Directional tests in Gaussian graphical models

TL;DR

This paper introduces directional tests for Gaussian graphical models, demonstrating their exactness for chordal graphs and superior performance over traditional methods in complex scenarios, with applications in veterinary and genetic data analysis.

Contribution

It develops a novel directional testing framework for covariance selection in Gaussian graphical models, with proven exactness for chordal graphs and effective approximation for non-chordal graphs.

Findings

Exact control of test size for chordal graphs

Overperformance of the saddlepoint approximation in non-chordal graphs

Successful application to veterinary and genetic datasets

Abstract

Directional tests to compare incomplete undirected graphs are developed in the general context of covariance selection for Gaussian graphical models. The exactness of the underlying saddlepoint approximation is proved for chordal graphs and leads to exact control of the size of the tests, given that the only approximation error involved is due to the numerical calculation of two scalar integrals. Although exactness is not guaranteed for non-chordal graphs, the ability of the saddlepoint approximation to control the relative error leads the directional test to overperform its competitors even in these cases. The accuracy of our proposal is verified by simulation experiments under challenging scenarios, where inference via standard asymptotic approximations to the likelihood ratio test and some of its higher-order modifications fails. The directional approach is used to illustrate the assessment of Markovian dependencies in a dataset from a veterinary trial on cattle. A second example with microarray data shows how to select the graph structure related to genetic anomalies due to acute lymphocytic leukemia.