Calculation of MAX test P value with geometric characterization of

2x3 table tests

2010 Joint Statistical Meetings Vancouver, Canada2010/08/05

Ryo YamadaKyoto Univ. Kyoto Japan

Case-Controlx

MM,Mm,mm genotypes

Genetic models

DominantRecessive

Somewhere between them

Contingency tables test with ellipse and sphere

• N categories x M categories

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

２ｘ３ =6 vectors

２ｘ４ =8 vectors

• 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

Any test of 1 df is a plane tangent to the ellipsoid or the spher

e• Tables with the same Psn’s chi-sq are ellipsoid/sphere.• Tables with the same stat value of test of 1 df are a plan

e.

Any test of 1 df = a vector in the space

a

b

• In the spherized coordinate, the radius to the tangent point is perpendicular to the plane.

• In the coordinate with ellipsoid, NOT perpendicular.

Test Vector

Tangent point to the smaller

sphere.

MAX3 test for 2x3 table• 3 tests of 1 df in matrix-form and vector-form

– Find tangent points for three test planes.

dom

recadd

The contour lines of MAX3 test stats are

combinations of 3 parallel line sets

To which model fit most?It depends where the observed

table locates.

DominantDominant

Recessive

Recessive

Additive

Additive

MAX testAny model between dominant and recessive • Two sets of parallel lines with arcs

Arc

P value Probability of tables with a statistic

value equal to or more than the observed table

Pr(θ): The probability further than the contour line in the direction

How to integrate?

• For 2x3 tables (df=2),

• Sum of the probabilities in evenly spaced multiple directions gives the good approximation of the integral.

Multiple tests of 1 df on a table with higher dimensions

How to integrate?

• For NxM tables (df=(N-1)x(M-1)),

• Sum of the probabilities in evenly spaced multiple directions gives the good approximation of the integral.

How to integrate?

• For NxM tables (df=(N-1)x(M-1)),

• Sum of the probabilities in evenly spaced multiple directions gives the good approximation of the integral.

How to integrate?

• For NxM tables (df=(N-1)x(M-1)),

• Sum of the probabilities in randomly spaced multiple directions gives the good approximation of the integral.

How to integrate?

• For NxM tables (df=(N-1)x(M-1)),

• Sum of the probabilities in randomly spaced multiple directions gives the good approximation of the integral.

Random directions are easy to be sampled in the “spherized” coordinat

e.

Random directions are easy to be sampled in the “spherized” coordinat

e.

Random directions are easy to be sampled in the “spherized” coordinat

e.

Random directions are easy to be sampled in the “spherized” coordinat

e.

Random directions are easy to be sampled in the “spherized” coordinat

e.

Random directions are easy to be sampled in the “spherized” coordinat

e.

P-value estimation using random points on the sphere fits well with

the permutation method

Black : Permutation

Red : Sphere method

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