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
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(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 → [email protected]