A rotation test for behavioural point-process data

A common problem in animal behaviour is determining whether the rate at which a certain behavioural event occurs is affected by an environmental or other factor. In the example considered later in this paper, the event is a vocalization by an individual sperm whale and the factor is the operation or nonoperation of an underwater sound source. A typical experiment to test for such effects involves observing animals during control and treatment periods and recording the times of the events that occur in each. In statistical terminology, the data arising from such an experimentdthe times at which events of a specified type occurdrepresent a point process (Cox & Lewis 1978). Events in a point process are treated as having no duration. Although this is not strictly correct for behavioural events, the approximation is reasonable when the duration of events is small in relation to the interval between them. In some cases, under the null hypothesis of no treatment effect, behavioural events can be assumed to follow a stationary Poisson process. Under this model, the intervals between successive events are independent and, conditional on their number, the events are uniformly distributed over the observation period. As described below, when the Poisson assumption is valid, a statistical test to determine whether event rate changes under treatment can be based on the binomial distribution. In many cases, however, the Poisson model has been shown to be invalid for behavioural events. This is the case, for example, when events occur in bouts (Slater & Lester 1982; Sibly et al. 1990; Haccou & Meelis 1992). As illustrated below, when behavioural events do not follow a Poisson process, the binomial test can give misleading results. A number of methods are available to test whether a point process is Poisson based on the uniformity result mentioned above (Stephens 1986). If a point process cannot be assumed to be Poisson, one option is to use a test that is valid under a particular alternative to the Poisson model. Unfortunately, although it is often easy to show that a point process is not Poisson, it can be difficult to specify an appropriate alternative model. The purpose of this paper is to describe and illustrate the use of a simple nonparametric method that can be used to analyse behavioural point-process data even if the process generating the data is unknown.