Summary: Time to event data have long been important in many applied fields. Many models and analysis methods have been developed for this type of data in which each sample unit experiences at most a single end-of-life event.
In contrast, many applications involve repeated events, where a subject or sampling unit experiences more than one event. There is growing interest in the analysis of recurrent events data, also called repeated events and recurrence data. This type of data arises in many fields. For example, the repair history of manufactured items can be modeled as recurrent events. In medical studies, the times of recurrent disease episodes in patients can also be modeled as recurrent events. In this paper we focus on medical applications (e.g. seizures, heart attacks, cancerous tumors, etc.). However, our proposed methodologies can be applied to other areas as well.
For analyzing recurrence data, the first and perhaps most important step is to model the expected number of events, and sometimes this can be facilitated by modeling the cumulative intensity function or its derivative, the intensity rate function. One particular recurrent events scenario involves patients experiencing events according to a common intensity rate, and then a treatment may be applied. Assuming the treatment to be effective, the patients would be expected to follow a different intensity rate after receiving the treatment. Further, the treatment might be effective for a limited amount of time, so that a third rate would govern arrivals of the recurrent events after the effects of the treatment wore out. In this paper we model the intensity rate for such scenarios. In particular we allow models for the intensity rate, post-treatment, to be piecewise constant. Two estimators of the location of this change are proposed. Properties of the estimators are discussed. An example is studied for illustrative purposes.