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-average models such as GEE are quite complex (see, e.g., 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 nonparametric components (nonparametric link and nonparametric 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 Müller, 2005). Taking the idea of modeling with nonparametric components one step further, the Functional Data Analysis (FDA) approach to longitudinal data provides an alternative nonparametric method for the modeling of individual trajectories. The underlying idea is to view observed longitudinal trajectories as a sample of random functions, which are not parametrically specified. The observed measurements for an individual then correspond to the values of the random trajectory, corrupted by measurement error. A primary objective is to reduce the high dimension of the trajectories – considered to be elements of an infinite-dimensional function space – to finite dimension. One goal is to predict individual trajectories from the measurements made for a subject, borrowing strength from the entire sample of subjects. The necessary dimension reduction or regularization step can be implemented in various ways. For the analysis of longitudinal data, with its typically sparse and irregular measurements per subject, the method of Functional Principal Component Analysis (FPCA), which will be reviewed in Section 1.5, has been recently proposed (Yao, Müller and Wang, 2005ab), extending previous work by James (2002). Other regularization methods that have proven useful in FDA include smoothing splines (Ke and Wang, 2001), B-splines (Rice and Wu, 2000) or P-splines (Yao and Lee, 2006) (see Chapters 10 and 11). The classical theory and applications of FDA have been developed for densely sampled or fully observed trajectories that in addition are sampled without noise. This setting is not conducive to applications in longitudinal studies, due to the common occurrence of irregular and sparse measurement times, often due to missing data. Excellent overviews