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

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.     

Brightness of different hues is a single psychophysical ratio scale of intensity

Recent studies based on testable behavioral axioms have concluded that psychological scales of subjective intensive attributes involving the ears and eyes form ratio scales. These studies have shown that a certain commutativity of proportion property must hold under either successive increases or successive decreases, with all other independent dimensions fixed. However, until recently limited attention has been paid to whether such subjective intensity scales differ when a dimension independent of intensity, such as frequency or wavelength (e.g., pitch in audition, hue in vision) is varied. Using a simple and favorably tested theoretical model for global psychophysics, Luce, Steingrimsson, and Narens (2010) arrived at a necessary and sufficient cross-frequency, commutativity condition for there to exist a common intensity ratio scale. Here we show that brightness--already established to be a ratio-scalable dimension--and hue satisfy the same conditions.     

Predictions about bisymmetry and cross-modal matches from global theories of subjective intensities

The article first summarizes the assumptions of Luce (2004, 2008) for inherently binary (2-D) stimuli (e.g., the ears and eyes) that lead to a "p-additive," order-preserving psychophysical representation. Next, a somewhat parallel theory for unary (1-D) signals is developed for intensity attributes such as linear extent, vibration to finger, and money. The 3rd section studies the property of bisymmetry in these 2 cases. For the 2-D case and the nontrivial p-additive forms, Proposition 3 shows that bisymmetry implies commutativity of the presentations. Bisymmetry has been empirically well sustained, whereas commutativity has been rejected for loudness, brightness, and perceived contrast, thus implying that pure additivity must obtain in the 2-D context. By contrast, bisymmetry and commutativity are automatically satisfied by the p-additive 1-D theory. The 4th section explores the resulting complex of cross-modal predictions. For the additive 1-D case and the 2-D case, the predictions are power functions. For the nonadditive 1-D cases, other relations are predicted (see Table 2). Some parameter estimation issues are taken up in Appendices B and C.     

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

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?C532, 2002, Psychological Review, 111, 446?C454, 2004, Psychological Review, 115, 601, 2008, Psychological Review, 119, 373?C387, 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.