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Sparse directed acyclic graphs incorporating the covariates

Author

Listed:
  • Xiao Guo

    (Northwest University)

  • Hai Zhang

    (Northwest University
    Macau University of Science and Technology)

Abstract

Directed acyclic graphs (DAGs) have been widely used to model the causal relationships among variables using multivariate data. However, covariates are often available together with these data which may influence the underlying causal network. Motivated by such kind of data, in this paper, we incorporate the covariates directly into the DAGs to model the dependency relationships among nodal variables. Specifically, the causal strengths are assumed to be a linear function of the covariates, which enhances the interpretability and flexibility of the model. We fit the model in the $$l_1$$ l 1 penalized maximum likelihood framework and employ a coordinate descent based algorithm to solve the resulting optimization problem. The consistency of the estimator are also established under the regime where the order of nodal variables are known. Finally, we evaluate the performance of the proposed method through a series of simulations and a lung cancer data example.

Suggested Citation

  • Xiao Guo & Hai Zhang, 2020. "Sparse directed acyclic graphs incorporating the covariates," Statistical Papers, Springer, vol. 61(5), pages 2119-2148, October.
  • Handle: RePEc:spr:stpapr:v:61:y:2020:i:5:d:10.1007_s00362-018-1027-8
    DOI: 10.1007/s00362-018-1027-8
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    References listed on IDEAS

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    1. Yang Ni & Francesco C. Stingo & Veerabhadran Baladandayuthapani, 2017. "Sparse Multi-Dimensional Graphical Models: A Unified Bayesian Framework," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 779-793, April.
    2. Min Jin Ha & Wei Sun & Jichun Xie, 2016. "PenPC : A two-step approach to estimate the skeletons of high-dimensional directed acyclic graphs," Biometrics, The International Biometric Society, vol. 72(1), pages 146-155, March.
    3. Mengjie Chen & Zhao Ren & Hongyu Zhao & Harrison Zhou, 2016. "Asymptotically Normal and Efficient Estimation of Covariate-Adjusted Gaussian Graphical Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(513), pages 394-406, March.
    4. Lam, Clifford & Fan, Jianqing, 2009. "Sparsistency and rates of convergence in large covariance matrix estimation," LSE Research Online Documents on Economics 31540, London School of Economics and Political Science, LSE Library.
    5. J. Peters & P. Bühlmann, 2014. "Identifiability of Gaussian structural equation models with equal error variances," Biometrika, Biometrika Trust, vol. 101(1), pages 219-228.
    6. Ali Shojaie & Alexandra Jauhiainen & Michael Kallitsis & George Michailidis, 2014. "Inferring Regulatory Networks by Combining Perturbation Screens and Steady State Gene Expression Profiles," PLOS ONE, Public Library of Science, vol. 9(2), pages 1-16, February.
    7. Chenlei Leng & Cheng Yong Tang, 2012. "Sparse Matrix Graphical Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1187-1200, September.
    8. 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.
    9. Friedman, Jerome H. & Hastie, Trevor & Tibshirani, Rob, 2010. "Regularization Paths for Generalized Linear Models via Coordinate Descent," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 33(i01).
    10. Peng, Jie & Wang, Pei & Zhou, Nengfeng & Zhu, Ji, 2009. "Partial Correlation Estimation by Joint Sparse Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 735-746.
    11. 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.
    12. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
    13. Sung Won Han & Gong Chen & Myun-Seok Cheon & Hua Zhong, 2016. "Estimation of Directed Acyclic Graphs Through Two-Stage Adaptive Lasso for Gene Network Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(515), pages 1004-1019, July.
    14. Fei Fu & Qing Zhou, 2013. "Learning Sparse Causal Gaussian Networks With Experimental Intervention: Regularization and Coordinate Descent," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 288-300, March.
    15. Ali Shojaie & George Michailidis, 2010. "Penalized likelihood methods for estimation of sparse high-dimensional directed acyclic graphs," Biometrika, Biometrika Trust, vol. 97(3), pages 519-538.
    16. Cai, Tony & Liu, Weidong & Luo, Xi, 2011. "A Constrained â„“1 Minimization Approach to Sparse Precision Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 594-607.
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    Cited by:

    1. Mingao Yuan & Fan Yang & Zuofeng Shang, 2022. "Hypothesis testing in sparse weighted stochastic block model," Statistical Papers, Springer, vol. 63(4), pages 1051-1073, August.

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