Transcript Slides

FastANOVA: an Efficient Algorithm
for Genome-Wide Association Study
Xiang Zhang Fei Zou Wei Wang
University of North Carolina at Chapel Hill
Speaker: Xiang Zhang
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Genotype-phenotype association study
• Goal: finding genetic factors causing phenotypic difference
Mouse genome
Phenotype variation
http://www.bcgsc.ca
http://www.jax.org/
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Genotype-phenotype association study
Chrom1
bp3,568,717
Chrom6
bp120,323,342
• Single Nucleotide Polymorphism…… A A A C G …… A A T C C ……
 Mutation of a single nucleotide
(A,C,T,G)
 The most abundant source of
genotypic variation
 Server as genetic markers of
locations in the genome
 High throughput genotyping -thousands to millions of SNPs
…… A A A C G …… A A T C C ……
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…… A A A C G …… A A T C G ……
…… A A A C G …… A A T C G ……
…… A A T C G …… A A T C C ……
…… A A T C G …… A A T C C ……
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…… A A T C G …… A A T C C ……
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…… A A T C G …… A A T C C ……
Thousands to millions of SNPs
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Genotype-phenotype association study
• Genotype
Phenotype
value
SNPs
 SNPs can be represented as binary
{0,1} (e.g. inbred mouse strains)
• Quantitative phenotypes
 Body weight, blood pressure, tumor
size, cancer susceptibility, ……
• Question
 Which SNPs are the most highly
associated with the phenotype?
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A simple example: single marker
association study
• Partition individuals into groups
according to genotype of a SNP
• Do a statistic (t, ANOVA) test
• Repeat for each SNP
Phenotype
value
SNPs
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Two-locus association mapping
• Many phenotypes are complex traits
 Due to the joint effect of multiple genes
 Single marker approach may not suffice
• Consider SNP-SNP interactions
 Four possible genotype combinations for each SNP-pair:
00, 01, 10, 11
 Split mice into four groups according to the genotype of
each SNP-pair
 Do statistic test for each SNP-pair
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Statistical issue
• Multiple test problem
 , the family-wise error
  1  (1   )n
 Do n tests with Type I error
'
rate  is
'
• Example
 Performing 20 tests with Type I error=0.05, family-wise
error rate = 0.64
 64% probability to get at least one spurious result
• Solution
 permutation test
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Permutation test
• K permutations of
phenotype values
• For each permutation, find
the maximum test value
• Given Type I error α, the
critical value Fα is αK-th
largest value among K
maximum values
• SNP-pairs whose test
values are greater than Fα
are significant
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Genome-wide association study
• What’s GWA?
 Simple Idea: search for the associations in the
whole genome
• Hard to implement
 Enormous search space: 10,000 SNPs and
1,000 permutations, number of SNP-pairs
need to be tested: 5 ×1010
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Preliminary: ANOVA test and F-statistic
• ANOVA test
 To determine whether the group means
are significantly different
 Partition Total sum of squares into
Between-group sum of squares and
Within-group sum of squares
SST
SSB
SSW
• F-statistic
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SNPs {X1, X2, …, XN},
a quantitative phenotype Y
Single SNP test -- F(Xi, Y)
SNP-pair test -- F(XiXj, Y)
SSB
F  C
SSW
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Problem Formalization
• Dataset: M individuals, N SNPs {X1, X2, …, XN}, a quantitative
phenotype Y, and its K permutations {Y1, Y2, …, Yk}.
• Maximum ANOVA test (F-statistic) value of permutation Yk
FYk = max {F(XiXj, Yk)|1≤i<j≤N}
• Problem 1: Given Type I error threshold α, find critical value
Fα, which is αK-th largest value among {FYk|1≤k≤K}
• Problem 2: Given the threshold Fα, find all significant SNPpairs such that F(XiXj, Y)≥ Fα
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Brute force approach
• Problem 1: Permutation test to find critical value
 For permutation Yk, test all SNP-pairs to find the
maximum test value FYk
 Repeat for all permutations
 Report αK-th largest value in {FYk|1≤k≤K}
• Problem 2: Finding significant SNP-pairs
 For phenotype Y, test all SNP-pairs and report the SNPpairs whose test values are above Fα
Problem 1 is more demanding due to large number of permutations
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Overview of FastANOVA
• Goal: Scale large permutation test to genome-wide
• Question: Do we have to perform ANOVA tests for
every SNP-pair and repeat for all permutations?
• Idea:
 Develop an upper bound: to filter out SNP-pairs having no
chance to become significant (all nodes on the same level
of the search tree, no sub-tree pruning, how?)
 Efficiently compute the upper bound: calculate the upper
bound for a group of SNP-pairs together (possible?)
 Identify redundant computations in the permutation tests
(reuse computations, how?)
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The upper bound
• For any SNP-pair (XiXj)
equivalent
SSB (XiXj, Y) ≥θ
F(XiXj, Y) ≥ Fα
• Bound on SSB
Fixed for given Fα
SSB ( X i X j , Y )  SSB ( X i , Y )  R1  R2
Need to be greater than θ for
(XiXj) to be significant
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The upper bound
Given Xi ,Xj , and Y
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Xj
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na  min{#X j  1, # X j  0 | X i  0}
nb  min{# X j  1, # X j  0 | X i  1}
SSB ( X i X j , Y )  SSB ( X i , Y )  R1  R2
na  2
Constant
nb  1
f(na) f(nb)
Only depend on the
genotype of Xj
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Applying the upper bound
For a given Xi , let AP= {(XiXj)|i+1≤j≤N}.
Index the SNP-pairs in AP in the 2D space of (na , nb).
(X1X3)
(X1X5)
(X1X6)
X1 X2 X3 X4 X5 X6
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Key properties
SSB ( X i X j , Y )  SSB ( X i , Y )  R1  R2
f(na) f(nb)
• Maximum possible size: (M / 4  1)2
• Many SNP-pairs share the same entry
• All SNP-pairs in the same entry have the
same upper bound
• The indexing structure does not depend
on the phenotype permutations
Same upper bound value
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Schema of FastANOVA
(for permutation test)
• For each Xi , index the SNP-pairs
{(XiXj)|i+1≤j≤N} in the 2D space of (na, nb)
• For each permutation, find the candidate
SNP-pairs by accessing the indexing
structure
 Candidates are SNP-pairs whose upper bounds
are above the threshold.
 The dynamic threshold is the maximum test
value found so far.
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Complexity of FastANOVA
• Time complexity
 FastANOVA: O(N2M + KNM2 +CM)
 Brute force: O(KN2M)
• Space complexity
 O((N+K)M)
N = # SNPs
M << N
M = # individuals
K = # permutations
C = # candidates
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Brute force v.s. FastANOVA
Two orders of magnitude faster than the brute force alternative
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runtime (sec.)
runtime (sec.)
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brute force approach
FastANOVA
brute force approach
FastANOVA
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type I error threshold
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10k
18k
26k
34k
number of SNPs
42k
#SNPs = 44k, #individuals = 26,
phenotype: metabolism (water intake)
SNP and phenotype data available at http://www.jax.org
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Pruning power of the bound
100.0%
99.9%
99.9%
99.8%
99.8%
99.7%
99.7%
99.6%
0.05
0.04
0.03
0.02
type I error threshold
0.01
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Runtime of each component
One time cost
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Future work
• Association study involving more than two SNPs
 Computationally much more demanding
 Three loci VS. two loci: in the order of number of SNPs
• Association study for heterozygous case
 SNPs are encoded as ternary variables {0, 1, 2}
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Thank You !
Questions?
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