# Simple Reaction Time (Card, Moran & Newell, Ex 9, 1983, p.66)

Example 9. "A user sits before a computer display terminal. Whenever any symbol appears, he is to press the space bar. What is the time between signal and response?" (CMN, 1983, p.66)

Card, Moran & Newell (CMN) use their Model Human Processor - a simplified engineering model of the human perceptual-cognitive-motor system - to calculate the user response time RT. They assume that the user is "... in some state of attention to the display ...". The total response time is the sum of tau-p, tau-c, and tau-m. These are the cycle times of the hypothetical perceptual processor, the cognitive processor, and the motor processor, respectively. CMN (p.26, p.66, p.433f) provide the reader with interval-based cycle times according to the template tau-x = mean-x[low-x ~ high-x]. The meaning thereof is that the values for tau-x range from low-x to high-x with mean-x.

The tau-intervals are:

tau-p = 100 [50 ~ 200]   ; cycle time of perceptual processor

tau-c =   70 [25 ~ 170]   ; cycle time of cognitive processor

tau-m =  70 [30 ~ 100]   ; cycle time of motor processor

RT = tau-p + tau-c + tau-m = 240 [105 ~ 470] = Middleman [Fastman ~ Slowman].

# CHURCH-Code for CMN's Simple Reaction Time Model

The CHURCH program simulates the interval-specifications provided by CMN with gamma distributions. Tha tau's are distributed according to the related gamma distribution. They are added together to form the random variable RT. Then the distribution of RT is plotted. In this example we have no observational constraint, so that there is no need for a Bayesian inference.

We try to make the vague interval semantics of CMN more concrete. We approximate the intervals tau-x = Middleman [Fastman ~ Slowman] by gamma(a, b) distributions with mean m=a*b and standard deviation s = sqrt(a)*b. Gamma distributions seem to be useful because the interval generating distributions are skewed. This is indicated by the fact, that the tau-means are not identical to the interval centers.

Solving towards a and b we get a = (m/s)^2 and b = (s^2)/m. The intervals are then defined by tau-x = m[m - 2s ~ m + 2s].

The simulation runs show that the CMN intervals can be sufficiently well approximated by a 2s or 2-sigma rule. The interval width is 2*2s or 2*2sigma of the underlying gamma(a, b)-distribution.