Longitudinal Parameters

  • Hans-Georg Müller
Part of the Lecture Notes in Statistics book series (LNS, volume 46)


In biomedical settings, a common problem is the comparison and description of samples of curves. Assuming there are N subjects and nj 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 gj 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).


Peak Size Invariant Shape Local Bandwidth High Order Kernel Random Coefficient Regression Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Hans-Georg Müller
    • 1
    • 2
  1. 1.Institute of Medical StatisticsUniversity of Erlangen-NürnbergErlangenFederal Republic of Germany
  2. 2.Division of StatisticsUniversity of CaliforniaDavisUSA

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