The document proposes a new method for modifying the Cochran-Armitage trend test to address population structure in genome-wide association studies (GWAS). Existing methods control for population structure using genomic control or principal components, but have limitations. The new method imposes a common correlation matrix based on a kinship matrix and estimates effect sizes while accounting for population structure, allowing p-values to be reordered to improve power over other methods. Simulations show the new method controls type 1 error rates similarly to genomic control but can have greater power, especially when genotyping errors are present.

1. Modifying the
Cochran-Armitage
trend test to
address population
Modifying the structure in GWAS
Gary K. Chen
Department of
Cochran-Armitage trend test to Preventive
Medicine
USC
address population structure in 1. Background
2. Proposed
GWAS Method
3. Simulations
Gary K. Chen
Department of Preventive Medicine
USC
August 25, 2008

2. Modifying the
Outline Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
1. Background 1. Background
2. Proposed
Method
3. Simulations
2. Proposed Method
3. Simulations

3. Modifying the
Genome wide association studies of Cochran-Armitage
trend test to
address population
cases and controls structure in GWAS
Gary K. Chen
Department of
Interested in differences between cases and Preventive
Medicine
controls USC
Estimate the correlation between predictors 1. Background
2. Proposed
(e.g. genotypes) and outcomes (disease Method
status) 3. Simulations
Common methods: logistic regression,
Pearson’s χ2 , Cochran-Armitage trend test
Confounding can be a serious problem:
inflate type I errors
Some non-causal SNPs can be correlated to
case control status:
Population structure
Artifacts from sample preparation and/or
genotyping

4. Modifying the
Existing approaches: genomic Cochran-Armitage
trend test to
address population
control structure in GWAS
Gary K. Chen
Department of
Let T be a test statistic Preventive
Medicine
Estimate Var (T ) at some random markers USC
assumed to be unlinked to disease 1. Background
Define inflation factor as λ = (Rp1 p2 (T ) ))
var
(1+F 2. Proposed
Method
T now scaled by λ0.5 3. Simulations
Controls type I error to nominal rates
Can be anti-conservative (Marchini et al,
Nat. Gen. 2004)
New GCF method compares against F
instead of T distribution
P-values are not re-ordered. Other
approaches may yield more interesting
rankings.
Reference: Devin and Roeder, Biometrics 1999

5. Modifying the
Existing approaches: structured Cochran-Armitage
trend test to
address population
association structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
Parameters estimated by MCMC Gibbs USC
sampling 1. Background
Estimate P, describing population specific 2. Proposed
Method
allele frequencies 3. Simulations
Estimate Q, describing individual specific
admixture proportions
Significance tested through likelihood
ratio:
ˆ ˆ
Pr1 (C ;P1 ,Q)
Λ= ˆ ˆ
Pr0 (C ;P0 ,Q)
Computationally intensive
Reference: Pritchard et al, Genetics 2000

6. Modifying the
Existing approaches: principal Cochran-Armitage
trend test to
address population
components structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
Axes of variation (ancestry vectors) USC
computed by singular value decomposition 1. Background
Regress genotypes on ancestry vector. 2. Proposed
Method
Residuals are adjusted genotypes. 3. Simulations
Perform analogous regression with
phenotypes.
Method can be very sensitive to small
differences between case-controls
e.g. differences in genotyping errors
Can lead to power loss if researcher ignores
these effects
Reference: Price et al, Nat Gen 2006

7. Modifying the
An outline Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
1. Background 1. Background
2. Proposed
Method
3. Simulations
2. Proposed Method
3. Simulations

8. Modifying the
Our proposed method Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
Combines ideas from genomic control and
1. Background
principal components 2. Proposed
Method
A common correlation matrix is imposed 3. Simulations
on each SNP
However, p-values can be re-ordered when
structure is present
For SNP j: Yj = µj + βj Sj + Σj

9. Modifying the
A potential model for variance Cochran-Armitage
trend test to
address population
structure of SNP Sj structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
Beta-binomial model: Balding and USC
Nichols, 1995 1. Background
2. Proposed
Var (Sj ) = 2pj (1 − pj )k Method
3. Simulations
Given a population l = 1, 2, ..L
Diagonal of k:1 + Fl
Off-diagonal of k:2F or 0
m
Sj∗ Sj∗T s −2pˆ
ˆ
k= j=1 ∗
where sn,j = √ n,j j
M 2pj (1−pj )
ˆ ˆ
Ancestral freq pj difficult to estimate
ˆ
Can use half the sample mean as pj , but
ˆ
maybe biased

10. Modifying the
Variance structure for new method Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
For SNP j, Σj = σj2 K 1. Background
σj2 is variance of pooled sample 2. Proposed
Method
K is an empirically estimated kinship matrix 3. Simulations
Genotype correlation between subject m
and n
km,n element in K matrix:
M (snj −2pj )(smj −2pj )
ˆ ˆ
j=1 2pj (1−pj )
ˆ ˆ

11. Modifying the
Mean structure Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
1. Background
For SNP j, µj = C βj 2. Proposed
Method
µj is vector across N individuals 3. Simulations
C is Nx2 matrix
βj is a length 2 vector

12. Modifying the
Best Linear Unbiased Estimates Cochran-Armitage
trend test to
address population
(BLUE) structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
ˆ
βj = (C T K −1 C )C T K −1 Sj 1. Background
2. Proposed
ˆ ˆ
Vj = σ 2 (C T K −1 C )−1
Method
j 3. Simulations
ˆ SjT (K −1 −H)Sj
σj2 = N−2
H = K C (C T K −1 C )−1 C T K −1
−1
Assess significance with Wald statistic:
2
βˆ
j2
Tj = vˆ 2
j2

13. Modifying the
An outline Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
1. Background 1. Background
2. Proposed
Method
3. Simulations
2. Proposed Method
3. Simulations

14. Modifying the
Simulation Study Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
Goal: simulate up to 10 hidden
1. Background
sub-populations 2. Proposed
Simulate data for 100,000 SNPs Method
3. Simulations
Draw ancestral allele freq U ∼ [.1, .9]
Strata specific freq: Balding Nichols model
Beta ∼ (p 1−Fi , (1 − p) 1−Fi )
Fi Fi
Induce a 1% genotyping error in cases
(N ∼ (0, .01))

15. Modifying the
Empirical type I errors Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
1. Background
Alpha %Geno Arm. GC PC New 2. Proposed
Method
Error Test 3. Simulations
.05 0 .265 .047 .056 .050
1 .261 .047 .055 .050
−4
1e 0 .011 5e −5 7e −5 8e −5
1 .025 6e −5 2.3e −4 1.9e −4

16. Modifying the
Power Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
Alpha %Geno Arm. GC PC New 1. Background
Error Test 2. Proposed
Method
.05 0 .55 .31 .65 .65 3. Simulations
1 .75 .34 0 .40
−4
1e 0 .90 .40 .90 .95
1 .75 .41 0 .65

17. Modifying the
Summary Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Accounting for population structure Department of
Preventive
Medicine
improves power, reduces false positives USC
Methods need to be efficient 1. Background
2. Proposed
New method shares principles from GC Method
and PC 3. Simulations
Reranks p-values in contrast to GC
Can be more powerful than PC when
genotyping error is present
Caveat: markers should be mostly
unlinked
We can simulate more realistic scenarios (e.g.
LD)

18. Modifying the
Acknowledgements Cochran-Armitage
trend test to
address population
structure in GWAS
Gary K. Chen
Department of
Preventive
Medicine
USC
1. Background
2. Proposed
Method
Cyril S. Rakovski 3. Simulations
Daniel O. Stram