pcadapt has been developped to detect genetic markers involved in biological adaptation. pcadapt provides statistical tools for outlier detection based on Principal Component Analysis (PCA).
In the following, we show how the pcadapt package can perform genome scans for selection. The package contains two examples: geno3pops (genotype data) and pool3pops (Pool-seq data). The geno3pops data contain simulated genotypes for 1,500 diploid markers. A total of 150 individuals coming from three different populations were genotyped. Simulations were performed with simuPOP using a divergence model assuming that 200 SNPs confer a selective advantage. The pool3pops data contain simulated allele frequencies in 3 populations for 1,500 diploid markers. Allele frequencies have been computed based on the geno3pops dataset.
To run the package, you need to install it and to load it using the following command lines:
install.packages("pcadapt")
library(pcadapt)
Sections A-E explain how to use the package for genotype data. Section F additionnaly explains how to use the package for Pool-seq data, but previous sections should also be read. Section G explains advanced usage of the package.
A. Reading the genotype data
The geno3pops example data can be loaded using the following command.
data <- read4pcadapt(x="geno3pops",option="example")
print(dim(data))
## [1] 150 1500
For your own dataset, use also the read4pcadapt function. The current version of read4pcadapt supports two formats. The first format assumes that the genotype is a matrix with individuals in rows and genetic markers in columns (same format as LFMM). Data files should be tab-delimited or space-delimited. An example of genotype data containing 3 diploid individuals and 4 SNPs is given below. \[ 0 \ 0 \ 1 \ 2 \\ 0 \ 1 \ 2 \ 0 \\ 2 \ 1 \ 0 \ 1 \] Assuming the data file is called “mydata” and is available at “path_to_directory”, the command line to read the genotype data is
data <- read4pcadapt(x="path_to_directory/mydata")
If the genotype is a matrix with genetic markers in rows and individuals in columns, you can also use the read4pcadapt function. The command line to read the genotype data is
data <- read4pcadapt(x="path_to_directory/mydata",transpose=TRUE)
The second supported format is the standard .ped format. To read a .ped file, the argument x should contain the name of the file only without the .ped extension, and the option ped should be specified. If you read a .ped file, make sure the .map file is in the same directory. Assuming the data file is called “mydata.ped” and is available at “path_to_directory”, the command line is
data <- read4pcadapt(x="path_to_directory/mydata",option="ped")
NB : in both cases, standard read.table options header and sep can be specified in the read4pcadapt function. For .ped files, the separator for the .map file can also be specified using the option mapsep.
B. Choosing the number K of Principal Components
The pcadapt function performs two successive tasks. First, PCA is performed on the centered and scaled genotype matrix. The second stage consists of computing test statistics and p-values based on the correlations between SNPs and the first K principal components (PCs). To run the function pcadapt, the user should specify the number K of principal components to work with.
To choose K, principal component analysis should first be performed with a large enough number of principal components (e.g. K=20).
x <- pcadapt(data,K=20)
B.1. Scree plot
The ‘scree plot’ displays the percentage of variance that is explained by each PC. The ideal pattern in a scree plot is a steep curve followed by a bend and an almost horizontal line. The recommended value of K corresponds to the largest value of K before the plateau is reached. In the provided example, K=2 is the optimal choice of K. The plot functions displays a scree plot:
plot(x,option="screeplot")
By default, the number of principal components taken into account in the scree plot is set to K but it can be reduced via the argument num_pc.
plot(x,option="screeplot",num_pc=4)
B.2. Score plot
Another option to choose the number of PCs is based on the ‘score plot’ that displays population structure. The score plot displays the projections of the individuals onto the specified principal components. Using the score plot, the choice of K can be limited to the values of K that correspond to a relevant level of population structure.
When population labels are known, individuals of the same populations can be displayed with the same color using the pop argument, which should contain the list of indices of the populations of origin. In the geno3pops example, the first population is composed of the first 50 individuals, the second population of the next 50 individuals, and so on. Thus, a population file that can be provided to the argument pop should look like this:
poplist <- c(rep(1,50),rep(2,50),rep(3,50))
print(poplist)
## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## [71] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3
## [106] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [141] 3 3 3 3 3 3 3 3 3 3
If this field is left empty, the points will be displayed in black. By default, if the values of i and j are not specified, the projection is done onto the first two principal components.
plot(x,option="scores",pop=poplist)
Looking at population structure beyond K=2 confirms the results of the scree plot. The third and the fourth principal components do not ascertain population structure anymore.
plot(x,option="scores",i=3,j=4,pop=poplist)
NB: for .ped files, the plot function adds colors to the ‘score plot’ automatically provided that the data have been loaded with read4pcadapt.
C. Computing the test statistic based on PCA
For a given SNP, the test statistic is based on the loadings which are defined as the correlations between the SNP and the PCs. The test statistic for detecting outlier SNPs is the Mahalanobis distance, which is a multi-dimensional approach that measures how distant is a point from the mean. Denoting by \(\rho^j = (\rho_1^j,\dots,\rho_K^j)\) the vector of K correlations between the \(j\)-th SNP and the first K PCs, the Mahalanobis distance is defined as
\[D_j = \sqrt{(\rho^j - \bar{\rho})\Sigma^{-1}(\rho^j - \bar{\rho})}\]
where \(\bar{\rho}\) and \(\Sigma\) are robust estimates of the mean and of the covariance matrix. Once divided by a constant \(\lambda\) called the genomic inflation factor, the scaled distances \(D_j/\lambda\) should have a chi-square distribution with K degrees of freedom under the assumption that there are no outlier.
For the geno3pops data, it was found in section B that K=2 corresponds to the optimal choice of the number of PCs.
x <- pcadapt(data,K=2)
In addition to the number K of principal components to work with, the user can also set the parameter minmaf that corresponds to a threshold of minor allele frequency. By default, the parameter minmaf is set to 5%. P-values of SNPs with a minor allele frequency smaller than the threshold are not computed (NA is returned).
The object x returned by the function pcadapt contains numerical quantities obtained after performing a PCA on the genotype matrix.
summary(x)
## Length Class Mode
## stat 1500 -none- numeric
## pvalues 1500 -none- numeric
## maf 1500 -none- numeric
## gif 1 -none- numeric
## chi2_stat 1500 -none- numeric
## scores 300 -none- numeric
## loadings 3000 -none- numeric
## singular_values 2 -none- numeric
We assume in the following that there are n individuals and L markers.
stat is a vector of size L containing Mahalanobis distances by default.
pvalues is a vector containing L p-values.
maf is a vector of size L containing minor allele frequencies.
gif is a numerical value corresponding to the genomic inflation factor estimated from stat.
chi2_stat is a vector of size L containing the rescaled statistics stat/gif that follow a chi-squared distribution with K degrees of freedom.
scores is a (n,K) matrix corresponding to the projections of the individuals onto each PC.
loadings is a (L,K) matrix containing the correlations between each genetic marker and each PC.
singular_values is a vector containing the K ordered squared root of the proportion of variance explained by each PC.
All of these elements are accessible using the $ symbol. For example, the p-values are contained in x$pvalues.
E. Choosing a cutoff for outlier detection
To provide a list of outliers, we recommend to use the R package qvalue, transforming the p-values into q-values. To install and load the package, type the following command lines:
## try http if https is not available
source("https://bioconductor.org/biocLite.R")
biocLite("qvalue")
library(qvalue)
For a given \(\alpha\) (real valued number between \(0\) and \(1\)), SNPs with q-values less than \(\alpha\) will be considered as outliers with an expected false discovery rate bounded by \(\alpha\). The false discovery rate is defined as the percentage of false discoveries among the list of candidate SNPs. Here is an example of how to provide a list of candidate SNPs for the geno3pops data, for an expected false discovery rate lower than 10%:
qval <- qvalue(x$pvalues)$qvalues
alpha <- 0.1
outliers <- which(qval<alpha)
print(outliers)
## [1] 1 2 3 4 5 6 8 12 13 16 17 18 22 24
## [15] 25 26 27 28 30 31 32 34 36 37 42 44 45 46
## [29] 47 51 52 54 55 56 57 58 59 60 61 65 66 70
## [43] 73 75 76 77 78 80 81 82 83 84 85 86 87 88
## [57] 89 91 92 93 94 95 97 98 99 100 101 102 103 104
## [71] 105 106 107 110 111 112 113 114 116 117 118 119 120 121
## [85] 122 123 124 125 126 127 128 129 130 131 132 133 134 135
## [99] 137 138 139 141 142 143 144 145 146 147 149 150 525 625
## [113] 663 1406
F. Pool-seq Data
The package also handles Pool-seq data. A Pool-seq example is provided in the package and can be loaded as follows :
pooldata <- read4pcadapt("pool3pops",option="example",transpose=FALSE)
For your own dataset, asssuming the data file is called “mydata” and is available at “path_to_directory”, the command line to read the (n,L) frequency matrix data (where n is the number of populations and L is the number of genetic markers) is
pooldata <- read4pcadapt("path_to_directory/mydata",transpose=FALSE)
If the input data is (L,n), set transpose to TRUE to convert to the right format.
As for genotype data, the pcadapt function performs two successive tasks. First PCA is performed on the centered matrix of allele frequencies (not scaled). And the second stage consists of computing test statistics (Mahalanobis distances by default) and p-values based on the covariances between allele frequencies and the first K PCs. By default, the function assumes that the number K of PCs is equal to the number of populations minus one (n-1). The user can use a smaller number of PCs (K < n-1) by looking at the screeplot.
When calling the pcadapt function, make sure to specify data.type ="pool".
xpool <- pcadapt(pooldata,data.type="pool")
Plotting options mentioned in section D such as "manhattan", "qqplot", "scores", "screeplot", or "stat.distribution" are still valid for Pool-seq data:
plot(xpool,option="manhattan")
G. Advanced Usage
G.1. Other test statistics
The default option uses the Mahalanobis distance as a test statistic to seek for outlier SNPs, but other statistics can be computed based on the loadings. However except for power users, we recommend to use the Mahalanobis distance which provides the best rankings of SNPs in most cases we have investigated.
Communality
The communality statistic measures the proportion of variance explained by the first K PCs. When there are K+1 populations, the communality statistic provides a ranking similar to the widely-used Fst statistic. P-values are computed using a chi-square approximation (Duforet-Frebourg et al. 2015). The test based on the communality statistic can be performed by setting method to communality in the pcadapt function.
x_com <- pcadapt(data,K=2,method="communality")
The communality can be approximated by a constant times a chi-square distribution, allowing to compute p-values for each SNP.
plot(x_com,option="stat.distribution")
Euclidean
Instead of using the Mahalanobis distance, we can simply use Euclidean distance as a test statistic.
x_eucl <- pcadapt(data,K=2,method="euclidean")
plot(x_eucl,option="stat.distribution")
Component-wise
Another possibility (component-wise p-values) is to perform one genome scan for each principal component. The test statistics are the loadings, which correspond to the correlations between each PC and each SNP. P-values are computed by making a Gaussian approximation for each PC and by estimating the standard deviation of the null distribution.
x_cw <- pcadapt(data,K=2,method="componentwise")
summary(x_cw$pvalues)
## p1 p2
## Min. :0.0000001 Min. :0.0000
## 1st Qu.:0.1373472 1st Qu.:0.1568
## Median :0.3805793 Median :0.4068
## Mean :0.4138816 Mean :0.4326
## 3rd Qu.:0.6691068 3rd Qu.:0.7005
## Max. :0.9986491 Max. :0.9999
pcadapt returns K vectors of p-values (one for each principal component), all of them being accessible, using the $ symbol or the [] symbol. For example, typing x_cw$pvalues$p2 or x_cw$pvalues[,2] in the R console returns the list of p-values associated with the second principal component (provided that K is larger than or equal to 2).
The p-values are computed based on the matrix of loadings. The loadings of the neutral markers are assumed to follow a centered Gaussian distribution. The standard deviation of the Gaussian distribution is estimated using the Median Absolute Deviation.
To display the neutral distribution for the component-wise case, the value of K has to be specified.
plot(x_cw,option="stat.distribution",K=2)
G.2 Reading outputs from PCAdapt fast
For large datasets (more than 100,000 SNPs), computing PCA with the R package can be slow. In this case, the PCAdapt C software is faster than the R package.
Installation
The C software is currently available for UNIX OS or MAC OS and can be downloaded here. To install the software, proceed as follows :
- open a terminal and go to the directory where the archive has been downloaded.
- unzip the archive (a directory called ‘PCAdaptPackage’ should be created)
- type the following command lines successively:
cd PCAdaptPackage
make lapack
make
To run PCAdapt, use the following command line:
./PCAdapt fast -i genotype -o myoutput -K number_of_PCs
where genotype is the genotype matrix, myoutput is the name of the output file and number_of_PCs is the number of principal components that will be computed. For more details about the software, check out the manual.
Read the outputs with the R package pcadapt
To read outputs in R from the software PCAdapt, switch the argument PCAdapt to TRUE in the pcadapt function. The argument data should be the name of the output provided by PCAdapt without any extention. Assuming the file returned by the PCAdapt sofwtare is called “myoutput” and is available at “path_to_directory”, the command line should be :
x <- pcadapt(data="path_to_directory/myoutput",K=K,PCAdapt=TRUE)
