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

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

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

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

A surface normal represents the test of 1 df

Ellipse → Circle Easy handling

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

Relation between Pearson’s chi-sq and 1-df chi-sq gets simple Test Vector a b Tangent point to the smaller circle 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.

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

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

Complex categorical phenotypes Disease Genotype R1 R2 R3 R4 MM Mm mm total C1 + - 200 1260 1470 2930 C2 180 840 980 2000 C3 90 420 490 1000 C4 C5 270 3000 830 4200 4900 9930 Example. A disease is defined as: A disease is diagnosed when 3 or more out of 4 criteria are met. 5x3 table

Complex categorical phenotypes Genotype Stage MM Mm mm total 360 1680 2050 4090 1 270 1260 1470 3000 2 135 630 735 1500 3 90 420 490 1000 4 60 210 245 515 915 4200 4990 10105 #> 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) Example. A disease with ordered stages: A disease is diagnosed when 3 or more out of 4 criteria are met. 5x3 table

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

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

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.

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

Ex. df=3 Discrete MAX test Continusous MAX test 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

discrete MAX test continuous MAX test

How to construct df-dimensional expression

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

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

NxM vectors can be placed in df-dimensional space

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

Spherization = Eigenvalue decomposition

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

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

Comments and questions are wellcome → ryamada@genome.med.kyoto-u.ac.jp 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