Local estimation of smooth curves for longitudinal data

Longitudinal data are commonly analysed using mixed-effects models in which the population growth curve and individual subjects' growth curves are assumed to be known functions of time. Frequently, polynomial functions are assumed. In practice, however, polynomials may not fit the data and a mechanistic model that could suggest a non-linear function might not be known. Recent, more flexible approaches to these data approximate the underlying population mean curve or the individual subjects' curves using smoothing splines or kernel-based functions. I apply the local likelihood estimation method of Tibshirani and Hastie and estimate smooth population and individual growth curves by assuming that they are approximately linear or quadratic functions of time within overlapping neighbourhoods. This method requires neither complete data, nor that measurements are made at the same time points for each individual. For descriptive purposes, this approach is easy to implement with standard software. Inference for the resulting curve is facilitated by the theory of estimating equations. I illustrate the methods with data sets containing longitudinal measurements of serum neopterin in an AIDS clinical trial, measurements of ultrafiltration rates of high flux membrane dialysers for haemodialysis, and measurements of the volume of air expelled by individuals.