  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.
