Analysis of gap times based on panel count data with informative observation times and unknown start time

In biomedical studies, one is often interested in repeat events with longitudinal observations occurring only intermittently, resulting in panel count data. The first stage of labor, measured through unit-increments of cervical dilation in pregnant women, provides such an example. Obstetricians are interested in assessing the gap time distribution of per-unit increments of cervical dilation for better management of labor process. Typically, only intermittent medical examinations for cervical dilation occur after (already dilated) women get admitted to hospital. The observation frequency is very likely correlated to how fast/slow she dilates. Thus one could view such data as panel count data with informative observation times and unknown start time. Here, we propose semiparametric proportional rate models for the event process and the observation process, with a multiplicative subject-specific frailty variable capturing the correlation between the two processes. Inference procedures for the gap times between consecutive events are proposed when the start times are known as well when unknown, using likelihood based approach and estimating equations. The methodology is assessed through simulation study and through large sample property. A detailed analysis using the proposed methods is applied to data from two studies: the Collaborative Perinatal Project and the Consortium on Safe Labor.