  In biomedical studies, longitudinal processes are collected till
time-to-event, sometimes on nested timescales (example, days within months).
Most of the literature in joint modeling of longitudinal and time-to-event data
has focused on modeling the mean or dispersion of the longitudinal process with
the hazard for time-to-event. However, based on the motivating studies, it may
be of interest to investigate how the cycle-level {\it geometric features}
(such as the curvature, location and height of a peak), of a cyclical
longitudinal process is associated with the time-to-event being studied. We
propose a shared parameter joint model for a cyclical longitudinal process and
a discrete survival time, measured on nested timescales, where the
cycle-varying geometric feature is modeled through a linear mixed effects model
and a proportional hazards model for the discrete survival time. The proposed
approach allows for prediction of survival probabilities for future subjects
based on their available longitudinal measurements. Our proposed model and
approach is illustrated through simulation and analysis of Stress and
Time-to-Pregnancy, a component of Oxford Conception Study. A joint modeling
approach was used to assess whether the cycle-specific geometric features of
the lutenizing hormone measurements, such as its peak or its curvature, are
associated with time-to-pregnancy (TTP).
