Functional modeling of longitudinal data
Longitudinal studies are characterized by data records containing repeated measurements per subject, measured at various points on a suitable time axis. The aim is often to study change over time or time dynamics of biological phenomena such as growth, physiology, or pathogenesis. One is also interested in relating these time dynamics to certain predictors or responses. The classical analysis of longitudinal studies is based on parametric models, which often contain random eﬀects, such as the generalized linear mixed model (GLMM) (Chapter 4), or on marginal methods such as generalized estimating equations (GEE) (Chapter 3). The relationships between the subject-speciﬁc random-eﬀects models and the marginal, population-averaged models underlying methods such as GEE are quite complex (see Zeger, Liang, and Albert, 1988; Heagerty, 1999; Heagerty and Zeger, 2000; and the detailed discussion of this topic in Chapter 7). To a large extent, this non-compatibility
of various approaches is due to the parametric assumptions that are made in these models. These include the assumption of a parametric trend (linear or quadratic in the simplest cases) over time and of parametric link functions. Speciﬁc common additional assumptions are normality of the random eﬀects in a GLMM and a speciﬁc covariance structure (“working correlation”) in a GEE. Introducing non-parametric components (non-parametric link and non-parametric covariance structure) can ameliorate the diﬃculties of relating various longitudinal models to each other, as it increases the inherent ﬂexibility of the resulting longitudinal models substantially (see the estimated estimating equations approach in Chiou and Mu¨ller, 2005).