Modeling the redundant signals effect by specifying the hazard function

In the bimodal detection task, the observer must respond as soon as a signal is presented in either of two modalities (e.g., a tone or a flash). On single-signal trials, only one signal is presented; on redundant-signal trials, both signals are presented either simultaneously or with a short stimulus onset asynchrony (SOA). The common empirical finding is that response time (RT) is shorter for redundantsignal trials. A number of sensory processing models have been proposed for this redundant-signals effect (RSE). These models differ in terms of assumptions made with respect to the underlying stochastic mechanisms (e.g., Meijers & Eijkman, 1977; Nickerson, 1973; Raab, 1962). Recently, special efforts have been made to derive empirically testable properties from various models (e.g., Colonius, 1986; Diederich & Colonius, 1987; Gielen, Schmidt, & Van den Heuvel, 1983; Miller, 1982; Ulrich & Giray, 1986). In this note, we propose a particular way of dealing with some of these models in an integrated manner by using the concept of a hazard function. The hazard function is very closely related to the probability distribution (see definition below). It specifies, for any point in time, the tendency for the response to occur instantaneously given that it has not yet occurred. It will be shown that formulating models for the RSE in terms of the hazard function clarifies their underlying assumptions and facilitates derivation of their testable properties. This will be demonstrated with reference to a recent paper by Miller (1986), in which a number of different models for the RSE are discussed. The hazard function, a probabilistic concept originally developed in reliability theory (cf. Barlow & Proschan, 1975), has become a valuable tool in the analysis of RTs (see, e.g., Bloxom, 1984, 1985; Luce, 1986, p. 13ff; Townsend & Ashby, 1983).