  Wearable devices for continuous monitoring of electronic health increased
attention due to their richness in information. Often, inference is drawn from
features that quantify some summary of the data, leading to a loss of
information that could be useful when one utilizes the functional nature of the
response. When functional trajectories are observed repeated over time, it is
termed longitudinal functional data. This work is motivated by the interest to
assess the efficacy of a noninflammatory medication, meloxicam, on the daily
activity levels of household cats with a pre-existing condition of
osteoarthritis under a crossover design. These activity profiles are recorded
at a minute level by accelerometer over the entire study period. To this
aspect, we propose an orthogonal projection-based test pseudo generalized F
test for significance of the functional treatment effect under a functional
additive crossover model after adjusting for the carryover effect and other
baseline covariates. Under mild conditions, we derive the asymptotic null
distribution of the test statistic when the projection function for the
underlying Hilbert space is estimated from the data. In finite sample numerical
studies, the proposed test maintains the size, is powerful to detect the
significance of the smooth effect of meloxicam, and is very efficient compared
to bootstrap-based alternatives.
