Modelling Times Between Events with a Cured Fraction Using a First Hitting Time Regression Model with Individual Random Effects

Abstract

The empirical survival function of time-to-event data very often appears not to tend to zero. Thus there are long-term survivors, or a “cured fraction” of units which will apparently never experience the event of interest. This feature of the data can be incorporated into lifetime models in various ways, for example, by using mixture distributions to construct a more complex model. Alternatively, first hitting time (FHT) models can be used. One of the most attractive properties of a FHT model for lifetimes based on a latent Wiener process is that long-term survivors appear naturally—corresponding to failure of the process to reach the absorbing boundary—without the need to introduce special components to describe the phenomenon. FHT models have been extended recently in order to incorporate individual random effects into their drift and starting level parameters and also to be applicable in situations with recurrent events on the same unit with possible right censoring of the last stage. These models are extended here to allow censoring to occur at every intermediate stage. Issues of model selection are also considered. Finally, the proposed FHT regression model is fitted to a dataset consisting of the times of repeated applications for treatment made by drug users.