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.

Inference of directed relations given some unspecified interventions, that is, the target of each intervention is not known, is important yet challenging. For instance, it is of high interest to unravel the regulatory roles of genes with inherited genetic variants like singlenucleotide polymorphisms (SNPs), which can be unspecified interventions because of their regulatory function on some unknown genes. In this article, we test hypothesized directed relations with unspecified interventions. First, we derive conditions to yield an identifiable model. Unlike classical inference, hypothesis testing requires identifying ancestral relations and relevant interventions for each hypothesis-specific primary variable, referring to as causal discovery. Towards this end, we propose a peeling algorithm to establish a hierarchy of primary variables as nodes, starting with leaf nodes at the hierarchy's bottom, for which we derive a difference-of-convex (DC) algorithm for nonconvex minimization. Moreover, we prove that the peeling algorithm yields consistent causal discovery, and the DC algorithm is a low-order polynomial algorithm capable of finding a global minimizer almost surely under the data generating distribution. Second, we propose a modified likelihood ratio test, eliminating nuisance parameters to increase power. To enhance finite-sample performance, we integrate the modified likelihood ratio test with a data perturbation scheme by accounting for the uncertainty of identifying ancestral relations and relevant interventions. Also, we show that the distribution of a data-perturbation test statistic converges to the target distribution in high dimensions. Numerical examples demonstrate the utility and effectiveness of the proposed methods, including an application to infer gene regulatory networks.

Mediation analysis is a useful tool in biomedical research to investigate how molecular phenotypes, such as gene expression, mediate the effect of an exposure on health outcomes. However, commonly used mean-based total mediation effect measures may suffer from cancellation of component-wise mediation effects of opposite directions in the presence of high-dimensional omics mediators. To overcome this limitation, a variance-based R-squared total mediation effect measure has been recently proposed, which, nevertheless, relies on the computationally intensive nonparametric bootstrap for confidence interval estimation. In this work, we formulate a more efficient two-stage cross-fitted estimation procedure for the R-squared measure. To avoid potential bias, we perform iterative Sure Independence Screening (iSIS) in two subsamples to exclude the non-mediators, followed by ordinary least squares (OLS) regressions for the variance estimation. We then construct confidence intervals based on the newly-derived closed-form asymptotic distribution of the R-squared measure. Extensive simulation studies demonstrate that the proposed procedure is hundreds of times more computationally efficient than the resampling-based method with comparable coverage probability. Furthermore, when applied to the Framingham Heart Study, the proposed method replicated the established finding of gene expression mediating age-related variation in systolic blood pressure and discovered the role of gene expression profiles in the relationship between sex and high-density lipoprotein cholesterol. The proposed cross-fitted interval estimation procedure is implemented in R package RsqMed.

Estimating phenotype networks is a growing field in computational biology. It helps deepen the understanding of disease etiology and is useful in many applications. In this study, we present a method that constructs a phenotype network by assuming a Gaussian linear structure model embedding a directed acyclic graph (DAG). We utilize genetic variants as instrumental variables and show how our method only requires access to summary statistics from a genome-wide association study (GWAS) and a reference panel of genotype data. Besides estimation, a distinct feature of the method is its summary statistics-based likelihood ratio test on directed edges. We applied our method to estimate a causal network of 29 cardiovascular-related proteins and linked the estimated network to Alzheimer's disease (AD). A simulation study was conducted to demonstrate the effectiveness of this method. An R package implementing the proposed method and an R Shiny App for the visualization of the estimated protein network are being made available.