Introduction to Statistical Genetics (BST227) Lecture 6: Population Substructure in Association Studies

Transcription

1 Introduction to Statistical Genetics (BST227) Lecture 6: Population Substructure in Association Studies

2 Confounding in gene+c associa+on studies q What is it? q What is the effect? q How to detect it? q What to do about it?

3 What is it? A major threat to the validity of nonrandomized studies is confounding. What is confounding? What is a poten+al confounding variable for a gene+c study? What are some different kinds of popula+on substructures and what are they?

4 Popula+on Substructure Features of a popula+on which result in varia+on of allele and/or genotype frequencies across individuals in a popula+on

5 Popula+on Substructure: Stra+fica+on / Admixture / Inbreeding Popula+on stra+fica+on: dis+nct subgroups within a popula+on. Popula+on admixture: ma+ng among individuals of different gene+c origin over mul+ple genera+ons. Usually occult. Inbreeding: ma+ng between close rela+ves

6 Inbreeding Inbreeding implies a posi+ve probability that an individual inherits the exact same allele (A or a) from both parents, i.e. the parents had a common ancestor The inbreeding coefficient, F, measures the degree of inbreeding in a popula+on. F = p(a random individual inherits the same ancestral allele from both parents). Extreme form of inbreeding (F=1) is pure strains as used in breeding experiments. In randomly ma+ng popula+ons, F=0

7 Admixture Gila River American Indian Community Strong correla0on between Gm 3;5,13,14 allele frequency across strata of Indian heritage ( with number of great- grandparents) Indian Heritage Gm 3;5;13;14 % % Diabetes age 0 69% adjusted 18.5% 4 45% 28.6% 8.01% 39.2% Adapted from Knowler, 1988 Gm 3;5,13,14 allele frequency is a marker of Indian Heritage. Admixture usually creates a gradient of allele frequencies

8 What is the effect? Any form of population substructure causes HWD in population. From Chapter 3 with HWE, X is B(2,p), where X is the number of A alleles E(X) = 2p and var(x) = 2p(1-p) With inbreeding (HWD) E(X) = 2p and var(x) = 2p(1-p)(1+F) 0<F<1 is the Wahlund CoefLicient F is also the coeflicient of inbreeding F is also the correlation between alleles transmitted by parents This expression still assumes p is uniform in population.

9 Effect of Substructure Suppose we have K strata, each with allele frequency p k, k = 1, K, and HWE within each strata. Then E(X) =2p Where p is the average allele frequency in the popula+on. However, var(x) is bigger than binomial variance: var(x) = 2pq + 2var(p k ) (Chapter 3) Both inbreeding and popula+on stra+fica+on can cause variance infla+on. Almost all tests of associa+on are based on es+mated allele frequencies. Popula+on substructure has consequences for inference about sample propor+ons.

10 General Background on Test Sta+s+cs Many test sta+s+cs can be expressed as: Z = T/SE(T) where T is some sta+s+c of interest, and SE(T) means the square root of the es+mated variance of (T), the standard error. Under H0, E(T)=0, var(z)=1, Z~ N(0,1) or Z 2 ~χ2 on 1 df (approximately in large samples) Eg. Trend test: Allele test: T = n(x cases X controls ) / 2 T = n(p cases p controls )

11 Effect of Popula+on Substructure in Gene+c Associa+on Studies Popula+on substructure can 1) bias numerator (E(T) 0 under H0) 2) inflate the var(t) (so es+mated SE(T) is too small) 3) or do both Both of these problems will lead to increased false posi+ve rate. In gene+c studies, ohen argued that variance infla+on is the worst. Confounder biases the numerator and causes variance infla+on.

12 Consider Trend and Alleles Test Nota+on: X is the number of A alleles for one subject, p is propor+on of A alleles in the popula+on, X, p denote sample means and propor+ons in each group, SE is es+mated standard error. Assume r=s=n/2. The alleles test and the trend test have same T T = n(x cases X controls ) / 2 = n(p cases p controls ) SE(T) assuming HWE gives alleles test SE(T) without HWE gives the trend test TREND TEST CORRECTS FOR HWD (but not populaaon straaficaaon or admixture)

13 Consider Trend Test and Popula+on Stra+fica+on, 2 strata Consider 2 strata (K=2) where p 1 and p 2 are A allele frequencies in the two strata. Disease rates may also vary over strata: K 1, K 2 Under H0 : p 1cases = p 1controls = p 1 p 2cases = p 2controls = p 2 Thus there is no difference in allele frequencies within strata (Ho is true in both strata)

14 Problem with Popula+on Stra+fica+on If you do not sample cases and controls according to strata, then they will be unbalanced. Suppose K1>K2. If strata sizes are equal, expect more cases from strata 1 and more controls from strata. Ignoring strata in analysis leads to systema+c bias. Let c = propor+on of cases from strata 1 d = propor+on of controls from strata 1 Then under H 0 : E(T ) = n(x cases X controls ) / 2 = n(c d)(p 1 p 2 )

15 Bias in Numerator E(T )= n(c d)(p 1 p 2 ) To eliminate bias, need E(T) = 0 p 1 = p 2 or c = d E(c d) = S 1 S 2 (K 1 K 2 ) / (K(1 K)) K 1 =K 2 means c = d on average (K is the disease rate in the overall popula+on) The absence of varia+on in disease rate or the allele frequency over strata is sufficient to eliminate the bias.

16 Bias in Numerator General Formula for K strata (index by k): E(T ) = ncov(p k, K k ) / (K(1 K)) In general, need to have systema+c associa+on between disease rates and allele frequencies (K k and p k ) to get non-zero covariance. Although rare, strong covariance have been documented, and are more likely with a small number of strata, or admixtures of a small number of popula+ons.

17 Effect of Popula+on Substructure on Variance Trend test, under H 0 no difference in allele frequency, treat as a single sample and es+mate var(x) empirically. With admixture, var(x) = 2p(1-p)(1+F) F is Wahlund effect, coefficient of inbreeding or correla+on between parental alleles If no popula+on substructure, empirical variance will es+mate var(x) correctly. However, F accounts for correla+on of alleles WITHIN an individual, but ignores covariance induced by stra+fica+on.

18 Variance Infla+on for Trend Test Trend test T = n(x cases X controls ) / 2 Assumes no substructure but non-zero F Assumes subjects are independent, but alleles within subject are not For alleles test, (1+ F) is omined. Thus alleles test is bigger because variance is smaller. With popula+on stra+fica+on in case-control designs you have addi+onal variance infla+on. That is, the actual varia+on is bigger than what you assume because allele frequency varies by strata.

19 Variance Infla+on (Devlin and Roeder, 1999) True var(t) under stra+fica+on: Var(T ) = 2np(1 p)[1 F + nf(c d) 2 ] Variance infla+on: true Var(T with popula+on substructure) = λ var(t for simple trend test/allele test) Variance infla+on substan+ally larger than bias because it depends on n (Devlin et al, 2000) Variance infla+on increases with n, and does not depend on allele frequency. Similar result for alleles test: λ = [1 F + nf(c d) 2 ] Es+ma+on of λ is the basis for Genomic Control.

20 Variance Infla+on factor Es+mate λ to check whether λ > 1. Basis for genomic control. How to detect it? Q-Q plots Since p-values follow an uniform distribu+on between 0 and 1, we can plot the observed p-values vs quan+les of an uniform distribu+on.

21 What to do about it? Match or stra+fy on ethnic ancestry Self report may not be accurate Ethnicity [e.g. race ] may not be good surrogate for ancestry Difficult to match admixed ancestry subjects Use family-based controls Siblings (condi+onal logis+c) Case-parent pseudocontrols (TDT, FBAT etc.) Adjust using mul+ple unlinked markers Genomic control Es+mate infla+on factor and adjust accordingly Popula+on Components Infer popula+on substructure using con+nuous axes of varia+on Linear mixed effect models

22 Genomic Control (Devlin and Roeder, 1999) Idea: null markers should have correct chi-squared distribu+on if there is no substructure. Use data at mul+ple markers (should be unlinked and null) to es+mate λ. How es+mate λ? Let χ 2 1, χ 2 2 2,..., χ L denote the trend test calculated for the L markers. Then the χ 2 1, χ 2 2 2,..., χ L should look like a sample of 1 2 L chi-square random variables with one df. Sample var(χ 2 j ) = 2 if no variance infla+on. Could calculate var(χ 2 j ) and compare to 2.

23 Genomic Control (Devlin and Roeder, 1999) However, var is very sensi+ve to outliers, one bad test can ruin your es+mate, use ˆλ = median[χ 1 2, χ 2 2,..., χ L 2 ] / Use es+mated λ to adjust variance of Trend Test: χ 2 T / ˆλ 2 ~ approximately χ 1 under H 0 Assumes λ constant across loci, similar muta+on rates, limited selec+on at marker, limited varia+on in F across subpopula+ons Note, book uses 1/ λ.

24 How large a problem? (Freedman et al., 2004) Examined 11 case-control/cohort studies for evidence of excess varia+on (λ > 1) (Reich and Goldstein, 2001) No evidence of excess varia+on for original range of sample sizes ( cases) and number of SNP s (24-48) Expanded two case-cohort studies of prostate cancer

25 Prostate Cancer Studies (Freedman et al., 2004) Prostate cancer among African Americans Cases/Controls SNPs P-value (λ=1) Estimated λ original 90/69 48 < Expanded 469/ <.04 AIM* included 474/ <.0001 *SNP s chosen to discriminate ancestry are included

26 Prostate Cancer in African Americans (Freedman, et. Al) Removed 40/474 cases, 48/476 controls with self-report of non African American ancestry P-values for all 211 markers is 10-7 λ is 1.5 (extrapolated to 1000 cases and controls) Typical sized studies may not detect variance infla+on, and using self-reported heritage may not solve the problem Larger studies will likely have more substructure With GWAS data, different approaches can be used to adjust for substructure.

27 Second approach Principle Components: Designed for GWAS With a GWAS, get 500K or more markers; essen+ally enough to nail down an individual s gene+c signature (do not use imputed SNPs) Var-cov matrix of individual gene+c varia+on, denote by C Use PC to determine major axes or con+nuous axes of gene+c varia+on

28 Box 7.2 Principal component adjustment of association studies Notation: M... Number of SNPs N... Number of subjects X = (Zij) an M x N matrix of standardized genotypes coded for the additive model for the ith SNP in the jth proband, i.e, z ij = (X ij X i. ) / p i (1 p i ) where Pi estimates the population frequency of the ith SNP. The algorithm: Step 1: Compute the Variance-Covariance matrix for the probands as C = (X T X)/(N - 1). Step 2: Compute the eigenvalue decomposition of the covariance matrix. Step 3: Select the top K eigenvalues that are statistically significant. Step 4: Include the significant eigenvectors in the linear regression of the phenotype on marker, or use the significant eigenvectors to match cases and controls, and do a matched pair analysis.

29 Principle Components of C: Let j indexes people, i indexes SNP diag C j = var(g ij ): variance over snps for person j off diag C jj = cor(g ij,g jj ) for pairs j and j : high cor indicates that 2 individuals have very similar SNP profiles common gene+c background Eigenvalues and eigenvectors of C: Eigenvectors: linear combina+on of SNPs, maximize differences in people, or maximize variance in SNPs across people Eigenvalues: how much of total varia+on is explained by each eigenvector.

30 Case + + Control X N I... 0 u Q) > C Q) CJ) Q) t- -it + :++-PF-+ -ft* + + ; r , t \ + x + +: * +-t= * + + +x f=i"" X +++ +x -f -h + :5t X + - It- + X + X X X X > x + X + + X X X ++ X X + +,=I- X X + x +++ x +x + +x Xi- X + + X X X + X X X X X eigenvector 1

31 PCA on 1000 Genome Project

32 Principle Components: Designed for GWAS Let Z 1 j, Z 2 j,... denote the eigenvectors for the jth person (use only top 10 PCs or so) g(e(y X)) = a + bx + c 1 Z 1 + c 2 Z Can also use Z 1 j, Z 2 j,..., to cluster, or stratify individuals into homogeneous strata.

33 Linear mixed models PCA does not account for cryp+c relatedness between individuals, which may also inflate the tests Therefore, use linear mixed model: Y = XB + µ +ε Var(µ) = σ g 2 K µ is the random effect, which represents the heritable component of random varia+on K is the gene+c similarity/relatedness matrix according to pairwise genotype similarity of individuals, it is influenced by popula+on structure, family structure and cryp+c relatedness EMMAX, BOLT-LMM, etc

34 Consequences of Substructure If both strata allele frequencies AND disease rates vary, then E(T) 0 and have bias If only allele frequencies vary, but sample is unbalanced (c d) have variance inflation. This is biggest concern, problem increases as n increases. Using PC s may NOT fix variance inflation.

35 Summary q What is population substructure? q What is the effect? E(T) biased, Var(T) inflated. q How to detect it? QQ plot, Genomic control q What to do about it? Genomic control, PCA, linear mixed models. Active research on correcting population substructure for sequencing data. Reading: Price, AL, et al. New approaches to population stratification in genome-wide association studies Nat Rev Genet.