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.

翻译：现代纵向研究以多种结局作为主要终点，以理解疾病的复杂动态变化。尤其在临床试验中，多维度响应之间的联合变异在评估两组或多组间的差异特征时起着关键作用，而非基于单一结局进行推断。将纵向设计纳入稀疏观测函数数据的框架下，我们开发了一种基于投影的两样本显著性检验，用于识别典型多变量轮廓间的差异。该方法基于广泛采用的多变量函数主成分分析，在降维无限维多模态函数的同时，保留各分量间的动态相关性。该检验适用于响应变量的一大类（非平稳）协方差结构，并通过单一p值检测显著的组间差异，从而克服了因分别比较各分量均值而产生的多重p值调整问题。有限样本数值研究表明，与现有先进检验方法相比，该检验能维持I型误差，且具有检测显著组间差异的高效性。该检验被应用于针对阿尔茨海默病所致轻度认知障碍高风险个体的纵向TOMMORROW研究，以检测吡格列酮组与安慰剂组在认知测试得分上的差异。