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Human Psychophysical Functions, an Update: Methods for Identifying their form; Estimating their Parameters; and Evaluating the Effects of Important Predictors

Published online by Cambridge University Press:  01 January 2025

Diana E. Kornbrot
Diana E. Kornbrot*
Affiliation:
University of Hertfordshire
*
Correspondence should be sent to Diana E. Kornbrot, Department of Psychology, University of Hertfordshire College Lane Hatfield, Hertfordshire, AL10 9AB, UK. Email: d.e.kornbrot@herts.ac.uk
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Abstract

Stevens’ power law for the judgments of sensation has a long history in psychology and is used in many psychophysical investigations of the effects of predictors such as group or condition. Stevens’ formulation Ψ=aPn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varPsi = {aP}^{n}$$\end{document}, where Ψ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varPsi $$\end{document} is the psychological judgment, P is the physical intensity, and n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n$$\end{document} is the power law exponent, is usually tested by plotting log (Ψ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\varPsi )$$\end{document} against log (P). In some, but by no means all, studies, effects on the scale parameter, a\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a$$\end{document}, are also investigated. This two-parameter model is simple but known to be flawed, for at least some modalities. Specifically, three-parameter functions that include a threshold parameter produce a better fit for many data sets. In addition, direct non-linear computation of power laws often fit better than regressions of log-transformed variables. However, such potentially flawed methods continue to be used because of assumptions that the approximations are “close enough” as to not to make any difference to the conclusions drawn (or possibly through ignorance the errors in these assumptions). We investigate two modalities in detail: duration and roughness. We show that a three-parameter power law is the best fitting of several plausible models. Comparison between this model and the prevalent two parameter version of Stevens’ power law shows significant differences for the parameter estimates with at least medium effect sizes for duration.

Keywords

magnitude estimationmagnitude productionStevens’ power lawdurationroughnessindividual differences
Type
Original paper
Information
Psychometrika , Volume 81 , Issue 1 , March 2016 , pp. 201 - 216
Copyright
Copyright © 2014 The Psychometric Society

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Footnotes

Electronic supplementary material The online version of this article (doi:https://doi.org/10.1007/s11336-014-9418-9) contains supplementary material, which is available to authorized users.

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