1. 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. 2 phenotypes (Case, Control) x 3 genotypes (MM,Mm,mm) Multiple genetic models Dominant Recessive Additive

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 → Circle Easy 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
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.

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
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

13. 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,]
#[2,]
#[3,]
#[4,]
#[5,]
#>
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

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
下が、ステージ検定
> gmtOut$PowOut[1]
[1] e-25
>
> gmtOutc$PowOut[1]
[1] 0
Max chi-sq = , corrected P =

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

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
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 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

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 triangle 4 categories in a tetrahedron and 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

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

27. ...
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28. (N-1)(M-1) component test matrices of MAX test for NxM tables

29. 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;
Comments and questions are wellcome →
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