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).

翻译：在生物医学研究中，纵向过程常被收集至事件发生时间，有时基于嵌套时间尺度（例如，月份内的天数）。当前关于纵向数据与事件发生时间联合建模的文献大多聚焦于通过风险函数对纵向过程的均值或离散程度进行建模。然而，基于本研究的动机，探究周期性纵向过程中周期水平的几何特征（如峰值曲率、位置及高度）如何与所研究的事件发生时间相关联具有重要意义。本文提出一种共享参数联合模型，用以处理嵌套时间尺度上测量的周期性纵向过程与离散生存时间，其中周期变化的几何特征通过线性混合效应模型建模，离散生存时间则采用比例风险模型。该方法允许根据已观测的纵向测量值预测新个体的生存概率。通过模拟实验及牛津受孕研究中的“压力与怀孕时间”数据分项，我们验证了所提出模型与方法的有效性。联合建模方法被用于评估促黄体激素测量值的周期特异性几何特征（如峰值或曲率）是否与怀孕时间存在关联。