FUNCTIONAL AND LONGITUDINAL DATA ANALYSIS: PERSPECTIVES ON SMOOTHING

The perspectives and methods of functional data analysis and longitu- dinal data analysis for smoothing are contrasted and compared. Topics include kernel methods and random effects models for smoothing, basis function methods, and examination of correlates of curve shapes. Some directions in which method- ology might advance are identified. Until recently, functional data analysis (FDA) and longitudinal data anal- ysis (LDA) have been rather disjoint enterprises. Both are concerned with the analysis of data consisting of repeated measurements of objects over time. Mea- surements treated in the FDA literature typically are recorded by high frequency automatic sensing equipment, whereas those treated in the LDA literature are more typically sparsely, and often irregularly, spaced measurements on human or other biological subjects. The aims of the analysis are often somewhat different, partly because of different scientific subject matter (it is interesting to read and compare the introductions of the two classic texts of Diggle, Liang, and Zeger (1994) and Ramsay and Silverman (1997)). Those of FDA tend to be exploratory − to represent and display data in order to highlight interesting characteristics, perhaps as input for further analysis − whereas those of LDA have a stronger inferential component. This contrast can be seen in the use of estimated time- correlation functions in the two areas; correlation functions are used in the FDA literature in a descriptive manner to characterize time dependencies in the curves, whereas an important aim in the LDA literature of estimating these correlation functions is to draw valid inferences. Another important difference is that the LDA literature has had to pay much more attention to data that are missing because of a variety of mechanisms. Despite these differences in focus, there are many common aims, among them are the following: