Achieving Constant Hazard Rate in Psychophysics Experiments

In psychophysics, it is sometimes desirable for the temporal onset of an event to be unpredictable, e.g. to control possible effects of expectation and temporal attention in behavioral or neural data. One way of achieving temporal unpredictability is to make the hazard rate of the event constant. Roughly speaking, the hazard rate measures the odds that an event will occur now or later. If the hazard rate is constant throughout a temporal interval, then the time elapsed thus far provides the observer with no information for updating their expectation of when the event will occur. A constant hazard rate thus prevents the building of predictive anticipation that the event is about to occur even as time continues to unfold. Here we consider the question of how best to control hazard rates in psychophysics experiments with time considered as either a continuous or discrete variable. We argue that for computerized psychophysics experiments, it is both more appropriate and more convenient to model time as a discrete variable. We also consider that for practical reasons, experimenters may wish to impose upper bounds on the duration of random temporal intervals. We show how hazard rate can be exactly controlled for unbounded time intervals when time is treated as either a continuous or discrete variable by using the exponential and geometric distributions, respectively. We discuss complications in controlling hazard rate for bounded time intervals and present a solution yielding exact control of hazard rate for bounded time intervals with discrete time steps using a modified form of the geometric distribution. The latter approach is likely to be the most relevant for most psychophysics experiments that seek to control hazard rate.