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

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.

SUBMITTER: Zhu H

PROVIDER: S-EPMC2824554 | biostudies-literature | 2009

REPOSITORIES: biostudies-literature

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

Zhu Hongtu H Chen Yasheng Y Ibrahim Joseph G JG Li Yimei Y Hall Colin C Lin Weili W

Journal of the American Statistical Association 20090101 487

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 r ...[more]

PMID: 20174601

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Project description:BackgroundR package mbend was developed for bending symmetric non-positive-definite matrices to positive-definite (PD). Bending is a procedure of transforming non-PD matrices to PD. The covariance matrices used in multi-trait best linear unbiased prediction (BLUP) should be PD. Two bending methods are implemented in mbend. The first is an unweighted bending with small positive values in a descending order replacing negative eigenvalues (LRS14), and the second method is a weighted (precision-based) bending with a custom small positive value (ϵ) replacing smaller eigenvalues (HJ03). Weighted bending is beneficial, as it relaxes low precision elements to change and it reduces or prohibits the change in high precision elements. Therefore, a weighted version of LRS14 was developed in mbend. In cases where the precision of matrix elements is unknown, the package provides an unweighted version of HJ03. Another unweighted bending method (DB88) was tested, by which all eigenvalues are changed (eigenvalues less than ϵ replaced with 100 × ϵ), and it is originally designed for correlation matrices.ResultsDifferent bending procedures were conducted on a 5 × 5 covariance matrix (V), V converted to a correlation matrix (C) and an ill-conditioned 1000 × 1000 genomic relationship matrix (G). Considering weighted distance statistics between matrix elements before and after bending, weighting considerably improved the bending quality. For weighted and unweighted bending of V and C, HJ03-4 (HJ03, ϵ = 10-4) performed the best. HJ03-2 (HJ03, ϵ = 10-2) ranked better than LRS14 for V, but not for C. Though the differences were marginal, LRS14 performed the best for G. DB88-4 (DB88, ϵ = 10-4) was used for unweighted bending and it ranked the last. This method could perform considerably better with a lower ϵ.ConclusionsR package mbend provides necessary tools for transforming symmetric non-PD matrices to PD, using different methods and parameters. There were benefits in both weighted bending and small positive values in a descending order replacing negative eigenvalues. Thus, weighted LRS14 was implemented in mbend. Different bending methods might be preferable for different matrices, depending on the matrix type (covariance vs. correlation), number and the magnitude of negative eigenvalues, and the matrix size.

Project description:ObjectiveWe aimed to gain more insight into the pathomechanisms of metachromatic leukodystrophy (MLD), by comparing magnitude and direction of diffusion between patients and controls at diagnosis and during follow-up.MethodsFour late-infantile, 16 juvenile and 8 adult onset MLD patients [of which 13 considered eligible for hematopoietic cell transplantation (HCT)] and 47 controls were examined using diffusion tensor imaging. Fractional anisotropy (FA), mean diffusivity (MD), axial diffusivity (AD) and radial diffusivity (RD) were quantified and compared between groups using tract-based spatial statistics (TBSS). Diffusion measures were determined for normal-appearing white matter (NAWM), corpus callosum, thalamus (all based on subject-wise segmentation), and pyramidal tracts, determined with probabilistic tractography. Measures were compared between HCT-eligible patients, non-eligible patients and controls using general linear model and nonparametric permutation analyses (randomise) for TBSS data, considering family-wise error corrected p < 0.05 significant.ResultsThroughout white matter (WM), FA was decreased and MD and RD increased in both patient groups compared to controls, while AD was decreased in NAWM and corpus callosum. In the thalamus, no differences in FA were observed, but all diffusivities were increased in both patient groups. Differences were most pronounced between controls and patients non-eligible for HCT. Longitudinally (median follow-up 3.9 years), diffusion measures remained relatively stable for HCT-treated patients, but were progressively abnormal for non-eligible patients.InterpretationThe observed diffusion measures confirm that brain microstructure is changed in MLD, reflecting different pathological processes including loss of myelin and sulfatide accumulation. The observation of both increased and decreased AD probably reflects a balance between myelin and axonal loss vs. intracellular sulfatide storage in macrophages, depending on region and disease stage.

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

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

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