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Covariate-adjusted precision matrix estimation with an application in genetical genomics

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  • T. Tony Cai
  • Hongzhe Li
  • Weidong Liu
  • Jichun Xie

Abstract

Motivated by analysis of genetical genomics data, we introduce a sparse high-dimensional multivariate regression model for studying conditional independence relationships among a set of genes adjusting for possible genetic effects. The precision matrix in the model specifies a covariate-adjusted Gaussian graph, which presents the conditional dependence structure of gene expression after the confounding genetic effects on gene expression are taken into account. We present a covariate-adjusted precision matrix estimation method using a constrained ℓ 1 minimization, which can be easily implemented by linear programming. Asymptotic convergence rates in various matrix norms and sign consistency are established for the estimators of the regression coefficients and the precision matrix, allowing both the number of genes and the number of the genetic variants to diverge. Simulation shows that the proposed method results in significant improvements in both precision matrix estimation and graphical structure selection when compared to the standard Gaussian graphical model assuming constant means. The proposed method is applied to yeast genetical genomics data for the identification of the gene network among a set of genes in the mitogen-activated protein kinase pathway. Copyright 2013, Oxford University Press.

Suggested Citation

  • T. Tony Cai & Hongzhe Li & Weidong Liu & Jichun Xie, 2013. "Covariate-adjusted precision matrix estimation with an application in genetical genomics," Biometrika, Biometrika Trust, vol. 100(1), pages 139-156.
    • Handle: RePEc:oup:biomet:v:100:y:2013:i:1:p:139-156
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    File URL: http://hdl.handle.net/10.1093/biomet/ass058
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    Citations

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    Cited by:

    1. Shan Luo, 2020. "Variable selection in high-dimensional sparse multiresponse linear regression models," Statistical Papers, Springer, vol. 61(3), pages 1245-1267, June.
    2. Xiao Guo & Hai Zhang, 2020. "Sparse directed acyclic graphs incorporating the covariates," Statistical Papers, Springer, vol. 61(5), pages 2119-2148, October.
    3. Huihang Liu & Xinyu Zhang, 2023. "Frequentist model averaging for undirected Gaussian graphical models," Biometrics, The International Biometric Society, vol. 79(3), pages 2050-2062, September.
    4. Yang, Yuehan & Xia, Siwei & Yang, Hu, 2023. "Multivariate sparse Laplacian shrinkage for joint estimation of two graphical structures," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    5. Sihai D. Zhao & T. Tony Cai & Hongzhe Li, 2014. "More powerful genetic association testing via a new statistical framework for integrative genomics," Biometrics, The International Biometric Society, vol. 70(4), pages 881-890, December.
    6. Marco Molinari & Andrea Cremaschi & Maria De Iorio & Nishi Chaturvedi & Alun D. Hughes & Therese Tillin, 2022. "Bayesian nonparametric modelling of multiple graphs with an application to ethnic metabolic differences," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(5), pages 1181-1204, November.
    7. Jie Cheng & Elizaveta Levina & Pei Wang & Ji Zhu, 2014. "A sparse ising model with covariates," Biometrics, The International Biometric Society, vol. 70(4), pages 943-953, December.
    8. Guido Consonni & Luca La Rocca & Stefano Peluso, 2017. "Objective Bayes Covariate-Adjusted Sparse Graphical Model Selection," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 741-764, September.
    9. Fan, Xinyan & Zhang, Qingzhao & Ma, Shuangge & Fang, Kuangnan, 2021. "Conditional score matching for high-dimensional partial graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    10. Fan, Jianqing & Feng, Yang & Xia, Lucy, 2020. "A projection-based conditional dependence measure with applications to high-dimensional undirected graphical models," Journal of Econometrics, Elsevier, vol. 218(1), pages 119-139.
    11. Runmin Shi & Faming Liang & Qifan Song & Ye Luo & Malay Ghosh, 2018. "A Blockwise Consistency Method for Parameter Estimation of Complex Models," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 179-223, December.

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