Genetic Association Studies in the Presence of Population Structure and Admixture

Transcription

1 Genetic Association Studies in the Presence of Population Structure and Admixture Purushottam W. Laud and Nicholas M. Pajewski Division of Biostatistics Department of Population Health Medical College of Wisconsin A presentation in the BNP Workshop, INI, Cambridge, August 7, 2007

2 Outline Some genetics background Impact of structure on association studies Remedies A DPM model Clustering success Estimation success A model for admixture Detection of admixed individuals

3 Some Genetics Terminology Chromosome 1 A d GAATTAGTCAAGC Chromosome 2 a D Chromosome Locus / Polymorphism Allele Genotype Two alleles A and a yield three potential genotypes at each locus: AA, aa/aa, or aa Phenotype

4 Propagation, Inheritance and Diversity In a pair of homologous chromosomes, one is inherited from the mother and the other from the father Recombination can result in alleles that differ from those of parents Mechanisms leading to variability in the frequency of genetic variants in human populations Positive Selection Ancestral Admixture Genetic Drift Allele frequency assumptions Hardy-Weinberg Equilibrium - Chromosome independence Linkage Disequilibrium - Locus to locus independence

5 Wright s Fixation Index Wright s F ST - Measures the divergence between populations relative to the overall level of genetic diversity within a species (Wright 1951) Simulation mechanism {Balding & Nichols (1995)} : Let p denote ancestral allele frequency at a locus. This could be drawn from, say, Uniform(0.10,0.90) or fixed Then, π - the allele frequency at this locus is simulated as π Beta (rp, r(1 p)), r = (1 F ST ) F ST E(π) = p, V ar(π) = p(1 p)f ST

6 Common Study Designs Main goal is to determine if a phenotype is associated with a genotype at some locus or a combination of loci Phenotype of interest can be quantitative or binary Sampling design can be cross-sectional or case-control Disease presence/absence, sample of cases and sample of non-cases leads to 2 2 table with column totals fixed Generally, traditional analyses use linear models and contingency tables

7 Population Structure and Impact on Association If sampling from a homogeneous population with stable allele frequencies at loci of interest, traditional methods can appropriately determine association Variation in allele frequencies in human subpopulations can mislead these analyses A disease with higher prevalence in one subpopulation than the other will look spuriously associated with alleles that have higher prevalence in the first Attenuation of a genetic effect is also possible - e.g., high prevalence in a small subpopulation

8 Impact of Structure on Association Genetics Literature Considerable debate in the genetics community about the potential for spurious associations caused by population allele frequency differences (Wacholder et.al. 2000; Cardon and Palmer 2003; Freedmen et.al. 2004) Bidirectional effect - Structure can also mask or reverse genetic effects (Deng 2001) Examples Knowler et.al. (1988) - Type 2 diabetes in American Indians Campbell et.al. (2005) - Lactase (LCT) gene and tall/short status in European Americans

9 Remedies Transmission Disequilibrium Test - Spielman et al. (1993) Genomic Control - Devlin & Roeder (1999) Modeled Classification - Pritchard et al. (2000) Structural Association - Satten et al. (2001)

10 Modeled Classification Two-Stage Approach of Pritchard et.al. Assume we have a sample of N individuals, consisting of n 1 cases and n 0 controls Each subject is genotyped at L null marker loci and the candidate locus Assume each subject originates from one of K distinct subpopulations, where K is fixed. Notation Population pointer i = 1,..., N z i Genotype X (c) il l = 1,..., (L + 1) Genotype frequencies p klj j = 1, 2, 3 Admixture proportions q (i) k Z X P Q

11 Modeled Classification STRUCTURE Intermediate Goal: Inference on P and Z/Q based on the genotypic data X Pritchard et al construct a Markov Chain Monte Carlo (MCMC) algorithm to approximate the posterior distribution P r (P, Z/Q X) P r (Z/Q) P r (P ) P r (X P, Z/Q) Given Z and P, Pritchard et al then test H0 : No association between candidate allele frequencies and phenotype φ H1 : Candidate locus allele frequencies depend on phenotype φ

12 Modeled Classification Likelihood Ratio Statistic: STRAT Define Λ = P r 1 (X; ˆP ) 1, Q ) P r 0 (X; ˆP 0, Q where [ ] P r (i) 0 X (c) i1 = j P 0, φ = k [ ] P r (i) 1 X (c) i1 = j P 1, φ = k q (i) k p k1j q (i) k p[φ(i) ] k1j The maximized likelihoods under each hypothesis are estimated using the EM algorithm Significance levels for Λ are approximated through simulation

13 Modeled Classification Extensions and Other Approaches Falush et.al. (2003) - Extended Pritchard et.al. s original work to allow for linkage disequilibrium and correlated allele frequencies Zhang et.al. (2006) - Parametric Bayesian approach that incorporates the inherent variability in estimating K through Model Averaging Other methods include Frequentist approaches such as Satten et.al. (2001), Epstein et.al. (2007), and Price et.al (2006) and a Bayesian approach for quantitative traits by Hoggart et.al. (2003)

14 Model : Quantitative Trait Part Given a sample of N individuals, genotyped at L null marker loci and C candidates, Y i Normal (µ i, τ P ) µ i = β 0i + G i + γz i C G i = (β il W il + β 2l V il ) l=1 V and W are indicators of AA and aa respectively

15 Model : Genetic Marker Part Let θ il = logit(π il ), L(W i, V i ) = L+C l=1 [ 2 W il ] exp(θil (2V il + W il )) (1 + exp(θ il )) 2 β 0i, θ i1,..., θ i,l+c G G G α, G 0 Dirichlet Process (αg 0 ) α Gamma (α 0, λ 0 ) G 0 = L+C f(β 0 ) f(θ il ) l=1

16 Model Graph l=1,..,c G θ il θ il β 0i (W il, V il ) (W il, V il ) β 1l β 2l l=c+1,..,l τ P Y i i=1,...,n

17 Simulation of Posterior Methods for DPM Conjugate Setting Escobar (1994), Escobar and West (1995), Bush and MacEachern (1996) Split-Merge Samplers: Jain and Neal (2004), Green and Richardson (2001) Non-conjugate Setting No-Gaps algorithm: MacEachern and Muller (1998) Neal (2000), Walker and Damien (1998) Jain and Neal (2007) extended their Split-Merge algorithm to a conditionally conjugate setting

18 Simulation of Posterior Algorithm 5 of Neal (2000) Given K current clusters of distinct values 1. Update configuration indicator for each individual (s i ) by proposing a new cluster membership (s i ) R times P (s i = j s i ) = n i,j /(N + α 1) for j=1,..,k P (s i = K + 1 s i ) = α/(n + α 1) 2. Use Metropolis-Hastings update for proposed move [ L(Yi, W a(s i, s i, V i φ s i i ) = min ) ] L(Y i, W i, V i φ si ), 1 (1) 3. Update other parameters given current configuration. Combination of conjugate updates and Adaptive Rejection Sampling (ARS) (Gilks and Wild 1992)

19 Simulation of Posterior MCMC Scheme Given j=1,2,..,k distinct clusters in the DP, let φ j = (β 0j, θ j1,..., θ jl ), Update configuration using Neal s Algorithm, sampling from [s i (s i, φ)] for i=1,...,n For j=1,2,..,k, Sample [β 0j X j, β 11,.., β 1C, β 21,.., β 2C, τ P ] where X j = {Y i, W i, V i s i = j} (conjugate normal update) For l=1,...,l, update cluster specific allele frequencies, sampling from [θ jl W j, V j ] where (W j, V j ) denote genotype data for subjects in j th cluster (ARS)

20 Simulation of Posterior MCMC Scheme (continued) Update genetic effects at candidate loci, for l=1,2,..,c Sample [ β 1l Y, W, β 01,.., β 0K, β (1 l), β 21,.., β 2C, τ P ] Sample [ β 2l Y, V, β 01,.., β 0K, β (2 l), β 11,.., β 1C ] where β (r l) = β (r1),.., β (r,l 1), β (r,l+1),.., β (rc) (both are conjugate normal updates) Update phenotypic precision Sample [τ P Y, W, V, β 01,.., β 0K, β 11,.., β 1C, β 21,.., β 2C ] (conjugate gamma update)

21 A Two-Population Scenario Sample Size: 400 subjects with 500 markers Pop. 1 (50%) with F ST = 0.01 Pop. 2 (50%) with F ST = Simulated data characteristics The biggest allele frequency difference,δ l = π 1l π 2l, occurs at locus 254, where δ 254 = =.355 The smallest difference occurs at locus 223, δ 223 = =.000

22 Prior for K n=400 and α=1

23 Posterior Distribution for K Two populations scenario: n=400, L=500 and α=1

24 Predictive Density for π 254 π P OP = and π P OP = 0.358

25 Predictive Density for π 223 π P OP = and π P OP = 0.820

26 Predictive Density for π 109 π P OP = and π P OP = 0.598

27 A Four-Population High Heterogeneity Scenario Sample Size: 800 subjects at 1000 markers Population % of sample F (j) ST

28 Prior for K n=800 and α=1

29 Posterior for K 4 Populations - High Heterogeneity

30 Predictive Density for π 43 (20%) π (1) = 0.07 (40%) π (2) = 0.08 (30%) π (3) = 0.37 (10%) π (4) = 0.24

31 Predictive Density for π 47 (20%) π (1) = 0.34 (40%) π (2) = 0.39 (30%) π (3) = 0.42 (10%) π (4) = 0.34

32 A Four-Population Low Heterogeneity Scenario Sample Size: 800 subjects at 1000 markers Population % of sample F (j) ST

33 Posterior for K 4 Populations - Low Heterogeneity

34 Quantitative Trait Model Simulation Three scenarios where individuals originate from two ancestral populations Baseline trait mean differs between populations, with β 01 = and β 02 = Subjects genotyped at 750 markers including 5 candidates for association with the phenotype (generically denoted as loci 1-5) Scenarios I. n=700; Pop. 1 (50%) and Pop. 2 (50%) II. n=1000; Pop. 1 (50%) and Pop. 2 (50%) III. Same as I plus 300 admixed individuals

35 Simulation Setup Locus β 1l β 2l A l D l Y i Normal (µ i, τ P ) where 5 µ i = β 0i + (β 1l W il + β 2l V il ) τ P = 0.01 l=1 A l and D l = Additive / Dominance effects Falconer & MacKay (1996)

36 Generation of Admixed Ancestry Individuals Assume j=1,..,k subpopulations with associated allele frequencies π jl at each locus Let q i denote an admixture vector where q ij denotes the proportion of the i th admixed individual s genome that originates from the j th subpopulation. q i Dirichlet(α q,..., α q ) where α q was set at 1.0 Then the genotype at the l th locus is independently simulated as Binomial(2, K q ij π jl ) j=1

37 Naive Analysis Effect True ˆβ 95% CI* χ 2 2,0.99 = 9.28 β to β to β to β to β to β to 2.24 β to β to β to β to *Simultaneous CI using Bonferroni Correction

38 Scenario I Results Two Distinct Subpopulations (n=700) Parameter True ˆβ 95% Cred. Int.* χ 2 l β to β to β to β to 7.00 β to β to 3.33 β to β to β to β to 1.93 * denotes simultaneous 95% Credible Intervals

39 Scenario II Results Two Distinct Subpopulations (n=1000) Parameter True ˆβ 95% Cred. Int.* χ 2 l β to β to β to β to 7.35 β to β to 4.13 β to β to β to β to 4.64 * denotes simultaneous 95% Credible Intervals

40 Scenario III Results n=1000, Including 300 Admixed Individuals Parameter True ˆβ 95% Cred. Int.* χ 2 l β to β to β to β to 7.06 β to β to 2.31 β to β to β to β to 7.48 * denotes simultaneous 95% Credible Intervals

41 Comparative Results Parameter True N S I S II S III β β β β β β β β β β

42 Model for Admixture Inclusion of highly admixed individuals affects estimation The DPM construction above clusters individuals based on their entire set of genotypes at all L loci. It does not allow for differing ancestries across the genome How to handle admixed individuals more appropriately? Assume, for a moment, that we know which individuals are highly admixed. Divide the sample between N L low admixture subjects and N H high admixture subjects We then consider a separate parametric likelihood for the N H individuals that uses locus-specific information, effectively making an assignment at each locus for each admixed person to the clusters detected by the Dirichlet Process

43 Admixture Model Likelihood For the i th admixed subject, let X (c) il denote the allele at the l th locus on the c th chromosome, c = 1, 2. Let z (c) il denote a subpopulation assignment indicator for that allele Let q ij represent an N L dimensional admixture vector that denotes how much of that person s genome originates from the subpopulations represented by the low admixture sample. L H = N H i=1 L τ 1/2 P 2π exp l=1 c=1j=1 τ P 2 2 N L (q ij e θ jlx (c) il 1 + e θ jl ( ) N 2 L Y i q ij β 0j G i ) I(z (c) il j=1 =j)

44 Admixture Model Continued The model is then the same as before except for the addition of two prior specifications q i Dirichlet (α 1,..., α NH ) for i = 1,..., N H Pr(z (c) il = j)= 1 N L for j = 1,..., N L Note: Although the model is setup to assign alleles to specific low-admixture individuals, because of the inherent Dirichlet Process clustering, the calculations simplify to assigning admixed alleles to the current K distinct clusters

45 Admixture Model Graph l = C + 1,.., L j = 1,.., N L θ jl l = 1,.., C i = 1,.., N H X (c) il G q ij β 0j Y i X (c) θ jl il β 2l β 1l τ P

46 Scenario III Results With Hybrid Model Parameter True ˆβ 95% Cred. Int.* χ 2 l β to β to β to β to 8.17 β to β to 2.73 β to β to β to β to 6.33 * denotes simultaneous 95% Credible Intervals

47 Comparative Results Parameter True N S I S II S III S IIIA β β β β β β β β β β

48 Detecting Admixed Individuals Let s (g) i denote the cluster membership indicator for the i th individual at the g th MCMC iteration, then define Then let ηˆ (g) il = n ɛ (g+1) i = j:s (g) j L l=1 N (2V o jl + W jl ) =s (g) i 2n s (g) i ( ˆη (g+1) il ) 2 ˆη (g) il

49 Results on Detection

50 Results on Detection

51 Modifications for Case-Control Studies Binary Outcomes Simply involves switching to a logistic or probit likelihood. Case-Control Studies The extension to a case-control setting is also relatively straightforward Need to specify a retrospective likelihood such as that proposed by Seaman and Richardson (2001) Müller and Roeder (1997)

52 Extensions 1. Clustering by genome sections 2. Introducing dependence structure amongst neighboring loci, i.e. incorporating linkage disequilibrium patterns 3. Extending to Genomewide Association Studies - Removing candidate loci designation so that all markers are potentially phenotypically associated

53 References In a separate file