Biplots in Practice MICHAEL GREENACRE. Professor of Statistics at the Pompeu Fabra University. Chapter 13 Offprint
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1 Biplts in Practice MICHAEL GREENACRE Prfessr f Statistics at the Pmpeu Fabra University Chapter 13 Offprint CASE STUDY BIOMEDICINE Cmparing Cancer Types Accrding t Gene Epressin Arrays First published: September 2010 ISBN: Supprting websites: Michael Greenacre, 2010 Fundación BBVA, 2010
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3 CHAPTER 13 Case Study 1: Cmparing Cancer Types Accrding t Gene Epressin Arrays This first case study cntains many aspects f biplts treated in this bk. The cntet is a large data set f micrarray data frm tumur samples fund in children. This is a very wide data set in the sense that there are nly 63 samples but ver 2000 variables in the frm f genes epressed in the micrarray eperiments. The variables are n the same cntinuus scale and s the regular PCA biplt f Chapter 6 will be used t visualize the raw data. But because the samples are gruped we shall als apply the centrid biplt described in Chapter 11 t shw separatin f the tumur grups. There are tw additinal aspects t this case study. First, because f the large number f variables we will be interested in quantifying the cntributins f each ne t the biplts that we cnstruct, with a view t reducing the gene set t the mst imprtant nes. Secnd, an additinal sample f 20 tumurs is available, which can be used t test whether the biplt prvides a predictin rule capable f classifying these additinal tumurs crrectly. Cntents Data set cancer Principal cmpnent biplt Reducing the number f variables Centrid biplt all variables Centrid biplt reduced set f variables Classificatin f additinal samples Imprving predictin SUMMARY: This data set cancer is taken frm the bk The Elements f Statistical Learning (secnd editin) by Hastie, Tibshirani and Friedman and cnsists f a matri f 2308 genes (clumns) bserved n 63 samples (rws) see the Bibligraphy fr a link t the bk s website and accmpanying data sets. The data arise frm micrarray eperiments, a technlgy which has becme imprtant in genmic research, Data set cancer 129
4 BIPLOTS IN PRACTICE especially the relatin f genes t varius diseases. The samples are frm small, rund blue-cell tumurs fund in children. The genes are quantified by their epressin values, the lgarithm f the rati R/G, where R is the amunt f gene-specific RNA in the target sample that hybridizes t a particular (gene-specific) spt n the micrarray, and G is the crrespnding amunt f RNA frm a reference sample. The data set is called wide because f the large number f variables cmpared t the samples. The tumurs fall int fur majr types: (Ewing s sarcma), RMS (rhabdmysarcma) (neurblastma) and (Burkitt lymphma) in this data set f 63 samples there are 23, 20 RMS, 12 and 8 tumurs. There is an additinal data set f 20 samples frm these fur cancer types, which we will use later in the case study t test a classificatin rule predicting cancer type. Principal cmpnent biplt The basic data are all n a lgarithmic scale and d nt require further standardizatin. Ntice that these lgarithms f ratis are nt lg-ratis in the sense f Chapter 7, where the ratis are frmed frm all pairs f a set f bserved variables. Because there are 2308 variables we will nt use arrws t depict each ne, but grey dts Ehibit 13.1: PCA cntributin biplt f the data set cancer, shwing cnve hulls arund the fur grups and labels at their centrids. Grey dts indicate the 2308 genes RMS RMS
5 CASE STUDY 1: COMPARING CANCER TYPES ACCORDING TO GENE EXPRESSION ARRAYS Ehibit 13.2: Scree plt f the 63 eigenvalues in the PCA f the data set cancer, shwing the last ne equal t 0 (there are 62 dimensins in this wide data set) n a grey scale, where the darkness f the pint is related t the gene s cntributin t the slutin see Ehibit Because this is a PCA biplt with n differential weights n the variables, the highly cntributing genes will als be thse far frm the centre f the display. The PCA biplt des nt separate the cancer types very well, as seen by the large verlap f the fur grups. Of curse, this is nt the bjective f the PCA, which aims t maimize the between-sample dispersin, nt the between-grup dispersin. This sample-level biplt gives a first idea f hw the samples lie with respect t ne anther and is useful fr diagnsing unusual samples r variables, as well as sptting pssible errrs in the data. The dimensinality f this matri is 62, determined by the number f samples minus 1 in this wide case rather than the number f variables. The percentage f variance accunted fr by the tw-dimensinal slutin is 28.5%. It is useful t lk at the scree plt f the eigenvalues t try t assess the amunt f nise in the data (Ehibit 13.2). The ttal variance in this data set is equal t 982.0, with an average per dimensin f 982.0/62 = By this criterin the first 14 dimensins are abve average, althugh it is clear that the first tw d separate clearly frm the rest. 131
6 BIPLOTS IN PRACTICE Reducing the number f variables We have several tls at ur dispsitin nw t reduce the number f variables (genes) while keeping track f the effect this has n the visualizatin f the cancer samples. A pssible strategy is t reduce the gene set ne at a time, remving each time the gene that cntributes the least t the slutin. At each stage f the gene remval we measure the fllwing aspects, shwn in Ehibit 13.3: a. The ttal variance and the average ver the dimensins (the latter will be the frmer divided by 62 until the number f genes reduces belw 62, in which case the dimensinality is determined by the number f genes). b. The number f dimensins that are abve the average. c. The percentage f variance eplained by the tw-dimensinal slutin. d. The Prcrustes statistic n the cnfiguratin f sample pints, cmpared t the initial slutin (Ehibit 13.1) this will quantify hw much the cnfiguratin is changing. Ttal variance (Ehibit 13.3a) bviusly decreases as genes are remved the decrease is less at the start f the prcess when the genes f very minr cntributin t the slutin are remved. The number f dimensins greater than the average als decreases (Ehibit 13.3b) but still remains fairly high until the end f the re- Ehibit 13.3: Mnitring f fur statistics as the number f remved genes increases a) b) Ttal variance Number f dimensins Genes remved Genes remved c) d) % variance in 2-d Prcrustes Genes remved Genes remved 132
7 CASE STUDY 1: COMPARING CANCER TYPES ACCORDING TO GENE EXPRESSION ARRAYS RMS RMS Ehibit 13.4: PCA biplt f the reduced gene set (75 highcntributing genes, that is 2233 genes mitted), shwing ne set f genes (in dashed ellipse) at bttm right separating the grup centrids (indicated by the labels) and anther grup at bttm left that is separating the ttal sample int tw distinct grups (shwn in the green ellipses), independent f their cancer types mval prcess. The percentage f variance n the first tw aes increases as the nisy part f the data is remved (Ehibit 13.3c). Accrding t the Prcrustes analysis (Ehibit 13.3d) the tw-dimensinal cnfiguratin remains almst the same even when as many as 1500 genes are remved. We chse a slutin when the Prcrustes statistic reached 10%, when 2233 genes were remved, leaving nly 75 included in the PCA. Ntice the gradual change in the Prcrustes statistic (Ehibit 13.3d) up t this pint, then a relative stability in the cnfiguratin at abut 10% fllwed by mre dramatic changes. Ehibit 13.4 shws the biplt with the reduced set f genes. The spread f the fur grups, frm t RMS, is retained (see Ehibit 13.1), just slightly rtated. What is evident here is the emergence f tw grups f genes, ne at bttm right which is respnsible fr the separatin f the tumur grups, and anther grup at bttm left which separates the samples int tw clear clusters independent f their grups the nly eceptin is an RMS tumur suspended between the tw clusters. In rder t see the separatin f the tumur grups better and t identify which genes are determining the difference between them, a biplt f the grup centrids can be perfrmed, as described in Chapter 11 n discriminant Centrid biplt all variables 133
8 BIPLOTS IN PRACTICE Ehibit 13.5: Centrid biplt f the fur tumur grups, using all 2308 variables. The percentage f centrid variance displayed is 75.6%, with between-grup variance in the plane 88.6% f the ttal RMS RMS analysis (DA) biplts. Because there are fur centrids, the space they ccupy is three-dimensinal; hence the planar display invlves the lss f nly ne dimensin. Ehibit 13.5 shws the centrid (r DA) biplt based n all 2308 genes. The tumur grups are nw very well separated, and the separatin f the clusters bserved in Ehibit 13.4 is n lnger present. Of the ttal variance f the centrids in their full three-dimensinal space, 75.6% is represented in the biplt. Of the ttal variance f the 63 samples represented in this tw-dimensinal biplt, 88.6% is between-grup variance and 11.4% within-grup variance. Centrid biplt reduced set f variables Again, we are interested in reducing the number f genes t see which are the mst determinant in separating the grups. By applying the same step-by-step reductin in the number f genes, always remving the gene with the least cntributin t the grup differentiatin at each step, and by mnitring the percentage f variance displayed in the tw-dimensinal map as well as the prprtin f ttal planar variance accunted fr by the between-grup part. It turns ut that a 134
9 CASE STUDY 1: COMPARING CANCER TYPES ACCORDING TO GENE EXPRESSION ARRAYS RMS RMS Ehibit 13.6: Centrid biplt f the fur tumur grups, using 24 highest cntributing variables after stepwise remval. The percentage f centrid variance displayed is 94.9%, with betweengrup variance in the plane 90.5% f the ttal maimum f the latter percentage is reached when we have reduced the gene set t 24 genes, fr which the slutin is shwn in Ehibit The between-grup variance in the plane is 90.5% f the ttal, abut 3 percentage pints better than Ehibit There is nly 5.1% f the centrid variance in the third dimensin nw, as ppsed t 24.4% in Ehibit In Ehibit 13.6 we have thus achieved an ptimal separatin f the grups, while als reducing the residual variance in the centrids that is in the third dimensin. Ntice the lining up f the three tumur grups, and RMS frm tp left t bttm right, cinciding with the genes etending t the bttm right hand side: it will be these genes that distinguish these three grups, with increasing values frm t t RMS. On the ther hand the grup is situated at bttm left assciated with high values f the grup f genes at bttm left, and lw values f the single gene that ne finds at tp right. There is a grup f si genes at the bttm f the display that are separated frm the grup at bttm left, which n dubt nt nly separate frm the ther grups but als cntribute slightly t the left-t-right separatin f the ther three grups. 135
10 BIPLOTS IN PRACTICE Ehibit 13.7: The 20 additinal tumurs in the centrid slutin space fr all 2308 genes (upper biplt), and the reduced set f 24 genes (lwer biplt) a) RMS RMS RMS RMS RMS RMS b) RMS RMS RMS RMS RMS RMS
11 CASE STUDY 1: COMPARING CANCER TYPES ACCORDING TO GENE EXPRESSION ARRAYS In additin t the 63 samples studied up t nw, an additinal sample f 20 tumurs was available, and the type f tumur was knwn in each case. We can use ur results in Ehibits 13.5 and 13.6 abve t see whether accurate predictins f the tumur types are achieved in this test data set. We d this in a very simple way, just by situating the tumurs in the tw-dimensinal slutin space and cmputing their distances t the grup centrids, and then predicting the tumur type by the clsest centrid. Ehibit 13.7 shws the new tumurs in the slutin f Ehibit 13.5 using all 2308 genes (upper biplt) and then in the slutin f Ehibit 13.6 using the reduced set f 24 genes (lwer biplt). It is clear that in the upper biplt that fur f the si tumurs will be misclassified as RMS. In the lwer biplt, the new tumurs generally lie clser t their crrespnding centrids, with just tw tumurs being misclassified as, the ne n the right being nly a tiny bit clser (in the third significant digit) t the centrid than t the ne. It is a general principle that the eliminatin f irrelevant variables can imprve the predictive value f the slutin, and this is well illustrated here. Classificatin f additinal samples As a final remark, it is pssible t imprve the predictive quality f this centrid classificatin prcedure in tw ways. First, there is sme additinal variance in the centrids in the third dimensin, which we have ignred here. Calculating tumur-t-centrid distances in the full three-dimensinal space f the fur centrids will imprve the classificatin. Secnd, in the area knwn as statistical learning, a branch f machine learning, the small subset f genes used t define the predictr space wuld be chsen in a mre sphisticated way, using crss-validatin. This invlves dividing the training set f data (that is, ur initial sample f 63 tumurs) int 10 randm grups, say, and then using 9 ut f the 10 grups t determine the subset f variables that best predicts the mitted grup, and then repeating this prcess mitting each f the ther grups ne at a time. There wuld thus be 10 ways f predicting new bservatins, which we wuld apply in turn t the test set (the 20 additinal tumurs), btaining 10 predictins fr each new tumur frm which the final predictin is made by majrity vte. If these tw additinal imprvements are implemented in ur prcedure it turns ut that we can predict the grup membership f all 20 tumurs eactly. We have shwn hw biplts based n principal cmpnent analysis f bth individual-level and aggregate-level data can be used t identify natural grups f bservatins in a large data set as well as distinguish between eisting knwn grups. With respect t this data set which has a huge number f variables cmpared t bservatins: Imprving predictin SUMMARY 1. In bth the individual- and aggregate-level analyses, it is useful t reduce the number f variables t a smaller set that is the mst determinant in shwing 137
12 BIPLOTS IN PRACTICE respectively (i) the patterns in the individual-level data, and (ii) the separatin f the knwn grups. 2. One way f eliminating variables is t calculate each variable s cntributin t the slutin (a planar biplt in ur applicatin). The variable with the least cntributin is eliminated, and the prcedure is repeated ver and ver again until a small subset is fund. 3. We decided t stp the variable eliminatin prcess in the individual-level analysis when the Prcrustes statistic rse t 10% this was an ad hc decisin, but was based n bserving the evlutin f the Prcrustes statistic as variables were eliminated. This statistic increased very slightly and slwly up t this pint, but reducing the variables beynd this stage the slutin started t change dramatically 4. In the case f the aggregate-level analysis, we mnitred the rati f betweengrup variance t ttal variance in the lw-dimensinal slutin as variables were eliminated, and stpped when this reached a maimum. 5. In the centrid analysis, the eventual space based n the smaller set f variables can be used t classify new bservatins, by calculating their distances in the slutin t the centrids and then chsing the centrid that is clsest as the grup predictin. 138
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