Projection-based two-sample inference for sparsely observed multivariate functional data

Modern longitudinal studies collect multiple outcomes as the primary endpoints to understand the complex dynamics of the diseases. Oftentimes, especially in clinical trials, the joint variations among the multidimensional responses play a significant role in assessing the differential characteristics between two or more groups, rather than drawing inferences based on a single outcome. Enclosing the longitudinal design under the umbrella of sparsely observed functional data, we develop a projection-based two-sample significance test to identify the difference between the typical multivariate profiles. The methodology is built upon widely adopted multivariate functional principal component analysis to reduce the dimension of the infinite-dimensional multi-modal functions while preserving the dynamic correlation between the components. The test is applicable to a wide class of (non-stationary) covariance structures of the response, and it detects a significant group difference based on a single p-value, thereby overcoming the issue of adjusting for multiple p-values that arises due to comparing the means in each of components separately. Finite-sample numerical studies demonstrate that the test maintains the type-I error, and is powerful to detect significant group differences, compared to the state-of-the-art testing procedures. The test is carried out on the longitudinally designed TOMMORROW study of individuals at high risk of mild cognitive impairment due to Alzheimer's disease to detect differences in the cognitive test scores between the pioglitazone and the placebo groups.