Intrinsic regression models for positive-definite matrices with applications to diffusion tensor imaging

Hongtu Zhu, Yasheng Chen, Joseph G. Ibrahim, Yimei Li, Colin Hall, Weili Lin

Research output: Contribution to journalArticlepeer-review

47 Scopus citations


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.

Original languageEnglish
Pages (from-to)1203-1212
Number of pages10
JournalJournal of the American Statistical Association
Issue number487
StatePublished - 2009


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