Joint modeling of longitudinal functional feature and time to event: an application to fecundity studies

In many longitudinal studies, it is often of interest to investigate how the {\it geometric functional features} (such as the curvature, location and height of a peak), of a marker's measurement process is associated with the time to event being studied. We propose a joint model for certain geometric functional features of a longitudinal process and a time to event, making use of B-splines to smoothly approximate the infinite dimensional functional data. The proposed approach allows for prediction of survival probabilities for future subjects based on their available longitudinal measurements. We illustrate the performance of our proposed model on a prospective pregnancy study, namely Stress and Time to Pregnancy, a component of Oxford Conception Study, where hormonal measurements of luteinizing hormone (LH) and estrogen indicate timing of ovulation, and whether ovulation is going to occur, in a menstrual cycle. A joint modeling approach was used to assess whether the functional features of the hormonal measurements, such as the peak of the hormonal profile and its curvature, are associated with time to pregnancy. Our simulation studies indicate reasonable performance of the proposed approach.