Evaluating a model of global psychophysical judgments—I: Behavioral properties of summations and productions

The research presented is a partial empirical evaluation of the second author's proposed psychophysical theory [Luce (2002). Psychological Review, 109, 520-532; Luce (2004). Psychological Review, 111, 446-454]. The theory deals with the global percept of subjective intensity, in which there is a psychophysical function Ψ that maps pairs of physical intensities onto the positive real numbers and represents, in an explicit mathematical way, subjective summation and a form of ratio production. A number of behavioral properties have been shown to follow from these specific representations, and in the presence of certain plausible background assumptions these properties are also sufficient for the representations. In four auditory experiments, key behavioral properties of summation over the two ears and a form of generalized ratio production are evaluated empirically. Considerable support is reported for particular forms of Ψ for summations and ratio productions separately. A second article, Steingrimsson and Luce (Journal of Mathematical Psychology, in press), explores the behavioral properties that link summations and productions.     

A Simple Axiomatization of Binary Rank-Dependent Utility of Gains (Losses)

For binary gambles composed only of gains (losses) relative to a status quo, the rank-dependent utility model with a representation that is dense in intervals is shown to be equivalent to ten elementary properties plus event commutativity and a gamble partition assumption. The proof reduces to a (difficult) functional equation that has been solved by Aczél, Maksa, and Páles (in press).     

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.     

Empirical evaluation of a model of global psychophysical judgments: III. A form for the psychophysical function and intensity filtering

In part I, a concept of ratio estimation is defined and it is shown that if such estimates depend only upon the physical ratio of the signal to the reference signal, the psychophysical function must be a power function. Assuming the same exponents for each component, an invariance condition, equivalent to a sum of power functions, is studied empirically for binaural loudness. It is fully or partially sustained for 19 of 22 respondents. Since failures may be attributable to different exponents in the two ears, the ratio of the two exponents is estimated but that fails to explain the failures. Other possible explanations are suggested. In part II, an intensity filtering model is presented, accounting for the phenomenon where monaural loudness matches show a bias depending on the matching ear. We show (a) that the existence of such a bias does not alter the prior experimental results; and (b) assuming the power function, that five respondents attenuate the opposite ear and two enhance it.     

Evaluating a model of global psychophysical judgments—II: Behavioral properties linking summations and productions

Steingrimsson and Luce [Journal of Mathematical Psychology, in press] outlined the second author's proposed psychophysical theory [Luce (2002), Psychological Review, 109, 520-532; Luce (2004a) Psychological Review, 111, 446-454] and tested behavioral attributes that, separately, gave rise to two psychophysical functions, Ψ_⊕ and Ψ_°p. The function Ψ_⊕ maps pairs of physical intensities onto the positive real numbers and represents subjective summation, and the function Ψ_°p represents a form of ratio production. This article evaluates properties linking summation and production in such a way as to force Ψ_°p=Ψ_⊕=Ψ. These properties, which are a form of distributivity, are subjected to an empirical evaluation in three experiments. The testing strategy is carried out in the auditory domain and concerns the subjective perception of loudness. Considerable support is provided for the existence of a single function Ψ for both summation and ratio production.