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Published July 10, 2009 | Submitted + Published
Journal Article Open

Convex Rank Tests and Semigraphoids

Abstract

Convex rank tests are 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. Each class consists of the linear extensions of a partially ordered set specified by data. Our methods refine existing rank tests of nonparametric statistics, such as the sign test and the runs test, and are useful for exploratory analysis of ordinal data. We establish a bijection between convex rank tests and probabilistic conditional independence structures known as semigraphoids. The subclass of submodular rank tests is derived from faces of the cone of submodular functions or from Minkowski summands of the permutohedron. We enumerate all small instances of such rank tests. Of particular interest are graphical tests, which correspond to both graphical models and to graph associahedra.

Additional Information

© 2009 Society for Industrial and Applied Mathematics. Submitted: 16 February 2008. Accepted: 22 January 2009. Published online: 10 July 2009. This paper extends the note Geometry of Rank Tests, in Proceedings of the Probabilistic Graphical Models (PGM 3) conference, Prague, 2006. Research for this author [JM] was supported as part of the DARPA Program Fundamental Laws of Biology. Research for the second and fourth authors was supported as part of the DARPA Program Fundamental Laws of Biology. Research for the third author was supported by a Lucent Technologies Bell Labs Graduate Research Fellowship. Research for this author [OW] was supported by the Wipprecht Foundation. Our research on rank tests originated in discussions with Olivier Pourquié and Mary-Lee Dequéant as part of the DARPA Program Fundamental Laws of Biology. We thank Milan Studený and František Matúš for helpful comments.

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