Multivariate generalizations of jonckheere's test for ordered alternatives

In many dose-response studies, each of several independent groups of animals is treated with a different dose of a substance. Many response variables are then measured on each animal. The distributions of the response variables may be nonnormal, and Jonckheere's (1954) test for ordered alternatives in the one-way layout is sometimes used to test whether the level of a single variable increases with increasing dose. In some applications, however, it is important to consider a set of response variables simultaneously. For instance, an increase in each of certain enzymes in the blood serum may suggest liver damage. To test whether these enzyme levels increase with increasing dose, it may be preferable to consider these enzymes as a group, rather than individually.

I propose two multivariate generalizations of Jonckheere's univariate test. Each multivariate test statistic is a function of coordinate-wise Jonckheere statistics—one a sum, the other a quadratic form. The sum statistic can be used to test the alternative hypothesis that each variable is stochastically increasing with increasing dose. The quadratic form statistic is designed for the more general alternative hypothesis that each variable is stochastically ordered with increasing dose.

For each of these two alternatives, I also propose a multivariate generalization of a normal theory test described by Puri (1965). I examine the asymptotic distributions of the four test statistics under the null hypothesis and under translation alternatives and compare each distribution-free test to the corresponding normal theory test in terms of asymptotic relative efficiency.

The multivariate Jonckheere tests are illustrated using does-response data from a subchronic toxicology study carried out by the National Toxicology Program. Four groups of ten male rats each were treated with increasing doses of vinylidene flouride, and the serum enzymes SDH, SGOT, and SGPT were measured. A comparison of univariate Jonckheere tests on each variable, bivariate tests on SDH and SGOT, and multivariate tests on all three variables gives insight into the behavior of the various procedures.