Intrinsic Regression Models for Positive-Definite Matrices With Applications to Diffusion Tensor Imaging

Hongtu Zhu H. Zhu is Associate Professor of Biostatistics , J. G. Ibrahim is Alumni Distinguished Professor of Biostatistics , and Y. Li is Ph.D. Student , 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 and W. Lin is Professor of Radiology , 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 . 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.

Yasheng Chen H. Zhu is Associate Professor of Biostatistics , J. G. Ibrahim is Alumni Distinguished Professor of Biostatistics , and Y. Li is Ph.D. Student , 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 and W. Lin is Professor of Radiology , 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 . 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.

Joseph G. Ibrahim H. Zhu is Associate Professor of Biostatistics , J. G. Ibrahim is Alumni Distinguished Professor of Biostatistics , and Y. Li is Ph.D. Student , 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 and W. Lin is Professor of Radiology , 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 . 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.

Yimei Li H. Zhu is Associate Professor of Biostatistics , J. G. Ibrahim is Alumni Distinguished Professor of Biostatistics , and Y. Li is Ph.D. Student , 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 and W. Lin is Professor of Radiology , 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 . 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.

Colin Hall H. Zhu is Associate Professor of Biostatistics , J. G. Ibrahim is Alumni Distinguished Professor of Biostatistics , and Y. Li is Ph.D. Student , 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 and W. Lin is Professor of Radiology , 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 . 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.

Weili Lin H. Zhu is Associate Professor of Biostatistics , J. G. Ibrahim is Alumni Distinguished Professor of Biostatistics , and Y. Li is Ph.D. Student , 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 and W. Lin is Professor of Radiology , 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 . 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.

Abstract

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.

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Intrinsic Regression Models for Positive-Definite Matrices With Applications to Diffusion Tensor Imaging

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