Theory and tests of the conjoint commutativity axiom for additive conjoint measurement☆
Highlights
► We prove that the conjoint commutativity property can replace the Thomsen condition in proving an additive conjoint representation. ► We show that it is an advantage for empirical testing over the Thomsen condition. ► We report data on the axiom for loudness and for brightness that favor the property.
Section snippets
Background
Several recent studies (Luce, 2004, Luce, 2008, Steingrimsson, 2009, Steingrimsson and Luce, 2005a) have focused on whether or not subjective measures of intensity over the two ears or two eyes satisfy the axioms for additive conjoint measurement which lead to an additive numerical representation. A great deal of the relevant literature such as Luce and Tukey (1964) was summarized in the Foundations of Measurement (FofM) (Krantz et al., 1971, Luce et al., 1990, Suppes et al., 1989). The gist of
Current tests of conjoint additivity
Additivity over sense organs has been studied in a variety of ways; here, we focus on its axiomatic evaluation. As summarized in Definition 7 of FofM I (p. 256), the test of additivity involves the evaluation of either double cancellation or the Thomsen condition. Double cancellation was explored by Levelt, Riemersma, and Bunt (1972), Falmagne (1976), Falmagne, Iverson, and Marcovici (1979), and Gigerenzer and Strube (1983) for loudness, by Schneider (1988) for loudness across critical bands,
Equivalence of conjoint commutativity to the Thomsen condition
As first pointed out by Gigerenzer and Strube (1983), in an article on random conjoint measurement, Falmagne (1976) introduced the following property, which he called the commutativity rule, and which we make more specific by speaking of conjoint commutativity. With the function1 defined by conjoint commutativity asserts that
Note that,
Discussion
We proved a theorem that shows that, under certain plausible structural assumptions of a binary conjoint structure, Assumptions A4 and A6 of Section 3, the conjoint commutativity property formulated by Falmagne (1976, which he called, somewhat ambiguously, the “commutative rule”) can play the role of double cancellation or the Thomsen condition in arriving at the additive conjoint representation. Conjoint commutativity has certain symmetric advantages over the existing axioms when it comes to
Acknowledgments
This research was supported in part by National Science Foundation grant BCS-0720288 and by the Air Force Office of Scientific Research grant FA9550-08-1-0468. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or of the Air Force. We thank Dr. Bruce Berg for unfettered access to his auditory laboratory. And finally we thank the Editor and three reviewers for
References (28)
A note on additive conjoint measurement
Journal of Mathematical Psychology
(1971)- et al.
Simultaneous conjoint measurement: a new type of fundamental measurement
Journal of Mathematical Psychology
(1964) - et al.
Evaluating a model of global psychophysical judgments: I. Behavioral properties of summations and productions
Journal of Mathematical Psychology
(2005) - et al.
Evaluating a model of global psychophysical judgments: II. Behavioral properties linking summations and productions
Journal of Mathematical Psychology
(2005) - et al.
Empirical evaluation of a model of global psychophysical judgments: III. A form for the psychophysical function and intensity filtering
Journal of Mathematical Psychology
(2006) - et al.
Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function
Journal of Mathematical Psychology
(2007) Ethical principles of psychologists and code of conduct
American Psychologist
(2002)- et al.
Empirical evaluation of axioms fundamental to Stevens’s ratio-scaling approach: I. Loudness production
Perception and Psychophysics
(2000) - et al.
Perceptual ratios, differences, and the underlying scale
Random conjoint measurement and loudness summation
Psychological Review
(1976)
Binaural loudness summation: probabilistic theory and data
Psychological Review
Are there limits to binaural additivity of loudness?
Journal of Experimental Psychology: Human Perception and Performance
Foundations of measurement, Vol. I
Binocular interactions in suprathreshold contrast perception
Perception & Psychophysics
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Portions of this material have also appeared in Conference Proceedings of the Meeting of International Society for Psychophysics (Steingrimsson & Luce, 2010), in which we referred to Conjoint Commutativity as the Commutativity Rule.