  Directional tests to compare nested parametric models are developed in the
general context of covariance selection for Gaussian graphical models. The
exactness of the underlying saddlepoint approximation leads to exceptional
accuracy of the proposed approach. This is verified by simulation experiments
with high-dimensional parameters of interest, where the accuracy of standard
asymptotic approximations to the likelihood ratio test and some of its
higher-order modifications fails. The directional p-value isused 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.
