Evaluating a model of global psychophysical judgments for brightness: II. Behavioral properties linking summations and productions

Steingrimsson (Attention, Perception, & Psychophysics, 71, 1916–1930, 2009) outlined Luce’s (Psychological Review, 109, 520–532 2002, 111, 446–454 2004) proposed psychophysical theory and tested, for brightness, behavioral properties that, separately, gave rise to two psychophysical functions, Ψ⊕ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Psi_{{ \circ_p}}}$$\end{document}. The function Ψ⊕ maps pairs of physical intensities onto positive real numbers and represents subjective summation, and the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Psi_{{ \circ_p}}}$$\end{document} represents a form of ratio production. This article, the second in a series expected to consist of three articles, tests the properties linking summation and production such that it forces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Psi_{{ \circ_p}}} = {\Psi_\oplus } = \Psi$$\end{document}. The properties tested are a form of distributivity and, in three experiments, were subjected to an empirical evaluation. Considerable support is provided for the existence of a single function Ψ for both summation and ratio production. The scope of this series of articles is to establish the theory as a descriptive model of binocular brightness perception

* Manuscript Empirical Evaluation of a Model of Global Psychophysical Judgments: IV. Forms for the Weighting Function

Understanding the psychological interpretation of numbers is of both practical and theoretical interest. In classical magnitude estimation, respondents match numbers to sensations and in magnitude production they select tones that stand in a prescribed numerical ratio to a given tone. The present work focusses on evaluating several possible, and related, forms for this numerical distortion function, or weighting function, W. The main form, of which a power function is a special case, is the Prelec exponential/power representation. Behavioral equivalents to power and to Prelec functions are described, tested, and rejected. Thus, either the mathematical form or the assumption of W (1) = 1 is wrong. Whereas, the axiomatic literature has focussed exclusively on the former inference, we explore the possibility that W (1) � = 1. Behavioral axioms are formulated in each case and experimentally tested. We conclude that most respondents satisfy a general power function and that those who do not, appear to satisfy the more general Prelec function. Key words: audition; power functions; multiplicative form; numerical distortion; psychophysical regression effect; Prelec functions; reduction invariance; ratio production and ratio estimation; weighting function; sequential effects

File: SL-1-6c.tex

The research presented is a partial empirical evaluation of the second author’s proposed psychophysical theory (Luce, 2002, 2003a,b). The theory deals with the global percept of subjective intensity in which there is a psychophysical function Ψ that maps pairs of physical intensities onto the positive real numbers and represents subjective summation and a form of ratio production. A number of behavioral properties have been shown to follow from these specific representations, and in the presence of structural conditions these properties are also sufficient for the representations. In four auditory experiments, key behavioral properties of summation over the two ears and a form of generalized ratio production are evaluated empirically. Despite a number of methodological issues, considerable support is found for particular forms of Ψ for summations and ratio productions separately. Steingrimsson and Luce (2003a) explore the behavioral properties that link summations and productions; Steingrimsson and Luce (2003b) addresses issues regarding the form of the psychophysical and weighting functions

Evaluating a model of global psychophysical judgments: III. Invariance and forms for the psychophysical and weighting functions

Steingrimsson and Luce (2003a) outlined the second author’s proposed psychophysical theory (Luce, 2002, 2003a,b) and tested behavioral attributes that, separately, gave rise to one psychophysical function, Ψ, that maps pairs of physical intensities onto the positive real numbers that represents subjective summation and a second one, Ψ ∗,that represents a form of ratio production. This article takes up properties linking summation and production in such a way as to force Ψ ∗ = Ψ. The properties are evaluated in a series of experiments. The testing strategy is carried out in the auditory domain and concerns the subjective perception of loudness. Considerable support is provided for theoretical forms of Ψ for both summations and ratio productions. 1 This article is based, in part, on the first author’s Ph. D. dissertation (Steingrimsson