1. Calculation of MAX test P value with geometric characterization of 2x3 table tests 2010 Joint Statistical Meetings Vancouver, Canada 2010/08/05 Ryo Yamada Kyoto Univ. Kyoto Japan

2. Case-Control x MM,Mm,mm genotypes Genetic models Dominant Recessive Somewhere between them

3. Contingency tables test with ellipse and sphere N categories x M categories

4. K categories are expressed as (K-1)-simplex or K-complete graph

5. 3 categories in a triangle 4 categories in a tetrahedron and so on

6. NxM vectors can be placed in df-dimensional space ２ｘ３ =6 vectors ２ｘ４ =8 vectors

7. Expected values determine shape of ellipsoid Pearson’s chi-sq values draws ellipsoid contour lines, which can be spherized Tables on a contour line have the same statistic value Spherization

8. Spherization = Eigenvalue decomposition Eigenvalue decomposition

9. Any test of 1 df is a plane tangent to the ellipsoid or the sphere Tables with the same Psn’s chi-sq are ellipsoid/sphere. Tables with the same stat value of test of 1 df are a plane.

10. Any test of 1 df = a vector in the space a b In the spherized coordinate, the radius to the tangent point is perpendicular to the plane. In the coordinate with ellipsoid, NOT perpendicular. Test Vector Tangent point to the smaller sphere.

11. MAX3 test for 2x3 table 3 tests of 1 df in matrix-form and vector-form –Find tangent points for three test planes. dom rec add

12. The contour lines of MAX3 test stats are combinations of 3 parallel line sets

13. To which model fit most? It depends where the observed table locates. Dominant Recessive Additive

14. MAX test Any model between dominant and recessive Two sets of parallel lines with arcs Arc

15. P value Probability of tables with a statistic value equal to or more than the observed table

16. Pr(θ): The probability further than the contour line in the direction

17. How to integrate? For 2x3 tables (df=2), Sum of the probabilities in evenly spaced multiple directions gives the good approximation of the integral.

18. Multiple tests of 1 df on a table with higher dimensions

19. How to integrate? For NxM tables (df=(N-1)x(M-1)), Sum of the probabilities in evenly spaced multiple directions gives the good approximation of the integral.

20. How to integrate? For NxM tables (df=(N-1)x(M-1)), Sum of the probabilities in evenly spaced multiple directions gives the good approximation of the integral.

21. How to integrate? For NxM tables (df=(N-1)x(M-1)), Sum of the probabilities in randomly spaced multiple directions gives the good approximation of the integral.

22. How to integrate? For NxM tables (df=(N-1)x(M-1)), Sum of the probabilities in randomly spaced multiple directions gives the good approximation of the integral.

23. Random directions are easy to be sampled in the “spherized” coordinate.

29. P-value estimation using random points on the sphere fits well with the permutation method Black : Permutation Red : Sphere method

30. Collaborators –Graduate school of Medicine, Kyoto University, Kyoto, Japan Takahisa Kawaguchi Katsura Hirosawa Meiko Takahashi Fumihiko Matsuda –Lab for Autoimmune Diseases, CGM, RIKEN, Yokohama, Japan Yukinori Okada Yuta Kochi Akari Suzuki Kazuhiko Yamamoto R code and web-based calculator of the method for 2x3 table presented are available at; http://www.genome.med.kyoto-u.ac.jp/wiki_tokyo/index.php/Estimate_of_P- value_of_MAX_for_2x3_tables Comments and questions are welcome → ryamada@genome.med.kyoto-u.ac.jpryamada@genome.med.kyoto-u.ac.jp