Detecting Multiethnic Rare Variants

Abstract

Genome-wide association studies have identified many common genetic variants which are associated with certain diseases. The identified common variants, however, explain only a small portion of the heritability of a complex disease phenotype. The missing heritability motivated researchers to test the hypothesis that rare variants influence common diseases. Next-generation sequencing technologies have made the studies of rare variants practicable. Quite a few statistical tests have been developed for exploiting the cumulative effect of a set of rare variants on a phenotype. The best-known sequence kernel association tests (SKATs) were developed for rare variants analysis of homogeneous genomes. In this chapter, we illustrate applications of the SKATs and offer several caveats regarding them. In particular, we address how to modify the SKATs to integrate local allele ancestries and calibrate the cryptic relatedness and population structure of admixed genomes.

1. Introduction

1.1. Background

Initially, genome-wide association studies (GWAS) aimed to localize common genetic risk factors for complex common diseases. It was believed that a large number of genotyped samples can provide sufficient power to detect common variants which have modest effects on a phenotype. Hundreds of GWAS have been performed for mapping genetic variants of common diseases, such as hypertension, bipolar disease, coronary artery disease, diabetes, and cancer [1–3]. Such studies have successfully identified thousands of genes which are significantly associated with hundreds of traits [4] (https://www.ebi.ac.uk/gwas/). Genome-wide association studies have greatly advanced our understanding of genetic mechanisms of many common diseases.

For a common disease phenotype, however, the significant common variants identified by GWAS account for only a small fraction of the heritability observed in family studies [5]. For example, height is known to be a heritable trait with estimated heritability around 0.8 from family and twin studies, which implies about 80% of the trait variation is attributable to genetic factors. Multiple GWAS on height [3, 6–9] identified hundreds of significant common variants, which together explain only 27.4% of height variation. The missing heritability may be potentially accounted for by many rare variants [5, 10–13]. With the publication of the 1000 Genomes Project [14], we entered the era of next-generation sequencing studies. Deep sequencing technologies have been providing more comprehensive and accurate descriptions of rare variants. By directly testing rare variants in candidate genes, next-generation sequencing studies have identified many rare variant associations for a range of common diseases, e.g., type I diabetes, sterol absorption, plasma levels of LDL-C and blood pressure [15–18].

Quite a few sequence association tests have been developed for exploiting the cumulative effect of a set of rare variants on a phenotype. Prominent population-based sequence association tests include the weighted sum test [19], the C-alpha score test [20], the Estimated REgression Coefficient (EREC) test [21], the variable threshold (VT) test [22], the Sequence Kernel Association Test (SKAT) [23], the SKAT-O [24], and the Smoothed Functional Principal Component Analysis (SFPCA) method [25]. As a combination of SKAT and burden test, SKAT_O cannot always outperform the burden test or SKAT (see Note 1). Population-based tests can be invalid and suboptimal in the presence of familial relatedness. Rare variants may arise from recent mutations in pedigrees [26–28]. Several family-based sequence association tests were developed, e.g., the sibpair and odds ratio weighted sum tests [29, 30], the famSFPCA [31], and the famSKAT [32]. These family-based association tests require clear relatedness information or adopt a conventional kinship estimate of cryptic relatedness, e.g., KING-Robust [33]. As detailed below, SKAT and famSKAT assume linear mixed-effect models to integrate variant sets and compute significance analytically.

1.2. The Sequence Kernel Association Test (SKAT)

SKAT [23] appears to be a most prevailing method for analyzing unrelated genomes. For individual i, let y_i be the trait value, g_ij be the copy number of the minor allele at the jth SNP, and x_ik be the value of the kth covariate, g_i = (g_i1, …, g_iL)′ and x_i = (x_i1, …, x_im)^′. For a quantitative trait, SKAT assumes

$$y_i={\alpha }_0+x_i^{\text{'}}\alpha +g_i^{\text{'}}\beta +{\epsilon }_i,$$

where, and ε_i ~ N(0; σ²) is random error. For a binary trait (i.e., y_i = 1 for a case, and 0 for a control), SKAT assumes.

$$\text{logit}P\left(y_i=1\right)={\alpha }_0+x_i^{\text{'}}\alpha +g_i^{\text{'}}\beta .$$

SKAT allows arbitrary sizes and directions of variant effects. For both equations, α₀ is an intercept term, α=(α₁,…α_m)′ is the column vector of regression coefficients for the m covariates, and β =(β₁,…,β_L)′ is the column vector of regression coefficients for the L test variants. Let H₀:β =0 be the null hypothesis of no association between the test variants and trait. SKAT assumes that each β_i follows an arbitrary distribution with mean 0 and variance w_jτ, where τ(≥0) is a variance component and w_j(≤0) is a prespecified weight for variant j. H₀: β =0 is equivalent to H₀:τ=0. SKAT utilizes the variance-component score statistic

$$Q={\left(y-\widehat{\mu }\right)}^{\text{'}}K\left(y-\widehat{\mu }\right),$$

where K =GWG′, G[g₁,…,g_N ]′, W =diag(w₁,….,w_L), ${\widehat{\mu }}_i={\widehat{\alpha }}_0+{\mathit{x}}_{\mathit{i}}^{\text{'}}\widehat{\mathit{\alpha }}$ for a dichotomous trait, and ${\widehat{\mu }}_i={\text{logit}}^{-1}\left({\widehat{\alpha }}_0+{\mathit{x}}_{\mathit{i}}^{\text{'}}\widehat{\mathit{\alpha }}\right)$ for a dichotomous trait, ${\widehat{\alpha }}_0$ and $\widehat{\alpha }$ are hypothesis by regressing y on the covariates only. The default weight is $\sqrt{w_j}=Beta\left({\text{MAF}}_j;1,25\right)$ where MAF_j is the frequency of the minor allele at the jth SNP, which can be evaluated from the entire sample. The default weighting scheme is not always optimal (see Note 2). When analyzing admixed genomes, for example, ancestries of variants should be incorporated for better weighting schemes (see Note 3). For a binary trait, statistic Q collapses to the C-alpha test statistic T [20] when all the w_j are set to be 1 and all covariates are excluded.

Under the null hypothesis, Q follows a mixture of chi-square distributions. To be specific,

$$Q\sim {\sum }_{j=1}^L{\lambda }_j{\chi }_{1,j}^2,$$

Where the ${\chi }_{1,j}^2$ are independent ${\chi }_1^2$ variables, and the λ_j are the eigenvalues of the matrix ${\mathit{P}}_0^{1/2}\mathit{K}{\mathit{P}}_0^{1/2}$, ${\mathit{P}}_0=\mathit{V}-\mathit{V}\mathit{X}{\left({\mathit{X}}^{\text{'}}\mathit{V}X\right)}^{-1}{X}^{\text{'}}V$ and X=[1,(x₁,…,x_n)′] is an n×(m+1) covariate matrix. For quantitative phenotypes, $\mathit{V}={\widehat{\sigma }}_0^2\mathit{I},{\widehat{\sigma }}_0^2$ is the is the estimator of σ² under the null hypothesis. For binary phenotypes, $\mathit{V}=diag\left({\widehat{\mu }}_1\left(1-{\widehat{\mu }}_1\right),\mathrm{\dots },{\widehat{\mu }}_n\left(1-{\widehat{\mu }}_n\right)\right)$. Therefore, the P-value of Q can be closely approximated with the computationally efficient Davies’ method [34]. The SKAT package depends on R version 2.13.0 or above. In Subheading 2.1, we illustrate how to install and run this package to scan homogeneous genomes.

1.3. The Family-Based SKAT

When analyzing family data, SKAT has inflated type I error if the relatedness between family members is ignored. To calibrate familial correlation, Chen et al. [32] extended SKAT to the family-based SKAT (famSKAT) for rare variant association analysis with quantitative traits in family data. Compared to SKAT, famSKAT has a different form of test statistic and null distribution, but is equivalent to SKAT when there is no familial correlation. In famSKAT, the vector of a quantitative trait is assumed to follow a linear mixed effects model

$$y=X\alpha +G\beta +\delta +\epsilon ,$$

where X is an n×(m+1) covariate matrix, α is a (m+1)×1 vector consisting of fixed effects parameters (an intercept and m coefficients for covariates), G is an n×L genotype matrix for L rare genetic variants of interest, β is an L×1 vector for the random effects of rare variants, δ is an n×1 vector for the random effects of familial correlation, and ε is an n×1 vector for the error. The vector of errors ε and the random effects β and δ are assumed to be normally distributed and uncorrelated with each other. To be specific, we assume

$$\beta \sim \mathcal{N}\left(0,\tau W\right),\delta \sim \mathcal{N}\left(0,{\sigma }_G^2\Phi \right),\epsilon \sim \mathcal{N}\left(0,{\sigma }_E^{2}I\right),$$

where W is the prespecified diagonal weight matrix for the rare variants, Φ is twice.

the kinship matrix of size n×n obtained from family information only, I is the identity.

matrix of size n × n, and $\left(\tau ,{\sigma }_G^2,{\sigma }_E^2\right)$ are corresponding variance component parameters. Under these assumptions, testing H₀:τ=0 versus H₁:τ>0 is equivalent to testing H₀:β=0 versus H₁:β ≠ 0. The famSKAT statistic is

$$Q={(y-X\widehat{\beta })}^{\text{'}}{\widehat{\Sigma }}^{-1}\text{GW}{G}^{\text{'}}{\widehat{\Sigma }}^{-1}\left(y-X\widehat{\beta }\right),$$

Where ${}^{\widehat{\Sigma }}={\widehat{\sigma }}_G^2\mathit{\Phi }+{\widehat{\sigma }}_E^2\mathit{I}\text{,}\widehat{\mathit{\beta }}={({\mathit{X}}^{\text{'}}{\widehat{\Sigma }}^{-1}\mathit{X})}^{-1}{\mathit{X}}^{\text{'}}{\widehat{\Sigma }}^{-1}\mathit{y}$, and $\left({\widehat{\sigma }}_G^2,{\widehat{\sigma }}_E^2\right)$ are maximum likelihood estimators of $\left({\widehat{\sigma }}_G^2,{\widehat{\sigma }}_E^2\right)$ the null linear mixed effects model y = Xα + δ + ε. Under the null hypothesis,

$$Q\sim {\sum }_{j=1}^L{\lambda }_j{z}_j^2,$$

where the $z_j^2$ are independent ${\chi }_1^2$ variables, and the λ_j are the eigenvalues of the matrix $W^{1/2}{G}^{\text{'}}{\widehat{\Sigma }}^{-1}({}^{\widehat{\Sigma }}-\mathit{X}{\left({\mathit{X}}^{\text{'}}{\widehat{\Sigma }}^{-1}X\right)}^{-1}{\mathit{X}}^{\text{'}}){\widehat{\Sigma }}^{-1}\mathit{G}{\mathit{W}}^{1/2}$. Therefore, the P-value of Q can be closely approximated with the computationally efficient Davies’ method [34]. In Subheading 2.2, we illustrate how to install and run this package to scan homogeneous genomes.

1.4. Rare Variants Analysis of Admixed Genomes

Sequence association tests were originally designed for gene-based association analysis of homogeneous genomes. These methods do not explicitly model the particular information resources or confounders of admixed genomes. Current admixed genomes are formed as various mosaics of two or more ancestral genomes. For example, genomes of African Americans often have ancestral genomic segments from Europeans and West Africans. On average, admixture regions extend over several megabases in current admixed genomes [35]. Most genetic variants have different frequencies in different ancestral populations and, thus, variants in an identical admixture region are associated with their ancestral origins—local ancestries [36]. Genetic data on admixed individuals offer distinctive advantages for localizing admixture blocks that harbor causal variants which exhibit different frequencies between ancestral populations [37]. Population structure [38] and related-ness [39] are two essential confounders in genetic association analysis of admixed genomes. Population structure is due to differences in genetic ancestry among samples; and cryptic relatedness is due to distant relatedness among samples with no known family relationships. Rare variants can show a stratification that is systematically different from, and typically stronger than, common variants [40]. Accounting for population structure is more challenging when family structure or cryptic relatedness is also present [41].

Local ancestry captures the cumulative effect on the phenotype of causal variants in the entire ancestral block. The local ancestry weighted dosage test [42] was specifically designed for identifying rare variant associations in admixed populations. However, this test only allows for unrelated subjects and a binary disease. It can be invalid and suboptimal in the presence of cryptic relatedness. Rare variant association methods have not been explicitly optimized for admixed genomes. Therefore, in Subheading 2.3, we illustrate how to modify famSKAT to perform gene-based association analysis of admixed genomes.

2. Methods

2.1. The SKAT R Package

This package aggregates individual score statistics of SNPs in a set and efficiently computes the set-level P-value. It requires R version 2.13.0 or above. For installation, just run install.packages(“SKAT”) and then require(SKAT). In this section, we illustrate how to run this package to scan homogeneous genomes. For such a purpose, we perform gene-wise association analysis on the data of genotypes of chromosome 22 and primary phenotypes from the Genetics of Alcoholism (COGA) study. After routine quality control, the trimmed dataset comprises drinking symptoms and genotypes of 14,720 SNPs on 991 unrelated whites.

Step 1. Formatting the data.

Four plain input files are needed as detailed below. The first file is Geno_chr22.txt, containing a 991×14,720 white-spaced matrix of genotypic scores. The PLINK commands --recode and --recodeA can generate Gen- oChr22.txt from PLINK format COGA data:

./plink --file COGA --chr 22 --recode --out COGAChr22 --noweb
./plink --file COGAChr22 --recodeA --out GenoChr22.txt --noweb

Geno_chr22.txt is derived by removing the first six columns of GenoChr22.txt (FID, IID, PAT, MAT, SEX, PHENOTYPE). At each SNP, the default reference allele is the minor allele; missing genotypes are recorded as NA. This step recodes genotypic scores as shown below:

0	0	0	1	0	…
2	1	2	2	2	…
0	NA	2	1	2	…
…	…	…	…	…	…

The second file, Gene_chr22.txt, comprises basic information on chromosome-wide genes. Except for the header name line, each row is for one gene, including gene name, start position, end position and chromosome index:

Gene	Start	End	Chr
ACO2	40195074	40254938	22
ACR	49523517	49530592	22
ADM2	49266877	49271731	22
……

The third file, SNP_chr22.txt, comprises basic information of the SNPs on chromosome 22:

Chr	rs_ID	Pos
22	rs2334386	14430353
22	rs2334336	14519442
22	rs12163493	14809328
……

The last file, Pheno_Cov.txt, comprises individual trait values. The trait values are the residuals of drinking symptoms after adjusting for sex, age, and the first ten principal components (PCs). In this file, the trait values are organized one person per row:

−1.31541709

−0.095934426

−0.717596737

……

The four 4 plain files are loaded into the R platform and saved into COGA_Chr22.RDATA by running the following R command line:

save.image(“COGA_Chr22.RData”);

Step 2. Running the SKAT package.

The COGA_Chr22.RDATA needs to be loaded into R for gene-wise association analyses. The SKAT outputs the corresponding statistics, estimated parameters and gene-wise P-values (which are usually of particular interest). Running the following R command lines saves gene-wise P-values of the genes on chromosome 22 to array SKAT_p:

load(“COGA_Chr22.RData”);
SKAT_p<-array()
for (i in 1:dim(Gene_chr22)[1]) {
ID<-which((Geno_chr22$Pos<=Gene_chr22$End[i])&(Geno_chr22
$Pos>=Gene_chr22$Start[i]))
Z<-as.matrix(Geno_chr22[,ID])
pheno<-Pheno_Cov
obj<-SKAT_Null_Model(pheno~1, out_type=“C”);
SKAT_p[i]<-SKAT(Z, obj)$p.value
}

2.2. The famSKAT R Package

The famSKAT R code can be downloaded from https://cdn1.sphharvard.edu/wp-content/uploads/sites/1589/2014/07/famSKAT_v1.8_04052013.txt. In addition to the inputs of SKAT, the kinship matrix needs to be calculated in order to run famSKAT. This matrix can be generated by using the makekinship(.) function in the R package kinship. The input of this makekinship(.) function is a plain file, i.e., Fam.txt. This file comprises the family ID, individual ID, paternal ID and maternal ID of each subject, which can be extracted from the first four columns of the ped file in PLINK format, e.g., COGA.ped. Gene-wise P-values for chromosome 22 can be derived by running the following R code.

install.packages(“kinship”);
require(kinship);
kin_matrix<-makekinship(Fam[,1], Fam[,2], Fam[,3], Fam[,4])
famSKAT_p<-array()
for (i in 1:dim(Gene_chr22)[1]){
ID<-which((Geno_chr22$Pos<=Gene_chr22$End[i])&(Geno_chr22 $Pos>=Gene_chr22$Start[i]))
Z<-as.matrix(Geno_chr22[,ID])
pheno<-Pheno_Cov
famSKAT_p[i]<-famSKAT(phenotype=pheno,genotypes=Z,id=Fam[,2], fullkins¼kin_matrix)$pvalue
}

2.3. Admixed Sequence Association Analysis

As in Chapter 21, we use the data on African Americans in the Maywood cohort study for the purpose of illustration.

2.3.1. Ancestry Deconvolution

As detailed in Chapter 21, SNP-wise ancestries of African American are inferred from the genotype data by the ELAI software [43]. For example, the local ancestry scores for Chromosome 22 are saved in the output file AdmWood701_Chr22.ps21.txt. Each row is for one individual. Every two columns are for one SNP. The odd columns comprise CEU local ancestry scores. For each individual, we define the global ancestry (Global_A) as the average of genome-wide SNP-wise European ancestry scores. We adopt global ancestry rather than PCs to represent population structure (see Note 4). To integrate gene-wide ancestries, we extract SNP-wise European ancestry scores and save them in file Chr22_Local.txt. Then, we load the Chr22_Local.txt file into R and round individual scores to 0/1/2 format:

Local<-read.table(“Chr22_Local.txt”)
Local_Chr22<-round(Local)

SNPs within an admixture block share identical local ancestry. We assume no ancestry-switch point within a gene. Hence, the following R code can be applied to extract local ancestry information for a test gene:

A<-as.matrix(Local_Chr22[, ID1])
C<-duplicated(t(A))
New<-A[, !C]
if (dim(as.matrix(New))[2]==1){
Local_A<-New
}
if (dim(as.matrix(New))[2]!=1){
Count<-array(0,dim=c(dim(as.matrix(New))[2],dim(A)[2]))
for (j in 1:dim(A)[2]){
for (k in 1: dim(as.matrix(New))[2]){
Count[k,j]<-ifelse(sum(A[,j]==New[,k])==701,1,0)
}
}
ID<-which(rowSums(Count)==max(rowSums(Count)))
Local_A<-New[,ID]
}

Here, “ID1” comprises the IDs of SNPs in the test gene. The number “701” in the ifelse() function is the sample size of the dataset. Due to possible inference errors, different local ancestry scores may occur within a test gene. When this is the case, we choose the local ancestry shared by most SNPs within the gene to represent the local ancestry of the gene.

2.3.2. Inference of Cryptic Relatedness

For admixed genomes, we adopt the REAP algorithm [44] to infer cryptic relatedness. This algorithm accounts for population structure and ancestry-related assortative mating. Thus it provides more accurate relatedness inference for admixed genomes (see Note 5). The REAP package can be freely downloaded from http://faculty.washington.edu/tathornt/software/REAP/download.html.

Four input files are needed to run the REAP package. The first two files are Data_701_qc.tped and Data_701_qc.tfam format files that can be generated by running the PLINK command:

./plink --file Data_701_qc --recode12 --output-missing-genotype 0
--transpose --out Data_701_qc

The third file, Wood_Ancestry.txt, is the individual ancestry file that comprises four columns. The first two columns are family ID and individual ID. The third column gives the global European ancestry proportions. The fourth column gives the global African ancestry proportions.

The fourth file, Wood_freq.txt, has two columns. To generate this file, a reference allele needs to be assigned for each SNP. It can be the minor allele in CEU, for example. The first column of Wood_freq.txt contains the frequencies of the reference alleles in European ancestry; and the second column contains the frequencies of the reference alleles in African ancestry.

Kinship can be inferred by running the following command line:

./REAP-gData_701_qc.tped-pData_701_qc.tfam-aWood_Ancestry.txt-
f Wood_freq.txt -k 2 -t 0.025 -r 2

One of the output files, REAP_Kincoef_matrix.txt, contains the kinship matrix information.

2.3.3. Running famSKAT

After deriving individual global ancestries, local ancestries, and the kinship matrix, running the following R code yields the P-values for genes on chromosome 22:

kin_matrix<- read.table(“REAP_Kincoef_matrix.txt”)
famSKAT_p<-array()
for (i in 1:dim(Gene_chr22)[1]){
ID<-which((Geno_chr22$Pos<=Gene_chr22$End[i])&(Geno_chr22
$Pos>=Gene_chr22$Start[i]))
Z<- as.matrix(cbind(Geno_chr22[,ID], Local_A))
pheno<-Pheno_Cov
famSKAT_p[i]<-famSKAT(phenotype=pheno,genotypes=Z,id=Fam
[,2],fullkins=kin_matrix, covariates=Global_A)$pvalue
}

In such an analysis, gene-wide local ancestry is taken as a surrogate for the cumulative effect of the gene-wide variants; the global ancestry is adjusted as a fixed effect, and the relatedness is adjusted as a random effect.