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Likelihood Ratio Tests for a Large Directed Acyclic Graph

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  • Chunlin Li
  • Xiaotong Shen
  • Wei Pan

Abstract

Inference of directional pairwise relations between interacting units in a directed acyclic graph (DAG), such as a regulatory gene network, is common in practice, imposing challenges because of lack of inferential tools. For example, inferring a specific gene pathway of a regulatory gene network is biologically important. Yet, frequentist inference of directionality of connections remains largely unexplored for regulatory models. In this article, we propose constrained likelihood ratio tests for inference of the connectivity as well as directionality subject to nonconvex acyclicity constraints in a Gaussian directed graphical model. Particularly, we derive the asymptotic distributions of the constrained likelihood ratios in a high-dimensional situation. For testing of connectivity, the asymptotic distribution is either chi-squared or normal depending on if the number of testable links in a DAG model is small. For testing of directionality, the asymptotic distribution is the minimum of d independent chi-squared variables with one-degree of freedom or a generalized Gamma distribution depending on if d is small, where d is number of breakpoints in a hypothesized pathway. Moreover, we develop a computational method to perform the proposed tests, which integrates an alternating direction method of multipliers and difference convex programming. Finally, the power analysis and simulations suggest that the tests achieve the desired objectives of inference. An analysis of an Alzheimer’s disease gene expression dataset illustrates the utility of the proposed method to infer a directed pathway in a gene network.

Suggested Citation

  • Chunlin Li & Xiaotong Shen & Wei Pan, 2020. "Likelihood Ratio Tests for a Large Directed Acyclic Graph," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(531), pages 1304-1319, July.
  • Handle: RePEc:taf:jnlasa:v:115:y:2020:i:531:p:1304-1319
    DOI: 10.1080/01621459.2019.1623042
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    Cited by:

    1. Haoran Xue & Wei Pan, 2020. "Inferring causal direction between two traits in the presence of horizontal pleiotropy with GWAS summary data," PLOS Genetics, Public Library of Science, vol. 16(11), pages 1-30, November.
    2. Shi, Chengchun & Zhou, Yunzhe & Li, Lexin, 2023. "Testing directed acyclic graph via structural, supervised and generative adversarial learning," LSE Research Online Documents on Economics 119446, London School of Economics and Political Science, LSE Library.
    3. Kuang‐Yao Lee & Lexin Li, 2022. "Functional structural equation model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 600-629, April.

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