Smoothing splines for longitudinal data

Abstract

In a longitudinal data model with fixed and random effects, polynomials are used to model the fixed effects and smoothing polynomial splines are used to model the within-subject random effect curves. The splines are generated by modelling the data for each subject as observations of an integrated random walk with observational error. The initial conditions for each subject's deviation from the fixed effect curve are assumed to have zero mean and arbitrary covariance matrix which is estimated by maximum likelihood, producing an empirical Bayes estimate. This is in contrast to modelling a single curve using a diffuse prior. An example is presented using unbalanced longitudinal data from a pilot study in breast cancer patients.