Inhibition Theory - Theory

Theory

If one thinks of a non-negative continuous random variable T as representing the time until some event will take place then the hazard rate λ(t) for that random variable is defined to be the limiting value of the probability that the event will take place in a small interval, given the event has not occurred before time t, divided by Δt. Formally, the hazard rate is defined by the following limit:

The hazard rate λ(t) can also be written in terms of the density function or probability density function f(t) and the distribution function or cumulative distribution function F(t):

The transition rates λ₁(t), from state 1 to state 0, and λ₀(t), from state 0 to state 1, depend on inhibition Y(t): λ₁(t) = l₁(Y(t)) and λ₀(t) = l₀(Y(t)), where l₁ is a non-decreasing function and l₀ is a non-increasing function. Note, that l₁ and l₀ are dependent on Y, whereas Y is dependent on T. Specification of the functions l₁ and l₀ leads to the various inhibition models. What can be observed in the test are the actual reaction times. A reaction time is the sum of a series of alternating distraction times and attention times, which both cannot be observed. However, it is nevertheless possible to estimate from the observable reaction times some properties of the latent process of distraction times and attention times, such as the average distraction time, the average attention time and the ratio a₁/a₀. In order to be able to simulate the consecutive reaction times, inhibition theory has been specified into various inhibition models. One is the so-called beta inhibition model. In the beta-inhibition model, it is assumed that the inhibition Y(t) oscillates between two boundaries which are 0 and M (M for Maximum), where M is positive. In this model l₁ and l₀ are as follows:

and

both with c₀ > 0 and c₁ > 0. Note that, according to the first assumption, as y goes to M (during an interval), l₁(y) goes to infinity and this forces a transition to a state of rest before the inhibition can reach M. Note further that, according to the second assumption, as y goes to zero (during a distraction), l₀(y) goes to infinity and this forces a transition to a state of work before the inhibition can reach zero. For a work interval starting at t₀ with inhibition level y₀=Y(t₀) the transition rate at time t₀+t is given by λ₁(t) = l₁(y₀+a₁t). For a non-work interval starting at t₀ with inhibition level y₀=Y(t₀) the transition rate is given by λ₀(t) = l₀(y₀-a₀t). Therefore

and

The model has Y fluctuating in the interval between 0 and M. The stationary distribution of Y/M in this model is a beta distribution (reason to call it the beta inhibition model).

The total real working time until the conclusion of the task (or the task unit in case of a repetition of equivalent unit tasks, such as is the case in the Attention Concentration Test is referred to as A. The average stationary response time E(T) may written as

.

For M goes to infinity λ₁(t) = c₁. This model is known as the gamma - or Poisson inhibition model (see Smit and van der Ven, 1995).