David H. Krantz

Contemporary Developments in Mathematical Psychology

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The use of statistical heuristics in everyday inductive reasoning

In reasoning about everyday problems, people use statistical heuristics, that is, judgmental tools that are rough intuitive equivalents of statistical principles. Statistical heuristics have improved historically and they improve ontogenetically. Use of statistical heuristics is more likely when (a) the sample space and the sampling process are clear, (b) the role of chance in producing events is clear, or (c) the culture specifies statistical reasoning as normative for the events. Perhaps because statistical procedures are part of peoples intuitive equipment to begin with, training in statistics has a marked impact on reasoning. Training increases both the likelihood that people will take a statistical approach to a given problem and the quality of the statistical solutions. These empirical findings have important normative implications.

A theory of magnitude estimation and cross-modality matching ☆

Abstract The generalizations that have emerged from “ratio scaling” of sensory continua include: consistency among various methods, consistency with changes in modulus, and the power law. These generalizations pose difficulties for the commonly held view (called here mapping theory ) that “ratio scaling” judgments are mediatedby mappings of stimuli into sensations. The main difficulty is that the ratio-like consistency properties of cross-modality matching cannot be accounted for by essentially nonnumerical sensations, and hence must be treated by a process theory which assumes that cross-modality matching is mediated by magnitude estimation. An alternative process theory ( relation theory ) assumes that “ratio scaling” judgments are not mediated by a property of single stimuli (sensation) but rather by a property of pairs of stimuli. The perceived relations of pairs (called sensation “ratios”) are assumed to share a qualitative property of numerical ratios. This axiom leads to a measurement representation by ratios of psychophysical functions (sensation functions). These psychophysical functions can be obtained directly by magnitude estimation provided that mental estimation of length ratios and perceived sensation “ratios” of length pairs are ordered alike (in fact, both correspond roughly to physical length ratios). The assumed ordering of stimulus pairs by sensation “ratios” leads to a simple account of the various empirical consistency generalizations. Such an ordering may also be related to other pair orderings, such as those based on discriminability measures and on perceived sensation “differences.”

OPPONENT-PROCESS ADDITIVITY-I: RED/GREEN EQUILIBRIA'

A red/green equilibrium light is one which appears neither reddish nor greenish (i.e. either uniquely yellow, uniquely blue, or achromatic). A subset of spectral and nonspectral red/green equilibria was determined for several luminance levels, in order to test whether the set of all such equilibria is closed under linear color-mixture operations. The spectral loci ofequilibrium yellow and blue showedeither no variation or visually insignificant varia- tion over a range of l-2 log,, unit. There were no trends that were repeatable across observers. We con- cluded that spectral red/green equilibria are closed under scalar multiplication; consequently they are in- variant hues relative to the Bezold-Briicke shift. The additive mixture of yellow and blue equilibrium wavelengths, in any luminance ratio, is also an equilibrium light. Small changes of the yellowish component of a mixture toward redness or greeness must be compensated by predictable changes of the bluish component of the mixture toward greenness or red- ness. We concluded that yellow and blue equilibria are complementary relative to an equilibrium white; that desaturation of a yellow or blue equilibrium light with such a white produces no Abney hue shift; and that the set of red/green equilibria is closed under general linear operations. One consequence is that the red/green chromatic-response function, measured by the Jameson-Hurvich technique of cancellation to equilibrium, is a linear function of the individuals color-matching coor- dinates. A second consequence of linear closure of equilibria is a strong constraint on the class of combina- tion rules by which receptor outputs are recoded into the red/green opponent process.

Conditional Expected Utility

This chapter focuses on conditional expected utility. Unlike most theories of measurement, which can have both physical and behavioral interpretations, the theory of expected utility is devoted explicitly to the problem of making decisions when their consequences are uncertain. It is the most familiar example of a theory of measurement in the social sciences. This chapter illustrates the ideas underlying the theory. The basic entities of the theory are a set of uncertain alternatives and an individuals ordering of them according to his personal preferences. Each specific gamble prescribes a particular contingency between events and their consequences, but numerous other gambles can be constructed from the same chance events and consequences. The set of possible consequences can include many different types of things: the gain or loss of money, the receipt of commodities or commodity bundles, the presentation of emotional stimuli of various sorts, etc.

Opponent process additivity--II. Yellow/blue equilibria and nonlinear models

Abstract A yellow/blue equilibrium light is one which appears neither yellowish nor bluish (i.e. uniquely red, uniquely green, or achromatic). The spectral locus of the monochromatic greenish equilibrium (around 500 nm) shows little, if any, variation over a luminance range of 2 log10 units. Reddish equilibria are extraspectral, involving mixtures of short- and long-wave light. Their wavelength composition is noninvariant with luminance: a reddish equilibrium light turns bluish-red if luminance is increased with wavelength composition constant. The additive mixture of the reddish and greenish equilibria is again a yellow/blue equilibrium light. We conclude that yellow/blue equilibrium can be described as the zeroing of a nonlinear functional, which is, however, approximately linear in the short-wavelength (“blue”) and middle-wavelength (“green”) cone responses and nonlinear only in the long-wavelength (“red”) cone response. The “red” cones contribute to yellowness, but via a compressive function of luminance. This effect works against the direction of the Bezold-Brucke hue shift. The Jameson-Hurvich yellow/blue chromatic-response function is only approximately correct: the relative values of yellow/blue chromatic response for an equal energy spectrum must vary somewhat with the energy level.

The dimensional representation and the metric structure of similarity data

Abstract A set of ordinal assumptions, formulated in terms of a given multidimensional stimulus set, is shown to yield essentially unique additive difference measurement of dissimilarity, or psychological distance. According to this model, dissimilarity judgments between multidimensional objects are regarded as composed of two independent processes: an intradimensional subtractive process, and an interdimensional additive process. Although the additive difference measurement model generalizes traditional metric models, the conditions under which it satisfies the metric axims impose severe restrictions on the measurement scales. The implications of the results for the representation of similarity data by metric and/or dimensional models are discussed.

Similarity of rectangles: An analysis of subjective dimensions☆

Two defining properties of psychological dimensions (intradimensional subtractivity and interdimensional additivity) are introduced and their consequences, formulated in terms of an ordinal dissimilarity scale, are derived. These consequences are investigated using dissimilarity judgments between rectangles to determine which of two alternative dimensional structures area (A) and shape (S), or width (W) and height (H), satisfies additivity and/or subtractivity. The results show that neither dimensional structure is acceptable, although A x S provides a better account for the data of most Ss than does W x H. Tests of relative straightness show that A is the least “curved” of the four attributes. Methodological and substantive implications of the study are discussed. Much work in psychology is based on the assumption that stimuli (e.g., color patches, words, geometric figures) are perceived and evaluated in a dimensionally organized fashion. In speaking of hue, saturation, and brightness as dimensions of color space, or of potency as a dimension of semantic space, it is typically assumed that these dimensions serve as organizing principles in the perception and the evaluation of colors or words. Despite its intuitive appeal, the precise content of this assumption is far from clear, particularly because the notion of psychological dimension is used in the literature in several different senses. How, then, does one test the hypothesis that a particular variable (specified physically or inferred via some model) acts like a subjective dimension ? To answer this question, one needs a theory that specifies the formal properties of psychological dimensions-properties that are

The Poggendorff illusion: Amputations, rotations, and other perturbations·

Studies of the Poggendorff illusion (a transversal interrupted by parallel lines) showed that illusory effects increased linearly with increasing separation between the parallels, increased in inverse proportion to the tangent of the angle of intersection between transversal and parallels, decreased whenever line segments (other than a transversal segment) were omitted, decreasing to zero when the segment of a parallel forming the obtuse angle with the transversal was omitted, and varied systematically with the tilt of the whole display, approaching zero when the transversal was oriented in a horizontal or vertical position. Hypothesis: The Poggendorff illusion involves at least three kinds of effects on the perceived orientation of a segment: distortion by other segments (especially a segment intersecting at an obtuse angle), stability of vertical and Horizontal orientations, and assimilation towards vertical or horizontal.

Color Measurement and Color Theory: I. Representation Theorem for Grassmann Structures

Abstract For trichromatic color measurement, the empirically based structure consists of the set of colored lights, with its operations of additive mixture and scalar multiplication, and the binary relation of metameric matching. The representing numerical structure is a vector space. The important axioms are Grassmanns laws. The vector representation is constructed in a canonical or coordinate-free manner, mainly using Grassmanns additivity law. Trichromacy is used only to fix the dimensionality. Color theories attempt to get a more unique homomorphism by enriching the basic empirical structure with new empirical relations, subject to new axioms. Examples of such enriching relations include: discriminability or dissimilarity ordering of color pairs; dichromatic matching relations; and unidimensional matching relations, or codes. Representation theorems for the latter two examples are based on Grassmann-type laws also. The relationship between a Grassmann structure and its unidimensional Grassmann codes is modeled by the relationship between a vector space and its dual space of linear functionals. Dual spaces are used to clarify theorems relating to the three-pigment hypothesis and to reduction dichromacy.