  There is increasing interest in modeling high-dimensional longitudinal
outcomes in applications such as developmental neuroimaging research. Growth
curve model offers a useful tool to capture both the mean growth pattern across
individuals, as well as the dynamic changes of outcomes over time within each
individual. However, when the number of outcomes is large, it becomes
challenging and often infeasible to tackle the large covariance matrix of the
random effects involved in the model. In this article, we propose a
high-dimensional response growth curve model, with three novel components: a
low-rank factor model structure that substantially reduces the number of
parameters in the large covariance matrix, a re-parameterization formulation
coupled with a sparsity penalty that selects important fixed and random effect
terms, and a computational trick that turns the inversion of a large matrix
into the inversion of a stack of small matrices and thus considerably speeds up
the computation. We develop an efficient expectation-maximization type
estimation algorithm, and demonstrate the competitive performance of the
proposed method through both simulations and a longitudinal study of brain
structural connectivity in association with human immunodeficiency virus.
