Terminal behavior of recurrent marker processes with informative failure events

Recurrent event processes with marker measurements are mostly and largely studied with forward time models starting from an initial event. Interestingly, the processes could exhibit important terminal behavior during a time period before occurrence of the failure event. A natural and direct way to study recurrent events prior to a failure event is to align the processes using the failure event as the time origin and to examine the terminal behavior by a backward time model. We will discuss both nonparametric estimation and regression modeling of backward recurrent marker processes by counting time backward from the failure event.  For nonparametric estimation, a risk-set reweighted estimator is studied. In addition, a three-level semiparametric regression model is studied for jointly modeling the time to a failure event, the backward recurrent event process, and the marker observed at the time of each backward recurrent event. The first level is a proportional hazards model for the failure time, the second level is a proportional rate model for the recurrent events occurring before the failure event, and the third level is a proportional mean model for the marker given the occurrence of a recurrent event backward in time. By jointly modeling the three components, estimating equations can be constructed for marked counting processes toestimate the target parameters in the three-level regression models.