{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T07:20:08Z","timestamp":1790666408923,"version":"4.1.0"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1137\/080715822","type":"journal-article","created":{"date-parts":[[2009,7,10]],"date-time":"2009-07-10T18:15:45Z","timestamp":1247249745000},"page":"1117-1134","source":"Crossref","is-referenced-by-count":48,"title":["Convex Rank Tests and Semigraphoids"],"prefix":"10.1137","volume":"23","author":[{"given":"Jason","family":"Morton","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lior","family":"Pachter","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anne","family":"Shiu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bernd","family":"Sturmfels","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Oliver","family":"Wienand","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,7,10]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1993.1020"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00383444"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2006.09.020"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"K. Brown,\n                      Buildings\n                      , Springer, New York, 1989.","DOI":"10.1007\/978-1-4612-1019-1"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2005.08.010"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1080\/00031305.1983.10483126"},{"key":"R7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1979.tb01052.x","volume":"41","author":"Dawid A.","year":"1979","journal-title":"J. Roy. Statist. Soc. Ser. B"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9868.2007.00636.x"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"M. Drton, B. Sturmfels, and S. Sullivant,\n                      Lectures on Algebraic Statistics\n                      , Oberwolfach Seminars, Vol. 40, Birkauser, Basel 2009.","DOI":"10.1007\/978-3-7643-8905-5"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"E. Gawrilow and M. Joswig,\n                      Polymake: A framework for analyzing convex polytopes\n                      , in Polytopes \u2013 Combinatorics and Computation, G. Kalai and G. M. Ziegler, eds., Birkh\u00e4user, Basel, 2000, pp. 43\u201374.","DOI":"10.1007\/978-3-0348-8438-9_2"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548307008838"},{"key":"R12","first-page":"189","author":"Mat\u00fa\u0161 F.","year":"1992","journal-title":"in Proceedings of the Eleventh Prague Conference on Inform. Theory, Stat. Dec. Functions and Random Proc., Academia"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.2307\/3214910"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(03)00155-9"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.2202\/1544-6115.1286"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"J. Pearl,\n                      Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference\n                      , Morgan Kaufman, San Mateo, CA, 1988.","DOI":"10.1016\/B978-0-08-051489-5.50008-4"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.2307\/2984124"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"A. Postnikov,\n                      Permutohedra, Associahedra, and Beyond\n                      , International Mathematics Research Notices, 2005, pp. 1026\u20131106.","DOI":"10.1093\/imrn\/rnn153"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.4171\/dm\/248"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"R. P. Stanley,\n                      Enumerative Combinatorics\n                      , Vol. I, Cambridge University, London, 1997.","DOI":"10.1017\/CBO9780511805967"},{"key":"R21","unstructured":"J. Stembridge,\n                      Maple Packages for Symmetric Functions, Posets, Root Systems, and Finite Coxeter Groups\n                      , www.math.lsa.umich.edu\/~jrs\/maple.html."},{"key":"R22","first-page":"72","volume":"25","author":"Studen\u00fd M.","year":"1990","journal-title":"Kybernetika (Prague)"},{"key":"R23","unstructured":"M. Studen\u00fd, RR. Bouckaert, and T. Kocka,\n                      Extreme Supermodular Set Functions Over Five Variables\n                      , Research report 1977, Institute of Information Theory and Automation, Prague, 2000."},{"key":"R24","unstructured":"M. Studen\u00fd,\n                      Probabilistic Conditional Independence Structures\n                      , Springer Ser. Inform. Sci. Stat., Springer, London, 2005."},{"key":"R25","first-page":"175","volume":"1","author":"Tits J.","year":"1968","journal-title":"Symposia Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0082-0725","issn-type":"print"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1534\/genetics.104.036830"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1093\/bioinformatics\/bti549"},{"key":"R28","doi-asserted-by":"crossref","unstructured":"G. Ziegler,\n                      Lectures on Polytopes\n                      , Grad. Texts in Math. 152, Springer, New York, 1995.","DOI":"10.1007\/978-1-4613-8431-1"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080715822","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:26:06Z","timestamp":1787322366000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080715822"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080715822"],"URL":"https:\/\/doi.org\/10.1137\/080715822","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}