Likelihood Ratio Tests for a Large Directed Acyclic Graph

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.

An Adaptive Test on High-dimensional Parameters in Generalized Linear Models

“…Also, the sparsity assumption regarding q 0 may be relaxed. For example, the sparsity assumption is q 0 = o(n/ log p) in a directed graphical model with TLP constraint ( Li et al, 2019 ). More importantly, the sparsity assumption might not be essential for our proposed method.…”

Penalized mediation models for multivariate data

“…In contrast, when this prior belief is not known, an alternative approach is to treat all variables equal and use Gaussian graphical models to “learn” the network connections. Recent developments of penalized directed acyclic Gaussian graphical models provide a general approach to select a broader class of models than SEMs (Li et al, 2020; Yuan et al, 2019). However, when the assumptions of a SEM are correct, our approach would be more powerful, more computationally efficient, and more likely to result in biologically correct models than a general graphical model.…”

Inferring causal direction between two traits in the presence of horizontal pleiotropy with GWAS summary data

“…to determine between two traits which is the mediator and which is the outcome) lack and are urgently needed while a bi-directional mediation analysis would be problematic as recently reported [59]. In addition, we have not compared with structure equation modeling [60][61][62], Bayesian network [63] and latent causal variable [64] approaches, mainly because of their lack of statistical significance testing or handling hidden confounding. Finally, we have pointed out a strong connection between our proposed CD-Egger method and MR Egger regression in dealing with horizontal pleiotropy.…”