed noise. The parameters of sample problems are estimated using maximum likelihood, nonlinear least squares and minimum chi-square criteria. These methods are compared with respect to bias in and the variance of the estimators.

The Integration of the Multivariate Normal Density Function for the Triangular Method

The model involved in the triangular method is presented, which leads to the need for evaluating a multidimensional integral of the multidimensional normal density function over an irregular region. Work done on the numerical evaluation of this integral is discussed.

Noncentral and central chi-square, f and beta distribution functions as special cases of the distribution function of an indefinite quadratic form

The distribution functions of central and noncentral chi-square, F and β random variables are expressed as special cases of the distribution function of an indefinite quadratic form. Some of these distribution functions, particularly the doubly noncentral cases, have traditionally involved fairly complicated expressions. The form presented in this paper is computationally straightforward and is attractive because it presents these related distributions as special cases of a single equation.

The relative sensitivities of same-different and identification judgment models to perceptual dependence

Probabilistic models of same-different and identification judgments are compared (within each paradigm) with regard to their sensitivity to perceptual dependence or the degree to which the underlying psychological dimensions are correlated. Three same-different judgment models are compared. One is a step function or decision bound model and the other two are probabilistic variants of a similarity model proposed by Shepard. Three types of identification models are compared: decision bound models, a probabilistic multidimensional scaling model, and probabilistic models based on the Shepard-Luce choice rule. The decision bound models were found to be most sensitive to perceptual dependence, especially when there is considerable distributional overlap. The same-different model based on the city-block metric and an exponential decay similarity function, and the corresponding identification model were found to be particularly insensitive to perceptual dependence. These results suggest that if a Shepard-type similarity function accurately describes behavior, then under typical experimental conditions it should be difficult to see the effects of perceptual dependence. This result provides strong support for a perceptualindependence assumption when using these models. These theoretical results may also play an important role in studying different decision rules employed at different stages of identification training.

Hypothesis Testing for Equivalence Defined on Symmetric Open Intervals

There are many industrial applications for which it is desirable to know whether one product can act as a substitute for another. Examples include product modifications when ingredients change, substitution of generic drugs for brand-name drugs, and modifications of products in response to government regulations. In addition, some companies develop products that are direct substitutes for those of their competitors and make advertising claims concerning their equivalency. Using an open interval within which to define equivalence, exact and approximate methods for testing a null hypothesis of non equivalence are given. In each case, examples are provided. Comparisons are made between these novel methods and existing methods, including the “two one-sided tests” (TOST) method.