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General MAX test for complicated categorical phenotypes and genotypes

General MAX test for complicated categorical phenotypes and genotypes. ASHG Washington D.C., USA 2010/11/02-06 Ryo Yamada, Takahisa Kawaguchi Kyoto Univ. Kyoto, Japan. 2 phenotypes (Case, Control) x 3 genotypes (MM,Mm,mm) Multiple genetic models Dominant Recessive Additive.

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General MAX test for complicated categorical phenotypes and genotypes

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  1. General MAX test for complicated categoricalphenotypes and genotypes ASHG Washington D.C., USA 2010/11/02-06 Ryo Yamada, Takahisa Kawaguchi Kyoto Univ. Kyoto, Japan

  2. 2 phenotypes (Case, Control)x3 genotypes (MM,Mm,mm)Multiple genetic modelsDominantRecessiveAdditive

  3. 6 cells in 2x3 table are placed as 6 vectors on a plane

  4. Tables with the same Pearson’s chi-sq value draw an ellipse contour

  5. Tables with the same chi-sq value for 1 df test on 2x3 table draw a parallel line as a contour.

  6. A surface normal represents the test of 1 df

  7. Ellipse → CircleEasy handling

  8. Parallel lines and surface normal of test of 1 df rotate Spherization

  9. Relation between Pearson’s chi-sq and 1-df chi-sq gets simple • In the circular coordinate, the radius to the tangent point is perpendicular to the plane. • In the coordinate with ellipse, the radius is NOT perpendicular to the tangent point. Test Vector a b Tangent point to the smaller circle

  10. Surface normals of three genetic models in “spherized coordinate” Test expression in table form Test expression in table form dom add rec

  11. MAX3 test and MAX test • Two sets of parallel lines with arcs make the test contours for the MAX test MAX3 MAX Arc

  12. Complex categorical phenotypes • Example. A disease is defined as: • A disease is diagnosed when 3 or more out of 4 criteria are met. • 5x3 table

  13. Complex categorical phenotypes • Example. A disease with ordered stages: • A disease is diagnosed when 3 or more out of 4 criteria are met. • 5x3 table #> O # [,1] [,2] [,3] #[1,] 20 40 20 #[2,] 1000 2000 500 #[3,] 1000 2000 500 #[4,] 100 120 120 #[5,] 50 90 30 #> Ts<-MaxTables(O) Ts<-matrix(c(1,1,0,1,1,0,1,1,0,0,0,0,1,1,0, 1,1,0,1,1,0,0,0,0,1,1,0,1,1,0, 1,1,0,0,0,0,1,1,0,1,1,0,1,1,0, 0,0,0,1,1,0,1,1,0,1,1,0,1,1,0, 1,0,0,1,0,0,1,0,0,0,0,0,1,0,0, 1,0,0,1,0,0,0,0,0,1,0,0,1,0,0, 1,0,0,0,0,0,1,0,0,1,0,0,1,0,0, 0,0,0,1,0,0,1,0,0,1,0,0,1,0,0), ncol=N*M,byrow=TRUE)

  14. 黒がテーブル、赤は自由度＝自由度緑は自由度＝１青は観察テーブル左は通常スケール、右は対数スケール黒がテーブル、赤は自由度＝自由度緑は自由度＝１青は観察テーブル左は通常スケール、右は対数スケール • 下が、ステージ検定 > gmtOut$PowOut[1] [1] 1.988130e-25 > > gmtOutc$PowOut[1] [1] 0 > > Max chi-sq = 12.745, corrected P = 0.0029

  15. 同じテーブルをMaxVectorsで • > gmtOutd$PowOut[1] • [1] 0.003005323

  16. How to generalize MAX test defined for 2x3 tables, to NxM tables? • Space • df : 2 → (N-1)(M-1) • 1-df tests • Expression in NxM table should be defined. • Their geometric counterparts are surface normals in df-space.

  17. discrete MAX test continuous MAX test • Discrete MAX test • The model consists of the set of surface normals. • Continuous MAX test • The model is the area that the surface normals demarcate.

  18. Ex. df=3 • The Tips of green triangles are the surface normals for discrete model • Green triangles on the surface are the area of continuous model • Black dots : Observed tables • Red arcs the shortest path from observed table to the model • The arcs concentrate into the tips in “discrete MAX test” • The arcs reaches to the edges of the model area or the tips of the area Discrete MAX test Continusous MAX test

  19. discrete MAX test continuous MAX test

  20. How to construct df-dimensional expression

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

  22. 3 categories in a triangle4 categories in a tetrahedronand so on

  23. NxM vectors can be placed in df-dimensional space

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

  25. Spherization = Eigenvalue decomposition Eigenvalue decomposition

  26. Spherization-based P-value estimation for general MAX test fits well with the permutation method Black : Permutation Red : Sphere method

  27. ... ... ... ... ... ...

  28. (N-1)(M-1) component test matrices of MAX test for NxM tables

  29. 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 wellcome → ryamada@genome.med.kyoto-u.ac.jp • 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

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