Longitudinal Parameters

Abstract

In biomedical settings, a common problem is the comparison and description of samples of curves. Assuming there are N subjects and n_j measurements are made for the j-th subject, we might describe the situation by the following model:

$$ {\text{Y}}_{{\text{ij }}} - {\text{g}}_{\text{j}} {\text{(t}}_{{\text{ij}}} {\text{) + }}\varepsilon _{{\text{ij}}} ,{\text{ j = 1,}}...{\text{,N, i = 1,}}...{\text{,n}}_{\text{j}} {\text{ }}, $$

, where the functions g_j are assumed to be random processes. More specific assumptions and the problem of estimating a “longitudinal average” curve (to be distinguished from a cross-sectional ordinary average curve which would not represent a “typical” time course since e.g. peaks would be unreasonably broadened) are discussed e.g. in Müller and Ihm (1985). Another approach to deal with samples of curves is shape-invariant modelling (Lawton, Sylvestre and Maggio, 1972; Stützle et al, 1980; Kneip and Gasser, 1986), where under minimal assumptions of some “invariant shapes” constituting a curve by different scaling, the nonparametric shapes as well as the scaling parameters are found by iterative improvement of the model, pooling at each step residuals which belong to “corresponding” times over the sample of curves and estimating the model improvement by a spline function. Other related proposals have been made, e.g. principal components techniques for stochastic processes have been tried (Castro, Lawton and Sylvestre, 1986).