  In many modern applications, a dependent functional response is observed for
each subject over repeated time, leading to longitudinal functional data. In
this paper, we propose a novel statistical procedure to test whether the mean
function varies over time. Our approach relies on reducing the dimension of the
response using data-driven orthogonal projections and it employs a
likelihood-based hypothesis testing. We investigate the methodology
theoretically and discuss a computationally efficient implementation. The
proposed test maintains the type I error rate, and shows excellent power to
detect departures from the null hypothesis in finite sample simulation studies.
We apply our method to the longitudinal diffusion tensor imaging study of
multiple sclerosis (MS) patients to formally assess whether the brain's health
tissue, as summarized by fractional anisotropy (FA) profile, degrades over time
during the study period.
