  We propose novel optimal designs for longitudinal data for the common
situation where the resources for longitudinal data collection are limited, by
determining the optimal locations in time where measurements should be taken.
As for all optimal designs, some prior information is needed to implement the
proposed optimal designs. We demonstrate that this prior information may come
from a pilot longitudinal study that has irregularly measured and noisy
measurements, where for each subject one has available a small random number of
repeated measurements that are randomly located on the domain. A second
possibility of interest is that a pilot study consists of densely measured
functional data and one intends to take only a few measurements at
strategically placed locations in the domain for the future collection of
similar data. We construct optimal designs by targeting two criteria: (a)
Optimal designs to recover the unknown underlying smooth random trajectory for
each subject from a few optimally placed measurements such that squared
prediction errors are minimized; (b) Optimal designs that minimize prediction
errors for functional linear regression with functional or longitudinal
predictors and scalar responses, again from a few optimally placed
measurements. The proposed optimal designs address the need for sparse data
collection when planning longitudinal studies, by taking advantage of the close
connections between longitudinal and functional data analysis. We demonstrate
in simulations that the proposed designs perform considerably better than
randomly chosen design points and include a motivating data example from the
Baltimore longitudinal study of aging. The proposed designs are shown to have
an asymptotic optimality property.
