  Researchers are often interested in drawing inferences regarding the order
between two experimental groups on the basis of multivariate response data.
Since standard multivariate methods are designed for two-sided alternatives,
they may not be ideal for testing for order between two groups. In this article
we introduce the notion of the linear stochastic order and investigate its
properties. Statistical theory and methodology are developed to both estimate
the direction which best separates two arbitrary ordered distributions and to
test for order between the two groups. The new methodology generalizes Roy's
classical largest root test to the nonparametric setting and is applicable to
random vectors with discrete and/or continuous components. The proposed
methodology is illustrated using data obtained from a 90-day pre-chronic rodent
cancer bioassay study conducted by the National Toxicology Program (NTP).
