Similarity of rectangles

Krantz and Tversky found that neither (log-) height (y) and width (x), nor area (x + y) and shape (x ? y) qualify as “subjective dimensions of rectangles” because both pairs violate the decomposability condition for their dissimilarity data. However, the data suggest a nonlinear transformation of x, y into a pair of subjective dimensions u(x, y), v(x, y) for which decomposability should be approximately satisfied. An explicit statement of this mapping is given.     

Occasion setting is specific to the CS-US association

In Experiment 1, rats were trained on a discrimination in which one occasion setter, A, signaled that one cue (conditioned stimulus, CS), x, would be followed by one outcome, p (unconditioned stimulus, US), and a second CS, y, by a different outcome, q (x → p and y → q); a second occasion setter, B signalled the reverse CS-outcome relations (x → q and y → p). In a subsequent stage, the animals were divided into two groups, and trained as before, except that both A and B were presented in compound with a novel occasion setter, C. For Group S (same) the CS-outcome relations following A and B were identical to those in the pretraining stage, whereas in Group D (different) they were reversed. In a subsequent test, stimulus C was shown to be a more effective occasion setter in Group D than in Group S. In Experiment 2, rats were trained on a negative occasion-setting discrimination in which CS x signaled outcome p, and y outcome q; when x and y were signaled by the occasion setter A then no outcome followed (x → p, y → q, A:x-, and A:y-). In a subsequent stage, A was now trained as a positive occasion setter, signaling reinforcement of x and y. In Group S, x and y signaled the same outcomes as in the prior training stage (x-, y-, A:x → p, and A:y → q), whereas in Group D they signaled the opposite outcomes (x, y, A:x → q, and A:y → p); more efficient test performance was seen in the latter group. These results suggest that the each occasion setter conveyed information about the specific combination of CS and US paired in its presence (i.e., x → p and y → q, or x → no p and y → no q). These results are consistent with the suggestion that occasion setters operate, at least in part, on a specific CS-US association.     

Affine representations in psychophysics and the near-miss to Weber’s law

A theoretical account for the near-miss to Weber’s law in the form of a power function, with a special emphasis on the interpretation of the exponent, was proposed by Falmagne [Falmagne, J.-C. (1985). Elements of psychophysical theory. New York: Oxford University Press] within the framework of a subtractive representation, P(x,y)=F(u(x)−g(y)). In this paper, we examine a more general affine representation, P(x,y)=F(u(x)h(y)+g(y)). We first obtain a uniqueness theorem for the affine representation. We then study the conditions that force an affine representation to degenerate to a subtractive one. Part of that study involves the case for which two different affine representations co-exist for the same data. We also show that the balance condition P(x,y)+P(y,x)=1 constrains an affine representation to be a special kind of subtractive representation, a Fechnerian one. We further show that Falmagne’s power law takes on a special form for a so-called weakly balanced system of probabilities, in which case the affine representation is Fechnerian. Finally, following Iverson [Iverson, G.J. (2006a). Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences. Journal of Mathematical Psychology, 50, 271-282], we generalize the Fechner method to construct the sensory scales in a weakly balanced affine representation by integrating (derivatives of) just noticeable differences.     

Testing for selectivity in the dependence of random variables on external factors

Random variables A and B, whose joint distribution depends on factors (x,y), are selectively influenced by x and y, respectively, if A and B can be represented as functions of, respectively, (x,S_A,C) and (y,S_B,C), where S_A,S_B,C are stochastically independent and do not depend on (x,y). Selective influence implies selective dependence of marginal distributions on the respective factors: thus no parameter of A may depend on y. But parameters characterizing stochastic interdependence of A and B, such as their mixed moments, are generally functions of both x and y. We derive two simple necessary conditions for selective dependence of (A,B) on (x,y), which can be used to conduct a potential infinity of selectiveness tests. One condition is that, for any factor values x,x^′ and y,y^′,

s_xy≤s_xy^′+s_x^′y^′+s_x^′y,     

On minima of discrimination functions

A discrimination function ψ(x,y) assigns a measure of discriminability to stimulus pairs x,y (e.g., the probability with which they are judged to be different in a same-different judgment scheme). If for every x there is a single y least discriminable from x, then this y is called the point of subjective equality (PSE) for x, and the dependence h(x) of the PSE for x on x is called a PSE function. The PSE function g(y) is defined in a symmetrically opposite way. If the graphs of the two PSE functions coincide (i.e., g≡h⁻¹), the function is said to satisfy the Regular Minimality law. The minimum level functions are restrictions of ψ to the graphs of the PSE functions. The conjunction of two characteristics of ψ, (1) whether it complies with Regular Minimality, and (2) whether the minimum level functions are constant, has consequences for possible models of perceptual discrimination. By a series of simple theorems and counterexamples, we establish set-theoretic, topological, and analytic properties of ψ which allow one to relate to each other these two characteristics of ψ.     

The circumplex: A slightly stronger than ordinal approach

For some proximity matrices, multidimensional scaling yields a roughly circular configuration of the stimuli. Being not symmetric, a row-conditional matrix is not fit for such an analysis. However, suppose its proximities are all different within rows. Calling {{x,y},{x,z}} a conjoint pair of unordered pairs of stimuli, let {x,y}→{x,z} mean that row x shows a stronger proximity for {x,y} than for {x,z}. We have a cyclic permutation π of the set of stimuli characterize a subset of the conjoint pairs. If the arcs {x,y}→{x,z} between the pairs thus characterized are in a specific sense monotone with π, the matrix determines π uniquely, and is, in that sense, a circumplex with π as underlying cycle. In the strongest of the 3 circumplexes thus obtained, → has circular paths. We give examples of analyses of, in particular, conditional proximities by these concepts, and implications for the analysis of presumably circumplical proximities. Circumplexes whose underlying permutation is multi-cyclic are touched.     

Some counterexamples in multidimensional scaling

Given a set X with elements x, y,… which has a partial order < on the pairs of the Cartesian product X², one may seek a distance function ? on such pairs (x, y) which satisfies ?(x₁, y₁) < ?(x₂, y₂) precisely when (x₁, y₁) < (x₂, y₂), and even demand a metric space (X, ?) with some such compatible ? which has an isometric imbedding into a finite-dimensional Euclidean space or a separable Hilbert space. We exhibit here systems (X, <) which cannot meet the latter demand. The space of real m-tuples (ξ₁,…,ξ_m) with either the “city-block” norm Σ_i ∥ξ_i∥ or the “dominance” norm max_i, ∥ξ_i∥ cannot possibly become a subset of any finite-dimensional Euclidean space. The set of real sequences (ξ₁, ξ₂,…) with finitely many nonzero elements and the supremum norm sup_i, ∥ξ_i∥ cannot even become a subset of any separable Hilbert space.     

Resolution time and the coding of arithmetic relations on supraliminally different visual extents

Comparison time for pairs of vertical-line stimuli, sufficiently different that they can be errorlessly discriminated with respect to visual extent, was examined as a function of arithmetic relations (physical ratio and difference) on members of the pair. Arithmetic relations are coded very precisely by judgment time: Responses slow as stimulus ratios approach one with difference fixed, and as stimulus differences approach zero with ratio fixed. Most models which assume a simple (Difference or Ratio) resolution rule operating on independent sensations require judgment time to depend on either ratios or on differences but not on both. Further tests showed both an index based on median judgment times and a confusion index based on pairs of observed judgment times, satisfied the requirements for a Positive Difference Structure. One representation of these data, which remains acceptable through all analyses, is a Difference resolution rule operating on sensations determined by a power psychophysical function with β < 1. Specifically, L(x, y) = F{ψ(x) ? ψ(y)} + R, where L(x, y) is the judgment time with the stimulus pair x and y, ψ(x) = Ax^β + C, R is a positive constant, and F is a continuous monotone decreasing function.     

Psychophysics without physics: extension of Fechnerian scaling from continuous to discrete and discrete-continuous stimulus spaces

The computation of subjective (Fechnerian) distances from discrimination probabilities involves cumulation of appropriately transformed psychometric increments along smooth arcs (in continuous stimulus spaces) or chains of stimuli (in discrete spaces). In a space where any two stimuli that are each other's points of subjective equality are given identical physical labels, psychometric increments are positive differences ψ(x,y)-ψ(x,x) and ψ(y,x)-ψ(x,x), where x≠y and ψ is the probability of judging two stimuli different. In continuous stimulus spaces the appropriate monotone transformation of these increments (called overall psychometric transformation) is determined uniquely in the vicinity of zero, and its extension to larger values of its argument is immaterial. In discrete stimulus spaces, however, Fechnerian distances critically depend on this extension. We show that if overall psychometric transformation is assumed (A) to be the same for a sufficiently rich class of discrete stimulus spaces, (B) to ensure the validity of the Second Main Theorem of Fechnerian Scaling in this class of spaces, and (C) to agree in the vicinity of zero with one of the possible transformations in continuous spaces, then this transformation can only be identity. This result is generalized to the broad class of “discrete-continuous” stimulus spaces, of which continuous and discrete spaces are proper subclasses.     

A system for automatic recording and analysis of motor activity in rats

We describe the design and evaluation of an electronic system for the automatic recording of motor activity in rats. The device continually locates the position of a rat inside a transparent acrylic cube (50 cm/side) with infrared sensors arranged on its walls so as to correspond to the x-, y-, and z-axes. The system is governed by two microcontrollers. The raw data are saved in a text file within a secure digital memory card, and offline analyses are performed with a library of programs that automatically compute several parameters based on the sequence of coordinates and the time of occurrence of each movement. Four analyses can be made at specified time intervals: traveled distance (cm), movement speed (cm/s), time spent in vertical exploration (s), and thigmotaxis (%). In addition, three analyses are made for the total duration of the experiment: time spent at each x–y coordinate pair (min), time spent on vertical exploration at each x–y coordinate pair (s), and frequency distribution of vertical exploration episodes of distinct durations. User profiles of frequently analyzed parameters may be created and saved for future experimental analyses, thus obtaining a full set of analyses for a group of rats in a short time. The performance of the developed system was assessed by recording the spontaneous motor activity of six rats, while their behaviors were simultaneously videotaped for manual analysis by two trained observers. A high and significant correlation was found between the values measured by the electronic system and by the observers.     

Direction dependence analysis: A framework to test the direction of effects in linear models with an implementation in SPSS

In nonexperimental data, at least three possible explanations exist for the association of two variables x and y: (1) x is the cause of y, (2) y is the cause of x, or (3) an unmeasured confounder is present. Statistical tests that identify which of the three explanatory models fits best would be a useful adjunct to the use of theory alone. The present article introduces one such statistical method, direction dependence analysis (DDA), which assesses the relative plausibility of the three explanatory models on the basis of higher-moment information about the variables (i.e., skewness and kurtosis). DDA involves the evaluation of three properties of the data: (1) the observed distributions of the variables, (2) the residual distributions of the competing models, and (3) the independence properties of the predictors and residuals of the competing models. When the observed variables are nonnormally distributed, we show that DDA components can be used to uniquely identify each explanatory model. Statistical inference methods for model selection are presented, and macros to implement DDA in SPSS are provided. An empirical example is given to illustrate the approach. Conceptual and empirical considerations are discussed for best-practice applications in psychological data, and sample size recommendations based on previous simulation studies are provided.     

Spatial cuing in a stereoscopic display: Evidence for a “depth-blind” attentional spotlight

This experiment explored whether attentional selection observed in a spatial cuing task is based on a representation that includes depth information or not. Targets were presented inside placeholders appearing at the samex,y location on a stereoscopic display, but on different depth planes, or at differentx,y locations on the same depth plane. A peripheral precue produced significant cuing effects in the latter but not in the former condition. In a control experiment, significant cuing effects were found for targets appearing at differentx,y coordinates within the fovea, confirming that the lack of cuing effects in the depth condition was not due to foveal presentation. Together, the results suggest that spatial selection in spatial cuing tasks operates on a representation that does not include depth information.     

Positive-difference structures and bilinear utility functions

Let (M₁, f), (M₂, g) be mixture sets and let ? be a binary preference relation on M₁ × M₂. By using the concept of positive-difference structures, necessary and sufficient conditions are given for the existence of a real-valued utility function u on M₁ × M₂ which represents ? and possesses the bilinearity property

$\text{u(?(α, x}{}_1\text{,x}{}_2\text{),g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{αu(x}{}_1\text{, g(βy}{}_1\text{, y}{}_2\text{))}\text{+}\text{(1 ? α) u(x}{}_2\text{, g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{βu(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_1\text{)}\text{+}\text{(1 ? β) u(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_2\text{)}$

, for all α, β ∈ [0, 1], all x₁, x₂ ∈ M₁ and all y₁, y₂ ∈ M₂. Moreover, uniqueness up to positive linear transformations can be proved for those utility functions. Finally an outline is given of applications of these results in expected utility theory.     

Subtraction and the solution of open sentence problems

This paper is concerned with the processes used by children in solving open sentence problems of the form x + u = y (Type 1) and u + x = y (Type 2). Three models for reaction times to these problems are proposed. The first assumes they are solved by an incrementing process, the second assumes a decrementing process, while the third assumes that the subject increments or decrements, depending on which is quickest. Two experiments designed to evaluate these models are reported. It is shown, by means of a series of regression analyses that the third model gives the best account of the success latencies to Type 1 problems. This model predicts that times will be a linear function of the minimum of x and y ? x. It is also shown that none of the models give an adequate account of the latency data for Type 2 problems. Some possible reasons for this difference are discussed, together with some evidence that indicates that Type 1 problems and ordinary subtraction problems are solved by the same process.     

Scope Processing in Chinese

The standard view maintains that quantifier scope interpretation results from an interaction between different modules: the syntax, the semantics as well as the pragmatics. Thus, by examining the mechanism of quantifier scope interpretation, we will certainly gain some insight into how these different modules interact with one another. To observe it, two experiments, an offline judgment task and an eye-tracking experiment, were conducted to investigate the interpretation of doubly quantified sentences in Chinese, like Mei-ge qiangdao dou qiang-le yi-ge yinhang (Every robber robbed a bank). According to current literature, doubly quantified sentences in Chinese like the above are unambiguous, which can only be interpreted as ‘for every robber x, there is a bank y, such that x robbed y–surface scope reading), contrary to their ambiguous English counterparts, which also allow the interpretation that ‘there is a bank y, such that for every robber x, x robbed y–inverse scope reading). Specifically, three questions were examined, that is, (i) What is the initial reading of doubly quantified sentences in Chinese? (ii) Whether inverse scope interpretation can be available if appropriate contexts are provided? (iii) What are the processing time courses engaged in quantifier scope interpretation? The results showed that (i) Initially, the language processor computes the surface scope representation and the inverse scope representation in parallel, thus, doubly quantified sentences in Chinese are ambiguous; (ii) The discourse information is not employed in initial processing of relative scope, it serves to evaluate the two representations in reanalysis; (iii) The lexical information of verbs affects their scope-taking patterns. We suggest that these findings provide evidence for the Modular Model, one of the major contenders in the literature on sentence processing.     

On the scales of measurement

Let $\text{X}$ = 〈X, ≧, R₁, R₂…〉 be a relational structure, 〈$\text{X, ≧}$〉 be a Dedekind complete, totally ordered set, and n be a nonnegative integer. $\text{X}$ is said to satisfy n-point homogeneity if and only if for each x₁,…, x_n, y₁,…, y_n such that x₁ ? x₂ ? … ? x_n and y₁ ? y₂ … ? y_n, there exists an automorphism α of $\text{X}$ such that α(x₁) = y_i. $\text{X}$ is said to satisfy n-point uniqueness if and only if for all automorphisms β and γ of $\text{X}$, if β and γ agree at n distinct points of $\text{X}$, then β and γ are identical. It is shown that if $\text{X}$ satisfies n-point homogeneity and n-point uniqueness, then n ≦ 2, and for the case n = 1, $\text{X}$ is ratio scalable, and for the case n = 2, interval scalable. This result is very general and may in part provide an explanation of why so few scale types have arisen in science. The cases of 0-point homogeneity and infinite point homogeneity are also discussed.     

Nondecomposable conjoint measurement for bisymmetric structures

This paper discusses two “nondecomposable” conjoint measurement representations for an asymmetric binary relation ? on a product set A × X, namely (a, x) ? (b, y) iff f₁(a) + g₁(a)g₂(x) > f₁(b) + g₁(b)g₂(y), and (a, x) ? (b, y) iff f₁(a) + f₂(x) + g₁(a)g₂(x) > f₁(b) + f₂(y) + g₁(b)g₂(y). Difficulties in developing axioms for ? on A × X which imply these representations in a general formulation have led to their examination from the standpoint of bisymmetric structures based on applications of a binary operation to A × X. Depending on context, the binary operation may refer to concatenation, extensive or intensive averaging, gambles based on an uncertain chance event, or to some other interpretable process. Independence axioms which are necessary and sufficient for the special representations within the context of bisymmetric structures are presented.     

Completely nonmetric multidimensional scaling

Most of the distance models underlying multidimensional scaling assume that if a stimulus y is between the stimuli x and z on each dimension, then x and z should be the farthest apart of the three stimuli. An iterative algorithm is described that uses only this betweenness prediction to infer the ordering of a set of stimuli on each of one or two dimensions. Applied to previously published semantic similarity data, this algorithm produced two-dimensional configurations that were similar in appearance to Euclidean configurations but generally involved fewer violations of the betweenness prediction.     

The general psychophysical differential equation: A comparison of three specifications

The general psychophysical differential equation, dy/dx = W₂(y)/W₁(x), with the solution y = f(x), where x and y are subjective variables and W₁ and W₂ their subjective Weber functions, is (a) compared with a corresponding functional equation, and (b) studied from a stochastic point of view by error calculus, Methods for evaluating and handling divergences are proposed and illustrated for a number of combinations of Weber functions. It is shown that either the differential: and the functional equations have the same solution or the difference between the solutions is negligible compared to empirical scatter. The error calculus gives the same result: either no error at all or a negligible one.     

Interval representations for interval orders and semiorders

This paper discusses two types of real interval representations for interval orders and semiorders ? on a set X of arbitrary cardinality. In each type, each x in X is mapped into a real interval F(x). The first model is: x ? y iff a < b for all a in F(x) and all b in F(y). The second is: x ? y iff sup F(x) < infF(y). Necessary and sufficient countability conditions are presented for the second model for interval orders and for semiorders; simpler sets of these conditions are shown to be sufficient for the first model. Some special properties for the representations are noted, including two monotonicity properties for the semiorder representation.