## Overview

Principal Components Analysis (PCA) is commonly applied to genome-wide SNP genotype data from samples in genetic studies for population structure (i.e. ancestry) inference. PCA takes genotype values at hundreds of thousands of SNPs as input and performs a dimension reduction to principal components (PCs) that best reflect the variability of the data. Typically the top PCs calculated from the genotype data reflect population structure among the sample individuals. However, when a sample contains familial relatives, either known or unknown (cryptic), the top PCs obtained from a standard PCA are often confounded by this family structure and reflect clusters of close relatives.

The PC-AiR method is used to perform a PCA for the detection of population structure that is robust to possible familial relatives in the sample. Unlike a standard PCA, PC-AiR accounts for relatedness (known or cryptic) in the sample and identifies PCs that accurately capture population structure and not family structure. In order to accomplish this, PC-AiR uses measures of pairwise relatedness (kinship coefficients) and measures of pairwise ancestry divergence to identify an ancestry representative subset of mutually unrelated individuals. A standard PCA is performed on this “unrelated subset” of individuals, and PC values for the excluded “related subset” of indivdiuals are predicted from genetic similarity.

The functions in the GENESIS package read genotype data from a GenotypeData class object as created by the GWASTools package. Through the use of GWASTools, a GenotypeData class object can easily be created from:

• an R matrix of SNP genotype data
• a GDS file

Example R code for creating a GenotypeData object is presented below. Much more detail can be found in the GWASTools package reference manual.

### R Matrix

geno <- MatrixGenotypeReader(genotype = genotype, snpID = snpID, chromosome = chromosome,
position = position, scanID = scanID)
genoData <- GenotypeData(geno)
• genotype is a matrix of genotype values coded as 0 / 1 / 2, where rows index SNPs and columns index samples
• snpID is an integer vector of unique SNP IDs
• chromosome is an integer vector specifying the chromosome of each SNP
• position is an integer vector specifying the position of each SNP
• scanID is a vector of unique individual IDs

### GDS files

geno <- GdsGenotypeReader(filename = "genotype.gds")
genoData <- GenotypeData(geno)
• filename is the file path to the GDS object

## HapMap Data

To demonstrate a PC-AiR analysis with the GENESIS package, we analyze SNP data from the Mexican Americans in Los Angeles, California (MXL) and African American individuals in the southwestern USA (ASW) population samples of HapMap 3. Mexican Americans and African Americans have a diverse ancestral background, and familial relatives are present in these data, so a PC-AiR analysis is necessary for accurate population structure inference. Genotype data at a subset of 20K autosomal SNPs for 173 individuals are provided as a GDS file.

# read in GDS data
gdsfile <- system.file("extdata", "HapMap_ASW_MXL_geno.gds", package="GENESIS")
# create a GenotypeData class object
HapMap_genoData <- GenotypeData(HapMap_geno)
HapMap_genoData
## An object of class GenotypeData
##  | data:
## +    [  ] *
## |--+ sample.id   { Int32,factor 173 ZIP(40.90%) } *
## |--+ snp.id   { Int32 20000 ZIP(34.64%) }
## |--+ snp.position   { Int32 20000 ZIP(34.64%) }
## |--+ snp.chromosome   { Int32 20000 ZIP(0.13%) }
## |--+ genotype   { Bit2 20000x173 } *
##  | SNP Annotation:
## NULL
##  | Scan Annotation:
## NULL

## Pairwise Measures of Ancestry Divergence

It is possible to identify a subset of mutually unrelated individuals in a sample based solely on pairwise measures of genetic relatedness (i.e. kinship coefficients). However, in order to obtain accurate population structure inference for the entire sample, it is important that the ancestries of all individuals in the sample are represented by at least one individual in this subset. In order to identify a mutually unrelated and ancestry representative subset of individuals, PC-AiR also utilizes measures of ancestry divergence. These measures are calculated using the KING-robust kinship coefficient estimator (Manichaikul et al., 2010), which provides systematically negative estimates for unrelated pairs of individuals with different ancestry. The number of negative pairwise estimates that an individual has provides information regarding how different that individual’s ancestry is from the rest of the sample, which helps to prioritize individuals that should be kept in the ancestry representative subset.

The relevant output from the KING software is two text files with the file extensions .kin0 and .kin. The king2mat function can be used to extract the kinship coefficients (which we use as divergence measures) from this output and create a matrix usable by the GENESIS functions. The iids input of the king2mat function should be used to ensure that the output matrix orders individuals in the same way as the GenotypeData object.

# read individual IDs from GenotypeData object
iids <- getScanID(HapMap_genoData)
head(iids)
## [1] NA19919 NA19916 NA19835 NA20282 NA19703 NA19902
## 173 Levels: NA19625 NA19649 NA19650 NA19651 NA19652 NA19653 ... NA20412
# create matrix of KING estimates
KINGmat <- king2mat(file.kin0 = system.file("extdata", "MXL_ASW.kin0", package="GENESIS"),
file.kin = system.file("extdata", "MXL_ASW.kin", package="GENESIS"),
iids = iids)
## Reading .kin0 file...
## Determining Unique Individual IDs from KING Output...
## Checking Provided Individual IDs
## Adding Kinship Entries from .kin0 file...
## Adding Kinship Entries from .kin file...
KINGmat[1:6,1:6]
##         NA19919 NA19916 NA19835 NA20282 NA19703 NA19902
## NA19919  0.5000 -0.0009 -0.0059 -0.0080  0.0014 -0.0001
## NA19916 -0.0009  0.5000 -0.0063 -0.0150 -0.0039 -0.0053
## NA19835 -0.0059 -0.0063  0.5000 -0.0094 -0.0104 -0.0007
## NA20282 -0.0080 -0.0150 -0.0094  0.5000 -0.0134 -0.0112
## NA19703  0.0014 -0.0039 -0.0104 -0.0134  0.5000 -0.0047
## NA19902 -0.0001 -0.0053 -0.0007 -0.0112 -0.0047  0.5000

The column and row names of the matrix are the individual IDs, and each off-diagonal entry is the KING-robust estimate for the specified pair of individuals.

Alternative to running the KING software, the snpgdsIBDKING function from the SNPRelate package can be used to calculate the KING-robust estimates directly from a GDS file. The ouput of this function contains a matrix of pairwise estimates, which can be used by the GENESIS functions.

## Running PC-AiR

The PC-AiR algorithm requires pairwise measures of both kinship and ancestry divergence in order to partition the sample into an “unrelated subset” and a “related subset.” The kinship coefficient estimates are used to identify relatives, as only one individual from a set of relatives can be included in the “unrelated subset,” and the rest must be assigned to the “related subset.” The ancestry divergence measures calculated from KING-robust are used to help select which individual from a set of relatives has the most unique ancestry and should be given priority for inclusion in the “unrelated subset.”

The KING-robust estimates read in above are always used as measures of ancestry divergence for unrelated pairs of individuals, but they can also be used as measures of kinship for relatives (NOTE: they may be biased measures of kinship for admixed relatives with different ancestry). The pcair function performs the PC-AiR analysis.

# run PC-AiR
mypcair <- pcair(genoData = HapMap_genoData, kinMat = KINGmat, divMat = KINGmat)
## Partitioning Samples into Related and Unrelated Sets...
## Unrelated Set: 98 Samples
## Related Set: 75 Samples
## Running Analysis with 20000 SNPs ...
## Computing Genetic Correlation Matrix for the Unrelated Set: Block 1 of 2 ...
## Computing Genetic Correlation Matrix for the Unrelated Set: Block 2 of 2 ...
## Performing PCA on the Unrelated Set...
## Predicting PC Values for the Related Set...
## Concatenating Results...
• genoData is a GenotypeData class object
• kinMat is a matrix of pairwise kinship coefficient estimates
• divMat is a matrix of pairwise measures of ancestry divergence (KING-robust estimates)

If better estimates of kinship coefficients are available, then the kinMat input can be replaced with a similar matrix of these estimates. The divMat input should always remain as the KING-robust estimates.

### Reference Population Samples

As PCA is an unsupervised method, it is often difficult to identify what population structure each of the top PCs represents. In order to provide some understanding of the inferred structure, it is sometimes recommended to include reference population samples of known ancestry in the analysis. If the data set contains such reference population samples, we may want to use only those individuals as the “unrelated subset” for performing the PCA and predict values for all other sample individuals. This can be accomplished by setting the input unrel.set equal to a vector, IDs, of the individual IDs for the reference population samples.

mypcair <- pcair(genoData = HapMap_genoData, unrel.set = IDs)

If, instead, we want to make sure that these reference population samples are included in the “unrelated subset,” but also allow for the PC-AiR algorithm to select additional individuals from the sample to be a part of the “unrelated subset,” then this can be accomplished by using the inputs kinMat, divMat, and unrel.set together.

mypcair <- pcair(genoData = HapMap_genoData, kinMat = KINGmat, divMat = KINGmat, unrel.set = IDs)

This will force individuals specified with unrel.set into the “unrelated subset,” move any of their relatives into the “related subset,” and then perform the PC-AiR partitioning algorithm on the remaining samples.

### Partition a Sample without Running PCA

It may be of interest to partition a sample into an ancestry representative “unrelated subset” of individuals and a “related subset” of individuals without performing a PCA. The pcairPartition function, which is called within the pcair function, enables the user to do this.

part <- pcairPartition(kinMat = KINGmat, divMat = KINGmat)

The output contains two vectors that give the individual IDs for the “unrelated subset” and the “related subset.”

head(part$unrels); head(part$rels)
## [1] "NA19916" "NA19835" "NA19703" "NA19901" "NA19908" "NA19914"
## [1] "NA19919" "NA20282" "NA19902" "NA20287" "NA20335" "NA19713"

As with the pcair function, the input unrel.set can be used to specify certain individuals that must be part of the “unrelated subset.”

## Output from PC-AiR

An object returned from the pcair function has class pcair. A summary method for an object of class pcair is provided to obtain a quick overview of the results.

summary(mypcair)
## Call:
## pcair(genoData = HapMap_genoData, kinMat = KINGmat, divMat = KINGmat)
##
## PCA Method: PC-AiR
##
## Sample Size: 173
## Unrelated Set: 98 Samples
## Related Set: 75 Samples
##
## Kinship Threshold: 0.025
## Divergence Threshold: -0.025
##
## Principal Components Returned: 10
## Eigenvalues: 9.643 1.911 1.305 1.239 1.217 1.197 1.174 1.158 1.148 1.136 ...
##
## MAF Filter: 0.05
## SNPs Used: 19451

The output provides the total sample size along with the number of individuals assigned to the unrelated and related subsets, as well as the threshold values used for determining which pairs of individuals were related or ancestrally divergent. The eigenvalues for the top PCs are also shown, which can assist in determining the number of PCs that reflect structure. The minor allele frequency (MAF) filter used for excluding SNPs is also specified, along with the total number of SNPs analyzed after this filtering. Details describing how to modify the analysis parameters and the available output of the function can be found with the command help(pcair).

### Plotting PC-AiR PCs

The GENESIS package also provides a plot method for an object of class pcair to quickly visualize pairs of PCs. Each point in one of these PC pairs plots represents a sample individual. These plots help to visualize population structure in the sample and identify clusters of individuals with similar ancestry.

# plot top 2 PCs
plot(mypcair)
# plot PCs 3 and 4
plot(mypcair, vx = 3, vy = 4)

The default is to plot PC values as black dots and blue pluses for individuals in the “unrelated subset” and “related subsets” respectively. The plotting colors and characters, as well as other standard plotting parameters, can be changed with the standard input to the plot function.

## References

• Conomos M.P., Miller M., & Thornton T. Robust Inference of Population Structure for Ancestry Prediction and Correction of Stratification in the Presence of Relatedness. (Accepted to Genetic Epidemiology).

• Gogarten, S. M., Bhangale, T., Conomos, M. P., Laurie, C. A., McHugh, C. P., Painter, I., … & Laurie, C. C. (2012). GWASTools: an R/Bioconductor package for quality control and analysis of Genome-Wide Association Studies. Bioinformatics, 28(24), 3329-3331.

• Manichaikul, A., Mychaleckyj, J.C., Rich, S.S., Daly, K., Sale, M., & Chen, W.M. (2010). Robust relationship inference in genome-wide association studies. Bioinformatics, 26(22), 2867-2873.