EigenGWAS: finding loci under selection through genome-wide association studies of eigenvectors in structured populations

Abstract

We develop a novel approach to identify regions of the genome underlying population genetic differentiation in any genetic data where the underlying population structure is unknown, or where the interest is assessing divergence along a gradient. By combining the statistical framework for genome-wide association studies (GWASs) with eigenvector decomposition (EigenGWAS), which is commonly used in population genetics to characterize the structure of genetic data, loci under selection can be identified without a requirement for discrete populations. We show through theory and simulation that our approach can identify regions under selection along gradients of ancestry, and in real data we confirm this by demonstrating LCT to be under selection between HapMap CEU–TSI cohorts, and we then validate this selection signal across European countries in the POPRES samples. HERC2 was also found to be differentiated between both the CEU–TSI cohort and within the POPRES sample, reflecting the likely anthropological differences in skin and hair colour between northern and southern European populations. Controlling for population stratification is of great importance in any quantitative genetic study and our approach also provides a simple, fast and accurate way of predicting principal components in independent samples. With ever increasing sample sizes across many fields, this approach is likely to be greatly utilized to gain individual-level eigenvectors avoiding the computational challenges associated with conducting singular value decomposition in large data sets. We have developed freely available software, Genetic Analysis Repository (GEAR), to facilitate the application of the methods.

Introduction

In population genetics, eigenvectors have been routinely used to quantify genetic differentiation across populations and to infer demographic history (Cavalli-Sforza et al., 1996; Novembre et al., 2008; Reich et al., 2009). More recently, eigenvectors are commonly used as covariates in genome-wide association studies (GWASs) to adjust for population stratification (Price et al., 2006). Eigenvectors are usually estimated for each individual (individual-level eigenvectors, involving the inversion of a N × N matrix, where N is sample size). Theoretical studies have suggested that individual-level primary eigenvectors are measures of population differentiation reflecting F_st among subpopulations (Patterson et al., 2006; McVean, 2009; Bryc et al., 2013) and can be interpreted as the divergence of individuals from their most recent common ancestor. Eigenvectors can also be estimated for each single-nucleotide polymorphism (SNP; SNP-level eigenvectors that involve inversion of a M × M matrix; M is the number of SNPs) and these SNP-level eigenvectors can be interpreted as F_st metrics of each SNP (Weir, 1996). SNP-level eigenvectors from a reference population are useful for revealing the population structure of independent samples (Zhu et al., 2008) as they can be used to project, or predict, the eigenvector values of individuals. However, because of high-dimensional nature of GWAS data (commonly expressed as M>>N), direct estimation of SNP-level eigenvectors is nearly impossible when using millions of SNPs.

Singular value decomposition (SVD) enables SNP-level eigenvalues to be obtained in a computationally efficient manner for any set of genotype data (Chen et al., 2013); however, it is not possible to determine the SNPs that contribute most to the leading eigenvector, or to test whether specific SNPs are differentiated along the genetic gradient described by the eigenvector. Here, we propose an alternative, simple, fast approach for the estimation of SNP-level eigenvectors. By using individual-level eigenvectors as phenotypes in a linear regression, we demonstrate that the regression coefficients generated by single-SNP regression are equivalent to SVD SNP effects as proposed by Chen et al. (2013). As the single-SNP regression resembles the popular single-marker GWAS method, as implemented in PLINK (Purcell et al., 2007), we call this method EigenGWAS. We show that the EigenGWAS framework represents an alternative way for identifying regions under selection along gradients of ancestry.

Materials and methods

HapMap3 samples

HapMap3 samples were collected globally to represent genetic diversity of human population (Altshuler et al., 2010). HapMap3 contains representative samples from many continents: CEU and TSI represent population from north and south Europe, CHB and JPT from East Asia, and CHD Chinese from Denver, Colorado. Loci with palindrome alleles (A/T alleles or G/C alleles) were excluded, and 919 133 HapMap3 SNPs were used for the analysis.

1000 Genomes project

1000 Genomes project samples were used as a prediction set for projecting eigenvectors (The 1000 Genomes Project Consortium, 2012). We selected the Puerto Rico cohort (105 samples) and the Pakistan cohort (Punjabi from Lahore, Pakistan, 95 samples) for analysis.

POPRES samples

POPRES (Nelson et al., 2008) is a reference population for over 6000 samples from Asian, African and European nations. In this study, we selected 2466 European descendants. The POPRES genotype sample was imputed to a 1000 Genomes reference panel (The 1000 Genomes Project Consortium, 2012). Imputation for the POPRES was performed in two stages. First, the target data were haplotyped using HAPI-UR (Williams et al., 2012). Second, Impute2 was used to impute the haplotypes to the 1000 Genomes reference panel (Howie et al., 2011). We then selected SNPs that were present across all data sets at an imputation information score of >0.8. A full imputation procedure is described in https://github.com/CNSGenomics/impute-pipe. After quality control and removing loci with palindromic alleles (A/T alleles or G/C alleles), 643 995 SNPs for POPRES remained. In addition, we also conducted the analysis using nonimputed 234 127 common markers between POPRES and HapMap3. As the results between these two data sets were very similar, this report focussed on the results from 643 995 SNPs that were more informative.

Simulation scheme I: null model without population structure

A total of 2000 unrelated samples with 500 000 biallelic markers, which were in linkage equilibrium to each other, were simulated. The minor allele frequencies ranged from 0.01 to 0.5, and Hardy–Weinberg equilibrium was assumed for each locus. All individuals were simulated from a homogeneous population, with no population stratification. In order to calculate F_st at each locus, we divided the sample into subpopulations based upon eigenvectors that were estimated from a genetic relationship matrix calculated using all 500 000 markers (see below).

Simulation scheme II: null model with population structure

In general, this simulation scheme was followed Price et al., 2006. A total of 2000 unrelated samples with 10 000 biallelic markers, which were in linkage equilibrium to each other, were generated. For each marker, its ancestral allele frequency was sampled from a uniform distribution between 0.05 and 0.95, and its frequency in a subpopulation was sampled from β-distribution with parameters and . The β-distribution had mean of P and sampling variance of P(1−P)F_st. Once the allele frequency for a subpopulation over a locus was determined as P_s, individuals were generated from a binomial distribution Binomil(2,P_s). It agreed with the quantity that measures the genetic distance between a pair of subpopulations (Cavalli-Sforza et al., 1996).

Calculating individual-level eigenvectors

We assume that there is a reference sample consisting of N unrelated individuals and M markers. is a vector of the i^th individual's genotypes along M loci, with x the number of the reference alleles. An N × N genetic relatedness (correlation) matrix A (matrix in bold font) for each pair of individuals is defined as , in which f_l is the frequency of the reference allele. The principal component analysis (PCA) is then implemented on the A matrix (Price et al., 2006), generating -, which is a N × K (K⩽N) matrix, in which E_k is the eigenvector corresponding to the k^th largest eigenvector.

Unified framework for BLUP, SVD and EigenGWAS

Theoretically, PCA can also be implemented on a M × M matrix, but this is often infeasible because the M × M matrix is very large. However, for individual i, eigenvector k can also be written as:

in which β_k is a M × 1 SNP-level vector of the SNP effects on E_k, and x_i is the genotype of the i^th individual across M loci. In the text below, we denote individual-level eigenvector as eigenvector (N × 1 vector), and SNP-level eigenvector (M × 1) as SNP effects.

We review three possible methods to estimate β given eigenvectors. The first method is best linear unbiased prediction (BLUP) that is commonly used in animal breeding and recently has been introduced to human genetics for prediction (Henderson, 1975; Goddard et al., 2009). The second method is to convert an individual-level eigenvector to SNP-level eigenvector using SVD, as proposed by Chen et al. (2013). The third method is the approach outlined here, EigenGWAS, that is a single-marker regression, as commonly used in GWAS analysis.

Methods 1 and 2: BLUP and SVD

For a quantitative trait, y=μ+βX+e, in which y is the phenotype, μ is the grand mean, β is the vector for additive effects, X is the genotype matrix and e is the residual. Without loss of generality, the BLUP equation can be expressed as:

in which is the estimates of the SNP effects, is the standardized genotype matrix, V is the variance covariance with and y is the trait of interest (Henderson, 1975). Replacing y with individual-level eigenvector (E_k), Equation (2) can be written as

in which β_k is the BLUP estimate of the SNP effects and E_k is the k^th eigenvector estimated from the reference sample. The V matrix can be replaced with A because the eigenvector has no residual error (that is, h²=1). This method has also been proposed as an equivalent computing algorithm for genomic predictions (Maier et al., 2015).

In addition, the connection between PCA and SVD can be established through the transformation between the N × N matrix to the M × M matrix (McVean, 2009). Let A=PDP⁻¹, in which D is a N × N diagonal matrix with λ_k, P is N × N matrix with the eigenvectors. B=X^T(PDP⁻¹)⁻¹P=X^TPD⁻¹, in which B is M × N matrix. This is equivalent to the equation used in Chen et al. (2013) where B^T=D⁻¹(X^TP)^T. Thus, eigenvector transformation can be viewed as a special case of BLUP in which the heritability is 1 (Equation (3)). However, under SVD another analysis step is then required to evaluate the significance of the estimated SNP effect. In an EigenGWAS framework an empirical P-value is produced when estimating the regression coefficient.

Method 3: estimating SNP effects on eigenvectors with EigenGWAS

Given the realized genetic relationship matrix A, for unrelated homogeneous (i.i.d.) samples, E(A_ij)=0 (i ≠ j), and consequently E(A)=I, an identity matrix. Because of sampling variance of the genetic relationship matrix A, the off diagonal is a number slightly different from zero even for unrelated samples (Chen, 2014). If we replace the matrix with its mathematical expectation—the identity matrix—Equation (3) can be further reduced to , equivalent to single-marker regression E_k=a + bx + e, as implemented in PLINK (Purcell et al., 2007). Furthermore, standardization for X is not required because it will not affect P-value. Thus, SNP effects can be estimated using the single-marker regression that is computationally much easier in practice and is implemented in many software packages. Each SNP effect, , is estimated independently, and the P-value of each marker can be estimated that requires additional steps in BLUP and SVD.

We summarize the properties and their transformation of SVD, BLUP and EigenGWAS as below:

1. E _k is determined by the A matrix, or in another words, it is determined by the genotypes completely. If we consider each E _k is the trait of interest—a quantitative trait—its heritability is 1.

2. h ²=1. SVD and BLUP are both computational tools in converting a vector from N × N matrix to a M × M matrix. SVD is a special case to BLUP when h ²=1 for BLUP.

3. h ²=1 and E( A )= I . When these two conditions are set, BLUP is further reduced to single-marker association studies that is EigenGWAS as suggested in this study.

Recently, in an independent work Galinsky et al. (2016) introduced an approximation to find the proper scaling for SNP effects (‘SNP weight' in Galinsky's terminology) estimated from SVD in order to produce accurate P-values. In our EigenGWAS framework, P-values for individual-level SNP eigenvector are automatically generated. In practice, it is conceptually easier to conduct EigenGWAS on eigenvectors than to conduct BLUP/SVD. In addition, if computational speed is of concern, EigenGWAS can be easily parallelized for each chromosome, each region or even each locus.

Interpretation for EigenGWAS signals

We can write a linear regression model E_k=a + βx + e, in which E_k is standardized. Assuming that a sample has two subdivisions that have sample size n₁ and n₂, , and the sampling variance for β is . The χ² test for β is

in which is Nei's estimator of genetic difference for a biallelic locus (Nei, 1973). Furthermore, , in which for a pair of subpopulations as defined in Weir (1996). Hence, .

As the proportion of the variance explained by the largest eigenvalue is equal to F_st^W in PCA (McVean, 2009), , in which characterizes the average divergence for a pair of subpopulations. When the test statistic, Equation (4), is adjusted by the largest eigenvalue λ₁, an equivalent technique in GWAS for the correction of population stratification, . Hence,

after the adjustment of the largest eigenvalue, the EigenGWAS test statistic on E₁ immunes of population stratification, at least for a divergent sample.

For a locus under selection, it should have a greater F_st than the background divergence. Hence, the statistical power for detecting whether a locus is under selection is determined by the strength of selection, and this can be defined as the ratio between F_st of a particular locus and the average divergent in the sample. It is analogous to consider the χ² test with noncentrality parameter (NCP), . Otherwise specified, in this study F_st is referred to the one defined in Weir (1996).

Validation and prediction for population structure

Once β_k is estimated, it is straightforward to get genealogical profile for an independent target sample. In general, it is equivalent to genomic prediction, and the theory for prediction can be applied (Daetwyler et al., 2008; Dudbridge, 2013). The predicted genealogical score can be generated as

in which _k is the predicted k^th eigenvector, is the estimated SNP effects and X is the genotype for the target sample. We focus on the correlation between the predicted eigenvectors and the direct eigenvectors, and thus it does not matter whether X or is used.

In contrast to conventional prediction studies that focus on a metric phenotype of interest, prediction of population structure is focussed on a ‘latent' variable. This latent variable is the genetic structure of population that is shaped by allele frequency and linkage disequilibrium of markers. Thus, expectations of prediction accuracy differ from what has been established for conventional prediction (Daetwyler et al., 2008; Dudbridge, 2013) . We therefore assess prediction of accuracy for E₁ across markers when using different prediction thresholding (Purcell et al., 2009).

Here we proposed an equation for prediction accuracy, especially for E₁

when there is no heritability, the predictor can be simplified to , meaning that as the number of markers increases prediction accuracy should rapidly reach 1. Here the h² is interpreted as the genetic difference in the source population, or real ancestry informative markers. For a homogeneous population, the genetic difference is large because of genetic drift, and h² ≈ 0.

For this study, the genetic relationship matrix, PCA and BLUP estimation were conducted using GCTA software (Yang et al., 2011a). Single-marker GWAS was conducted using PLINK (Purcell et al., 2007) or GEAR (https://github.com/gc5k/GEAR/wiki/EigenGWAS; https://github.com/gc5k/GEAR/wiki/ProPC).

Results

Properties of the estimating SNP effects for eigenvectors

We applied EigenGWAS to the HapMap cohort, a known structured population. Eigenvectors were estimated via PCA based on the A matrix using all 919 133 SNPs. We conducted EigenGWAS for HapMap, using E_k, the k^th eigenvector, as the phenotype and investigated the performance of EigenGWAS from E₁ to E₁₀. From E₁ to E₁₀, we found 546 716 significant signals (231 677 quasi-independent signals after clumping) on E₁ and gradually reduced to 236 (163 after clumping) selection signals on E₁₀ (Figure 1). The large numbers of genome-wide significant loci are likely because HapMap3 comprised samples from different ethnicities, and these loci can be interpreted as ancestry informative marker (AIM). For each E_k, its associated eigenvalue was highly correlated with the λ_GC, the genomic inflation factor that is commonly used in adjusting population stratification for GWAS (Devlin and Roeder, 1999), resulted from its EigenGWAS. The top five eigenvalues associated to HapMap samples were 100.14, 47.66, 7.168, 5.92 and 4.40, and the corresponding λ_GC of EigenGWAS were 103.72, 44.69, 6.47, 5.17, and 3.96, respectively (Table 1). The large eigenvalues observed were consistent with previous theory that the magnitude of eigenvalues indicate structured population (Patterson et al., 2006). The connection between λ_GC and eigenvalues provides a straightforward interpretation: a large λ_GC indicates underlying population structure (Equation (5)). Therefore, correction for λ_GC will filter out signals due to population stratification, allowing loci under selection to be identified. For example, after correction for λ_GC, the number of GWAS hits was reduced for each EigenGWAS. For HapMap, EigenGWAS hits on E₁ dropped from 548 716 to 6 loci only, and for POPRES from 10 885 to 152, indicating that AIMs were largely driven by the genetic drift. These observations agreed well with our theory (see Materials and methods).

Figure 1.

Manhattan plots for EigenGWAS for top 10 eigenvectors for HapMap. Using E_i as the phenotype, the single-marker association was conducted for nearly 919 133 markers. The left panel illustrates from E₁ to E₅ and the right panel from E₆ to E₁₀. The horizontal lines indicate genome-wide significance after Bonferroni correction.

Table 1. EigenGWAS signals for HapMap and POPRES.

	HapMap				POPRES
Eigenvector (Ei)	Eigenvalue	GWAS λGCa	No. of GWAS hitsb	No. after clumpingc	Eigenvalue	GWAS λGCa	No. of GWAS hitsb	No. after clumpingc
1	100.135	103.715	546 716 (6)	231 677	5.104	5.005	10 885 (152)	3004
2	47.658	44.686	382 867 (7)	161 022	2.207	1.929	1254 (802)	289
3	7.168	6.471	33 317 (36)	15 344	2.157	1.910	1201 (713)	340
4	5.923	5.173	21 935 (51)	12 401	2.077	1.464	1353 (1,074)	331
5	4.402	3.964	9554 (66)	4727	1.971	1.866	781 (273)	76
6	2.449	1.982	1113 (36)	567	1.871	1.295	1162 (878)	111
7	2.285	1.986	593 (22)	389	1.843	1.337	1239 (1027)	130
8	2.107	1.742	236 (6)	171	1.818	1.486	1259 (940)	152
9	2.056	1.729	268 (3)	174	1.807	1.503	1701 (1086)	113
10	2.0217	1.661	236 (14)	163	1.798	1.492	1639 (1287)	90

Abbreviation: GWAS, genome-wide association study.

HapMap has 988 samples and 919 133 single-nucleotide polymorphisms (SNPs); its GWAS hits had P-values <5.44e−08 given α=0.05. POPRES has 2466 European samples and 643 995 SNPs; its GWAS hits had P-value <7.76e−08 given α=0.05.

^aλ_GC was calculated as the ratio between the median of observed χ² from EigenGWAS to the median of χ² value that is 0.455.

^bThe GWAS hits were counted without λ_GC correction, and with λ_GC correction in parentheses.

^cAfter clumping, the reported numbers were quasi-independent GWAS hits. Within 250K bp and linkage disequilibrium of r² >0.5 only the most significant GWAS hit was counted as a GWAS hit (see PLINK —clump default option). See Supplementary Tables 2 and 3 for pruned POPRES results.

We demonstrate theoretically that for EigenGWAS, the estimated SNP effects using single-marker GWAS are approximately equivalent to the estimates from BLUP, and the correlation between the estimates from these two methods was very high (greater than R² >0.98 on average; Figure 2), even in HapMap samples that consist of a mix of ethnicities where the A matrix is non-zero for off-diagonal elements (Supplementary Figure 1). This confirms that our EigenGWAS approach provides an accurate representation of the SNP effects on eigenvalues.

Figure 2.

Linear correlation for the SNP effects estimated using EigenGWAS and BLUP for HapMap3. The x axis represents EigenGWAS estimation for SNP effects, and the y axis represents BLUP estimation for SNP effects. The left panel illustrates from E₁ to E₅ and the right panel from E₆ to E₁₀. As illustrated at top left in each plot, the correlation, measured in R², is nearly 1.

We also conducted EigenGWAS on the POPRES samples, from which we selected 2466 European samples. On E₁, there were 10 885 (3004 quasi-independent signals after clumping) genome-wide significant signals, and reduced to 1639 (90 after clumping) on E₁₀ (Table 1). As in the HapMap sample, we observed a concordance between eigenvalues and λ_GC in POPRES. The top five eigenvalues were 5.104, 2.207, 2.157, 2.077 and 1.971, and their associated EigenGWAS λ_GC were 5.005, 1.929, 1.910, 1.464 and 1.866, respectively (Table 1), indicating population structure. The genetic relationship matrix estimated from the POPRES data resembled a diagonal matrix that had off-diagonal elements close to zero, suggesting that POPRES is a more homogenous samples as compared with HapMap (Supplementary Figure 1). Correlations between the estimates from EigenGWAS and BLUP were high, with an average of >0.999 from E₁ to E₁₀ (Supplementary Figure 2), close to one as expected.

The χ² statistics of the estimated SNP effects on eigenvectors from EigenGWAS were correlated with F_st for each SNP, such as on HapMap samples E₁ (Supplementary Figure 3), consistent with previous established relationship between eigenvectors and F_st (Patterson et al., 2006; McVean, 2009). Using naive threshold of E_k > 0, 2466 POPRES samples were divided into nearly two even groups that would be served as two subgroups in calculating F_st. E₁ > 0 split the POPRES samples into North and South Europe; samples from UK, Ireland, Germany, Austria and Australia were in one group, and samples from Italy, Spain and Portugal were in the other group; samples from Switzerland and France were nearly evenly split into two groups. F_st for each SNP was consequently calculated based on these two groups. For every eigenvector until E₁₀, we observed strong correlations between F_st and the χ² test statistics for EigenGWAS signals (Figure 3), and the averaged correlation was 0.925 (s.d., 0.067). For example, the correlation was 0.89 (P-value <1e−16) between χ² test statistics and F_st for E₁ in POPRES (Supplementary Table 1). This correlation is consistent with our theory where F_st has a strong linear relationship with its EigenGWAS χ² test statistic.

Figure 3.

The correlation between F_st and for EigenGWAS SNP effects for POPRES. For each eigenvector, upon E_i >0 or E_i ⩽0, POPRES samples were split into two groups, upon which F_st was calculated for each locus. The correlation, at top left in each plot, was measured in R².

We also validated our results in the simulation scheme I, in which there was neither selection nor population stratification. Given 2000 simulated samples, each of which had 500 000 unlinked SNPs, the EigenGWAS showed few GWAS signals, 2 genome-wide significant signals on E₂ (Supplementary Figure 4). As expected, λ_GC ranged from ∼1.124 to 1.130, with a mean of 1.127 for EigenGWAS on the top 10 eigenvectors, indicating little population stratification for the simulated data. After correction for λ_GC, no loci were significant, indicating that the type-I error rate was controlled. After splitting the samples into two groups depending on E_i > 0, the correlation between χ² test statistics and F_st is ∼0.67 from E₁ to E₁₀ (Supplementary Figure 5). Furthermore, as expected, after correction for λ_GC, the P-values followed a uniformed distribution for EigenGWAS from E₁ to E₁₀ (Supplementary Figure 6).

Furthermore, we also validated the theory in the simulation scheme II, in which there was population stratification. We wanted to know whether the adjustment of the test statistic with the greatest eigenvalue could render the distribution of the test statistics immunes of population stratification. Given various sample sizes for two subdivisions, after the adjustment for the test statistic with the largest eigenvalue, the test statistic followed the null distribution that was the χ² distribution of 1 degree of freedom (Supplementary Figure 7), indicating a well control of population stratification after correction.

The statistical power of EigenGWAS was also evaluated. As demonstrated, the power of EigenGWAS in detecting a locus under selection was determined by the ratio between the specific F_st of a locus and the averaged population stratification in the sample (Supplementary Figure 8).

Using EigenGWAS to identify loci under selection in structured populations

We propose EigenGWAS as a method of finding loci differentiated among populations, or across a gradiant of ancestry. Intuitively, every EigenGWAS hit is an AIM that differs in allele frequency along an eigenvector because of genetic drift or selection. A locus under selection should be more differed across populations than genetic drift can bring out. Thus, correction for λ_GC controls for background population structure, providing a test of whether an AIM shows greater allelic differentiation than expected under the process of genetic drift.

We pooled together CEU (112 individuals) and TSI (88 individuals) that represent Northwestern and Southern European populations in HapMap. EigenGWAS was conducted on E₁ that partitioned CEU and TSI into two groups accurately using E₁ > 0 as threshold (Supplementary Figure 9). We corrected for λ_GC, which was 1.723, for CEU and TSI. Adjustment for λ_GC significantly reduced population stratification (Supplementary Figure 10), and was consequently possible to filter out the baseline difference between these two cohorts. After correction, we found evidence of selection at the lactose persistence locus, LCT (P-value=1.21e−20, Figure 4). Because of hitchhiking effect, the region near LCT also showed divergent allele frequencies. For example, the DARS gene, 0.15 M away from LCT, was also significantly associated with E₁ (P-values=1.51e−23). HERC2 was slightly below genome-wide significance level (P-value=8.22e−08), indicating that anthropological difference reflected geographic locations of two cohorts but not under selection as strong as LCT. For the TSI and CEU example, given 200 individuals and 919 133 markers, it used ∼6 min and 1.9G RAM to complete the following three steps involved in EigenGWAS, developed in Java: (1) generate correlation matrix for 200 individuals using 919 133 markers; (2) estimate the first eigenvector, 919 133 elements, from the correlation matrix; and (3) run the linear regression model for every marker.

Figure 4.

EigenGWAS for CEU (112 samples) and TSI (88 samples) from HapMap. (a) Manhattan plot for EigenGWAS on E₁ without correction for λ_GC. When there was no correction, we found LCT on chromosome 2, MICA on chromosome 6 (HMC region), HIF1A on chromosome 14 and HERC2 on chromosome 15. The line in the middle was genome-wide significant level at α=0.05 given multiple correction. (b) Manhattan plot for EigenGWAS on E₁ with λ_GC correction; LCT was still significant, and HERC2 was slightly below whole genome-wide significance level. The genome-wide significance threshold was P-value=5.44e−08 for α=0.05.

We then conducted EigenGWAS in the POPRES sample by treating E₁ as a quantitative trait, and calculated the approximate F_st for each SNP given two groups split by the threshold of E₁ > 0 (Supplementary Figure 11). Given 643 995 SNPs, the genome-wide threshold was P-value <7.76e−08 for the significance level of α=0.05 (Figure 5). As expected for POPRES, λ_GC=5.00 indicated substantial population stratification. Correcting for λ_GC systematically reduced the EigenGWAS χ² test statistics (Supplementary Figure 12), and we replicated the significance of LCT (P-value=1.23e−22) and DARS (P-value=8.99e−22) (Table 2), suggesting selection at these regions. HERC2 was also replicated with P-value 8.15e−09, and with F_st of 0.041. The results were consistent after SNP pruning for POPRES (Supplementary Tables 2 and 3).

Figure 5.

EigenGWAS for POPRES samples on eigenvector 1. (a) Manhattan plot for EigenGWAS without correction for λ_GC. (b) After correction for λ_GC, we found LCT on chromosome 2, SLC44A4 on chromosome 6 and HERC2 on chromosome 15. The genome-wide significance level was P-value=7.76e−08 given α=0.05.

Table 2. Gene discovery using EigenGWAS.

Gene	Lead SNP	Position	Allele	P-valuea	MAF (TSI:CEU)	Fstb	Annotation
CEU and TSI samples
LCT	rs6719488	2:135817629	G/T	6.68e−34 (1.21e−20)	0.733:0.206	0.558	Lactose persistent locus
DARS	rs13404551	2:135964425	C/T	8.18e−39 (1.51e−23)	0.756:0.206	0.604	Genetic hitchhiking because of LCT
MICA	rs2256175	6:31412672	T/C	8.94e−10 (2.60e−6)	0.665:0.360	0.183	MHC class I polypeptide-related sequence A
HIF1A	rs2256205	14:61670944	A/G	1.51e−10 (8.86e−7)	0.464:0.179	0.192	HIF-1A thus plays an essential role in embryonic vascularization, tumour angiogenesis and pathophysiology of ischaemic disease.
HERC2	rs8039195	15:26189679	C/T	2.75e−12 (8.22e−08)	0.403:0.122	0.212	Genetic variations in this gene are associated with skin/hair/eye pigmentation variability
Gene	Lead SNP	Position	Allele	P-valuea	MAF (Southern Europeans: Northern Europeans)	Fstb	Annotation
POPRES European samples
LCT	rs3754686	2:135817629	T/C	3.30e−106 (1.23e−22)	0.514:0.279	0.110
DARS	rs13404551	2:135964425	C/T	6.32e−102 (8.99e−22)	0.518:0.293	0.106
SLC4A4	rs605203	6:31819235	C/A	8.94e−44 (5.77e−10)	0.214:0.343	0.040	Defects in this gene can cause sialidosis, a lysosomal storage disease
HERC2	rs1667394	15:26189679	C/T	3.90e−38 (8.15e−09)	0.276:0.173	0.041

Abbreviations: GWAS, genome-wide association study; MAF, minor allele frequency; SNP, single-nucleotide polymorphism.

The P-value cutoff for CEU and TSI was 5.44e−08 (919 133 SNPs) and for POPRES was 7.76e−08 (643 995 SNPs) at genome-wide significance level of α=0.05. λ_GC=1.725 for CEU and TSI and λ_GC=5.00 for POPRES.

^aP-values without λ_GC correction, and with λ_GC correction in parentheses.

^bF_st is calculated by partitioning the sample into two groups upon E₁ >0. For TSI and CEU set, partitioning on E₁ perfectly separated TSI (88 samples) and CEU (112 samples). For POPRES, partitioning on E₁ separated southern European population (1092 samples) and northern European population (1374 samples).

Prediction accuracy for projected eigenvector

We investigated three aspects of EigenGWAS prediction: (1) the number of loci needed to achieve high accuracy for the projected eigenvectors; (2) the required sample size of the training set; (3) the importance of matching the population structure between the training and the test sets.

Using the POPRES samples, we split 5% (125 individuals), 10% (250 individuals), 20% (500 individuals), 30% (750 individuals), 40% (1000 individuals) and 50% (1250 individuals) of the sample as the training set, and used the remainder of the samples as the test set. Eigenvectors were estimated using all markers in each training set. As predicted by our theory (Equation (7)), the prediction accuracy of the projected eigenvector was consistent with in which N_e=1000 for E₁ empirically. If only 100 and 1000 random SNPs were sampled as predictors, the expected maximal is R²=0.091 and 0.5, respectively, and accuracy reached almost 1 if >100 000 SNPs were sampled. In agreement with our theory (Figure 6), if the number of predictors were too small, the prediction accuracy was poor, with prediction accuracy increasing with the addition of more markers for E₁. When the sample size of the discovery was ⩾1000, maximal prediction accuracy was achieved, as predicted in our theory. Therefore, a discovery with a sample size of >1000 should be sufficient to predict the first eigenvector of an independent set, provided that population structure is the same across the discovery and prediction samples (Figure 6). In contrast, the prediction accuracy for prediction eigenvectors decreased (Figure 6) quickly for eigenvectors other than E₁. For example, the prediction accuracy for E₂ was below R² <0.2 and R² <0.15 for E₃. For E₄–E₁₀, the prediction accuracy dropped down to nearly zero. This is consistent with the top 2–3 eigenvectors explaining the majority of variation (McVean, 2009), if the training and the test sets had their population structure matched.

Figure 6.

Prediction accuracy of the projected eigenvectors for POPRES samples. Given 2466 POPRES samples, the data were split to 5:95%, 10:90%, 20:80%, 30:70%, 40:60% and 50:50%, as training and test sets. The left columns represent prediction accuracy (R²) using randomly selected numbers (100, 1000, 10 000, 100 000, all) of markers, and the 95% confidence interval were calculated from 30 replication for resampling given number of markers. In contrast, the right columns represent the predicted accuracy for 8 P-value thresholds (1e−6, 1e−5, 1e−4, 1e−3, 1e−2, 1e−1, 0.5 and 1) for EigenGWAS SNPs.

If EigenGWAS SNPs of low P-value were likely to be AIMs, we would hypothesize that AIM markers would be more efficient in giving high accuracy for the predicted eigenvectors (Figure 6). For E₁, the prediction accuracy reached 1 more quickly by using markers selected by P-value thresholds. The prediction accuracy for projected E₂ was dependent upon the threshold. For projected E₂ given a 50:50 split of POPRES sample, applying the threshold of P-value <1e−6 (927 SNPs), R²=0.136, as high as using all markers. For other projected eigenvectors, the pattern of accuracy did not change much after applying P-value thresholds because, in general, the prediction accuracy was low. This indicated that eigenvectors other than the first two eigenvectors capture little replicable population structure in POPRES.

In practice, the training and the test set may not match perfectly on population structure, and this will likely lead to a reduction in prediction accuracy. To demonstrate this, we split the POPRES samples into two sets: pooling Swiss (991 samples) and French (96 samples) samples into one group (SF), and the rest of the samples into the other group (NSF). We used SF as the training and the NSF as the testing. As SF was almost an average of North European and South European gene flow, making a less stratified population, its EigenGWAS effects would be consequently small and less ‘heritable'. When using all SNPs effects estimated from SF set, the observed prediction accuracy for NSF set was R²=0.33 and 0.005 for E₁ and E₂, respectively. These results indicate that a matched training and test set is important for prediction accuracy of the projected eigenvectors.

Ancestry information may still be elucidated well even if the training set and the test set do not match well in their population structure. Using HapMap3 as the training set, we also tried to infer the ancestry of the Puerto Rican cohort (105 individuals) and Pakistani cohort (95 individuals) from 1000 Genomes project (The 1000 Genomes Project Consortium, 2012). A total of 907 614 common SNPs were found between HapMap3 and 1000 Genomes project. As illustrated, using these SNPs between HapMap3 and 1000 Genome projects, the projected eigenvectors accurately revealed the demographic history of Puerto Rican cohort, an admixture of African and European gene flows, and Pakistan cohort, an admixture of Asian and European gene flows (Figure 7).

Figure 7.

Projected eigenvectors for Puerdo Rican cohort (PUR) and Pakistan cohort (PJL) in 1000 Genomes project. The training set was HapMap3 samples build on 919 133 SNPs. The eigenvectors 1 and 2 were generated on the 907 614 common SNPs. PUR showed an admixture of African and European gene flows, and PJL Asian and European gene flows.

As a negative control, we replicated the prediction study for simulated data used in the previous section. The simulated data were split to two equal sample size. As there was no population structure in the simulated data, the prediction accuracy was poor, R² <0.01 from E₁ to E₁₀. This demonstrates that prediction can be used to validate whether population structure exists within a genotype sample.

We concluded that to achieve high prediction accuracy of projected eigenvectors for independent samples, there are several conditions to be met: (1) the training set should harbour sufficient population stratification; (2) the sample size of the training should be sufficiently large; (3) the test sets should be as concordant as possible in its population structure; (4) when there is no real population structure, the prediction accuracy is very low close to zero; and (5) depending on the population, high prediction was largely achievable for the projected E₁.

Discussion

Eigenvectors have been routinely employed in population genetics, and various approaches have been proposed to offer interpretation and efficient algorithms (Patterson et al., 2006; Rokhlin et al., 2009; McVean, 2009; Chen et al., 2013; Galinsky et al., 2016). In this study, we created a GWAS framework for studying and validating population structure, and offer an interpretation of the GWAS signals for eigenvectors within this framework. Without prior information about the structure/grouping of the sample, the EigenGWAS framework (least square) identifies ancestry informative markers and loci under selection across gradients of ancestry. Although EigenGWAS resembles the conventional GWAS, it should be noted that EigenGWAS has its phenotype generated from the genotypes via PCA. Hence, EigenGWAS is a GWAS but does not use phenotypes in conventional sense. The effects estimated in EigenGWAS can be alternatively estimated via BLUP or SVD (Equation (3)), but two major differences should be noticed: (1) BLUP or SVD will be computationally infeasible when there are many markers involved, whereas EigenGWAS reduces the computation into simple regression and largely avoids this problem; (2) it is not obviously how to obtain the P-values for BLUP or SVD estimates, but EigenGWAS makes it possible to conduct a standard z-score test for an estimated SNP effect.

We integrated SVD, BLUP and single-marker regression into a unified framework for the estimation of SNP-level eigenvectors. SVD is a special case of BLUP when heritability is of 1 for the trait and the target phenotype is an eigenvector. Furthermore, the BLUP is equivalent to the commonly used GWAS method for estimating SNP effects. As demonstrated, the correlation between BLUP and GWAS is almost 1 for the estimated SNP effects. EigenGWAS offers an alternative way in estimating F_st that can replace conventional F_st when population labels are unknown, populations are admixed or differentiation occurs across a gradient. As demonstrated for CEU–TSI samples, EigenGWAS brings out nearly identical estimation of F_st compared with conventional estimation.

Different from conventional GWAS, which requires conventional phenotypes, the proposed EigenGWAS provides a novel method for finding loci under selection based on eigenvectors that are generated from the genotype data itself. An EigenGWAS hit may reflect the consequence of process and thus additional evidence is needed to differentiate selection from drift. LCT is a known locus under selection that differs in its allele frequency as indicated by F_st statistic between Northern and Southern Europeans (Bersaglieri et al., 2004). We replicated the significance of LCT in CEU and TSI samples and POPRES European samples. DARS has been found in association with hypomyelination with brainstem and spinal cord involvement and leg spasticity (Taft et al., 2013). In addition, we also found HERC2 locus independently that may indicate the existence of anthropological difference in certain characters, such as hair, skin or eye colour across European nations (Voight et al., 2006; Visser et al., 2012).

Although by definition selection and genetic drift are different biological processes, both lead to allele frequency differentiation across populations and it is often difficult to tear them apart. In this study, with and without adjustment for λ_GC, EigenGWAS offers a straightforward way to filter out population stratification. For example, with adjustment for λ_GC, LCT and DARS were still significant in both EigenGWAS, whereas HERC2 was only significant in POPRES. If adjustment for λ_GC removed the average genetic drift since the most recent common ancestor for the whole sample, it might indicate that HERC2 reflected the anthropological difference between subsamples but not under selection as strong as that for LCT. Nevertheless, LCT was differentiated because of selection that was on top of genetic drift, and for DARS, it might be significant because of hitchhiking effect. Hence, LCT, DARS and HERC2 were significant in EigenGWAS for different mechanisms.

In EigenGWAS application, it provides a clear scenario that λ_GC is necessary if genetic drift/population stratification should be filtered out. It has been debated whether correction for λ_GC is necessary for GWAS (Yang et al., 2011b). If the inflation is because of population stratification, as initially λ_GC introduced, it seems necessary to control for it. In contrast, if it is because of polygenic genetic architecture, then correction for λ_GC will be a overkilling for GWAS signals. Interestingly, Patterson et al. (2006) found that the top eigenvalues reflect population stratification, and in our study we found λ_GC from EigenGWAS was numerically so similar to its corresponding eigenvalues. It in another aspect indicates that λ_GC captures population stratification. Hence, in concept and implementation, the correction for λ_GC is technically reasonable. Of note, Galinsky et al. (2016). also proposed a similar procedure to filter out population stratification in a study similar to ours, but we believe our framework is much easier to understand and implement in practice.

Once we have EigenGWAS SNP effects estimated, it is straightforward to project those effects onto an independent sample. The prediction of population structure was that of recent studies (Chen et al., 2013). We found that the prediction accuracy for the top eigenvector could be as high as almost 1. Given a training set of ∼1000 samples, the prediction accuracy could be very high if there were a reasonable number of common markers in the order of 100 000. This number, which needs to be available in both reference set and the target set, is achievable. Further investigation may be needed to check whether this number of markers is related to effective number or markers after correction for linkage disequilibrium for GWAS data. When the population structure of the test sample resembles the training sample, high accuracy will be achieved for the leading projected eigenvectors. Therefore, this approach is likely to be extremely beneficial for extremely large samples, such as UK Biobank samples and 23andMe, both of which have more than half million samples where direct eigenvector analysis may be infeasible. Our results suggest that sampling ∼1000 individuals from the whole sample as the training set and subsequently project EigenGWAS SNP effects to the reminding samples will be sufficient to reach a reasonable high resolution of the population structure.

Many improvements to the inference of ancestry using projected eigenvectors have been suggested (Chen et al., 2013). As the concordance of population structure between the training and test sets is often unknown (population structure, upon from genetic or social–cultural perspectives, its definition can be difficult or controversial), improvement of the inference of ancestry may or may not be achieved dependent upon the scale of the precision required for a sample. However, for classification of samples at ethnicity level, projected eigenvectors are likely to have high accuracy, as demonstrated in the Puerto Rican cohort and the Pakistani cohort. Therefore, when identifying ethnic outliers, using projected eigenvectors from HapMap is likely to be sufficient in practice.

Eigenvector analysis of GWAS data is an important well-utilized data technique, and here we show that its interpretation depends on many factors, such as proportion of different subpopulations and F_st between subpopulations. Our EigenGWAS approach provides intuitive interpretation of population structure, enabling AIMs to be identified, and potentially loci under selection to be identified. To facilitate the use of projected eigenvectors, we provide estimated SNP effects from HapMap samples and POPRES and software that can largely reduce the logistics involved in conventional way in generating eigenvectors, such as reference allele match, and strand flips.

The authors declare no conflict of interest.