Functional Latent Feature Models for Data with Longitudinal Covariate Processes
We consider a joint model approach to study the association between nonparametric latent features of longitudinal processes and a primary endpoint. Our modeling strategy is closely related to generalized functional linear models (GFLM), but has several marked differences. We argue that the key assumption in the commonly used GFLM approach that the estimation variation in eigenfunctions is negligible for all eigen-components in the model is not necessarily true and is purely determined by the nature of data. The assumption could fail even when the estimated eigenvalues seem to die out sufficiently fast. We propose estimation procedures and corresponding supportive theory that allow one to perform investigation regardless the validity of this key assumption. Our approach takes into account the estimation uncertainty embedded in the estimated eigen-system and allows users to have a thorough understanding of where the estimation uncertainty/variation lies so that the choice of a final model and the future research plan can be made accordingly. To the best of our knowledge, the theoretical property we developed is the first that takes into account the uncertainty of the estimated eigen-components in the resulting parametric estimators. The techniques we developed here could be adopted by other estimators that use estimated eigenfunctions or eigenvalues. Numerical performances of the proposed approach are evaluated in simulations and through a study that investigated the impacts of body mass index and systolic blood pressure readings during selected periods of adulthood on hypertension status later in life.
Authors: Erning Li, Yehua Li, Nae-Yuh Wang, and Naisyin Wang