Zhu Hongtu, Chen Yasheng, Ibrahim Joseph G, Li Yimei, Hall Colin, Lin Weili
H. Zhu is Associate Professor of Biostatistics (
J Am Stat Assoc. 2009;104(487):1203-1212. doi: 10.1198/jasa.2009.tm08096.
The aim of this paper is to develop an intrinsic regression model for the analysis of positive-definite matrices as responses in a Riemannian manifold and their association with a set of covariates, such as age and gender, in a Euclidean space. The primary motivation and application of the proposed methodology is in medical imaging. Because the set of positive-definite matrices do not form a vector space, directly applying classical multivariate regression may be inadequate in establishing the relationship between positive-definite matrices and covariates of interest, such as age and gender, in real applications. Our intrinsic regression model, which is a semiparametric model, uses a link function to map from the Euclidean space of covariates to the Riemannian manifold of positive-definite matrices. We develop an estimation procedure to calculate parameter estimates and establish their limiting distributions. We develop score statistics to test linear hypotheses on unknown parameters and develop a test procedure based on a resampling method to simultaneously assess the statistical significance of linear hypotheses across a large region of interest. Simulation studies are used to demonstrate the methodology and examine the finite sample performance of the test procedure for controlling the family-wise error rate. We apply our methods to the detection of statistical significance of diagnostic effects on the integrity of white matter in a diffusion tensor study of human immunodeficiency virus. Supplemental materials for this article are available online.
本文旨在开发一种内在回归模型，用于分析黎曼流形中作为响应的正定矩阵及其与欧几里得空间中一组协变量（如年龄和性别）的关联。所提出方法的主要动机和应用在于医学成像。由于正定矩阵集不构成向量空间，在实际应用中直接应用经典多元回归可能不足以建立正定矩阵与感兴趣的协变量（如年龄和性别）之间的关系。我们的内在回归模型是一种半参数模型，它使用一个链接函数从协变量的欧几里得空间映射到正定矩阵的黎曼流形。我们开发了一种估计程序来计算参数估计值并确定其极限分布。我们开发了得分统计量来检验关于未知参数的线性假设，并基于重采样方法开发了一种检验程序，以同时评估在大感兴趣区域内线性假设的统计显著性。模拟研究用于演示该方法，并检验控制家族性错误率检验程序的有限样本性能。我们将我们的方法应用于在人类免疫缺陷病毒的扩散张量研究中检测对白质完整性的诊断效应的统计显著性。本文的补充材料可在线获取。
