  Uncovering the heterogeneity in the disease progression of Alzheimer's is a
key factor to disease understanding and treatment development, so that
interventions can be tailored to target the subgroups that will benefit most
from the treatment, which is an important goal of precision medicine. However,
in practice, one top methodological challenge hindering the heterogeneity
investigation is that the true subgroup membership of each individual is often
unknown. In this article, we aim to identify latent subgroups of individuals
who share a common disorder progress over time, to predict latent subgroup
memberships, and to estimate and infer the heterogeneous trajectories among the
subgroups. To achieve these goals, we apply a concave fusion learning method
proposed in Ma and Huang (2017) and Ma et al. (2019) to conduct subgroup
analysis for longitudinal trajectories of the Alzheimer's disease data. The
heterogeneous trajectories are represented by subject-specific unknown
functions which are approximated by B-splines. The concave fusion method can
simultaneously estimate the spline coefficients and merge them together for the
subjects belonging to the same subgroup to automatically identify subgroups and
recover the heterogeneous trajectories. The resulting estimator of the disease
trajectory of each subgroup is supported by an asymptotic distribution. It
provides a sound theoretical basis for further conducting statistical inference
in subgroup analysis..
