### 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.

Original language | English (US) |
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Journal | Journal of the American Statistical Association |

DOIs | |

State | Published - Jan 1 2019 |

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### Keywords

- Directed acyclic graph
- Gene network
- High-dimensional inference
- L0-regularization
- Nonconvex minimization

### Cite this

**Likelihood Ratio Tests for a Large Directed Acyclic Graph.** / Li, Chunlin; Shen, Xiaotong; Pan, Wei.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Likelihood Ratio Tests for a Large Directed Acyclic Graph

AU - Li, Chunlin

AU - Shen, Xiaotong

AU - Pan, Wei

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Directed acyclic graph

KW - Gene network

KW - High-dimensional inference

KW - L0-regularization

KW - Nonconvex minimization

UR - http://www.scopus.com/inward/record.url?scp=85068067376&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068067376&partnerID=8YFLogxK

U2 - 10.1080/01621459.2019.1623042

DO - 10.1080/01621459.2019.1623042

M3 - Article

AN - SCOPUS:85068067376

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

ER -