Theory and tests of the conjoint commutativity axiom for additive conjoint measurement

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Abstract

The empirical study of the axioms underlying additive conjoint measurement initially focused mostly on the double cancellation axiom. That axiom was shown to exhibit redundant features that made its statistical evaluation a major challenge. The special case of double cancellation where inequalities are replaced by indifferences–the Thomsen condition–turned out in the full axiomatic context to be equivalent to the double cancellation property but without exhibiting the redundancies of double cancellation. However, it too has some undesirable features when it comes to its empirical evaluation, the chief among them being a certain statistical asymmetry in estimates used to evaluate it, namely two interlocked hypotheses and a single conclusion. Nevertheless, thinking we had no choice, we evaluated the Thomsen condition for both loudness and brightness and, in agreement with other lines of research, we found more support for conjoint additivity than not. However, we commented on the difficulties we had encountered in evaluating it. Thus we sought a more symmetric replacement, which as Gigerenzer and Strube (1983) first noted, is found in the conjoint commutativity axiom proposed by Falmagne (1976, who called it the “commutative rule”). It turns out that, in the presence of the usual structural and other necessary assumptions of additive conjoint measurement, we can show that conjoint commutativity is equivalent to the Thomsen condition, a result that seems to have been overlooked in the literature. We subjected this property to empirical evaluation for both loudness and brightness. In contrast to Gigerenzer and Strube (1983), our data show support for the conjoint commutativity in both domains and thus for conjoint additivity.

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 function1mp,q defined by b=mp,q(a)iff (a,p)(b,q), conjoint commutativity asserts that mp,q[mr,s(a)]=mr,s[mp,q(a)].

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

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