General MAX test for complicated categorical phenotypes and

genotypes

ASHGWashington D.C., USA

2010/11/02-06Ryo Yamada, Takahisa Kawaguchi

Kyoto Univ. Kyoto, Japan

2 phenotypes (Case, Control)x

3 genotypes (MM,Mm,mm)

Multiple genetic models

DominantRecessiveAdditive

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 → CircleEasy 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

a

b

• 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

Tangent point to the smaller

circle

Surface normals of three genetic models in “spherized

coordinate”

dom

add

Test expression in table form Test expression in table form

rec

MAX3 test and MAX test

• Two sets of parallel lines with arcs make the test contours for the MAX test

Arc

MAX3 MAX

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

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 - + + + 90 420 490 1000

C5 + + + + 　 270 1260 1470 3000

total 830 4200 4900 9930

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

Phenotype Genotype

Stage MM Mm mm total

0 360 1680 2050 4090

1 270 1260 1470 3000

2 135 630 735 1500

3 90 420 490 1000

4 60 210 245 515

total 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<-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)

Ts<-MaxTables(O)

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

Max chi-sq = 12.745, corrected P = 0.0029

> gmtOut$PowOut[1][1] 1.988130e-25> > gmtOutc$PowOut[1][1] 0> >

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

• 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.

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

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 triangle4 categories in a tetrahedron

and so on

NxM vectors can be placed in df-dimensional space

• 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

Spherization

Spherization = Eigenvalue decomposition

Eigenvalue decomposition

Spherization-based P-value estimation for general MAX test fits well with the permutation method

Black : Permutation

Red : Sphere method

...

......

...

... ...

(N-1)(M-1) component test matrices

of MAX test for NxM tables

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