Covariance Regression Analysis

Zou, Tao; Lan, Wei; Wang, Hansheng; Tsai, Chih‐Ling

doi:10.1080/01621459.2015.1131699

Cited by 58 publications

(41 citation statements)

References 42 publications

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“…Firm and market-level information, such as industry type, firm's cash flow, stock size, and book-to-market ratio, plays an essential role in determining the synchronicity. Quantifying such association is an important topic in financial theory [43].…”

Section: Introductionmentioning

confidence: 99%

Principal regression for high dimensional covariance matrices

Zhao

Caffo

Luo

2021

Electron. J. Statist.

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This manuscript presents an approach to perform generalized linear regression with multiple high dimensional covariance matrices as the outcome. In many areas of study, such as resting-state functional magnetic resonance imaging (fMRI) studies, this type of regression can be utilized to characterize variation in the covariance matrices across units. Model parameters are estimated by maximizing a likelihood formulation of a generalized linear model, conditioning on a well-conditioned linear shrinkage estimator for multiple covariance matrices, where the shrinkage coefficients are proposed to be shared across matrices. Theoretical studies demonstrate that the proposed covariance matrix estimator is optimal achieving the uniformly minimum quadratic loss asymptotically among all linear combinations of the identity matrix and the sample covariance matrix. Under certain regularity conditions, the proposed estimator of the model parameters is consistent. The superior performance of the proposed approach over existing methods is illustrated through simulation studies. Implemented to a resting-state fMRI study acquired from the Alzheimer's Disease Neuroimaging Initiative, the proposed approach identified a brain network within which functional connectivity is significantly associated with Apolipoprotein E ε4, a strong genetic marker for Alzheimer's disease.

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Section: Introductionmentioning

confidence: 99%

Principal regression for high dimensional covariance matrices

Zhao

Caffo

Luo

2021

Electron. J. Statist.

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“…Covariance regression methods (Hoff & Niu, ; Zou, Lan, Wang, & Tsai, ) offer a potentially more flexible approach, with the ability to handle continuous predictor variables. Methods such as Zou et al () are ultimately sensitive to the property that similarity in predictors corresponds to similarity in responses and thus are similar to kernel methods such as SKAT (Ionita‐Laza, Lee, Makarov, Buxbaum, & Lin, ). The special case of a linear kernel and a restricted set of predictors in Zou et al () has some similarities to our testing approach, but the authors focus on efficiency and comparison of maximum likelihood and other fitting approaches.…”

Section: Introductionmentioning

confidence: 99%

“…Methods such as Zou et al () are ultimately sensitive to the property that similarity in predictors corresponds to similarity in responses and thus are similar to kernel methods such as SKAT (Ionita‐Laza, Lee, Makarov, Buxbaum, & Lin, ). The special case of a linear kernel and a restricted set of predictors in Zou et al () has some similarities to our testing approach, but the authors focus on efficiency and comparison of maximum likelihood and other fitting approaches. The regression method of Hoff and Niu () for a predictor denoted y, which may be continuous, models the covariance as Σ y =Ψ+ Byy T B T and for a positive definite p × p matrix Ψ maintains a positive definite covariance function.…”

Section: Introductionmentioning

confidence: 99%

“…Covariance regression methods (Hoff & Niu, 2012;Zou, Lan, Wang, & Tsai, 2017) offer a potentially more flexible approach, with the ability to handle continuous predictor variables. Methods such as Zou et al (2017) are ultimately sensitive to the property that similarity in predictors corresponds to similarity in responses and thus are similar to kernel methods such as SKAT (Ionita-Laza, Lee, Makarov, Buxbaum, & Lin, 2013).…”

mentioning

confidence: 99%

“…The special case of a linear kernel and a restricted set of predictors in Zou et al (2017) has some similarities to our testing approach, but the authors focus on efficiency and comparison of maximum likelihood and other fitting approaches. The regression method of Hoff and Niu (2012) for a predictor denoted y, which may be continuous, models the covariance as Σ y = Ψ + Byy T B T and for a positive definite p × p matrix Ψ maintains a positive definite covariance function.…”

mentioning

confidence: 99%

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Set‐based differential covariance testing for genomics

Zhou

2019

Stat

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The problem of detecting the changes in covariance for a single pair of genomic features has been studied in some detail but may be limited in importance or general applicability. For testing equality of covariance matrices of a set of features, many methods have been limited to the two‐sample problem and involve varying assumptions on the number of features p versus the sample size n. More general covariance regression approaches are appealing but have been insufficiently structured to provide interpretable testing. To address these deficiencies, we propose a simple uniform framework to test association of covariance matrices with an experimental variable, whether discrete or continuous. We describe four different summary statistics, to ensure power and flexibility under various alternatives, including a new “connectivity” statistic that is sensitive to the changes in overall covariance magnitude. For continuous experimental variables, a natural individual “risk score” is associated with several of the statistics. We establish asymptotic results applicable to both continuous and discrete responses, with relatively mild conditions and allowing for situations where p>n. We also show that the proposed statistics are permutationally equivalent to some existing methods in the two‐sample special case. We demonstrate the power and utility of our approaches via simulation and analysis of real data. The R package CorDiff is published on R CRAN.

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FEMA: Fast and efficient mixed‐effects algorithm for large sample whole‐brain imaging data

Parekh,

Fan,

Frei

et al. 2024

Human Brain Mapping

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The linear mixed‐effects model (LME) is a versatile approach to account for dependence among observations. Many large‐scale neuroimaging datasets with complex designs have increased the need for LME; however LME has seldom been used in whole‐brain imaging analyses due to its heavy computational requirements. In this paper, we introduce a fast and efficient mixed‐effects algorithm (FEMA) that makes whole‐brain vertex‐wise, voxel‐wise, and connectome‐wide LME analyses in large samples possible. We validate FEMA with extensive simulations, showing that the estimates of the fixed effects are equivalent to standard maximum likelihood estimates but obtained with orders of magnitude improvement in computational speed. We demonstrate the applicability of FEMA by studying the cross‐sectional and longitudinal effects of age on region‐of‐interest level and vertex‐wise cortical thickness, as well as connectome‐wide functional connectivity values derived from resting state functional MRI, using longitudinal imaging data from the Adolescent Brain Cognitive DevelopmentSM Study release 4.0. Our analyses reveal distinct spatial patterns for the annualized changes in vertex‐wise cortical thickness and connectome‐wide connectivity values in early adolescence, highlighting a critical time of brain maturation. The simulations and application to real data show that FEMA enables advanced investigation of the relationships between large numbers of neuroimaging metrics and variables of interest while considering complex study designs, including repeated measures and family structures, in a fast and efficient manner. The source code for FEMA is available via: https://github.com/cmig-research-group/cmig_tools/.

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Covariance Regression Analysis

Cited by 58 publications

References 42 publications

Principal regression for high dimensional covariance matrices

Principal regression for high dimensional covariance matrices

Set‐based differential covariance testing for genomics

FEMA: Fast and efficient mixed‐effects algorithm for large sample whole‐brain imaging data

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