Geometry of rank tests

Details Geometry of rank tests Geometry of rank tests

2006-05-07

arxiv.org/abs/math/0605173v2

Mathematics

Statistics Theory

8 pages, 4 figures. See also http://bio.math.berkeley.edu/ranktests/. v2: Expanded proofs, revised after reviewer comments

Scientific paper

We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. These permutations are the linear extensions of partially ordered sets specified by the data. Our methods refine rank tests of non-parametric statistics, such as the sign test and the runs test, and are useful for the exploratory analysis of ordinal data. Convex rank tests correspond to probabilistic conditional independence structures known as semi-graphoids. Submodular rank tests are classified by the faces of the cone of submodular functions, or by Minkowski summands of the permutohedron. We enumerate all small instances of such rank tests. Graphical tests correspond to both graphical models and to graph associahedra, and they have excellent statistical and algorithmic properties.

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