TY - JOUR
T1 - Intrinsic regression models for positive-definite matrices with applications to diffusion tensor imaging
AU - Zhu, Hongtu
AU - Chen, Yasheng
AU - Ibrahim, Joseph G.
AU - Li, Yimei
AU - Hall, Colin
AU - Lin, Weili
N1 - Funding Information:
H. Zhu is Associate Professor of Biostatistics (E-mail: [email protected]), J. G. Ibrahim is Alumni Distinguished Professor of Biostatistics (E-mail: [email protected]), and Y. Li is Ph.D. Student (E-mail: liyimei@email. unc.edu), Department of Biostatistics and Biomedical Research Imaging Center, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7420. Y. Chen is Research Fellow (E-mail: [email protected]) and W. Lin is Professor of Radiology (E-mail: [email protected]), Department of Radiology, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599. C. Hall is Professor of Neurology, Department of Neurology, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 (E-mail: [email protected]). This work was supported in part by NSF grants SES-06-43663 and BCS-08-26844 and NIH grants UL1-RR025747-01, R01MH086633, and R21 AG033387 to Dr. Zhu, NIH grants GM 70335 and CA 74015 to Dr. Ibrahim and NIH grants R01NS055754 to Dr. Lin. We thank the Editor, an associate editor, and two referees for helpful suggestions, which have improved substantially the present form of this article.
PY - 2009
Y1 - 2009
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=70349774788&partnerID=8YFLogxK
U2 - 10.1198/jasa.2009.tm08096
DO - 10.1198/jasa.2009.tm08096
M3 - Article
C2 - 20174601
AN - SCOPUS:70349774788
SN - 0162-1459
VL - 104
SP - 1203
EP - 1212
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
IS - 487
ER -