On Rare Variants in Principal Component Analysis of Population Stratification

doi:10.21203/rs.2.9120/v1

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On Rare Variants in Principal Component Analysis of Population Stratification

https://doi.org/10.21203/rs.2.9120/v1

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published 17 Mar, 2020

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Background

Population stratification is a known confounder of genome-wide association studies, as it can lead to false positive or false negative results. Principal component analysis (PCA) method is widely applied in the analysis of population structure with common variants. It is still controversial whether this method is effective when using rare variants to distinguish population stratification.

Results

In this work, we derive a mathematical expectation of the genetic relationship matrix. Variance and covariance elements of the expected matrix depend explicitly on allele frequencies of the genetic markers used in the PCA analysis. We show analytically that inter-population variance is contained in K principal components (PCs) and mostly in the largest K-1 PCs, where K is the number of populations in the samples. When allele frequencies become small, the ratio of the inter-population variance in the K PCs to the intra-population variance abates, the portion of variance explained by the K PCs decreases, and the distance among populations diminishes. As a result, the PCA of population stratification performs worse with rare variants than with common ones. The results are validated in the analysis of the 1000 Genomes Project data. The ratio of the two variance components is 93.85, variance explained by the largest five PCs is 17.09%, and population distance is 444.38 when using with common variants with allele frequencies between 0.4 and 0.5. However, the ratio, percentage and distance decrease to 1.83, 0.74% and 17.83, respectively, with rare variants of frequencies between 0.0001 and 0.01.

Conclusions

Although the vast majority of rare variants are population-specific and may have richer ancestry information than common ones, new methods must be developed to exploit the information effectively.

Internal Medicine Specialties

rare variants

population stratification

principal component analysis

single nucleotide polymorphism

Genome-wide association studies (GWAS) [1] have identified a considerable number of sequence variants, such as single nucleotide polymorphisms (SNPs), associated with human diseases or traits. Population stratification—allele frequencies of genetic markers of the studied samples having significant differences owing to systematic ancestry differences—can cause false positive or false negative results in case-control as well as cohort studies [2, 3]. Association mapping based on rare variants are much more susceptible to subtle effects of population stratification and therefore, more likely to yield false positive results [4]. From a population genetics point of view, exploring population structure is important for understanding the evolutionary history of populations. Many methods and software are available to study the population stratification, such as the principal component analysis (PCA) implemented in EIGENSOFT [5, 6], the multidimensional scaling analysis in PLINK,7 the clustering analysis in STRUCTURE [8, 9], and fastSTRUCTURE [10]. Recently, controlling population stratification in the association analysis using linear mixed models [11, 12, 13, 14] was also suggested. Based on a large number of common variants in linkage equilibrium whose minor allele frequencies (MAFs) are larger than 5%, the PCA of population structure is widely applied in GWAS and has become the gold standard of analysis.

With the advance of high-throughput sequencing technology, as well as the enormous reduction of the cost, it is capable and affordable in genetic studies to detect additional low-frequency and rare variants (MAF < 1%) [15]. Furthermore, existing sequencing data suggest that the vast majority of rare variants are population-specific. In the 1000 Genomes Project data [16, 17], there are a total of 77 million biallelic SNPs, among which 65 million are rare and 52 million are polymorphic in a single population. It seems that rare variants are more informative in distinguishing population structure than common ones. However, the efficacy of using rare variants in population stratification analysis remains controversial.

A number of efforts have been made concerning the use of low-frequency and rare variants in population stratification analysis. Baye et al. illustrated that more fine substructure can be detected using rare variants and suggested that more SNPs are required to account for a similar level of population structure using rare variants compared to common ones [18]. Siu et al. showed that rare variants have a much higher power to identify population substructure than common variants [19]. In contrast, Zhang et al. demonstrated that PCAs based on common and less-frequency SNPs perform better than those based on rare ones in separating European and African samples [20]. The authors further concluded that there is little added value for PCA of population stratification with rare variants only [21]. All existing work was based on analysis of genotype data from the 1000 Genomes Project with known population structure.

In this work, we investigate how rare variants affect PCA of population stratification from a theoretical perspective. We derive mathematical expectation of the genetic relationship matrix (GRM) [22]. The GRM is commonly computed from the observed genotypes and additional eigen-decomposition in the analysis of population stratification. Elements of the expected genetic relationship matrix (EGRM), however, depend explicitly on the allele frequencies of the markers used. We show that inter-population variance is contained in K principal components (PCs) and mostly in the largest K-1 PCs, where K is the number of populations in the sample. When the MAFs are smaller, the ratio of the inter-population variance contained in the K principal components (PCs) to the intra-population variance abates, the portion of variance explained by the K PCs decreases, and the distance among populations diminishes. The PCA of population stratification performs worse with rare variants than with common ones. The results are further validated in the analysis of the 1000 Genomes Project data with 2504 individuals from five continental populations.

Genetic relationship matrix

In the scenario where genotype data of individuals is sampled from K populations, there areindividuals in population k and the number of individuals in the total population is. We have M SNPs, whose frequencies of their reference alleles in population k are . Let X be the genotype matrix of dimension. The element X(n,m) is coded as the number of the reference allele of SNP m for individual n, where and . Typically, the number of individuals is less than the number of markers, i.e. We assume that all SNPs are under the Hardy-Weinberg equilibrium in each population. The GRM can be calculated as, where each entry of Y is a normalized version of the coded genotype in X, i.e. for and . Here, and denote the genotypic mean and standard deviation of SNP m in the total population, respectively. It can be shown that and relate to the population structure and allele frequencies as follows (Supplemental Text S1).

The reference-allele frequency of SNP m in the total population can be found as , where . The GRM is of dimension , whose diagonal elements are genotypic variance of individuals and off-diagonal elements are genotypic covariance between two individuals. It should be noted that genotypes follow mixed binomial distributions, and elements of are sample variances and covariance computed from the genotype data. The PCA analysis of population stratification is based on the eigen-analysis of the observed GRM .

In practice, and are unknown, and therefore sample mean and sample standard deviation or some other quantities similar are used instead. Usually, is used for the centralization in (2). In EIGENSOFT, is adopted for the normalization in (2), while is employed in GCTA [22]. Different estimates of the allele frequencywere suggested as well [5, 6]. In the following theoretical derivations, we assume that and are known for the sake of simplicity. This will bring about clear connections between variance and covariance elements of the EGRM and allele frequencies of the SNPs used in the analysis. The connections further provide clues and insights for understanding the effect of rare variants on the PCA of population stratification.

Expected genetic relationship matrix

We assume that all individuals are unrelated and all SNPs are in linkage equilibrium. When the number of markers M goes large, the sample variance and covariance elements in the GRM will converge to their mathematical expectations in probability due to the law of large numbers. We denote the EGRM as Z, which is the expectation of the GRM . Without loss of generality, we assume that individuals are ordered according to their population memberships. As such, the first rows and columns correspond to individuals from population 1, the next rows and columns are from population 2, and so on. Thus, the EGRM can be partitioned into a block matrix consisting of submatrices.

Diagonal sub-matrices of the EGRM Z have the following structure.

Here, diagonal elements of the submatrix are of the mathematical form which is the genotypic variance of individuals from population k. All off-diagonal elements share the form which is the genotypic covariance between two individuals from population k.

The off-diagonal sub-matrices of the EGRM Z are structured as follows

Elements of share the value which is the genotypic covariance between one individual from population k and one from population l. Details of the derivations are presented in Supplemental Text S2.

The EGRM Z, the mathematical expectation of GRM Z, depends only on the population sizesand allele frequencies of the M SNPs in K populations ,. Here, we treat the SNP number M and the allele frequencies as fixed numbers. A theoretical formulation of the PCA that considers allele frequencies in different populations as a random vector was proposed in Ma and Amos, 2010 [23]. Although based on different assumptions, a genotypic variance-covariance matrix of the same structure was attained, whereas elements of the EGRM are expressed as explicit functions of allele frequencies. This allows for the investigation of the effect of rare variants on the analysis of population stratification.

Rare variants on the eigenvalues

Carrying out eigen-decomposition on the EGRM, it can be shown that there are eigenvalues of value , for . The sum of the eigenvalues is (11)

Clearly, variations in the PCs are entirely intra-population variance of the K populations. The sum of the other K eigenvalues is, (12) where represents the inter-population variance component and stands for the intra-population variance component. Here, the intra-population covariance of the EGRM factor in the K PCs as the inter-population variance after the eigen-decomposition. Note that all inter-population variance is contained in the K PCs. Hence, it is sufficient to conduct the population stratification analysis based on the K PCs alone.

Given a set of SNPs, the divergence among the K populations can be measured by the ratio of the two variance components, i.e. (13)

Its normalized version can be defined as (14)

The larger the FPC is, the more divergence among the populations.

Note that , terms in are quadratic functions of , , . However, terms in are linear and quadratic functions of the frequencies. Therefore, FPC will decrease when the allele frequencies of the SNPs become smaller. As a result, instead of improving the population stratification analysis, using rare variants will deteriorate the analysis performance. Meanwhile, since decreases faster than , the K eigenvalues will be closer to the other eigenvalues when allele frequencies of the SNPs become smaller. Thus, the percent of variance explained by the K PCs will be smaller.

It can be shown that among the K eigenvalues, are of large values and one small. When intra-population variance of the K populations are equal, all inter-population variance is contained in the largest eigenvalues. In addition, when the sample size is large and the portions of populations remain, any inter-population variance contained in the small eigenvalue is negligible, almost all information on the population structure is contained in the largest PCs.

For cases with two populations, it can be shown that the two eigenvalues are.

When inter-population variance of the two populations are equal, i.e. , we have.

That is, all information on the population structure is contained in the largest PC. All proofs are presented in Supplemental Text S3.

Rare variants on the population distance

Suppose that , are the eigenvectors associated with the K eigenvalues containing inter-population variance. We can represent each individual as a point in the K-dimension space. Vector consists of coordinates of N individuals in the k-th dimension. Average value represents center of the total population in the k-th dimension, where is the vector of dimension and with each element as 1. Due to the structure of Z, individuals from the same population share the same coordinates in the K-dimension space, and the common points denote the representative points of the populations, or centers of the populations.23 We define (15) which measures the population divergence as the sum of squared distances among populations. The proof is shown in Supplemental Text S4. Note that the second term in (15) is an average intra-population variance. As explained earlier that when allele frequencies become smaller, the K eigenvalues decrease. Hence, the population distance is smaller when using rare SNPs compared to common ones.

The 1000 Genomes Project data

We used genotype data from the 1000 Genomes Project to validate our theoretical results. Genotype data used in this work was obtained from Phase 3 version 5a of the 1000 Genomes Project [16, 17], which contains 84.4 million genetic markers and 2504 individuals with ancestry from Europe (EUR), East Asia (EAS), South Asia (SAS), West Africa (AFR) and Americas (AMR). We extracted biallelic SNPs that are polymorphic in the total population. In summary, there are 77,279,863 SNPs; 5,261,820 are common (0.1<MAF≤0.5), 6,770,457 are low-frequency (0.01<MAF≤0.1), and 65,247,586 are rare (0.0001<MAF≤0.01). Genotype data were converted to PLINK format with VCFtools [24]. The SNPs were divided into six frequency bins according to their MAFs, as shown in Table 1. For each bin, we randomly sampled approximately 100,000 SNPs using PLINK for the population stratification analyses. PCAs were carried out, with GRMs computed by EIGENSOFT and PCAs on EGRMs conducted using GCTA. Default parameters were used when analyzing with EIGENSOFT, which excluded 68 and 116 outliers in the analyses of the data from frequency bin 5 and 6, respectively.

Theoretical and empirical EGRMs

To calculate the theoretical results (5)-(10), we computed MAFs of the SNPs with PLINK. Values of variance and covariance , , , were calculated as in (7), (8), and (10), respectively, in which was computed with (3) and , . Values of and for the five populations with SNPs from the six frequency bins are presented in Table 2. Absolute values of the inter-population covariance are much smaller and the results are shown in Supplemental Tables S1-6.

To obtain the empirical values of variance , as well as covariance and , we first computed GRMs with SNPs from the six bins using EIGENSOFT. Each GRM included variance and covariance terms of N individuals based on the observed genotype data. Empirical value of was computed as the average variance of the individuals from population k. The empirical value of is the average covariance of pairs of individuals from population k. Lastly, the value of is the average covariance of pairs of individuals, one from population k and one from population l. The results of and are shown in Table 2, and those of are presented in Supplemental Tables S1-6.

We can see that across the six frequency bins, theoretical values of , , and predicted by (7), (8), and (10), respectively, are close to their empirical values. When MAFs of the SNPs become smaller, intra-population covariance decreases. For example, was 0.2 for EAS with SNPs whose MAFs are between 0.4 and 0.5, which reduced to 0.003 in the sixth bin that included rare SNPs only. A similar pattern can be observed for the other four populations. FPC was estimated by (13) for the six bins, where empirical values of and were used. The FPC decreases from 93.85 in bin 1 to 55.01 in bin 5, and further to 1.83 in bin 6. Thus the divergence among the populations is much larger when measured by common SNPs than by rare ones.

PCAs of the 1000 Genomes Project data

With genotypes of SNPs from each frequency bin, we carried out PCAs of population stratification by EIGENSOFT, which was essentially based on the eigen-analysis of the observed GRMs. Scatter plots of the largest three PCs are shown in Figures 1 and 2, where eigenvectors were scaled by square roots of their corresponding eigenvalues.

From Figures 1 and 2, we can see patterns of population structure computed with common and less-frequency SNPs. For example, Figures 1a-e and 2a-e displayed similar patterns, whereas the scatter plots based on rare SNPs differed significantly. For example, AMR and SAS are separated mostly by the third PC with common SNPs, while they are distinguished by the second PC with rare ones. The third PC from rare SNPs reveals mostly substructure of AFR, likely because more rare SNPs are polymorphic in AFR than in other populations. Portions of variance explained by the largest five PCs decrease from 17.09% in bin 1 to 10.41% in bin 5, and it falls dramatically to 0.74% with rare SNPs only. As a result, the five populations are more closely distributed around the origin in Figures 1f and 2f, compared with those in Figures 1a-e and 2a-e. Clearly, common variants show much better performance in dissecting the population structure than rare variants do.

PCAs of EGRMs

For each frequency bin, we also constructed a EGRM with structure as described in (5), (6), and (9), whose variance and covariance elements were their theoretical values calculated by (7), (8), and (10), respectively. We conducted PCAs of the EGRMs using GCTA, and scatter plots of the largest three PCs shown in Figures 1 and 2. Similarly, coordinates were scaled by square roots of their eigenvalues.

Upon comparing the representative points in Figures 1 and 2, we can see that distances between populations decrease as the SNPs change from common to rare. Sum of the squared distance was calculated for the six frequency bins by (15), where , were the eigenvalues of the EGRM Z and , , were their theoretical values. The decreases from 444.38 in bin 1 to 254.10 in bin 5, and further to 17.83 in bin 6.

In addition, when portions of variance explained by the PCs become small, deviations between the representative points of the populations and true centers of the populations can be observed. This is particularly evident in the scatter plots with rare SNPs. In the PCAs of a single population, such deviations are more obvious when percents of variance explained by the largest PCs are much smaller.

We demonstrated that all information about the population structure is contained in K PCs. Genotypic variance explained by the K PCs can be further decomposed into the intra-population variance and inter-population variance . Note that M SNPs contributed to and in an additive manner. For a fixed number of SNPs, using a combination of common and rare SNPs will result in a lower FPC compared with using common SNPs only and therefore result in worse performance. In the case where there is one SNP, the ratio can be further reduced to, where is the allele frequency in the population k, and the frequency in the total population. Although the FPC looks similar to the classical Wright’s fixation index FST [25], the two quantities are different in nature. FST gauges the proportion of genetic variance due to the population stratification. Our FPC is motivated from a statistical point of view, which measures the population divergence as the statistical effect size of a one-way analysis of variance problem. When keeping the portion of each population in the total sample, the FPC increases as a linear function of the total population size. This is to be expected since it becomes easier to detect the differences among populations when sizes of the total sample are larger.

In GWAS, it is common practice to conduct population stratification analyses using a large number of random markers [26], which usually yields satisfactory results. As shown in this work, the capability of dissecting population structure depends on the allele frequencies of markers used in the analyses, and common variants perform much better than rare ones. This is not much of a concern for GWAS because SNP panels implemented in the genotyping platforms are mostly of common ones. In sequencing studies, however, the majority of the called variants are rare, and selecting SNPs randomly will yield a large portion of rare SNPs, which will deteriorate the analysis performance. Therefore, it is necessary to restrict the selection to only the common SNPs when analyzing population stratification with sequencing data. This would also be true for controlling population stratification based on the linear mixed models [11, 12, 13, 14].

In this work, we assumed that and are known constants in (2) in order to simplify the theoretical derivations. Our results are approximations of those when estimates of the two quantities are used. When sample size N is large, variations associated with and are much smaller than those with the genotype data. Therefore, the mathematical expectations are largely taken with respect to the genotypes and difference between the two sets of results would be small. As shown in Table 2, the predicted values of the EGRM are close to their empirical values in the 1000 Genome Project data. We carried out additional simulation studies to evaluate the effect of lacking knowledge on and . We randomly chose one SNP from each of the six frequency bins (Supplemental Table S7). Based on their MAFs observed in the five populations of the 1000 Genomes Project, we simulated genotypes of five populations each with 500 individuals. Values of and were computed with the assumption of known population structure and MAF information, and theoretical values of and were then calculated. For comparison, we first estimated and from the simulated genotype data. Y(n, m) were normalized with and , and and were obtained as averages of sample variance and covariance from 1000 replicates. The two sets of results are presented in Supplemental Table S8 and the differences between the two sets of results are negligible except for small differences in the results with the rare SNP.

Despite the fact that the vast majority of rare variants are population-specific, we showed performance of the PCA of population stratification is better when based on common SNPs rather than rare ones. On the other hand, the PCA results with rare SNPs do reveal a population structure that differs from that of common SNPs. Existing methods may not exploit ancestry information embedded in the rare variants efficiently, and different approaches from those applied to common variants should be developed [26].

To quantify population divergence as a function of allele frequencies of genetic markers used in the PCA analysis, we derived the expected genetic relationship matrix. Our theoretical results as well as the analyses of the 1000 Genomes Project data showed that employing rare variants yields smaller ratio of the inter-population variance to the intra-population variance in the K PCs, smaller portion of variance explained by the K PCs, and smaller distance among populations than those using common variants. Although the PCA method is a powerful tool to analyze population stratification using common variants, new methods must be developed to exploit the ancestral information contained in the rare variants.

GWAS: Genome-wide association study

SNP: Single Nucleotide Polymorphism

PCA: Principal Component Analysis

PC: Principal Component

MAF: Minor Allele Frequency

Ethics approval and consent to participate

Not applicable

Consent for publication

Not applicable

Availability of data and materials

The datasets analysed during the current study are available at http://www.internationalgenome.org/.

Competing Interests

The authors declare no competing interests.

Funding

This work was supported by the national Thousand Youth Talents Plan.

Author’s contributions

Shengqing Ma: conceived the concept, conducted the analyses, and drafted the manuscript. Gang Shi: conceived the concept, supervised the work, reviewed and revised the manuscript.

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Table 1. Summary of SNPs from the 1000 Genomes Project data.

MAF Pop (N)	Common SNP				Low-frequency SNP	Rare SNP
	(0.4,0.5]	(0.3,0.4]	(0.2,0.3]	(0.1,0.2]	(0.01,0.1]	(0. 0001,0.01]
	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6
EUR (503)	995,352	1,048,669	1,190,239	1,581,788	3,717,490	13,531,139
EAS (504)	970,359	1,010,549	1,130,961	1,440,178	2,982,582	14,189,976
AMR (347)	1,004,970	1,068,395	1,234,095	1,613,443	4,827,083	16,092,172
SAS (489)	1,001,330	1,077,620	1,239,727	1,626,183	3,989,855	15,562,799
AFR (661)	981,929	1,097,944	1,436,771	2,403,901	8,852,607	24,044,176
Total (2504)	1,023,570	1,105,365	1,308,728	1,824,157	6,770,457	65,247,586

MAF: Minor Allele Frequency. Pop: Population. EUR: European. EAS: East Asian. AMR: American. SAS: South Asian. AFR: African.

Table 2. Theoretical and empirical values of the variance and covariance elements of EGRMs.

MAF	EUR	EAS	AMR	SAS	AFR
(0.4,0.5]	1.06/0.97	1.11/1.02	1.04/0.97	1.04/0.96	1.15/1.05
(0.3,0.4]	1.06/0.97	1.11/1.02	1.04/0.97	1.04/0.96	1.14/1.05
(0.2,0.3]	1.05/0.96	1.09/1.00	1.03/0.97	1.03/0.96	1.17/1.07
(0.1,0.2]	0.99/0.91	1.01/0.93	0.98/0.93	0.99/0.92	1.32/1.22
(0.01,0.1]	0.61/0.57	0.50/0.47	0.73/0.70	0.56/0.53	2.50/2.37
(0.0001,0.01]	0.71/0.71	0.94/0.94	0.82/0.82	0.98/0.98	1.46/1.46
(0.4,0.5]	0.13/0.11	0.22/0.20	0.08/0.07	0.08/0.07	0.30/0.28
(0.3,0.4]	0.13/0.11	0.22/0.20	0.07/0.06	0.08/0.07	0.29/0.26
(0.2,0.3]	0.12/0.11	0.21/0.19	0.07/0.06	0.08/0.07	0.27/0.24
(0.1,0.2]	0.11/0.10	0.18/0.17	0.07/0.06	0.08/0.07	0.27/0.25
(0.01,0.1]	0.06/0.05	0.08/0.07	0.04/0.03	0.05/0.04	0.25/0.23
(0.0001,0.01]	0.004/0.002	0.005/0.003	0.004/0.002	0.005/0.003	0.011/0.008

The first values are theoretical values of the variance and covariance, and second values are empirical values.

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On Rare Variants in Principal Component Analysis of Population Stratification

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