Transcript JSMvancouver2010 - Statistical Genetics, Kyoto University

Calculation of MAX test P value
with geometric characterization of
2x3
table tests
2010 Joint Statistical Meetings Vancouver, Canada
2010/08/05
Ryo Yamada
Kyoto Univ. Kyoto Japan
Case-Control
x
MM,Mm,mm genotypes
Genetic models
Dominant
Recessive
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 triangle
4 categories in a tetrahedron
and so on
NxM vectors can be placed
in df-dimensional space
２ｘ４=8 vectors
２ｘ３=6 vectors
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
Eigenvalue
decomposition
Any test of 1 df is a plane
tangent to the ellipsoid or the
sphere
• Tables with the same Psn’s chi-sq are ellipsoid/sphere.
• Tables with the same stat value of test of 1 df are a plane.
Any test of 1 df = a vector in the space
Test Vector
a
b
Tangent point
to the smaller
sphere.
• In the spherized coordinate, the radius to the tangent
point is perpendicular to the plane.
• In the coordinate with ellipsoid, NOT perpendicular.
MAX3 test for 2x3 table
• 3 tests of 1 df in matrix-form and vector-form
– Find tangent points for three test planes.
add
dom
rec
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.
Additive
Recessive
Dominant
Dominant
Recessive
Additive
MAX test
Any 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”
coordinate.
Random directions are easy to be
sampled in the “spherized”
coordinate.
Random directions are easy to be
sampled in the “spherized”
coordinate.
Random directions are easy to be
sampled in the “spherized”
coordinate.
Random directions are easy to be
sampled in the “spherized”
coordinate.
Random directions are easy to be
sampled in the “spherized”
coordinate.
P-value estimation using random
points on the sphere fits well with
the permutation method
Black : Permutation
Red : Sphere method
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_Pvalue_of_MAX_for_2x3_tables
Comments and questions are welcome → [email protected]
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Collaborators
– Graduate school of Medicine, Kyoto University, Kyoto, Japan
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Takahisa Kawaguchi
Katsura Hirosawa
Meiko Takahashi
Fumihiko Matsuda
– Lab for Autoimmune Diseases, CGM, RIKEN, Yokohama, Japan
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Yukinori Okada
Yuta Kochi
Akari Suzuki
Kazuhiko Yamamoto