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.     

Multidimensional Fechnerian Scaling: Perceptual Separability

A new definition of the perceptual separability of stimulus dimensions is given in terms of discrimination probabilities. Omitting technical details, stimulus dimensions are considered separable if the following two conditions are met: (a) the probability of discriminating two sufficiently close stimuli is computable from the probabilities with which one discriminates the projections of these stimuli on the coordinate axes; (b) the psychometric differential for discriminating two sufficiently close stimuli that differ in one coordinate only does not depend on the value of their matched coordinates (the psychometric differential is the difference between the probability of discriminating a comparison stimulus from a reference stimulus and the probability with which the reference is discriminated from itself). Thus defined perceptual separability is analyzed within the framework of the regular variation version of multidimensional Fechnerian scaling. The result of this analysis is that the Fechnerian metric of a stimulus space with perceptually separable dimensions has the structure of a Minkowski power metric with respect to these dimensions. The exponent of this metric equals the psychometric order of the stimulus space, or 1, whichever is greater.     

The effects of response-reinforcer contingency on time allocation in the open field

The present experiment explores the effects of the response (1-sec occupancy of a target area in an open field)—reinforcer (intracranial stimulation) contingency on time allocation in the open field in rats. The probability of reinforcement given a response (X) and the probability of reinforcement given the absence of a response (Y) were varied randomly across sessions within a subject. The following (X, Y) values were utilized: (.05, 0), (.15, 0), (.25, 0), (.15, .05), and (.15, .15). The results of this experiment indicate that rate of acquisition of time allocation preference is uniformly rapid during all contingency treatments wherein Y = 0 and is negatively related to the value of Y when X = .15. The relationship between the asymptote of the time allocation acquisition function and the value of X (when Y = 0) is positively sloped and negatively accelerated, while the relationship between asymptote and the value of Y (when X = .15) is negatively sloped with zero acceleration. Proposed contingency metrics are evaluated.     

Sources and influence of perceptual variance: Comment on Dzhafarov's Regular Minimality Principle

Dzhafarov [(2002). Multidimensional Fechnerian scaling: Pairwise comparisons, regular minimality, and nonconstant self-similarity. Journal of Mathematical Psychology, 46, 583-608] claims that Regular Minimality (RM) is a fundamental property of “same-different” discrimination probabilities and supports his claim with some empirical evidence. The key feature of RM is that the mapping, h, between two observation areas based on minimum discrimination probability is invertible. Dzhafarov [(2003a). Thurstonian-type representations for “same-different” discriminations: Deterministic decisions and independent images. Journal of Mathematical Psychology, 47, 184-204; (2003b). Thurstonian-type representations for “same-different” discriminations: Probabilistic decisions and interdependent images. Journal of Mathematical Psychology, 47, 229-243] also demonstrates that well-behaved Thurstonian models of “same-different” judgments are incompatible with RM and Nonconstant Self-Similarity (NCSS). There is extensive empirical support for the latter. Stimulus and neural sources of perceptual noise are discussed and two points are made:

Point 1: Models that require discrimination probabilities for noisy stimuli to possess the property that h is invertible would be too restrictive.

Point 2: In the absence of stimulus noise, violations of RM may be so subtle that their detection would be unlikely.

     

A new definition of well-behaved discrimination functions

A discrimination function shows the probability or degree with which stimuli are discriminated from each other when presented in pairs. In a previous publication [Kujala, J.V., & Dzhafarov, E.N. (2008). On minima of discrimination functions. Journal of Mathematical Psychology, 52, 116–127] we introduced a condition under which the conformity of a discrimination function with the law of Regular Minimality (which says, essentially, that “being least discriminable from” is a symmetric relation) implies the constancy of the function’s minima (i.e., the same level of discriminability of every stimulus from the stimulus least discriminable from it). This condition, referred to as “well-behavedness,” turns out to be unnecessarily restrictive. In this note we give a significantly more general definition of well-behavedness, applicable to all Hausdorff arc-connected stimulus spaces. The definition employs the notion of the smallest transitively and topologically closed extension of a relation. We provide a transfinite-recursive construction for this notion and illustrate it by examples.     

The conjunction fallacy: a misunderstanding about conjunction?

It is easy to construct pairs of sentences X, Y that lead many people to ascribe higher probability to the conjunction X-and-Y than to the conjuncts X, Y. Whether an error is thereby committed depends on reasoners’ interpretation of the expressions “probability” and “and.” We report two experiments designed to clarify the normative status of typical responses to conjunction problems.     

Pseudocontingencies derived from categorically organized memory representations

Pseudocontingencies (PCs) allow for inferences about the contingency between two variables X and Y when the conditions for genuine contingency assessment are not met. Even when joint observations X _i and Y _i about the same reference objects i are not available or are detached in time or space, the correlation r(X _i ,Y _i ) is readily inferred from base rates. Inferred correlations are positive (negative) if X and Y base rates are skewed in the same (different) directions. Such PC inferences afford useful proxies for actually existing contingencies. While previous studies have focused on PCs due to environmental base rates, the present research highlights memory organization as a natural source of PC effects. When information about two attributes X and Y is represented in a hierarchically organized categorical memory code, as category-wise base rates p(X) and p(Y), the reconstruction of item-level information from category base rates will naturally produce PC effects. Three experiments support this contention. When the yes base rates of two respondents in four questionnaire subscales (categories) were correlated, recalled and predicted item-level responses were correlated in the same direction, even when the original responses to specific items within categories were correlated in the opposite direction.     

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.     

Correlations between rats' spatial location and intracranial stimulation administration affects rate of acquisition and asymptotic level of time allocation preference in the open field

The present experiment explores the effects of the response (1-sec occupancy of a target area in an open field)-reinforcer (intracranial stimulation) contingency on time allocation in the open field in rats. The probability of reinforcement given response (X) and the probability of reinforcement given nonresponse (Y) were varied randomly across sessions within a subject. The 21 contingency treatments explored included all possible combinations of values (0, .1, .2, .3, .4, .5) of X and Y such that X ≥ Y. The results indicate that rate of acquisition and asymptotic level of time allocation preference to the target area are negatively related to the value of Y (for any given value of X). Variations in X (for any given value of Y) were less effective. Evaluation of proposed contingency metrics revealed that the Weber fraction (X − Y)/X most closely approximates performance, and that the value of the difference detection threshold derived from the Weber fraction is a constant.     

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.     

Multidimensional Fechnerian Scaling: Pairwise Comparisons, Regular Minimality, and Nonconstant Self-Similarity

Stimuli presented pairwise for same-different judgments belong to two distinct observation areas (different time intervals and/or locations). To reflect this fact the underlying assumptions of multidimensional Fechnerian scaling (MDFS) have to be modified, the most important modification being the inclusion of the requirement that the discrimination probability functions possess regular minima. This means that the probability with which a fixed stimulus in one observation area (a reference) is discriminated from stimuli belonging to another observation area reaches its minimum when the two stimuli are identical (following, if necessary, an appropriate transformation of the stimulus measurements in one of the two observation areas). The remaining modifications of the underlying assumptions are rather straightforward, their main outcome being that each of the two observation areas has its own Fechnerian metric induced by a metric function obtained in accordance with the regular variation version of MDFS. It turns out that the regular minimality requirement, when combined with the empirical fact of nonconstant self-similarity (which means that the minimum level of the discrimination probability function for a fixed reference stimulus is generally different for different reference stimuli), imposes rigid constraints on the interdependence between discrimination probabilities and metric functions within each of the observation areas and on the interdependence between Fechnerian metrics and metric functions belonging to different observation areas. In particular, it turns out that the psychometric order of the stimulus space cannot exceed 1.     

The psychophysics of sugar concentration discrimination and contrast evaluation in bumblebees

The capacity to discriminate between choice options is crucial for a decision-maker to avoid unprofitable options. The physical properties of rewards are presumed to be represented on context-dependent, nonlinear cognitive scales that may systematically influence reward expectation and thus choice behavior. In this study, we investigated the discrimination performance of free-flying bumblebee workers (Bombus impatiens) in a choice between sucrose solutions with different concentrations. We conducted two-alternative free choice experiments on two B. impatiens colonies containing some electronically tagged bumblebees foraging at an array of computer-automated artificial flowers that recorded individual choices. We mimicked natural foraging conditions by allowing uncertainty in the probability of reward delivery while maintaining certainty in reward concentration. We used a Bayesian approach to fit psychometric functions, relating the strength of preference for the higher concentration option to the relative intensity of the presented stimuli. Psychometric analysis was performed on visitation data from individually marked bumblebees and pooled data from unmarked individuals. Bumblebees preferred the more concentrated sugar solutions at high stimulus intensities and showed no preference at low stimulus intensities. The obtained psychometric function is consistent with reward evaluation based on perceived concentration contrast between choices. We found no evidence that bumblebees reduce reward expectations upon experiencing non-rewarded visits. We compare psychometric function parameters between the bumblebee B. impatiens and the flower bat Glossophaga commissarisi and discuss the relevance of psychophysics for pollinator-exerted selection pressures on plants.     

The application of the first order system transfer function for fitting the 3-arm radial maze learning curve

A mathematical model is described based on the first order system transfer function in the form Y=B3∗exp(−B2∗(X−1))+B4∗(1−exp(−B2∗(X−1))), where X is the learning session number; Y is the quantity of errors, B2 is the learning rate, B3 is resistance to learning and B4 is ability to learn. The model is tested in a light-dark discrimination learning task in a 3-arm radial maze using Wistar and albino rats. The model provided good fits of experimental data under acquisition and reacquisition, and was able to detect strain differences among Wistar and albino rats. The model was compared to Rescorla-Wagner, and was found to be mutually complementary. Comparisons with Tulving’s logarithmic function and Valentine’s hyperbola and the arc cotangent functions are also provided. Our model is valid for fitting averaged group data, if averaging is applied to a subgroup of subjects possessing individual learning curves of an exponential shape.     

Multidimensional Fechnerian Scaling: Basics

Fechnerian scaling is a theory of how a certain (Fechnerian) metric can be computed in a continuous stimulus space of arbitrary dimensionality from the shapes of psychometric (discrimination probability) functions taken in small vicinities of stimuli at which these functions reach their minima. This theory is rigorously derived in this paper from three assumptions about psychometric functions: (1) that they are continuous and have single minima around which they increase in all directions; (2) that any two stimulus differences from these minimum points that correspond to equal rises in discrimination probabilities are comeasurable in the small (i.e., asymptotically proportional), with a continuous coefficient of proportionality; and (3) that oppositely directed stimulus differences from a minimum point that correspond to equal rises in discrimination probabilities are equal in the small. A Fechnerian metric derived from these assumptions is an internal (or generalized Finsler) metric whose indicatrices are asymptotically similar to the horizontal cross-sections of the psychometric functions made just above their minima. Copyright 2001 Academic Press.     

A relational differential outcomes effect: pigeons can classify outcomes as “good” and “better”

In a conditional discrimination each of two sample stimuli indicates which of two comparison stimuli is correct. When correct choice following each conditional stimulus is followed by a different outcome (one kind of food following one, a different kind of food following the other) it often facilitates acquisition and improves memory. In transfer designs, in which two different conditional discriminations are followed by the same two differential outcomes, outcome expectation can be shown to be sufficient for comparison choice. That is, the samples from one conditional discrimination are matched to comparisons from the other conditional discrimination based on the common outcomes alone. In the present study we asked if for pigeons the relative value of the differential outcomes (higher versus lower value) can serve as the basis for comparison choice, independent of other characteristics of the outcomes and of differential sample responding. That is, would different outcomes that could be described as “good” and “better” form two stimulus classes. For one conditional discrimination, the differential outcomes involved differential probability of reinforcement for choice of the correct comparison stimulus (0.80 vs. 0.20 for correct choice of the two comparisons, respectively). For the other conditional discrimination, the differential outcomes involved differential responding to the two comparison stimuli (5 pecks vs. 20 pecks to the correct comparisons, respectively). On test trials, when conditional stimuli from the two conditional discriminations were interchanged and the relative value of the differential outcomes could serve as the only basis for comparison choice, we found positive transfer. The results indicate that relational attributes of outcomes can serve as effective cues for comparison choice.     

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.     

Affective,but hardly effective: a reply to Gauvin and Rejeski (2001)

Background and purpose: This paper is a reply to Gauvin and Rejeski’s rebuttal (Psychol. Sport Exerc. 2 (2001) 73) of a previously published conceptual and methodological critique (Psychol. Sport Exerc. 2 (2001) 1) of the Exercise-induced Feeling Inventory (EFI; J. Sport Exerc. Psychol. 15 (1993) 403).Methods: Our responses focus on (a) issues regarding scientific debates, (b) the necessity of psychometric scrutiny, (c) the ongoing search for a definition of “exercise-induced feeling states,” (d) concerns regarding the underrepresentation of the intended domain of content of the EFI and its appropriate uses, (e) the implications of inductive and deductive scale development, (f) several methodological issues, (g) the value of a circumplex model for exercise psychology research, and (h) the compatibility of categorical and dimensional models of affect.Results and conclusions: We maintain that the most important issues raised in the original critique of the EFI, such as the definition, the demarcation, and the structure of its intended domain of content, were not addressed in Gauvin and Rejeski’s rejoinder and remain unclear. Researchers are urged to contemplate the theoretical bases and to scrutinize the psychometric data of the available measures before making their selection.     

Dissimilarity cumulation theory and subjective metrics

We present a new mathematical notion, dissimilarity function, and based on it, a radical extension of Fechnerian Scaling, a theory dealing with the computation of subjective distances from pairwise discrimination probabilities. The new theory is applicable to all possible stimulus spaces subject to the following two assumptions: (A) that discrimination probabilities satisfy the Regular Minimality law and (B) that the canonical psychometric increments of the first and second kind are dissimilarity functions. A dissimilarity function Dab for pairs of stimuli in a canonical representation is defined by the following properties: (1) a≠b?Dab>0; (2) Daa=0; (3) If and , then ; and (4) for any sequence {a_nX_nb_n}_n∈N, where X_n is a chain of stimuli, Da_nX_nb_n→0?Da_nb_n→0. The expression DaXb refers to the dissimilarity value cumulated along successive links of the chain aXb. The subjective (Fechnerian) distance between a and b is defined as the infimum of DaXb+DbYa across all possible chains X and Y inserted between a and b.     

Do people reason rationally about causally related events? Markov violations,weak inferences,and failures of explaining away

Making judgments by relying on beliefs about the causal relationships between events is a fundamental capacity of everyday cognition. In the last decade, Causal Bayesian Networks have been proposed as a framework for modeling causal reasoning. Two experiments were conducted to provide comprehensive data sets with which to evaluate a variety of different types of judgments in comparison to the standard Bayesian networks calculations. Participants were introduced to a fictional system of three events and observed a set of learning trials that instantiated the multivariate distribution relating the three variables. We tested inferences on chains X₁ → Y → X₂, common cause structures X₁ ← Y → X₂, and common effect structures X₁ → Y ← X₂, on binary and numerical variables, and with high and intermediate causal strengths. We tested transitive inferences, inferences when one variable is irrelevant because it is blocked by an intervening variable (Markov Assumption), inferences from two variables to a middle variable, and inferences about the presence of one cause when the alternative cause was known to have occurred (the normative “explaining away” pattern). Compared to the normative account, in general, when the judgments should change, they change in the normative direction. However, we also discuss a few persistent violations of the standard normative model. In addition, we evaluate the relative success of 12 theoretical explanations for these deviations.     

Verification of algebra step problems: A chronometric study of human problem solving

A class of simple problem solving tasks requiring fast accurate solutions is introduced. In an experiment subjects memorized a mapping rule represented by lists of words labeled by cue words and made true/false decisions about conjunctions of propositions of the form “Y is in the list labeled by X”, written “X → Y”. Response times are analyzed using a “stage modeling” technique where problem solving algorithms are composed using a small set of psychological operations that have real time characteristics specified parametrically. The theoretical analysis shows that response time performance is adequately described in terms of the sequential application of elementary psychological operations. Unexpectedly, it was found that the proposition “X → YandX → Z” was verified as quickly as the apparently simpler “X → Y”. A case is presented for the modeling technique as applied to memory and problem solving tasks in terms of theoretical parsimony, statistical simplicity, and flexibility in investigative empirical research. Suggestions are made as to possible theoretical relations among fast problem solving, more complex and slower problem solving, and research in fundamental memory processes.