lmdme: Linear Models on Designed Multivariate Experiments in R

Transcription

1 lmdme: Linear Mdels n Designed Multivariate Experiments in R Cristóbal Fresn Universidad Católica de Córdba Mónica G Balzarini Universidad Nacinal de Córdba Elmer A Fernández Universidad Católica de Córdba Abstract This intrductin t linear mdel n designed multivariate experiments f the R package lmdme is a (slightly) mdified versin f Fresn et al. (2014), published in the Jurnal f Statistical Sftware. The lmdme package (Fresn and Fernández 2013a) decmpses analysis f variance (ANOVA) thrugh linear mdels n designed multivariate experiments, allwing ANOVAprincipal cmpnent analysis (APCA) and ANOVA-simultaneus cmpnent analysis (ASCA) in R (R Develpment Cre Team 2012). It als extends bth methds with the applicatin f partial least squares (PLS) thrugh the specificatin f a desired utput matrix. The package is freely available n the Bicnductr website (Gentleman et al. 2004), licensed under GNU general public license. ANOVA decmpsitin methds fr designed multivariate experiments are becming ppular in mics experiments (transcriptmics, metablmics, etc.), where measurements are perfrmed accrding t a predefined experimental design (Smilde et al. 2005), with several experimental factrs r including subject-specific clinical cvariates, such as thse present in current clinical genmic studies. ANOVA-PCA and ASCA are well-suited methds fr studying interactin patterns n multidimensinal datasets. Hwever, current R implementatin f APCA is nly available fr Spectra data in ChemSpec package (Hansn 2012), whereas ASCA (Nueda et al. 2007) is based n average calculatins n the indices f up t three design matrices. Thus, n statistical inference n estimated effects is prvided. Mrever, ASCA is nt available in R package frmat. Here, we present an R implementatin fr ANOVA decmpsitin with PCA/PLS analysis that allws the user t specify (thrugh a flexible frmula interface), almst any linear mdel with the assciated inference n the estimated effects, as well as t display functins t explre results bth f PCA and PLS. We describe the mdel, its implementatin and ne high-thrughput micrarray example applied t interactin pattern analysis. Keywrds: linear mdel, ANOVA decmpsitin, PCA, PLS, designed experiments, R.

2 2 lmdme: Linear Mdels n Designed Multivariate Experiments in R 1. Intrductin Current mics experiments (prtemics, transcriptmics, metablmics r genmics) are multivariate in nature. Mdern technlgy allws us t explre the whle genme r a big subset f the prteme, where each gene/prtein is in essence a variable explred t elucidate its relatinship with an utcme. In additin, these experiments are including an increasing number f experimental factrs (time, dse, etc.) frm design r subject-specific infrmatin, such as age, gender and linage, and are available fr analysis. Hence, t decipher experimental design r subject-specific patterns, sme multivariate appraches shuld be applied, with principal cmpnent analysis (PCA) and partial least squares regressin (PLS) being the mst cmmn. Hwever, it is knwn that wrking with raw data might mask infrmatin f interest. Therefre, analysis f variance (ANOVA)-based decmpsitin is becming ppular t split variability surces befre applying such multivariate appraches. Seminal wrks n genmics were that f De Haan et al. (2007) n ANOVA-PCA (APCA) and f Smilde et al. (2005) n ANOVA-SCA (ASCA) mdels. Hwever, t the best f ur knwledge R implementatin f APCA is nly available fr Spectra data, ChemSpec R package by Hansn (2012). Regarding ASCA, as there is n R package fr this mdel, it can nly be used by uplading script-functin files resulting frm a MATLAB cde translatin (Nueda et al. 2007). In additin, ASCA nly accepts up t three design matrices, which limits its use and makes it difficult. Mrever, cefficient estimatins are based n average calculatins using binary design matrices, withut any statistical inference ver them. Here, we prvide a flexible linear mdel-based decmpsitin framewrk. Almst any mdel can be specified, accrding t the experimental design, by means f a flexible frmula interface. Because cefficient estimatin is carried ut by means f maximum likelihd, statistical significance is naturally given. The framewrk als prvides bth capacity t perfrm PCA and PLS analysis n apprpriate ANOVA decmpsitin results, as well as graphical representatins. The implementatin is well-suited fr direct analysis f gene expressin matrices (variables n rws) frm high-thrughput data such as micrarray r RNA-seq experiments. Belw we prvide an examples t intrduce the user t the package applicatins, thrugh the explratin f interactin patterns n a micrarray experiment. 2. The mdel A detailed explanatin f ANOVA decmpsitin and multivariate analysis can be fund in Smilde et al. (2005) and Zwanenburg et al. (2011). Briefly and withut the lss f generality, let us assume a micrarray experiment where the expressin f (G 1, G 2,..., G g ) genes are arrayed in a chip. In this cntext, let us cnsider an experimental design with tw main factrs: A, with a levels (A 1, A 2,..., A i,..., A a ) and B, with b levels (B 1, B 2,..., B j,..., B b ), with replicates R 1, R 2,..., R k,..., R r fr each A B cmbinatin levels. After preprcessing steps are perfrmed as described in (Smyth 2004), each chip is represented by a clumn vectr f gene expressin measurements f g 1. Then, the whle experimental data is arranged int a g n expressin matrix (X), where n = a b r. In this data scheme, single gene measurements acrss the different treatment cmbinatins (A i B j ) are presented in a rw n the X matrix, as depicted in Figure 1. An equivalent X matrix structure needs t be btained fr 2D-DIGE r RNA-seq experiments and s frth.

3 Cristóbal Fresn, Mónica G Balzarini, Elmer A Fernández 3 Factr A i B j Treatments n=abr Micrarrays Level A a B 2 A 1 B 2 A a B 1 R A 1 B 1 1 R r G 1 G g A 1 B b A a B b G 1 Genes A A A A 1 B 1 a B b Replicates G g R 1 R r 1 B 1 a B b 1 n X ijk = µ + α i + β j + α i β j + ε ijk A) B) C) Figure 1: Data representatin f micrarray gene expressin. A) Genes are sptted n the chip. Then, expressin levels fr each cmbinatin f treatment factr levels AiBj and their replicates Rk can be measured n the chips, yielding a ttal f n = a b r micrarrays. B) Gene expressin f each chip (micrarray) is then interpreted as a clumn vectr f expressin levels. C) Then, these clumn vectrs will be cmbined by clumns prducing the experiment gene expressin matrix X. Expressin measurements under all treatment cmbinatins fr a gene are represented by the X matrix rws. Thus, measurements n a rw are subjected t the ANOVA mdel f (1).

4 4 lmdme: Linear Mdels n Designed Multivariate Experiments in R Regardless f data generatin, the ANOVA mdel fr each gene (rw) in X can be expressed as (1): x ijk = µ + α i + β j + α i β j + ε ijk (1) Where x ijk is the measured expressin fr sme gene, at cmbinatin ij f factrs A and B fr the k replicate; µ is the verall mean; α, β and α β are the main and interactin effects respectively; and the errr term ε ijk N(0, σ 2 ). In additin, (1) can als be expressed in matrix frm fr all genes int (2): X = X µ + X α + X β + X αβ + E = X l + E (2) l {µ,α,β,αβ} Where X l, E matrices are f dimensin g n and cntain the level means f the crrespnding l th term and the randm errr respectively. Hwever, in the cntext f linear mdels X l can als be written as a linear cmbinatin f tw matrix multiplicatins in the frm f (3): X = l {µ,α,β,αβ} X l + E = l {µ,α,β,αβ} B l Z T l + E = B µ Z T µ B αβ Z T αβ + E = µ1 + B α Z T α B αβ Z T αβ + E (3) Where B l and Z l are referenced in the literature as cefficient and mdel matrices f dimensins g m (l) and n m (l), respectively, and m (l) is the number f levels f factr l. The first term is usually called intercept, with B µ = µ and Z µ = 1 being f dimensin g 1 and n 1, respectively. In this example, all Z l are binary matrices, identifying whether a measurement belngs ( 1 ) r nt ( 0 ) t the crrespnding factr. In the implementatins prvided by Smilde et al. (2005) and Nueda et al. (2007), the estimatin f the cefficient matrices is based n calculatins f averages using the design matrix (up t three design matrices Z α,β,αβ ), t identify the average samples. In thery, these authrs fully decmpse the riginal matrix as shwn in (1). On the cntrary, in this package the mdel cefficients are estimated, iteratively, by the maximum likelihd apprach, using the lmfit functin prvided by limma package (Smyth et al. 2011). Cnsequently, three desirable features are als incrprated: 1. Flexible frmula interface t specify any ptential mdel. The user nly needs t prvide: i) The gene expressin matrix (X), ii) The experimental data.frame (design) with treatment structure, and iii) The mdel in a frmula style, just as in an rdinary lm R functin. Internal mdel.matrix call, will autmatically build the apprpriate Z matrices, vercming the cnstraint n factrial design size, and tedius mdel matrix definitins. 2. Hypthesis tests n cefficient B l matrices. A T test is autmatically carried ut fr the s th gene mdel, t test whether r nt the th cefficient is equal t zer, i.e., H 0 : b s = 0 vs H 1 : b s 0. In additin, an F test is perfrmed t simultaneusly determine whether r nt all b s are equal t zer. 3. Empirical Bayes crrectin can als be achieved thrugh the ebayes limma functin. It uses an empirical Bayes methd t shrink the rw/gene-wise sample variances twards a cmmn value and t augment the degrees f freedm fr the individual variances (Smyth 2004).

5 Cristóbal Fresn, Mónica G Balzarini, Elmer A Fernández 5 By cntrast, De Haan et al. (2007) estimate the main and interactin effects by verall mean subtractin. Hence, genes need t be treated as an additinal factr. Meanwhile, in Smilde et al. (2005) and Nueda et al. (2007) implementatins, the estimatins are btained n a geneby-gene basis, as in (1). Therefre, in a tw-way factr experiment, such as time xygen, De Haan s mdel includes tw additinal duble interactins and a triple interactin, because genes are treated as a factr, unlike the mdels f Smilde and Nueda The decmpsitin algrithm The ANOVA mdel (2) is decmpsed iteratively using (3), where in each step the l th cefficients ˆB l, Ê l matrices and ˆσ l 2 are estimated. Then, the particular term cntributin matrix ˆX l = ˆB l Zl is subtracted frm the preceding residuals t feed the next mdel, as depicted in (4): X = X µ + X α + X β + X αβ + E = l {µ,α,β,αβ} X l + E step µ : X = X µ + E µ X = ˆB µ Z µ + ʵ ʵ = X ˆB µ Z µ step α : E µ = X α + E α ʵ = ˆB α Z α + Êα Êα = ʵ ˆB α Z α.. step l : E l 1 = X l + E l Êl 1 = ˆB l Z l + Êl Êl = Êl 1 ˆB l Z l (4).. step αβ : E β = X αβ + E Êβ = ˆB αβ Z αβ + Ê Ê = Êβ ˆB αβ Z αβ Where the hat ( ) dentes estimated cefficients. In this implementatin, the first step always estimates the intercept term, i.e., frmula= 1 in R style, with ˆB µ = ˆµ and Z µ = 1. The fllwing mdels will nly include the l th factr withut the intercept, i.e., frmula= lth_term-1, where lth term stands fr α, β r αβ in this example. This prcedure is quite similar t the ne prpsed by Harringtn et al. (2005) PCA and PLS analyses These methds explain the variance/cvariance structure f a set f bservatins (e.g., genes) thrugh a few linear cmbinatins f variables (e.g., experimental cnditins). Bth methds can be applied t the l th ANOVA decmpsed step f (4) t deal with different aspects: ˆ PCA cncerns with the variance f a single matrix, usually with the main bjectives f reducing and interpreting data. Accrdingly, depending n the matrix t which it is applied, there are tw pssible methds: ASCA, when PCA is applied t cefficient matrix, ˆBl, (Smilde et al. 2005); and APCA when PCA is calculated n the residual, Ê l 1. The latter is cnceptually an ASCA and is usually applied t, X l + E, i.e., the mean factr matrix X l, plus the errr f the fully decmpsed mdel E f (1), as in De Haan et al. (2007). ˆ PLS nt nly generalizes but als cmbines features frm PCA and regressin t explre the cvariance structure between input and sme utput matrices, as described by Abdi

6 6 lmdme: Linear Mdels n Designed Multivariate Experiments in R and Williams (2010) and Shawe-Taylr and Cristianini (2004). It is particularly useful when ne r several dependent variables (utputs - O) must be predicted frm a large and ptentially highly crrelated set f independent variables (inputs). In ur implementatin, the input can be either the cefficient matrix ˆB l r the residual Ê l 1. Accrding t the chice, the respective utput matrix will be a diagnal O=diag(nrw( ˆB l )) r design matrix O = Z l. In additin, users can specify their wn utput matrix, O, t verify a particular hypthesis. Fr instance, in functinal genmics it culd be the Gene Ontlgy class matrix as used in gene set enrichment analysis (GSEA) by Subramanian et al. (2005). When wrking with the cefficient matrix, the user will nt have t wrry abut the expected number f cmpnents in X (rank f the matrix, given the number f replicates per treatment level), as suggested by Smilde et al. (2005), because the cmpnents are directly summarized in the cefficient ˆB l matrix. In additin, fr bth PCA/PLS, the lmdme package (Fresn and Fernández 2013a) als ffers different methds t visualize results, e.g., biplt, ladingplt and screeplt r leverage calculatin, in rder t filter ut rws/genes as in Tarazna et al. (2012). 3. Example In this sectin we prvide an verview f lmdme package by Fresn and Fernández (2013a). The example cnsists f an applicatin f the analysis f gene expressin interactin pattern, where we address: Hw t define the mdel, undertake ANOVA decmpsitin, perfrm PCA/PLS analysis and visualize the results. Frm here nwards, sme utputs were remved fr reasns f clarity and the examples were perfrmed with ptins(digits=4) Package verview The riginal data files fr the first example are available at Gene Expressin Omnibus (Edgar et al. 2002), with accessin GSE37761 and stemhypxia package (Fresn and Fernández 2013b) n the Bicnductr website. In this dataset, Prad-Lpez et al. (2010) studied differentiatin f human embrynic stem cells under hypxia cnditins. They measured gene expressin at different time pints under cntrlled xygen levels. This experiment has a typical tw-way ANOVA structure, where factr A stands fr time with a = 3 levels {0.5, 1, 5 days}, factr B stands fr xygen with b = 3 levels {1, 5, 21%} and r = 2 replicates, yielding a ttal f 18 samples. The remainder f the dataset was excluded in rder t have a balanced design, as suggested by Smilde et al. (2005) t fulfil rthgnality assumptins in ANOVA decmpsitin. First, we need t lad stemhypxia package t access R bjects calling data("stemhypxia"), which will then lad the experimental design and gene expressin intensities M. R> library("stemhypxia") R> data("stemhypxia") Nw we manipulate the design bject t maintain nly thse treatment levels which create a

7 Cristóbal Fresn, Mónica G Balzarini, Elmer A Fernández 7 balanced dataset. Then, we change rwnames(m) f each gene in M, with their crrespnding M$Gene_ID. R> timeindex<-design$time %in% c(0.5, 1, 5) R> xygenindex<-design$xygen %in% c(1, 5, 21) R> design<-design[timeindex & xygenindex, ] R> design$time<-as.factr(design$time) R> design$xygen<-as.factr(design$xygen) R> rwnames(m)<-m$gene_id R> M<-M[, clnames(m) %in% design$samplename] Nw we can explre micrarray gene expressin data present n the M matrix, with g = rws (individuals/genes) and n = 18 clumns (samples/micrarrays). In additin, the experimental design data.frame cntains main effect clumns (e.g., time and xygen) and the sample names (samplename). A brief summary f these bjects is shwn using head functin: R> head(design) time xygen samplename h_1_ h_1_ h_5_ h_5_ h_21_ h_21_2 R> head(m)[, 1:3] 12h_1_1 12h_1_2 12h_5_1 A_24_P A_32_P A_23_P A_24_P A_24_P A_32_P Once the preprcessing f the experiment data is cmpleted, library("lmdme") shuld be laded. This instructin will autmatically lad the required packages: limma (Smyth et al. 2011) and pls (Mevik et al. 2011). Once the data are laded, the ANOVA decmpsitin f Sectin 2.1 can be carried ut using (4) calling lmdme functin with the mdel frmula, actual data and experimental design. R> library("lmdme") R> fit<-lmdme(mdel=~time*xygen, data=m, design=design) R> fit

8 8 lmdme: Linear Mdels n Designed Multivariate Experiments in R lmdme bject: Data dimensin: x 18 Design (head): time xygen samplename h_1_ h_1_ h_5_ h_5_ h_21_ h_21_2 Mdel:~time * xygen Mdel decmpsitin: Step Names Frmula CefCls 1 1 (Intercept) ~ time ~ -1 + time xygen ~ -1 + xygen time:xygen ~ -1 + time:xygen 9 The results f lmdme will be stred inside the fit bject, which is an S4 R class. By invking the fit bject, a brief descriptin f the data and design used are shwn as well as the Mdel applied and a summary f the decmpsitin. This data.frame describes the applied Frmula and Names fr each Step, as well as the amunt f estimated cefficients fr each gene (CefCls). At this pint, we can chse thse subjects/genes in which at least ne interactin cefficient is statistically different frm zer (F test n the cefficients) with a threshld p value f and perfrm ASCA n the interactin cefficient term, and PLS against the identity matrix (default ptin). R> id<-f.p.values(fit, term="time:xygen")<0.001 R> decmpsitin(fit, decmpsitin="pca", type="cefficient", + term="time:xygen", subset=id, scale="rw") R> fit.plsr<-fit R> decmpsitin(fit.plsr, decmpsitin="plsr", type="cefficient", + term="time:xygen", subset=id, scale="rw") These instructins will perfrm ASCA and PLS decmpsitin ver the scale="rw" versin f the 305 selected subjects/genes (subset=id) n fit and fit.plsr bject, respectively. The results will be stred inside these bjects. In additin, we have explicitly indicated the decmpsitin type="cefficient" (default value) in rder t apply it t the cefficient matrix, n "time:xygen" interactin term ( ˆB αβ ). Nw, we can visualize the assciated biplts (see Figure 2 (a) and (b)). R> biplt(fit, xlabs="", expand=0.7) R> biplt(fit.plsr, which="ladings", xlabs="", + ylabs=clnames(cefficients(fit.plsr, term="time:xygen")), + var.axes=true)

9 Cristóbal Fresn, Mónica G Balzarini, Elmer A Fernández time:xygen PC1(50.61%) PC2(23.68%) time0.5:xygen1 time1:xygen1 time5:xygen1 time0.5:xygen5 time1:xygen5 time5:xygen5 time0.5:xygen21 time1:xygen21 time5:xygen21 (a) ANOVA simultaneus cmpnent analysis time:xygen Cmp 1 Cmp time0.5:xygen1 time1:xygen1 time5:xygen1 time0.5:xygen5 time1:xygen5 time5:xygen5 time0.5:xygen21 time1:xygen21 time5:xygen21 (b) ANOVA partial least squares Figure 2: Biplt n the decmpsed interactin cefficients (time xygen) n genes satisfying the F test with p value < Ntice that the interactin matrix in the ASCA mdel is f rank 9. Thus, 9 arrws are expected and the scre f the 305 selected subjects are prjected nt the space spanned by the first tw principal cmpnents in Figure 2(a). Fr visual clarity, xlabs are changed with the "" symbl, instead f using the rwnames(m) with manufacturer ids, and secnd axis with the expand=0.7 ptin t avid cutting ff lading labels. In additin, PLS biplt is mdified frm the default pls behavir t btain a graph similar t ASCA utput (which="ladings"). Accrdingly, ylabs is changed t match the crrespnding cefficients f the interactin term and var.axes is set t TRUE. The ASCA biplt f the first tw cmpnents (see Figure 2(a)), explain ver 70% f the cefficient variance. The genes are arranged in an elliptical shape. Thus, it can be bserved that sme genes tend t interact with different cmbinatins f time and xygen. A similar behavir is bserved in PLS biplt f Figure 2(b).

10 10 lmdme: Linear Mdels n Designed Multivariate Experiments in R The interactin effect n the fit bject can als be displayed using the ladingplt functin (see Figure 3). Fr every cmbinatin f tw cnsecutive levels f factrs (time and xygen), the figure shws an interactin effect n the first cmpnent, which explains 50.61% f the ttal variance f the time:xygen term. R> ladingplt(fit, term.x="time", term.y="xygen") time:xygen PC 1 Exp. Var. (50.61%) xygen=1 xygen=5 xygen= time Figure 3: ANOVA simultaneus cmpnent analysis ladingplt n genes satisfying the F test with p value < n the interactin cefficients (time xygen). In the case f an ANOVA-PCA/PLS analysis, the user nly needs t change the type="residuals" parameter in the decmpsitin functin and perfrm a similar explratin.

11 Cristóbal Fresn, Mónica G Balzarini, Elmer A Fernández 11 Acknwledgements Funding: This wrk was supprted by the Natinal Agency fr Prmting Science and Technlgy, Argentina (PICT00667/07 t E.A.F. and PICT BID t E.A.F.), Córdba Ministry f Science and Technlgy, Argentina (PID2008 t E.A.F and PIP2009 t M.G.B.), Cathlic University f Córdba, Argentina and Natinal Cuncil f Scientific and Technical Research (CONICET), Argentina. References Abdi H, Williams L (2010). Principal Cmpnent Analysis. Wiley Interdisciplinary Reviews: Cmputatinal Statistics, 2(4), De Haan J, Wehrens R, Bauerschmidt S, Piek E, Van Schaik R, Buydens L (2007). Interpretatin f ANOVA Mdels fr Micrarray Data Using PCA. Biinfrmatics, 23(2), Edgar R, Dmrachev M, Lash AE (2002). Gene Expressin Omnibus: NCBI Gene Expressin and Hybridizatin Array Data Repsitry. Nucleic Acids Research, 30(1), ISSN di: /nar/ URL Fresn C, Balzarini MG, Fernández EA (2014). lmdme: Linear Mdels n Designed Multivariate Experiments in R. Jurnal f Statistical Sftware, 56(7), URL Fresn C, Fernández EA (2013a). lmdme: Linear Mdel Decmpsitin fr Designed Multivariate Experiments. R package versin 1.2.1, URL packages/release/bic/html/lmdme.html. Fresn C, Fernández EA (2013b). stemhypxia: Differentiatin f Human Embrynic Stem Cells Under Hypxia Gene Expressin Dataset by Prad-Lpez et al. (2010). R package versin , URL html/stemhypxia.html. Gentleman RC, Carey VJ, Bates DM, thers (2004). Bicnductr: Open Sftware Develpment fr Cmputatinal Bilgy and Biinfrmatics. Genme Bilgy, 5, R80. URL Hansn BA (2012). ChemSpec: Explratry Chemmetrics fr Spectrscpy. R package versin , URL Harringtn PdB, Vieira N, Espinza J, Nien J, Rmer R, Yergey A (2005). Analysis f Variance Principal Cmpnent Analysis: A Sft Tl fr Prtemic Discvery. Analytica Chimica Acta, 544(1), Mevik BH, Wehrens R, Liland KH (2011). pls: Partial Least Squares and Principal Cmpnent Regressin. R package versin 2.3-0, URL pls.

12 12 lmdme: Linear Mdels n Designed Multivariate Experiments in R Nueda M, Cnesa A, Westerhuis J, Hefslt H, Smilde A, Talón M, Ferrer A (2007). Discvering Gene Expressin Patterns in Time Curse Micrarray Experiments by ANOVA SCA. Biinfrmatics, 23(14), Prad-Lpez S, Cnesa A, Armiñán A, Martínez-Lsa M, Escbed-Lucea C, Gandia C, Tarazna S, Melguiz D, Blesa D, Mntaner D, et al. (2010). Hypxia Prmtes Efficient Differentiatin f Human Embrynic Stem Cells t Functinal Endthelium. Stem Cells, 28(3), R Develpment Cre Team (2012). R: A Language and Envirnment fr Statistical Cmputing. R Fundatin fr Statistical Cmputing, Vienna, Austria. ISBN , URL http: // Shawe-Taylr J, Cristianini N (2004). Kernel Methds fr Pattern Analysis. Cambridge University Press. ISBN URL 9i0vg12lti4C. Smilde A, Jansen J, Hefslt H, Lamers R, Van Der Greef J, Timmerman M (2005). ANOVA-Simultaneus Cmpnent Analysis (ASCA): A New Tl fr Analysing Designed Metablmics Data. Biinfrmatics, 21(13), Smyth GK (2004). Linear Mdels and Empirical Bayes Methds fr Assessing Differential Expressin in Micrarray Experiments. Statistical Applicatins in Genetics and Mlecular Bilgy, 3(1), Article 3. Smyth GK, Ritchie M, Silver J, Wettenhall J, Thrne N, Langaas M, Ferkingstad E, Davy M, Pepin F, Chi D, McCarthy D, Wu D, Oshlack A, de Graaf C, Hu Y, Shi W, Phipsn B (2011). limma: Linear Mdels fr Micrarray Data. R package versin , URL Subramanian A, Tamay P, Mtha V, Mukherjee S, Ebert B, Gillette M, Paulvich A, Pmery S, Glub T, Lander E, et al. (2005). Gene Set Enrichment Analysis: A Knwledge-Based Apprach fr Interpreting Genme-Wide Expressin Prfiles. Prceedings f the Natinal Academy f Sciences f the United States f America, 102(43), Tarazna S, Prad-López S, Dpaz J, Ferrer A, Cnesa A (2012). Variable Selectin fr Multifactrial Genmic Data. Chemmetrics and Intelligent Labratry Systems, 110(1), Zwanenburg G, Hefslt H, Westerhuis J, Jansen J, Smilde A (2011). ANOVA-Principal Cmpnent Analysis and ANOVA-Simultaneus Cmpnent Analysis: A Cmparisn. Jurnal f Chemmetrics, 25(10),

13 Cristóbal Fresn, Mónica G Balzarini, Elmer A Fernández 13 Sessin Inf R> sessininf() R versin Patched ( r74967) Platfrm: x86_64-pc-linux-gnu (64-bit) Running under: Ubuntu LTS Matrix prducts: default BLAS: /hme/bicbuild/bbs-3.8-bic/r/lib/librblas.s LAPACK: /hme/bicbuild/bbs-3.8-bic/r/lib/librlapack.s lcale: [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C [3] LC_TIME=en_US.UTF-8 LC_COLLATE=C [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8 [7] LC_PAPER=en_US.UTF-8 LC_NAME=C [9] LC_ADDRESS=C LC_TELEPHONE=C [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C attached base packages: [1] stats graphics grdevices utils datasets methds [7] base ther attached packages: [1] lmdme_ pls_2.7-0 stemhypxia_ laded via a namespace (and nt attached): [1] cmpiler_3.5.1 limma_ tls_3.5.1 Affiliatin: Cristóbal Fresn & Elmer A Fernández Biscience Data Mining Grup Faculty f Engineering Universidad Católica de Córdba X5016DHK Córdba, Argentina cfresn@bdmg.cm.ar, efernandez@bdmg.cm.ar URL: Mónica G Balzarini Bimetry Department Faculty f Agrnmy Universidad Nacinal de Córdba

14 14 lmdme: Linear Mdels n Designed Multivariate Experiments in R X5000JVP Córdba, Argentina mbalzari@gmail.cm URL: