A new formulation is proposed for multivariate test, consisting of not only a hierarchy of numerous tests organised in a lattice taxonomy of properties that come from different combinations of multi-variates and represent different factors associated with the rejection of null hypothesis, but also by a theory of property-oriented rejection. Located on the bottom level of this taxonomy is a conventional formulation of multivariate test, featured by a property with the weakest collegiality and a rejection with the largest p value. From one level up to the next, the dimension of rejection increases, the collegiality of properties strengthen, and the p values reduce, until the top level that is featured by a property with the strongest collegiality and a rejection with the smallest p value. Instead of traversing all the combinations in the taxonomy, an easy implementation is developed to identify distinctive properties by the best first path (BFP) in a lattice taxonomy of an appropriate number of intrinsic factors that are obtained after decoupling second-order dependence cross multivariate statistics and discarding those non-distinctive components. Even away off this BFP, if needed, a particular combination of intrinsic factors may be conveniently tested in such a taxonomy too. Moreover, further improvement is made by considering some dependence of higher than second order, with the top level p value refined into one upper bound that is obtained by directional test. Furthermore, detailed implementations are also provided for applications to genome-scale sequencing and expression, with particular emphasis on multivariate phenotype-targeted test for expression profile analyses.