Testing for and against a stochastic ordering between multivariate multinomial populations

Tests for comparing a multivariate response on a control and on a treatment population are considered. It is assumed that each component of the response is a categorical variable with ordered categories. In some situations it may be believed a priori that the treatment has a nondecreasing effect on each component of the response, and in such cases a test of equality of the two populations with a stochastically ordered alternative is of interest. In other situations, the stochastic ordering assumption may be questioned and one would want to test it as a null hypothesis. The likelihood ratio tests, as well as some chi-square analogues, for both of these situations are studied in the one- and two-sample cases. In the latter testing situation, equality of the two populations is shown to be asymptotically least favorable within the stochastic ordering hypothesis. A numerical example is given to illustrate the use of these tests.