Deriving exact predictions from the cascade model.

Abstract

McClelland's (1979) cascade model is investigated, and it is shown that the model does not have a well-defined reaction time (RT) distribution function because it always predicts a nonzero probability that a response never occurs. By conditioning on the event that a response does occur, RT density and distribution functions are derived, thus allowing most RT statistics to be computed directly and eliminating the need for computer simulations. Using these results, an investigation of the model revealed that (a) it predicts mean RT additivity in most cases of pure insertion or selective influence; (b) it predicts only a very small increase in standard deviations as mean RT increases; and (c) it does not mimic the distribution of discrete-stage models that have a serial stage with an exponentially distributed duration. Recently, McClelland (1979) proposed a continuous-time linear systems model of simple cognitive processes based on sequential banks of parallel integrators. This model, referred t o by McClelland as the cascade model, exhibits some potentially very interesting properties. For example, McClelland argues that under certain conditions it mimics some of the reaction time (RT) additivities characteristic o f serial discrete-stage models. Unfortunately, however, rigorous empirical testing of the model is precluded because McClelland (1979) offers no method for computing any of the RT statistics it predicts. The format of this note is as follows: I will show that the model always predicts a nonzero probability that a response never occurs, which means, for example, that it always predicts infinite mean RTs. One way to circumvent this problem is to look only at trials on which a reponse does occur. By doing this it is possible to derive an RT probability density function predicted by the cascade model. From it, virtually any desired RT statistic can be accurately computed. Some of these (e.g., means and variances) will be examined, with particular regard to how well they correspond t o known empirical results. For example, it turns out