Flexible Models for Simple Longitudinal Data

We propose a new method for modelling simple longitudinal data. We aim to do this in a flexible manner (without restrictive assumptions about the shapes of individual trajectories), while exploiting structural similarities between the trajectories. Hierarchical models (such as linear mixed models, generalised additive mixed models and hierarchical generalised additive models) are commonly used to model longitudinal data, but fail to meet one or other of these requirements: either they make restrictive assumptions about the shape of individual trajectories, or fail to exploit structural similarities between trajectories. Functional principal components analysis promises to fulfil both requirements, and methods for functional principal components analysis have been developed for longitudinal data. However, we find that existing methods sometimes give poor-quality estimates of individual trajectories, particularly when the number of observations on each individual is small. We develop a new approach, which we call hierarchical modelling with functional principal components. Inference is conducted based on the full likelihood of all unknown quantities, with a penalty term to control the balance between fit to the data and smoothness of the trajectories. We run simulation studies to demonstrate that the new method substantially improves the quality of inference relative to existing methods across a range of examples, and apply the method to data on changes in body composition in adolescent girls.