Practical course using the software Introduction to multivariate analysis for bacterial GWAS using Thibaut Jombart (tjombart@imperial.ac.uk) Imperial College London MSc Modern Epidemiology / Public Health Abstract This practical illustrates how multivariate methods can be used for the analysis of bacterial genomic datasets. Principal Component Analysis (PCA, [7, 2, 3]) is introduced for assessing the diversity between sampled isolates. We also show how hierarchical clustering methods can be applied on principal components to identify groups of genetically related isolates. Discriminant Analysis of Principal Components (DAPC, [5]) is then used for identifying polymorphic sites associated with phenotypic traits such as bacterial resistance. While this tutorial uses simulated data, the procedures described are applicable to a wide range of Genome-Wide Association Studies (GWAS). 1
Contents 1 Introduction 3 1.1 Required packages............................ 3 1.2 Getting help................................ 3 1.3 The data................................. 4 2 First assessment of the genetic diversity 6 3 Identifying SNPs linked to bacterial resistance 12 2
1 Introduction 1.1 Required packages This practical requires a working version of [8] greater than or equal to 2.15.2. To check which version of R you are using, examine the welcome message displayed when starting R, or type: > R.version$version.string [1] "R version 2.15.2 (2012-10-26)" The practical relies on the package ade4 [1] for classical multivariate analyses (PCA) and on adegenet [4] for the DAPC. Both packages need to be installed, which may be tricky if you do not possess administrative rights on your computer. However, most systems still public areas where any user has read/write access. This is exploited by a simple hack allowing one to install packages without the administrative rights. To use it, simply type (while connected to the internet): > source("http://adegenet.r-forge.r-project.org/files/hacklib/hacklib.r") > hacklib() Then, the required packages can be installed using the usual procedure: > install.packages("ade4", dep=true) > install.packages("adegenet", dep=true) and loaded using: > library(ade4) > library(adegenet) 1.2 Getting help There are several ways of getting information about R functions, including some specific documentation sources for adegenet. The function help.search is used to look for help on a given topic. For instance: > help.search("hardy-weinberg") replies that there is a function HWE.test.genind in the adegenet package, and other similar functions in genetics and pegas. To get help for a given function, use?foo where foo is the function of interest. For instance: >?spca will open up the manpage of the spatial principal component analysis [6]. At the end of a manpage, an example section often shows how to use a function. This can be copied and pasted to the R console, or directly executed from the console using example. For further researches on R functions, RSiteSearch can be used to perform online researches using keywords in R s archives (mailing lists and manpages). adegenet has a few extra documentation sources. Information can be found from the adegenet website (http://adegenet.r-forge.r-project.org/), in the documents section, including several tutorials and a manual which compiles all manpages of the package, and a dedicated mailing list with searchable archives. To open the website from R, use: > adegenetweb() Tutorials ( vignettes in R s terminology) are also distributed with adegenet, and can be accessed using the command vignette. These can be listed using: 3
> vignette(package="adegenet") To open a vignette, for instance the tutorial on DAPC, simply use: > vignette("adegenet-dapc") Lastly, several mailing lists are available to find different kinds of information on R; to name a few: R-help: general questions about R. https://stat.ethz.ch/mailman/listinfo/r-help R-sig-genetics: genetics in R. https://stat.ethz.ch/mailman/listinfo/r-sig-genetics R-sig-phylo: phylogenetics in R. https://stat.ethz.ch/mailman/listinfo/r-sig-phylo adegenet forum: adegenet and multivariate analysis of genetic markers. https://lists.r-forge.r-project.org/cgi-bin/mailman/listinfo/adegenet-forum 1.3 The data The simulated data used in this practical are available online from the following address: http://adegenet.r-forge.r-project.org/files/simgwas/simgwas.rdata. The dataset is in R s binary format (extension RData), which uses compression to store data efficiently (the raw csv file would be more than 4MB). R objects can be loaded into R using load. The instruction url is required to load the data directly from the internet; as data are loaded, a new object simgwas appears in the R environment: > load(url("http://adegenet.r-forge.r-project.org/files/simgwas/simgwas.rdata")) > ls(pattern="sim") [1] "simgwas" > class(simgwas) [1] "list" > names(simgwas) [1] "snps" "phen" > class(simgwas$snps) [1] "matrix" > class(simgwas$phen) [1] "character" > dim(simgwas$snps) [1] 95 10000 > simgwas$snps[1:10,1:20] 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ind1 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 ind2 0 0 0 1 0 1 1 0 0 1 1 1 1 1 0 1 1 0 0 1 ind3 0 0 0 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 ind4 0 1 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 1 0 0 ind5 1 0 0 1 0 1 0 1 0 1 1 0 1 1 0 1 1 1 0 0 ind6 1 1 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 0 1 ind7 1 1 0 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 0 ind8 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 ind9 0 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 ind10 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 1 > print(object.size(simgwas$snps), unit="mb") 7.8 Mb > length(simgwas$phen) [1] 95 > simgwas$phen [1] "R" "S" "S" "S" "S" "S" "S" "S" "S" "S" "S" "R" "S" "S" "R" "S" "S" "S" "S" [20] "S" "S" "S" "R" "S" "S" "S" "S" "S" "S" "S" "R" "R" "S" "S" "S" "S" "R" "S" [39] "S" "R" "R" "R" "S" "S" "S" "R" "S" "S" "S" "S" "S" "S" "S" "S" "R" "R" "S" [58] "S" "S" "S" "S" "S" "S" "S" "S" "S" "S" "S" "S" "R" "R" "R" "S" "S" "R" "S" [77] "R" "R" "R" "R" "S" "S" "S" "S" "S" "S" "S" "S" "S" "S" "R" "S" "S" "R" "R" > table(simgwas$phen) R S 24 71 The object simgwas is a list with two components: $snps is a matrix of Single Nucleotide Polymorphism (SNPs) data, and $phen is the phenotype of the different sampled isolates. The SNPs data has a modest size by GWAS standards: only 95 isolates (in row) and 10000 SNPs (alleles coded as 0/1). To simplify further commands, we create the new objects snps and phen from simgwas: > snps <- simgwas$snps > phen <- factor(simgwas$phen) 5
2 First assessment of the genetic diversity Principal Component Analysis (PCA) is a very powerful tool for reducing the diversity contained in massively multivariate data into a few synthetic variables (the principal components PCs). There are several versions of PCA implemented in R. Here, we use dudi.pca from the ade4 package, specifying that variables should not be scaled (scale=false) to unit variances (this is only useful when variables have inherently different scales of variation, which is not the case here): > pca1 <- dudi.pca(snps, scale=false) PCA eigenvalues 0 5 10 15 20 25 30 The method displays a screeplot (barplot of eigenvalues) to help the user decide how many PCs should be retained. The general rule is to retain only the largest eigenvalues, after which non-structured variation results in smoothly decreasing eigenvalues. How many PCs would you retain here? > pca1 Duality diagramm class: pca dudi $call: dudi.pca(df = snps, scale = FALSE, scannf = FALSE, nf = 4) $nf: 4 axis-components saved $rank: 94 eigen values: 31.76 28.77 28.13 25.72 21.68... vector length mode content 1 $cw 10000 numeric column weights 2 $lw 95 numeric row weights 3 $eig 94 numeric eigen values data.frame nrow ncol content 6
1 $tab 95 10000 modified array 2 $li 95 4 row coordinates 3 $l1 95 4 row normed scores 4 $co 10000 4 column coordinates 5 $c1 10000 4 column normed scores other elements: cent norm The object pca1 contains various information. Most importantly: pca1$eig: contains the eigenvalues of the analysis, representing the amount of information contained in each PC. pca1$li: contains the principal components. pca1$c1: contains the principal axes (loadings of the variables). > head(pca1$eig) [1] 31.75594 28.77240 28.12875 25.72180 21.68303 21.36911 > head(pca1$li) Axis1 Axis2 Axis3 Axis4 ind1 3.606420-2.132999 9.622764-6.301912 ind2 1.912918-1.656548 8.734490-10.006055 ind3 2.316603-2.564638 9.324818-7.445660 ind4 2.490536-2.484711 8.819193-6.029816 ind5 2.448958-1.489571 8.576321-8.775661 ind6 2.938701-2.693103 10.876804-3.797021 > head(pca1$c1) CS1 CS2 CS3 CS4 X1 1.004273e-02 0.004291539-0.003509719-0.0092503284 X2-5.145732e-03-0.003539221-0.001470553 0.0075073374 X3-3.349998e-05-0.003362894 0.003797944 0.0013048886 X4-1.017829e-03 0.002489303-0.002323418-0.0007847613 X5 7.047362e-03 0.007801922-0.003475240 0.0057483659 X6-1.011989e-02 0.013435213 0.021812199-0.0321210072 Because of the large number of variables, the usual biplot (function scatter) is useless to visualize the results (try scatter(pca1) if unsure). We represent only PCs using s.label: > s.label(pca1$li, sub="pca - PC 1 and 2") > add.scatter.eig(pca1$eig,4,1,2, ratio=.3, posi="topleft") 7
Eigenvalues d = 5 ind41 ind47 ind34 ind39 ind44 ind45 ind50 ind36 ind43 ind32 ind33 ind49 ind35 ind46 ind38 ind40 ind42 ind37 ind31 ind48 ind75 ind63 ind79 ind58 ind55 ind76 ind64 ind59 ind54 ind69 ind70 ind78 ind60 ind72 ind52 ind56 ind51 ind53 ind61 ind62 ind67 ind80 ind74 ind65 ind68 ind66 ind71 ind73 ind57 ind77 ind15 ind26 ind22 ind29 ind5 ind9 ind10 ind16 ind24 ind19 ind7 ind8 ind3 ind21 ind4 ind30 ind12 ind6 ind20 ind23 ind25 ind13 ind17 ind27 ind11 ind18 ind28 ind14 PCA PC 1 and 2 ind89 ind81 ind90 ind82 ind86 ind88 ind91 ind94 ind83 ind87 ind92 ind84 ind85 ind95 ind93 What can you say about the genetic relationships between the isolates? Are there indications the existence of distinct lineages of bacteria? If so, how many lineages would you count? For a more quantitative assessment of this clustering, we derive squared Euclidean distances between isolates (function dist) and use hierarchical clustering with complete linkage (hclust) to define tight clusters: > D <- dist(pca1$li[,1:4])^2 > clust <- hclust(d, method="complete") > plot(clust, main="clustering (complete linkage) based on the first 4 PCs", cex=.4) 8
Clustering (complete linkage) based on the first 4 PCs ind19 ind26 ind18 ind25 ind17 ind29 ind22 ind28 ind27 ind21 ind30 ind20 ind23 ind16 ind24 ind93 ind86 ind87 ind91 ind92 ind81 ind89 ind82 ind83 ind94 ind85 ind88 ind95 ind84 ind90 ind15 ind6 ind14 ind7 ind3 ind10 ind1 ind4 ind11 ind12 ind2 ind5 ind9 ind8 ind13 ind40 ind48 ind37 ind31 ind46 ind42 ind49 ind47 ind32 ind43 ind41 ind39 ind33 ind44 ind35 ind36 ind38 ind50 ind34 ind45 ind51 ind67 ind68 ind61 ind74 ind63 ind78 ind53 ind73 ind57 ind66 ind56 ind80 ind60 ind52 ind70 ind62 ind65 ind71 ind77 ind59 ind75 ind54 ind72 ind69 ind58 ind79 ind76 ind55 ind64 Height 0 100 200 300 400 500 600 D hclust (*, "complete") How many clusters are there in the data? How does it compare to what you would have assessed based on the first two PCs of PCA? Bonus question: considering that the original data are profile of binary SNPs, what does the height represent in this dendrogram? You can define clusters based on the dendrogram clust using cutree: > pop <- factor(cutree(clust, k=5)) > pop ind1 ind2 ind3 ind4 ind5 ind6 ind7 ind8 ind9 ind10 ind11 ind12 ind13 1 1 1 1 1 1 1 1 1 1 1 1 1 ind14 ind15 ind16 ind17 ind18 ind19 ind20 ind21 ind22 ind23 ind24 ind25 ind26 1 1 2 2 2 2 2 2 2 2 2 2 2 ind27 ind28 ind29 ind30 ind31 ind32 ind33 ind34 ind35 ind36 ind37 ind38 ind39 2 2 2 2 3 3 3 3 3 3 3 3 3 ind40 ind41 ind42 ind43 ind44 ind45 ind46 ind47 ind48 ind49 ind50 ind51 ind52 3 3 3 3 3 3 3 3 3 3 3 4 4 ind53 ind54 ind55 ind56 ind57 ind58 ind59 ind60 ind61 ind62 ind63 ind64 ind65 4 4 4 4 4 4 4 4 4 4 4 4 4 ind66 ind67 ind68 ind69 ind70 ind71 ind72 ind73 ind74 ind75 ind76 ind77 ind78 4 4 4 4 4 4 4 4 4 4 4 4 4 ind79 ind80 ind81 ind82 ind83 ind84 ind85 ind86 ind87 ind88 ind89 ind90 ind91 4 4 5 5 5 5 5 5 5 5 5 5 5 ind92 ind93 ind94 ind95 5 5 5 5 Levels: 1 2 3 4 5 Now, we can represent these groups on top of the PCs using s.class (clusters are indicated by different colors and ellipses): > s.class(pca1$li, fac=pop, col=transp(funky(5)), cpoint=2, + sub="pca - axes 1 and 2") 9
d = 5 3 4 1 2 5 PCA axes 1 and 2 We do the same for PCs 3 and 4: > s.class(pca1$li, xax=3, yax=4, fac=pop, col=transp(funky(5)), + cpoint=2, sub="pca - axes 3 and 4") 10
d = 5 5 3 4 2 1 PCA axes 3 and 4 Are the clusters compatible with the results of the PCA? What is the meaning of the 3rd axis of the PCA? How many dimensions are needed to differentiate the 5 groups? 11
3 Identifying SNPs linked to bacterial resistance The data contained in phen indicate whether isolates are susceptible or resistant to a given antibiotic (S/R): > head(phen,10) [1] R S S S S S S S S S Levels: R S As we have done with genetic clusters previously, we can represent these two groups on the PCs to assess whether antibiotic resistance correlates to some components of the genetic diversity. > s.class(pca1$li, fac=phen, col=transp(c("royalblue","red")), cpoint=2, + sub="pca - axes 1 and 2") d = 5 R S PCA axes 1 and 2 > s.class(pca1$li, xax=3, yax=4, fac=phen, col=transp(c("royalblue","red")), + cpoint=2, sub="pca - axes 3 and 4") 12
d = 5 R S PCA axes 3 and 4 This visual assessment can be completed by a standard Chi-square test to check if there is an association between genetic clusters and resistance: > table(phen, pop) pop phen 1 2 3 4 5 R 3 1 7 10 3 S 12 14 13 20 12 > chisq.test(table(phen, pop), simulate=true) Pearson's Chi-squared test with simulated p-value (based on 2000 replicates) data: table(phen, pop) X-squared = 5.2267, df = NA, p-value = 0.2419 What do you conclude? Is antibiotic resistance correlated to the main genetic features of these isolates? It is important to keep in mind that PCA optimizes the representation of the overall genetic diversity, and does not explicitly look for distinctions between predefined groups of isolates. If only a few loci are correlated to bacterial resistance, PCA may well overlook these, especially if stronger structures such as separate lineages or populations are present. To look for combinations of SNPs correlated to a given partition of individuals, DAPC is much more appropriate. We apply the method using the function dapc, specifying the input data snps and the groups of individuals to distinguish (susceptible/resistant, phen). 13
> dapc1 <- dapc(snps, phen) The function asks for a number of principal components to retain for the dimensionreduction step (PCA, retain 30 PCs) and for the subsequent discriminant analysis (DA). For the latter, only one axis can be retained (the maximum number of axes in DA is always the number of groups minus 1). > dapc1 ######################################### # Discriminant Analysis of Principal Components # ######################################### class: dapc $call: dapc.data.frame(x = as.data.frame(x), grp =..1, n.pca = 30, n.da = 1) $n.pca: 30 first PCs of PCA used $n.da: 1 discriminant functions saved $var (proportion of conserved variance): 0.371 $eig (eigenvalues): 116.4 vector length content 1 $eig 1 eigenvalues 2 $grp 95 prior group assignment 3 $prior 2 prior group probabilities 4 $assign 95 posterior group assignment 5 $pca.cent 10000 centring vector of PCA 6 $pca.norm 10000 scaling vector of PCA 7 $pca.eig 94 eigenvalues of PCA data.frame nrow ncol content 1 $tab 95 30 retained PCs of PCA 2 $means 2 30 group means 3 $loadings 30 1 loadings of variables 4 $ind.coord 95 1 coordinates of individuals (principal components) 5 $grp.coord 2 1 coordinates of groups 6 $posterior 95 2 posterior membership probabilities 7 $pca.loadings 10000 30 PCA loadings of original variables 8 $var.contr 10000 1 contribution of original variables The function scatter can be used to visualize the results of DAPC. It produces usual plots of the principal components, using colors and ellipses to indicate groups. However, whenever only one axis has been retained, scatter plots the density of the individuals on the first principal component: > scatter(dapc1, bg="white", scree.da=false, scree.pca=true, + posi.pca="topright", col=c("royalblue","red"), + legend=true, posi.leg="topleft") 14
R S PCA eigenvalues Density 0.0 0.1 0.2 0.3 4 2 0 2 4 Discriminant function 1 The contribution of each variable to the separation of the two groups (susceptible/resistant) is stored in dapc1$var.contr; it can be visualized using loadingplot, which displays all contributions as bars and annotates variables with the largest contributions (see argument threshold in?loadingplot): > loadingplot(dapc1$var.contr) 15
0.000 0.001 0.002 0.003 0.004 0.005 0.006 Loading plot Variables Loadings 7 11 12 14 17 22 25 28 31 38 42 47 49 51 54 55 62 63 66 67 72 75 76 79 80 86 88 89 91 96 100 108 112 120 121 122 123 124 125 126 129 133 141 145 149 151 155 158 159 161 162 163 165 166 171 172 180 192 194 199 202 207 227 231 242 246 247 249 256 257 261 262 269 273 274 284 289 291 292 293 297 298 300 301 304 306 312 318 323 326 328 335 346 350 362 364 368 372 380 381 382 386 392 393 398 403 406 408 415 416 426 435 441 442 445 451 453 456 458 465 466 472 476 477 480 483 488 495 496 499 500 503 504 507 513 514 515 524 526 531 532 534 535 543 545 546 547 548 549 555 556 557 558 559 563 565 574 576 588 590 596 598 599 601 603 604 605 615 616 620 632 637 638 644 650 661 666 668 672 675 676 679 682 683 687 688 689 690 692 693 694 695 698 699 705 708 717 720 721 735 741 749 750 751 753 754 756 760 761 766 767 775 779 780 784 792 796 808 813 814 817 825 833 835 836 837 840 844 845 853 857 864 872 878 884 885 889 896 897 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9841 9846 9848 9852 9857 9858 9859 9860 9862 9864 9872 9874 9875 9877 9882 9883 9886 9890 9891 9893 9894 9902 9906 9911 9914 9915 9922 9923 9924 9927 9928 9929 9932 9933 9936 9939 9943 9945 9946 9947 9951 9953 9956 9958 9962 9971 9975 9978 9982 9983 9984 9985 9986 9990 9993 The function also invisibly returns information on the annotated variables. Recall loadingplot, specifying a higher threshold so that only the few outlying variables are retained, and store this result in an object called sel.snps. 16
Loading plot Loadings 0.000 0.001 0.002 0.003 0.004 0.005 0.006 7197 7199 7202 7206 7207 Variables The object should look like this: > sel.snps $threshold [1] 0.004 $var.names [1] "7197" "7199" "7202" "7206" "7207" $var.idx 7197 7199 7202 7206 7207 7197 7199 7202 7206 7207 $var.values 7197 7199 7202 7206 7207 0.004943821 0.004943821 0.004943821 0.004943821 0.004943821 Which SNPs are the most strongly correlated to antibiotic resistance? The following command derives allelic profiles of these SNPs for each isolate: > sel.profiles <- apply(snps[,sel.snps$var.idx],1,paste,collapse="-") > head(sel.profiles) ind1 ind2 ind3 ind4 ind5 ind6 "1-1-1-1-1" "0-0-0-0-0" "0-0-0-0-0" "0-0-0-0-0" "0-0-0-0-0" "0-0-0-0-0" > table(sel.profiles) sel.profiles 0-0-0-0-0 1-1-1-1-1 71 24 > head(cbind.data.frame(phen,sel.profiles),10) 17
phen sel.profiles ind1 R 1-1-1-1-1 ind2 S 0-0-0-0-0 ind3 S 0-0-0-0-0 ind4 S 0-0-0-0-0 ind5 S 0-0-0-0-0 ind6 S 0-0-0-0-0 ind7 S 0-0-0-0-0 ind8 S 0-0-0-0-0 ind9 S 0-0-0-0-0 ind10 S 0-0-0-0-0 > tail(cbind.data.frame(phen,sel.profiles),10) phen sel.profiles ind86 S 0-0-0-0-0 ind87 S 0-0-0-0-0 ind88 S 0-0-0-0-0 ind89 S 0-0-0-0-0 ind90 S 0-0-0-0-0 ind91 R 1-1-1-1-1 ind92 S 0-0-0-0-0 ind93 S 0-0-0-0-0 ind94 R 1-1-1-1-1 ind95 R 1-1-1-1-1 A contingency table between phenotype and SNPs profile can be created using table: > table(phen,sel.profiles) sel.profiles phen 0-0-0-0-0 1-1-1-1-1 R 0 24 S 71 0 What can you conclude on these SNPs? Assuming that their position in the dataset reflects their original position in the genome, would you think that each of these SNPs actually determines the antibiotic resistance? How would you address this question? 18
References [1] S. Dray and A.-B. Dufour. The ade4 package: implementing the duality diagram for ecologists. Journal of Statistical Software, 22(4):1 20, 2007. [2] H. Hotelling. Analysis of a complex of statistical variables into principal components. The Journal of Educational Psychology, 24:417 441, 1933. [3] H. Hotelling. Analysis of a complex of statistical variables into principal components (continued from september issue). The Journal of Educational Psychology, 24:498 520, 1933. [4] T. Jombart. adegenet: a R package for the multivariate analysis of genetic markers. Bioinformatics, 24:1403 1405, 2008. [5] T. Jombart, S. Devillard, and F. Balloux. Discriminant analysis of principal components: a new method for the analysis of genetically structured populations. BMC Genetics, 11(1):94, 2010. [6] T. Jombart, S. Devillard, A.-B. Dufour, and D. Pontier. Revealing cryptic spatial patterns in genetic variability by a new multivariate method. Heredity, 101:92 103, 2008. [7] K. Pearson. On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 2:559 572, 1901. [8] R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2011. ISBN 3-900051-07-0. 19