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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
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
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
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
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
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
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
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
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
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
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
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
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
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))
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
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
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)
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

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Kinship adjusted armitage trend test for ENDGAME meeting 2008

  • 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