on the speed, closing rate, and headway (distance) between pairs of motor vehicles traveling over this stretch of the highway. Sufficient data were collected for a simulation analysis of 35,689 pairs of vehicles. The simulation, a Monte Carlo analysis, was conducted in order to evaluate the potential effectiveness of different collision-warning systems. Without collision warning, Farber and Paley estimated a crash rate of 173 for every million lead vehicle stops. This gives a prior probability or base rate p of .000173. Note that this figure refers to impending collisions, or traffic events that would lead to a collision if the driver took no evasive action. However, because drivers took appropriate action, the data are filtered by human response, so that the base rate of events when the human takes no action is not known. As a first approximation, and in the absence of other data, we assume that the base rate of all impending collisions can be represented by this value. Given this base rate, a detection system with d′ = 5, would yield posterior odds of a true alarm of only I in 3. Even for an extraordinarily sensitive system, with a hit rate of .9999 and a false alarm rate of .0001 (d′ = 7.4), the posterior odds of a true alarm would be only about 2 in 3. These numbers attest to the powerful influence of the base rate on the true alarm rate.
How high should the posterior probability be? Certainly as high as possible; and it seems clear that P(S∣R) should be greater than .5 for people to attend to alarms. But how much greater? It is known that low posterior probabilities discourage user action ( Parasuraman & Riley, 1997) and also delay speed of response ( Casey, 1993; Getty, Swets, Pickett, & Gounthier, 1995). Assume that the minimum acceptable value of the posterior probability P(S∣R) = m lies somewhere between .5 and 1.0. The design question now is, given a system with accuracy s and a measured base rate of the collision event of p = b, what should the appropriate decision threshold (ßm) be to achieve a posterior probability of at least m? Equations , , and  can be used to answer this question. Since z[ ] and y[ ] represent non-linear functions, a simple, tractable analytical solution is not possible. But a graphical analysis easily points to the correct solution.
Given a specified base rate, our analysis shows how to set ß = ßm in order to optimize performance by achieving a high desired posterior probability m. Of course, the base rate will vary with collision type and with the definition of a impending collision event as a signal. As was mentioned earlier, many collision-warning systems are designed so that the driver can adjust or pre-set the warning distance to be relatively long or short.
The rapid growth of automation has led to a proliferation of warning and alarm systems. Unfortunately, many warning systems are prone to false alarms, and it is not clear that existing collision- warning systems can be used effectively by drivers to prevent crashes. The quantitative, analytical approach we have presented, based on SDT and Bayesian statistics, provides step-by-step procedures for setting alarm parameters to avoid these problems.
The first step is to specify a maximum false alarm rate for the system. The minimum decision threshold ßf that an alarm system with given sensitivity d' must be set at can then be computed. However, setting a minimum alarm decision threshold to achieve a maximum permissible false alarm rate is necessary but not sufficient for effective alarm performance. In addition, alarm parameters must be designed so that the posterior probability of a true alarm is relatively high for collision events associated with particular base rates of occurrence. Functions relating the required decision threshold ßm to achieve a posterior probability of at least m can be derived for alarm systems of given sensitivity d'.
The analysis we have presented provides a set of standards against which the performance of collision- warning systems can be tested. Although sensitive alarm systems with high detection rates and low false alarm