Predictions from a model of global psychophysics about differences between perceptual and physical matches

Abstract

A well-known phenomenon is that "matched" successive signals do not result in physical identity. This phenomenon has mostly been studied in terms of how much the second of two signals varies from the first, which is called the time-order error (TOE). Here, theoretical predictions led us to study the more general question of how much the matching signal differs from the standard signal, independent of the position of the matching signal as the first or second in a presentation. This we call non-equal matches (NEM). Using Luce's (Psychological Review, 109, 520-532, 2002, Psychological Review, 111, 446-454, 2004, Psychological Review, 115, 601, 2008, Psychological Review, 119, 373-387, 2012 global psychophysical theory, we predicted NEM when an intensity z is perceived to be "1 times a standard signal x." The theory predicts two different types of individual behaviors for the NEM, and these predictions were evaluated and confirmed in an experiment. We showed that the traditional definition of TOE precludes the observation, and thus the study, of the NEM phenomenon, and that the NEM effect is substantial enough to alter conclusions based on data that it affects. Furthermore, we demonstrated that the custom of averaging data over individuals clearly leads to quite misleading results. An important parameter in this modeling is a reference point that plays a central role in creating variability in the data, so that the key to obtaining regular data from respondents is to stabilize the reference point.Keywords Audition . Loudness . Weighting function . 2IFC . 2AFC . Matching . Method of adjustment . Non-equal matches . Time-order error . Mathematical modelingThe time-order error, or TOE, is reported by Stevens (1975) as a "constant error discovered long ago by Fechner . . .[where] on average the second of two equal stimuli tends to be judged greater than the first" (p. 139). For additional remarks on the TOE, see Appendix A.Hellström (2003) lamented how relatively little work had been carried out on the TOE phenomenon. Indeed, we were quite surprised to realize how dramatic these effects can be, and so we second Hellström's (2003) sentiment, adding that these effects are probably only ignored at the peril of arriving at erroneous conclusions. We came to realize that both the definition of the TOE and the prevailing practice of averaging over individual data has precluded the study of a much more fundamental phenomenon: When a respondent provides a perceptual "match" between two successively presented signals differing in intensity only, in general, the signals are not physically identical. We call this general phenomenon non-equal matches (NEM).Let x denote the signal that is presented as a standard that is to be matched, and let z denote the matching signal. Any respondent-selected signal is written in boldface (to emphasize that it is a random variable). With any method of matching, the standard x can be in either the temporally first or second interval. When it is first and z is presented second, we write ...

Empirical and Theoretical Studies of Psychophysical Phenomena

“…Special cases include the time-order error (TOE) in audition and the spaceorder error in vision. Steingrimsson and Luce (2012a) collected extensive data on NEM using a certain version of the method of adjustment (MA) and these data accorded well with (2). Psychologists' often used 2IFC method yielded data that are far less coherent than those from MA run in blocks.…”

What are we estimating when we fit Stevens’ power law?

“…The axiomatization of Narens (1996) included the multiplicative axiom, largely found to fail empirically. However, Luce's (2002Luce's ( , 2008 model of global psychophysics (see also Steingrimsson & Luce, 2007) later showed that, by abandoning the requirement W (1) = 1, the commutativity axiom is equivalent to obtaining ratio-scale measures. This axiom has been favorably evaluated in many experiments.…”

“…Our approach could give even more precise predictions on the behavior of Stevens' exponent with more knowledge about the shape and forms of both ψ and W . This could come pursuing the line of research drawn in several recent experiments by Luce and Steingrimsson (Steingrimsson and Luce, 2005a, 2005b, 2006, 2007.…”

“…In fact, removing the condition W (1) = 1 leads to show that commutativity is all that is needed to obtain ratio scalability. Steingrimsson and Luce (2007) presented a careful analysis of the various conditions, providing behavioral equivalents for W (p) being a power function or a so-called Prelec's (1998) exponential representation, in both cases with and without the restriction W (1) = 1. The experimental evidence reported in the same paper rejected the latter restriction, showing that most respondents either satisfied the general power form (with W (1) ̸ = 1) or the general Prelec's form (also with W (1) ̸ = 1).…”

“…We will also refer 1 Therein he also asserts something that is slightly more general than our Eq. (1), namely W (p) = p k with k > 0, and W (1) = 1 (more detail in Section 3 below and in Steingrimsson & Luce, 2007, in particular their Section 2.1.1).…”