A generalization of Piéron's law to include background intensity and latency distribution

Piéron's law, which formulates the effect of the signal intensity i on the mean or the median reaction time (MRT) as a power function, MRT=R+i-βC, in which R, β and C are positive parameters, is of much interest in the simple reaction time literature. In particular, consistent estimates of β are obtained. However, this law offers no information regarding the effect of the signal intensity on other aspects of the reaction time distribution. In addition, the background intensity, which also has some impact on the detectability of the signal, is not considered in Piéron's law. In this paper, we generalize Piéron's law based on the above comments. That is, we assume that both β and C are functions of the background intensity and of the percentile rank. We then investigate from a theoretical viewpoint some possible functional forms of β and C, with a special emphasis on the dependency of β on the background intensity. It is shown that the so-called “Fechner-Thurstone-type representation” forces β and C to be “separable.” If we further assume a near-miss to Weber's law-type representation, then the functional form of β can be determined.