Modelling longitudinal data with nonstationary covariance structures: Random coefficients models versus alternative models

An important theme of longitudinal data analysis in the past two decades has been the development and use of explicit parametric models for the data's variance-covariance structure. However, nonstationary covariance structures had not been analyzed in detail for longitudinal data mainly because the existing applications did not require their use. There has been a large amount of recently proposed models but most of them are second-order stationary. A few, however, are flexible enough to accommodate nonstationarity, that is, nonconstant variances and/or correlations which are not only a function of the elapsed time between measurements. We study some of these proposed models and compare them to the random coefficients models, evaluating the relative strengths and limitations of each model, emphasizing when it is inappropriate or unlikely to be useful. We present two examples to illustrate the fitting and comparison of the models and to demonstrate that nonstationary longitudinal data can be modelled effectively and, in some cases, quite parsimoniously. In these examples the antedependence models generally prove to be superior and the random coefficients models prove to be inferior.