[Paper Review] Searching for a source of difference in Gaussian graphical models

Vera Djordjilović, Monica Chiogna|arXiv (Cornell University)|Nov 6, 2018

Bioinformatics and Genomic NetworksBiochemistry, Genetics and Molecular Biology21 references3 citations

TL;DR

This paper proposes a linear-time method to identify the minimal seed set—variables driving differences between two distributions—in Gaussian graphical models using decomposable graphs. By testing conditional independence across cliques in a strong meta Markov model framework, the approach estimates a graphical seed set that explains all distributional differences, demonstrated on a leukemia gene network where BCR, ABL, and a third gene were identified as the source of perturbation.

ABSTRACT

In this work, we look at a two-sample problem within the framework of Gaussian graphical models. When the global hypothesis of equality of two distributions is rejected, the interest is usually in localizing the source of difference. Motivated by the idea that diseases can be seen as system perturbations, and by the need to distinguish between the origin of perturbation and components affected by the perturbation, we introduce the concept of a minimal seed set, and its graphical counterpart a graphical seed set. They intuitively consist of variables driving the difference between the two conditions. We propose a simple testing procedure, linear in the number of nodes, to estimate the graphical seed set from data, and study its finite sample behavior with a stimulation study. We illustrate our approach in the context of gene set analysis by means of a publicly available gene expression dataset.

Motivation & Objective

• To address the limitation of marginal testing in identifying the true origin of perturbation in multivariate data.
• To distinguish between variables that are primary sources of difference (seed set) and those affected by network propagation.
• To develop a computationally efficient, linear-time procedure for estimating the graphical seed set in decomposable Gaussian graphical models.
• To apply the method to gene regulatory networks, particularly in identifying the root cause of disease-related perturbations in chronic myeloid leukemia.
• To provide a conditional, joint-distribution-based alternative to univariate marginal testing in two-sample problems.

Proposed method

• The method is based on strong meta Markov models over decomposable undirected graphs, ensuring variation independence of conditional parameters.
• It decomposes the global null hypothesis of distributional equality into local tests over cliques in the graph.
• For each clique, the method tests equality of conditional distributions given the clique, using permutation-based p-values to control familywise error rate.
• The min-P method is applied across all 41 unique local hypotheses to determine a global significance threshold.
• The estimated graphical seed set consists of nodes in cliques that are significantly different across conditions, with minimal seed set inferred via subset minimization.
• The approach leverages the modular structure of decomposable graphs to simplify likelihood ratio testing to dependence on the seed set alone.

Experimental results

Research questions

• RQ1 Can we identify the minimal set of variables that drive the difference between two multivariate distributions in a graphical model framework?
• RQ2 How can we distinguish between the source of perturbation and downstream effects in a networked system?
• RQ3 What is a computationally efficient, scalable method for estimating the source of difference in high-dimensional Gaussian graphical models?
• RQ4 To what extent does a conditional, joint-distribution approach outperform marginal testing in detecting true perturbation origins?
• RQ5 How can we control error rates when testing multiple conditional hypotheses across a decomposable graph?

Key findings

• The global null hypothesis of equal distributions in the two ALL patient groups was rejected (p-value = 2.06×10⁻¹¹).
• Using the min-P method with 1,640 permutations, a significance threshold of 2.4×10⁻³ was determined for local hypothesis testing.
• The estimated graphical seed set was {25, 613, 6776}, corresponding to the ABL, BCR, and a third gene, which fully explain the difference between the two groups.
• The majority of genes in the network (white nodes in Figure 4) were not part of the seed set, indicating no propagation of the difference beyond the seed set.
• The method successfully localized the source of perturbation to three key genes, consistent with biological knowledge of BCR/ABL in chronic myeloid leukemia.
• The approach demonstrated that the likelihood ratio depends only on the seed set, confirming the theoretical foundation of the method.