Accurately specifying covariance structures is critical for valid inference in longitudinal and functional data analysis, particularly when data are sparsely observed. In this study, we develop a global goodness-of-fit test to assess parametric covariance structures in multivariate sparse functional data. Our contribution is twofold. First, we extend the univariate goodness-of-fit test proposed by Chen et al. (2019) to better accommodate sparse data by improving error variance estimation and applying positive semi-definite smoothing to covariance estimation. These corrections ensure appropriate Type I error control under sparse designs. Second, we introduce a multivariate extension of the improved test that jointly evaluates covariance structures across multiple outcomes, employing novel test statistics based on the maximum and $\ell_2$ norms to account for inter-outcome dependencies and enhance statistical power. Through extensive simulation studies, we demonstrate that the proposed methods maintain proper Type I error rates and achieve greater power than univariate tests with multiple testing adjustments. Applications to longitudinal neuroimaging and clinical data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) and the Parkinson's Progression Marker Initiative (PPMI) illustrate the practical utility of the proposed methods for evaluating covariance structures in sparse multivariate longitudinal data.

翻译：在纵向与函数数据分析中，准确设定协方差结构对于统计推断的有效性至关重要，尤其是在数据稀疏观测的情况下。本研究提出了一种全局拟合优度检验方法，用于评估多元稀疏函数数据中的参数化协方差结构。我们的贡献主要体现在两个方面。首先，我们对Chen等人（2019）提出的单变量拟合优度检验进行了扩展，通过改进误差方差估计以及对协方差估计施加半正定平滑处理，使其能更好地适应稀疏数据。这些修正确保了在稀疏设计下对第一类错误率的有效控制。其次，我们进一步提出了该改进检验的多元扩展版本，能够联合评估多个结局变量的协方差结构；该方法采用基于最大值范数与$\ell_2$范数的新颖检验统计量，以考虑结局变量间的依赖关系并提升统计功效。通过大量模拟研究，我们证明所提方法在维持适当第一类错误率的同时，相较于经过多重检验校正的单变量检验具有更高的统计功效。在阿尔茨海默病神经影像学倡议（ADNI）与帕金森病进展标志物倡议（PPMI）的纵向神经影像及临床数据中的应用，展示了所提方法在评估稀疏多元纵向数据协方差结构方面的实际效用。