Practical course using the software ————— Introduction to genetic data analysis in Thibaut Jombart ————— Abstract This practical course is meant as a short introduction to genetic data analysis using [10]. After describing the main types of data, we illus- trate how to perform some basic population genetics analyses, and then go through constructing trees from genetic distances and performing standard multivariate analyses. The practical uses mostly the packages adegenet [6], ape [9] and ade4 [1, 3, 2], but others like genetics [11] and hierfstat [5] are also required. 1
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Practical course using the software
—————Introduction to genetic data analysis in
Thibaut Jombart
—————
Abstract
This practical course is meant as a short introduction to genetic dataanalysis using [10]. After describing the main types of data, we illus-trate how to perform some basic population genetics analyses, and then gothrough constructing trees from genetic distances and performing standardmultivariate analyses. The practical uses mostly the packages adegenet [6],ape [9] and ade4 [1, 3, 2], but others like genetics [11] and hierfstat [5] arealso required.
1
Contents
1 Let’s start 31.1 Loading the packages . . . . . . . . . . . . . . . . . . . . . . . 31.2 How to get information? . . . . . . . . . . . . . . . . . . . . . 3
2 Handling the data 52.1 Data formats . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Importing / exporting data . . . . . . . . . . . . . . . . . . . 72.3 Manipulating data . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Basic population genetics 9
4 Making trees 114.1 Hierarchical clustering . . . . . . . . . . . . . . . . . . . . . . 114.2 ape and phylogenies . . . . . . . . . . . . . . . . . . . . . . . 13
5 Multivariate Analysis 155.1 ade4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Principal Component Analysis (PCA) . . . . . . . . . . . . . 165.3 Principal Coordinates Analysis (PCoA) . . . . . . . . . . . . 19
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1 Let’s start
1.1 Loading the packages
Before going further, we shall make sure that all we need is installed on thecomputer. Launch , and make sure that the version being used is at least2.12.0 by typing:
> R.version.string
[1] "R version 2.12.0 (2010-10-15)"
The next thing to do is check that relevant packages are installed. To loadan installed package, use the library instruction; for instance:
> library(adegenet)
loads adegenet if it is installed (and issues an error otherwise). To get theversion of a package, use:
> packageDescription("adegenet", fields = "Version")
[1] "1.2-9"
adegenet version should read 1.2-8 or greater.In case a package would not be installed, you can install it using in-
stall.packages. To install all the required dependencies, specify dep=TRUE.For instance, the following instruction should install adegenet with all its de-pendencies (it can take up to a few minutes, so don’t run it unless adegenetis not installed):
> install.packages("adegenet", dep = TRUE)
Using the previous instructions, load (and install if required) the pack-ages adegenet, ade4, ape, genetics, and hierfstat.
1.2 How to get information?
There are several ways of getting information about R in general, or aboutadegenet in particular. The function help.search is used to look for helpon a given topic. For instance:
> help.search("Hardy-Weinberg")
replies that there is a function HWE.test.genind in the adegenet package,other similar functions in genetics and pegas. To get help for a given func-tion, use ?foo where ‘foo’ is the function of interest. For instance (quotescan be removed):
> `?`(spca)
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will open the manpage of the spatial principal component analysis [7]. Atthe end of a manpage, an ‘example’ section often shows how to use a func-tion. This can be copied and pasted to the console, or directly executedfrom the console using example. For further questions concerning R, thefunction RSiteSearch is a powerful tool to make an online research usingkeywords in R’s archives (mailing lists and manpages).
adegenet has a few extra documentation sources. Information can befound from the website (http://adegenet.r-forge.r-project.org/), inthe ‘documents’ section, including two tutorials, a manual which includesall manpages of the package, and a dedicated mailing list with searchablearchives. To open the website from , use:
> adegenetWeb()
The same can be done for tutorials, using adegenetTutorial (see manpageto choose the tutorial to open).
You will also find a listing of the main functions of the package typing:
> `?`(adegenet)
Note that you can also browse help pages as html pages, using:
> help.start()
To go to the adegenet page, click ‘packages’, ‘adegenet’, and ‘adegenet-package’.
Lastly, several mailing lists are available to find different kinds of infor-mation on R; to name a few:
R-help (https://stat.ethz.ch/mailman/listinfo/r-help): general ques-tions about R
R-sig-genetics (https://stat.ethz.ch/mailman/listinfo/r-sig-genetics):genetics in R
adegenet forum (https://lists.r-forge.r-project.org/cgi-bin/mailman/listinfo/adegenet-forum): adegenet and multivariate analysis ofgenetic markers
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2 Handling the data
2.1 Data formats
Two principal types of genetic data can be handled in R. The first one is(preferably aligned) DNA sequences, and the second one is genetic markers.
DNA sequences can be used to calibrate models of evolution and computegenetic distances, which can in turn be used for phylogenetic reconstructionor in multivariate analyses. In , DNA sequences are best handled asDNAbin objects, in the ape package. See ?DNAbin for more details about thisclass. After loading ape, we load the dataset woodmouse:
> library(ape)> data(woodmouse)> woodmouse
15 DNA sequences in binary format stored in a matrix.
All sequences of same length: 965
Labels: No305 No304 No306 No0906S No0908S No0909S ...
Base composition:a c g t
0.307 0.261 0.126 0.306
Use str, unclass and attributes to explore the content of the DNAbin
object. Convert the dataset to a matrix of characters using as.character.What is the size of this new object (use object.size)? Compare it to theoriginal DNAbin object. Why is this class optimal for handling DNA se-quences?
Genetic markers are features of DNA sequences that vary between indi-viduals. There is not a single type of genetic marker: some are codominant(all alleles are visible), some are dominant (some alleles hide the presence ofothers); some have many alleles (like microsatellites), and others are most of-ten binary (like SNPs). In general, codominant (more informative) markersare the rule, and dominant markers become more and more rare.
Both types of markers can be handled in using the class genind
implemented in adegenet. This class uses the S4 system of class definitionand methods, with which users are generally not familiar. Load thedataset nancycats:
> library(adegenet)> data(nancycats)> nancycats
######################## Genind object ########################
- genotypes of individuals -
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S4 class: genind@call: genind(tab = truenames(nancycats)$tab, pop = truenames(nancycats)$pop)
@tab: 237 x 108 matrix of genotypes
@ind.names: vector of 237 individual [email protected]: vector of 9 locus [email protected]: number of alleles per [email protected]: locus factor for the 108 columns of @[email protected]: list of 9 components yielding allele names for each locus@ploidy: 2@type: codom
Optionnal contents:@pop: factor giving the population of each [email protected]: factor giving the population of each individual
@other: a list containing: xy
Usually, the slots of S4 objects can be accessed by ’@’, which replaces the$ operator used in S3 classes (e.g. in lists). For genind objects, both canbe used. Documentation of the class can be accessed by typing ?’genind-
class’. Using names and @ (or $), browse the content of nancycats; whatinformation is contained in this object?
A more simple dataset can be used to understand how information iscoded. Here, we create a simple dataset with two diploid individuals (inrows), and two markers (columns).
> dat <- data.frame(locus1 = c("A/T", "T/T"), locus2 = c("G/G",+ "C/G"))> dat
locus1 locus21 A/T G/G2 T/T C/G
> x <- df2genind(dat, sep = "/")> x
######################## Genind object ########################
- genotypes of individuals -
S4 class: genind@call: df2genind(X = dat, sep = "/")
@tab: 2 x 4 matrix of genotypes
@ind.names: vector of 2 individual [email protected]: vector of 2 locus [email protected]: number of alleles per [email protected]: locus factor for the 4 columns of @[email protected]: list of 2 components yielding allele names for each locus@ploidy: 2@type: codom
Optionnal contents:@pop: - empty -
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@pop.names: - empty -
@other: - empty -
Look at the @tab component of the genind object. To extract this tablewith original labels, use truenames. How is information stored? Does theEuclidean distance between the rows of this matrix make some biologicalsense? This is a canonical way of storing information for any dominantmarkers, irrespective of the degree of ploidy.
It sometimes happens that analyses are performed at a population level,rather than at an individual level. In such cases, genpop objects are used.These objects contain allele numbers per populations rather than relativefrequencies, but are otherwise similar to genind objects. genpop can beobtained from genind objects using genind2genpop. Use this function tocompute allele counts for the cat colonies of Nancy (data nancycats), andcompare the the structure of the obtained object to the genind object.
2.2 Importing / exporting data
DNA sequences with usual formats are read using read.dna from the apepackage.
Genetic markers come in a wider variety of formats. import2genind
can be used to import some of the most frequent, and less twisted formats.However, it is also possible to convert data from a data.frame to a genindusing df2genind. Using the example above, create a data.frame with twotetraploid (i.e. 4 alleles) individuals and two loci. Note that alleles can begiven any name, and do not need to be letters. Then, used df2genind toconvert it to a genind object. Use truenames and genind2df to check thatthe data conversion has worked.
It also happens that genetic markers are derived from DNA sequences.In such a case, only sites that vary between individuals are retained; theseare called Single Nucleotide Polymorphism (SNPs). They can be extractedfrom DNAbin objects using DNAbin2genind. Use this function to extract thepolymorphic sites from the woodmouse dataset. The locus names correspondto the position of the polymorphic site. Use this information to plot thedistribution of polymorphism along the DNA sequence; you should arrive atsomething along the lines of (using density with a width of 30):
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0 200 400 600 800 1000
0.00
00.
001
0.00
20.
003
0.00
4
density.default(x = x, width = 30)
Position on DNA sequence
Den
sity
of p
olym
orph
ism
(S
NP
s)
||||||||| ||| || || ||||||| ||||| || |||| || ||| || ||||| | | | | || ||||
What can we say about the distribution of SNPs on the DNA sequence?
2.3 Manipulating data
It is often useful to be able to manipulate genetic marker data in differentways. There is no particular manipulation for DNAbin objects, which areessentially matrices or lists. genind objects benefit from different facilitiesfor manipulating data. First, the [ operator can be used to subset individualsand alleles; this is done like for a matrix, and is based on the @tab slot. Forinstance, to retain only the first 10 individuals of the nancycats dataset,use:
> nancycats[1:10, ]
######################## Genind object ########################
- genotypes of individuals -
S4 class: genind@call: .local(x = x, i = i, j = j, drop = drop)
@tab: 10 x 108 matrix of genotypes
@ind.names: vector of 10 individual [email protected]: vector of 9 locus [email protected]: number of alleles per [email protected]: locus factor for the 108 columns of @[email protected]: list of 9 components yielding allele names for each locus@ploidy: 2@type: codom
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Optionnal contents:@pop: factor giving the population of each [email protected]: factor giving the population of each individual
@other: a list containing: xy
Data can be split by population using seppop, and by locus using seploc.Indication of the population is obtained or set using pop. repool can beused to gather the genotypes stored in different genind objects into a singlegenind. Reading the documentation, interprete the following command line:
> temp <- seppop(nancycats)> x <- repool(lapply(temp, function(e) e[sample(1:nrow(e$tab),+ 5, replace = FALSE)]))
What does the produced object contain? How can this be useful whenanalysing genetic data?
3 Basic population genetics
This section aims at introducing some basic population genetics tools. Thefirst one consists in testing random mating within groups by testing Hardy-Weinberg equilibirum (HWE). Using HWE.test.genind, test HWE in thecat colonies of the dataset nancycats. Since this corresponds to a lot of tests(1 per group and locus), as a first approximation, ask for returning only thep-values. Compute the number of significant departures from HWE (p.value< 0.001) per locus, and then per population (use apply), and representgraphically the results. You should arrive at (in less than 3 command lines):
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P01
P02
P03
P04
P05
P06
P07
P08
P09
P10
P11
P12
P13
P14
P15
P16
P17
By population
Pro
port
ion
of d
epar
ture
s fr
om H
WE
0.0
0.1
0.2
0.3
0.4
fca8
fca2
3
fca4
3
fca4
5
fca7
7
fca7
8
fca9
0
fca9
6
fca3
7By marker
Pro
port
ion
of d
epar
ture
s fr
om H
WE
0.00
0.02
0.04
0.06
0.08
0.10
What are your conclusions? What can we say about the hypothesis ofrandom mating within these cat colonies?
To investigate the data further, we can measure and test the significanceof the genetic differentiation of the cat colonies. Use fstat to compute
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the Fst of these populations. This is a measure of population differentia-tion, which can be roughly interpreted as the proportion of genetic varianceexplained by differences between groups. Values less than 0.05 are oftenconsidered as negligible.
Then, test the significance of the overall group differentiation usinggstat.randtest, and plot the results; are there differences between groups?
4 Making trees
Genetic data are often represented using trees. In this section, we won’ttackle phylogenetic reconstruction based on maximum likelihood or parsi-mony, which should be covered in a practical of its own. For this, see thewell-documented package phangorn. In the following, we illustrate brieflyhow to contruct tree representations of genetic distances.
4.1 Hierarchical clustering
Several algorithm of hierarchical clustering are implemented in the functionhclust. This function requires a distance matrix produced by the functiondist, or similar function (e.g. dist.dna, dist.genpop).
We will try to describe the genetic variability in the cat colonies of Nancyusing clustering methods. First, we need to compute a distance matrix. Re-place missing data in nancycats using na.replace, and then compute theEuclidean distances between individuals using dist on the table of relativeallele frequencies ($tab of the object in which NAs have been replaced). Usehclust to compute a clustering with complete linkage, and one with singlelinkage. How do they compare? How would you interprete these graphs?Can you assess the number of actual genetic clusters in these data?
As a comparison, repeat these analyses with the dataset microbov, whichcontains genotypes of various cattle breeds in France and Africa.
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Microbov − complete linkage
hclust (*, "complete")D
Hei
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Microbov − single linkage
hclust (*, "single")D
Hei
ght
Species in the microbov dataset are well differentiated. How good ishierarchical clustering at displaying genetic structures?
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4.2 ape and phylogenies
ape is the main package for phylogenetic analyses. It implements variousmethods for reconstructing trees based on genetic distances. It also providesnice graphical functions for plotting trees.
Use the genetic distances obtained for the nancycat data to computea Neighbour-Joining tree (nj). Then, plot the tree, specifying that youwant to plot an unrooted tree, and use tiplabels to represent colonies bycoloured dots. To define one color for each colony, use:
> myCol <- rainbow(17)[as.integer(pop(nancycats))]
Since some colours are very light, you are best using a grey background forthe plot. The final result should ressemble:
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For a comparison, repeat these analyses with the cattle breed data (microbov).To define one color for each breed of the dataset, use:
> myCol <- rainbow(15)[as.integer(pop(microbov))]
You should obtain a tree like this one:
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There are two species, Bos taurus and Bos indicus, in these cattle breeds;the species information is stored in microbov$other$spe. Use the same ap-proach to represent the species information on the tips of the tree. Is thisclassification reflected by a clear-cut structure on the tree?
The quality of a tree is unfortunately rarely questioned in terms of howwell the real distances are represented by the tree. The literature is evenfull of examples of analyses based on distances computed from trees, whilethe original distances that had generated the tree could have been used. Asimple way of assessing how good a tree is at representing a set of distancesis simply comparing the original distances to the reconstructed distances(sum of branch lengths between tips). The latter can be computed usingcophenetic. Use this function to plot both distances (microbov data), afterconverting the distances into vectors using as.vector; you should obtainthe following figure:
14
The red line indicates the regression of the original distance onto the recon-structed (function lm). Perform this regression, and use summary to computevarious statistics for this model, including the R2. What percentage of thevariance of the original distances is represented by the tree reconstruction?Is this satisfying? Shall we always rely on a tree representation of a distancematrix?
5 Multivariate Analysis
5.1 ade4
There are a few of packages implementing multivariate analyses in . Oneof the most complete is ade4, with which adegenet is interfaced. It hasbeen developped around the “duality diagram” representation [4, 2], whichemphasizes that most multivariate analyses can be unified under a singlegeneral algorithm. Alternatively, some statisticians refer to this algorithmas “generalized PCA”.
In ade4, multivariate analyses all use the same underlying algorithm(as.dudi) and produce an object with the class dudi (for“duality diagram”).Methods such as Principal Component Analysis (PCA) or Principal Coor-dinates Analysis (PCoA) are implemented by functions where the methods’abbreviation is preceeded by dudi; for instance, PCA is implemented bydudi.pca, while PCoA is implemented by dudi.pco.
Apart from a few specific methods like spatial PCA (spca, [7]) or Dis-criminant Analysis of Principal Components (dapc, [8]), multivariate meth-ods are not implemented in adegenet. Procedures from ade4 are used aftertransforming genind or genpop objects in an appropriate way, which oftenconsists in replacing missing data, and computing and standardizing allelefrequencies before analysis.
15
5.2 Principal Component Analysis (PCA)
PCA can be used on standardized allele frequencies, at the individual orpopulational level. The function scaleGen is used to standardize the dataand replace possible missing data. Use it on the microbov dataset to obtaina table of centred, but not scaled allele frequencies. Then, perform the PCAof these data using dudi.pca. Be careful to set the argument scale toFALSE to avoid erasing the scaling choice made in scaleGen.
Eigenvalues
0.0
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The function dudi.pca displays a barplot of eigenvalues and asks for anumber of retained principal components. In general, eigenvalues representthe amount of genetic diversity — as measured by the multivariate methodbeing used — represented by each principal component (PC). Here, eacheigenvalue is the variance of the corresponding PC.
Once you have selected the number of retained axes, your PCA shouldressemble:
Duality diagrammclass: pca dudi$call: dudi.pca(df = X, center = FALSE, scale = FALSE, scannf = FALSE,
nf = 3)
$nf: 3 axis-components saved$rank: 341eigen values: 1.27 0.5317 0.423 0.2853 0.2565 ...vector length mode content
1 $cw 373 numeric column weights2 $lw 704 numeric row weights
16
3 $eig 341 numeric eigen values
data.frame nrow ncol content1 $tab 704 373 modified array2 $li 704 3 row coordinates3 $l1 704 3 row normed scores4 $co 373 3 column coordinates5 $c1 373 3 column normed scoresother elements: cent norm
The printing of dudi object is not always explicit. The essential itemsfor the analyses we will use are:
? $eig: eigenvalues of the analysis
? $c1: principal axes, containing the loadings of the variables (alleles)
? $li: principal components, containing the scores of the individuals
The function scatter provides a simultaneous representation of individ-uals and variables. However, it does not provide very informative graphicswhen there are too many items to be plotted. To represent individuals only,use s.label, and use the argument clab to change the size of the labels;for instance:
> s.label(pcaX$li, clab = 0.75)> add.scatter.eig(pcaX$eig, 3, 1, 2)
d = 1
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Eigenvalues
This is better, but far from perfect yet. In this case, it would be better torepresent results by breed. This can be achieved using s.class. In additionto the coordinates (the same as in s.label), this function needs a factor
17
grouping the observations. This can be obtained easily using pop on theobject microbov. To be even fancier, you can define one color per groupusing rainbow (for other palettes, see ?rainbow):
> myCol <- rainbow(length(levels(pop(microbov))))
and then pass it as the color argument in s.class; the result should ressem-ble this:
d = 1
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Borgou
Zebu
Lagunaire
NDama Somba
Aubrac
Bazadais
BlondeAquitaine BretPieNoire Charolais
Gascon Limousin MaineAnjou Montbeliard
Salers
Eigenvalues
Represent the results for the axes 1-2, and then 2-3. How can you in-terprete these principal components? What is the major factor of geneticdifferentiation in these cattle breeds? What is the second one? What is thethird one?
In PCA, the eigenvalues indicate the variance of the corresponding prin-cipal component. Verify that this is indeed the case, for the first and sec-ond principal components. Note that this is also, up to a constant, themean squared Euclidean distance between individuals. This is because (forx ∈ Rn):
var(x) =
∑ni=1
∑nj=1(xi − xj)
2
2n(n− 1)
This again can be checked; assuming this analyis is called pcaX:
> pcaX$eig[1]
[1] 1.269978
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> pc1 <- pcaX$li[, 1]> n <- length(pc1)> 0.5 * mean(dist(pc1)^2) * ((n - 1)/n)
[1] 1.269978
We sometimes want to express eigenvalues as a percentage of the totalvariation in the data. What is the percentage of variance explained by thefirst three eigenvalues?
Allele contributions can sometimes be informative. The basic graphicsfor representing allele loadings is s.arrow. Use it to represent the resultsof the PCA; is this informative? An alternative is offered by loadingplot,which represents one axis at a time. Use it to represent the squared loadingsof the analysis; how does it compare to previous representation? The func-tion returns invisibly some information; adapting the arguments and savingthe returned value, identify the 2% alleles contributing most to showing thediversity within African breeds. You should find:
[1] "ETH152.197" "INRA32.178" "HEL13.182" "MM12.119" "CSRM60.093"[6] "TGLA227.079" "TGLA126.119" "TGLA122.150"
5.3 Principal Coordinates Analysis (PCoA)
We have just seen that the variance is, up to a constant, equal to the meansquared distance between all pairs of observations. Therefore, maximizingthe variance along an axis should be exactly the same as maximizing thesum of all squared Euclidean distances. This latter maximization is theobjective of Principal Coordinates Analysis (PCoA), also known as Met-ric Multidimensional Scaling (MDS). This method is implemented in ade4by dudi.pco. After scaling the relative allele frequencies of the microbov
dataset, perform this analysis.Represent the principal components ($li) of this analysis and compare
them to the PCs of the previous PCA. What is the correlation betweenthem? Compare the scatter plots of both analyses; are there any differ-ences? In practice, in which cases should we prefer PCA over PCoA, or thecontrary?
As an exercise, compute genetic similarities based on the proportion ofshared alleles (propShared) on the microbov data. Transform the similar-ities into a distance (no dedicated function for this), and perform a PCoAof the resulting distance. Is this distance Euclidean? If not, it needs tobe. Transform it into an Euclidean distance (using cailliez) and redo thePCoA. How do the results compare to the previous analyses?
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References
[1] D. Chessel, A-B. Dufour, and J. Thioulouse. The ade4 package-I- one-table methods. R News, 4:5–10, 2004.
[2] S. Dray and A.-B. Dufour. The ade4 package: implementing the dualitydiagram for ecologists. Journal of Statistical Software, 22(4):1–20, 2007.
[3] S. Dray, A.-B. Dufour, and D. Chessel. The ade4 package - II: Two-table and K-table methods. R News, 7:47–54, 2007.
[4] Y. Escoufier. The duality diagramm : a means of better practicalapplications. In P. Legendre and L. Legendre, editors, Development innumerical ecology, pages 139–156. NATO advanced Institute , Serie G.Springer Verlag, Berlin, 1987.
[5] J. Goudet. HIERFSTAT, a package for R to compute and test hierar-chical F-statistics. Molecular Ecology Notes, 5:184–186, 2005.
[6] T. Jombart. adegenet: a R package for the multivariate analysis ofgenetic markers. Bioinformatics, 24:1403–1405, 2008.
[7] T. Jombart, S. Devillard, A.-B. Dufour, and D. Pontier. Revealingcryptic spatial patterns in genetic variability by a new multivariatemethod. Heredity, 101:92–103, 2008.
[8] Thibaut Jombart, Sebastien Devillard, and Francois Balloux. Discrim-inant analysis of principal components: a new method for the analysisof genetically structured populations. BMC Genetics, 11(1):94, 2010.
[9] E. Paradis, J. Claude, and K. Strimmer. APE: analyses of phylogeneticsand evolution in R language. Bioinformatics, 20:289–290, 2004.
[10] R Development Core Team. R: A Language and Environment for Sta-tistical Computing. R Foundation for Statistical Computing, Vienna,Austria, 2009. ISBN 3-900051-07-0.
[11] G. R. Warnes. The genetics package. R News, 3(1):9–13, 2003.
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