Comparison of Parametric Rate Models for Gap Times Between Recurrent Events

Over the past two decades, substantial efforts have been made to develop survival models for gap times between recurrent events. An emerging approach involves considering rate models derived from a non-homogeneous Poisson process, thus allowing the conditional distribution of a gap time given the previous recurrence time to be deduced. Under this approach, some parametric rate models have been proposed, differing in their distributional assumptions on gap times. In particular, the extended exponential–Poisson, Weibull and extended Chen–Poisson distributions have been considered. Alternatively, a flexible rate model using restricted cubic splines is proposed here to capture complex non-monotonic rate shapes. Moreover, a comprehensive comparison of parametric rate models is presented. The maximum likelihood method is applied for parameter estimation in the presence of right-censoring. It is shown that some models include important special cases that allow testing of the independence assumption between a gap time and the previous recurrence time. The likelihood ratio test, as well as two information criteria, are discussed for model selection. Model fit is assessed using Cox–Snell residuals. Applications to two well-known clinical data sets illustrate the comparative performance of both the existing and proposed models, as well as their practical relevance.