The General - Gamma Distribution and * Reaction Times

The general-gamma distribution describes· input-output times in,.a multi-stage process consisting of exponential components whose timeconstants are all different. T~e distribution and its unique history are examined. A stochastic process that leads to it is presented. The conditional density (hazard) function is studied as a means for estimating parameters. Finally the multi-stage process model is applied to simple reaction times in an effort to reveal underlying detection rod response components. A system receives an input and grinds it through a series of discrete processing stages, then generates an output. There is no feedback, no parallel processing, no branching, no backward motion of any kind. The description sounds almost too simple to bother with. Yet something like this with a few added restrictions turns out to be a rather well known stochastic process whose behavior is not especially easy to work out. In fact, if we start with just the processing times and an uncertain knowledge of the size of the system, the task of unraveling it proves to be fairly tricky. Our interest in chain-like stochastic processes is aroused because they offer plausible models for important latency effects. For instance when an animal runs down an alley or learns to delay before responding, the duration of the latent period can be examined as a potential stochastic process. Bush & Mosteller's (1955) treatment of response times in an alley maze illustrates this type of application. Stochastic chains have also been used as models for reaction time; (see Christie & Luce, 1956; Stone, 1960; Restle, 1961; and LaBerge, 1962). It is clear that any latency which is thought to span an intervening and unobserved series of events can be a candidate for our interest, and the stubborn resistance that many of these latency effects offer to theoretical analysis shows the problem is not trivial. The term "general-gamma distribution" is used in this paper to describe the density function for input-output times in a mUlti-stage process whose time! constants are all different. It is an obvious extension of the ordinary gamma distribution generated by a process