Research Article Cancer Outlier Analysis Based on Mixture Modeling of Gene Expression Data

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1 Compuaional and Mahemaical Mehods in Medicine Volume 23, Aricle ID 6939, 8 pages hp://dx.doi.org/.55/23/6939 Research Aricle Cancer Oulier Analysis Based on Mixure Modeling of Gene Expression Daa Keia Mori,,2 Tomonori Oura, 3 Hisashi Noma, 4 and Shigeyuki Masui,4 Deparmen of Saisical Science, School of Mulidisciplinary Sciences, The Graduae Universiy for Advanced Sudies, -3 Midori-cho, Tachikawa, Tokyo , Japan 2 Clinical Trial Coordinaion Office, Shizuoka Cancer Cener, 7 Shimonagakubo, Nagaizumi-cho Suno-gun, Shizuoka , Japan 3 Asia-Pacific Saisical Sciences, Lilly Research Laboraories Developmen Cener of Excellence Asia Pacific, Eli Lilly Japan K. K. Sannomiya Plaza Building 7--5 Isogamidori, Chuo-ku, Kobe, Hyogo 65-86, Japan 4 Deparmen of Daa Science, The Insiue of Saisical Mahemaics, -3 Midori-cho, Tachikawa, Tokyo , Japan Correspondence should be addressed o Keia Mori; kmori@ism.ac.jp Received 3 January 23; Acceped 23 March 23 Academic Edior: Shino Eguchi Copyrigh 23 Keia Mori e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Molecular heerogeneiy of cancer, parially caused by various chromosomal aberraions or gene muaions, can yield subsanial heerogeneiy in gene expression profile in cancer samples. To deec cancer-relaed genes which are acive only in a subse of cancer samples or cancer ouliers, several mehods have been proposed in he conex of muliple esing. Such cancer oulier analyses will generally suffer from a serious lack of power, compared wih he sandard muliple esing seing where common acivaion of genes across all cancer samples is supposed. In his paper, we consider informaion sharing across genes and cancer samples, via a parameric normal mixure modeling of gene expression levels of cancer samples across genes afer a sandardizaion using he reference, normal sample daa. A gene-based saisic for gene selecion is developed on he basis of a poserior probabiliy of cancer oulier for each cancer sample. Some efficiency improvemen by using our mehod was demonsraed, even under seings wih misspecified, heavy-ailed -disribuions. An applicaion o a real daase from hemaologic malignancies is provided.. Inroducion Heerogeneiy of he expression of oncogenes wihin he same hisological cancers is considered o have significan implicaions for undersanding disease biology, idenifying risk groups, and opimizing paien reamen [, 2]. Recenly, Tomlins e al. [3] argued ha radiional analyical mehods, for example, a wo-sample -saisic, which search for common acivaion of genes across a class of cancer samples, will fail o deec cancer genes which show differenial expression in a subse of cancer samples or cancer ouliers. Theydeveloped he cancer oulier profile analysis () mehod o deec cancer genes wih such heerogeneous expression profiles wihin cancer samples and revealed subypes of prosae cancer paiens defined by recurren chromosomal aberraion. Inspired by he saisic, some auhors have proposed oher mehods for deecing cancer-relaed genes wih cancer oulier profiles in he framework of muliple esing [4 6]. However, such cancer oulier analyses will generally suffer from a serious lack of power because he analysis aemps o deec relaively small fracions of cancer ouliers; he signal conained in he daa is relaively limied, compared wih ha in he sandard muliple esing seing where common acivaion of cancer-relaed genes for all cancer samples is supposed. As informaion sharing across unis in he daa generally improves efficiency of he analysis, we propose a simple efficien mehod via informaion sharing across

2 2 Compuaional and Mahemaical Mehods in Medicine boh genes and cancer samples. Specifically, we propose a parameric normal mixure modeling of gene expression levels of cancer samples across genes afer a sandardizaion using he reference, normal sample daa. Then, a gene-based saisicforgeneselecionisproposedonhebasisofaposerior probabiliy of cancer oulier for each cancer sample. This poserior probabiliy iself is o provide a useful index o aid idenifying cancer ouliers for a seleced gene. This paper is organized as follows. Afer providing a brief summary of he exising muliple esing mehods for he cancer oulieranalysis in Secion2, we provide he proposed mehod in Secion 3. We assess performance of our mehods via simulaions in Secion 4. Anapplicaionoarealdaase from hemaologic malignancies is given in Secion 5.Finally, concluding remarks appear in Secion6. 2. Exising Muliple Tesing Mehods for Cancer Oulier Analysis We suppose a microarray sudy o deec cancer-relaed genes from a large pool of G genes based on heir gene expression levels measured for n samples, comprised of n samples from anormalclassandn samples from a cancer class. The gene expression daa considered here comprise normalized log raios from wo-color cdna arrays or normalized log signals from oligonucleoide arrays (e.g., Affymerix GeneChip). For gene g (g =,...,G), le x gi be he expression value for sample i (i =,...,n ) in he normal class and le y gj be ha for sample j (j =,...,n ) in he cancer class. The mos muliple esing mehods developed for analyzing cancer ouliers inend o a one-sided esing. Wihou loss of generaliy, we are ineresed in deecion of acivaed genes ha are overexpressed or upregulaed in a subse of cancer samples, ha is, cancer ouliers. For deecing cancer-relaed genes wih over- or underexpressions, one may perform wo one-sided ess separaely, one for deecing cancer-relaed genes wih overexpressions and he oher for deecing hose wih underexpressions. The radiional wo-sample -saisic for gene g is defined as g = y g x g s g, () where y g is he mean expression value in he cancer samples, x g ishemeanexpressionvalueinhenormalsamples, and s g is he usual pooled sandard error esimae for gene g (g =,...,G). The -saisic is efficien in deecing cancer-relaed genes on which mos cancer samples are acivaed,bumaynobeefficienforhosewihcanceroulier profiles. Tomlins e al. [3] defines he saisic as Copa g = q r (y gj : j n ) med g mad g, (2) where q r ( ) is he rh percenile of he expression level, med g ishemedianofexpressionvalues,andmad g is he median absolue deviaion of expression values in all of he samples: med g = median (x gi,y gj ; i=,...,n, j=,...,n ), mad g =.4826 median ( x gi med g, y gj med g ; i=,...,n, j=,...,n ). (3) The value of r in q r ( ), which represens a hreshold in deermining cancer oulier, is specified by he user, such as r=75, 9, or 95. Inseadofusingafixedr percenile value, approximaely equivalen o using he informaion from only one sample, he use of addiional oulier samples can be more efficien. Specifically, he saisic [4] is defined as g = i R g (y gj med g ). (4) mad g Here he se of cancer ouliers, R g, is heurisically idenified by R g ={j: y gj >q 75 (x gi,y gj : i=,...,n ; j=,...,n )+ IQR(x gi,y gj : i=,...,n ; j=,...,n )},whereiqr(d) is he inerquinile range of he daa D, IQR(D) = q 75 (D) q 25 (D). Wu [5] proposed he saisic hrough idenifying cancer ouliers relaive o he normal sample, raher han he pooled sample. Specifically, he saisic is defined as g = i O g (y gj med g,x ) mad, (5) g where O g ={j:y gj >q 75 (x gi : i =,...,n )+IQR(x gj :i=,...,n )}, med g,x = median(x gi ; i =,...,n ),med g,y = median(y gj ;j=,...,n ),and mad g =.4826 median ( x gi med g,x, y gj med g,y, i=,...,n, j=,...,n ). (6) As he,, and saisics are criicized because he ouliers are arbirarily defined, Lian [6] considers all possible values of he oulier hreshold. Specifically, for he ordered gene expressions for he cancer samples, y g y g2 y gn, he saisic is defined as {( j k ( y gj med g,x )/mad g g = max ) μ k } k n { σ }, k { } (7) where μ k =E j k z j and σ 2 k = Var j k z j for z > z 2 > > z n, he order saisics of n samples from he sandard normal disribuion. The sandardizaion in he parenhesisisomakedifferenvaluesofhesaisiccomparable for differen values of he oulier hreshold, k (k =,...,n ).

3 Compuaional and Mahemaical Mehods in Medicine 3 3. The Mehod 3.. Mixure Modeling of Gene Expression Daa. In order for informaion sharing across boh genes and cancer samples, we propose a simple parameric normal mixure modeling of gene expression daa of cancer samples. As he exising muliple esing mehods, for each gene, we consider sandardized gene expressions of he cancer samples based on he reference, normal sample daa, u gj = y gj x g s g,x, (8) where s g,x is he usual sandard error esimae wihin he normal samples for gene g (g =,...,G; j =,...,n ). Again, he sandardizaion inends o make all gene expression daa from he cancer samples comparable across genes. We hen assume he finie normal mixure model wih he hree componens, f(u gj )=π f (u gj )+π f (u gj )+π 2 f 2 (u gj ). (9) The densiy funcion f corresponds o he null componen wih no differenial expressions for he reference, normal sample daa. The densiies f and f 2 correspond o he nonnull componens (i.e., cancer ouliers) of underexpression and overexpression, respecively, for he normal sample daa. We specify normal disribuions, N(, 2 ), N(δ, 2 ), and N(δ 2, 2 ),forf, f,andf 2,respecively.π k represens he mixing proporion (k =,, 2), and π +π +π 2 =. We denoe Z gj,k as unobservable indicaor random variables, such ha Z gj,k =if he (sandardized) expression level, u gj, of cancer sample j on gene g belongs o he kh componen, and Z gj,k =oherwise (g =,...,G; j =,...,n ). We esimae he parameers, δ, δ 2,andπ s, via applying he EM algorihm o cope wih he unobservable indicaor variable Z gj,k in he mixure model (e.g., [7]) Saisics for Gene Selecion. The poserior probabiliy, w gj,k,haz gj,k =, ha is, he expression level u gj belongs o he kh componen, provides a basis for gene selecion, w gj,k = π k f k (u gj ) f(u gj ). () For deecing overexpressed genes, possibly wih a cancer oulier profile (as a one-sided esing), we propose o use he following gene-based saisic for gene selecion: S g = n j= ( w gj,2 ). () This saisic may correspond o one minus he poserior probabiliy ha none of samples are cancer ouliers wih overexpressions. We will selec genes wih greaes values of S g. Gene-based saisics for deecing underexpressed cancer-relaed genes can be similarly developed. In our framework, we can also derive a similar gene-based saisic for deecing under- or overexpressed genes (as a wo-sided esing). One has T g = n j= { (w gj, +w gj,2 )}. (2) I is imporan o noe ha he poserior probabiliies, w gj,k, hemselves can serve as a helpful index o aid idenifying cancer oulier samples for a paricular (seleced) gene. In conras, he exising cancer oulier mehods do no provide such an expression-level saisic for idenifying cancer oulier samples. Unlike he exising saisics for cancer oulier analysis, he saisic, S g, does no involve any paricular cancer oulier hreshold, so ha cancer-relaed genes wih various proporionsofcancerouliers(φ in Secion 4), even hose wih common acivaion across all cancer samples, could be deeced. However, as S g is a composie of he poserior probabiliies from all of he cancer samples, cancer-relaed genes wih smaller proporions of cancer oulier will be more difficul o be deeced because he saisic will be more dominaed by he poserior probabiliies from he cancer samples oher han cancer ouliers. The impac of he proporion of cancer oulier will be invesigaed in Secion Simulaion Sudy We conduced a simulaion sudy o assess he performance of our mehod in deecing cancer-relaed genes wih cancer oulier profiles. We considered a microarray sudy wih G= genes for n=4,8,or2samples,wherehefirs half of samples were from he normal class and he laer half from he cancer class, ha is, n = n = n/2.of noe, for a given n, he power of he analysis will improve as n increases because more precise esimaes of he mean and variance of he normal sample daa become available in he sandardizaion u gj before fiing he mixure model (9) o deec cancer-relaed genes. We generaed he gene expression levels for each gene from he sandard normal disribuion N(, 2 ) or he cenral -disribuion wih 2 degrees of freedom o assess he impac of deviaion from he normaliy assumpion. No ineracion across genes was assumed. We supposed ha G genes were divided ino he hree gene componens according o he mixure model (9), ha is, he null, underexpression, and overexpression componen. The mixing proporions were se o as π =.6, π =π 2 =.2. For each nonnull gene wih under- or overexpressions, he proporion of cancer ouliers in he cancer samples, φ, wasseobeφ =.,.3, or.5. We supposed a common difference or effec size in gene expression beween he cancer oulier samples and he oher samples (normal samples and nonoulier cancer samples) across nonnull genes andsehevalueofδ as 2. and ha of δ 2 as 2.. For each configuraion, we performed gene selecion based on he saisic,,,,,andheproposedonesided saisics, S g, for deecing overexpressed cancer-relaed genes.

4 4 Compuaional and Mahemaical Mehods in Medicine φ =. φ =.3 φ = n= n= n = Figure : ROC curves ha plo versus for normally disribued gene expression daa. We assessed he false discovery rae () and rue posiive rae (), defined as he proporion of false posiives in he se of significan genes and he proporion of seleced rue posiives in he se of all of he overexpressed genes (= Gπ 2 ), respecively. Noe ha he corresponds o average power in muliple esing (e.g., [8, 9]). We conduced 2 simulaions o obain average for a given value of for each mehod, as he esimaes of were highly sable for G = values of each saisic obained in a single simulaion run. Figures and 2 show ROC curves ha plo he and for various numbers of significan genes in muliple esing for normally disribued and -disribued gene expressions, respecively. For normally disribued gene expressions, he gene selecion based on he proposed saisic, S g, generally provided he greaes values of (for a given value of ). As is expeced,heproposedgeneselecionbasedons g provided greaer as φ increased. The gene selecion based on he -saisic provided he smalles values of, especially when he proporion of cancer ouliers is small, such as φ =., bu he improved for greaer values of he proporion, such as φ =.5,asisexpeced.Theand mehods performed wors among he mehods excep he es, especially for greaer values of φ, such as φ =.5. In paricular, he performance of he mehod was largely deerioraed for φ =.5. The and mehods generally provided comparable values, bu less han hose of he proposed mehod based on S g. For -disribued gene expressions, similar rends were observed. Again, he proposed mehod based on S g provided greaes in general, excep for he scenario wih n=4

5 Compuaional and Mahemaical Mehods in Medicine 5 φ =. φ =.3 φ = n= n= n = Figure 2: ROC curves ha plo versus for -disribued gene expression daa. and φ =., alhough he degree of is superioriy o he oher mehods, such as he and mehods, becomes smaller, compared wih he seings wih normally disribued gene expressions. The and mehods again provided he smalles values of, especially when φ is large, such as φ = Applicaion We illusrae how he proposed mehod can capure he heerogeneiy of cancer samples hrough is applicaion o a microarray gene expression daa from he myelodysplasic syndromes (MDSs) []. The MDSs are complex hemaologic malignancies wih heerogeneous clinicopahological feaures wih various chromosomal aberraions. In order o discover he heerogeneous clinicopahological feaures of MDSs, possibly including hose relaed o prognosis, we adoped he proposal using mixure disribuions mehod for 39 MDSs and 69 nonleukemias (samples from bone marrow mononuclear cells from nonleukemic condiions), which were regarded as cancer and normal samples, respecively. Here, following Mills e al. [], we removed samples of he chronic myelomonocyic leukemia disease ype from MDS samples. We firs adoped he RMA normalizaion [] o he raw daa (Raw daa (CEL files) downloaded from Gene Expression Omnibus daabase (GEO, hp:// accession number GSE56)). We make

6 6 Compuaional and Mahemaical Mehods in Medicine Table : The number of overlaps in seleced genes beween he gene selecion mehods in he example of hemaologic malignancies. Top 2 genes were seleced by each mehod. -saisic -saisic saisics using he log scales expression inensiies of each gene. As an iniial screening of genes relaed o cancer ouliers from a pool of G = 54, 675 candidae genes, we adoped he exising and proposal mehods. For each mehod, we seleced 2 op genes wih he greaes values of he saisic. The esimaes of he parameers in he mixure model (9) obained under an EM algorihm wih a convergence crierion ha are relaive changes of he parameers < 4 were as follows: π =.8, π 2 =.8, δ =.22,and δ 2 = Table summarizes he overlap in a number of seleced genes beween he gene selecion mehods. Generally, he,, and mehods had subsanial overlaps in he seleced genes. The degree of overlap can be explained by he affiniy among he mehods in erms of he used sandardizaion and oulier hresholds (see Secion 2). On he oher hand, i is ineresing ha he proposed mehod based on he gene-based saisic, S g, had inermediae overlaps wih all of he oher mehods. This would indicae ha he proposed mehod could deec cancer-relaed genes wih various cancer oulier profiles. Figure 3 shows hisograms of he sandardized expression levels wihin each class (normal and cancer) for hree genes ha were seleced by our mehod, bu no by he oher mehods. The proporion of cancer ouliers was relaively small for he firs wo genes (Figures 3(a) and 3(b)),bu large for he hird gene(figure 3(c)),which again indicaes ha our mehods can deec cancer-relaed genes wih various proporions of cancer ouliers. 6. Discussion In his paper, we have aemped o improve he efficiency of he cancer oulier analysis hrough informaion sharing across genes and cancer samples. In our simulaions, he proposed gene selecion based on a parameric normal mixure modeling of gene expression daa demonsraed some improvemen in efficiency for deecing cancer-relaed genes wih moderae o large proporions of cancer oulier (φ.3), even under seings wih heavy-ailed -disribuions. The proposed saisic would herefore be effecive for selecing cancer-relaed genes ha are involved in relaively major acivaion among cancer samples. Modificaion of he saisic for selecing cancer-relaed genes wih more minor acivaion (i.e., small φ) is a subjec for fuure research. Anoher imporan subjec would be he addiion of a gene-level mixure srucure, ha is, nonnull and null genes in erms of he associaion wih cancer, o provide a more formal basis for evaluaing false posiives and rue posiives in gene selecion. We have assumed he mixure srucure (9)wih he hree componens, f, f,andf 2,haiscommonacrossgenes. In some cases, he use of only one nonnull componen for a paricular direcion of differenial gene expression may be raher resricive for plausible, large heerogeneiy among cancer samples. Our mehod can be exended o involve muliple nonnull componens, possibly wih selecion of he number of nonnull componens based on model-selecion crieria, such as AIC and BIC [7]. Anoher resricion of our model is ha no ineracion or correlaion is assumed among genes. According o an invesigaion in he conex of mixure modeling of a gene-level saisic (e.g., [2]), he impac of correlaion is generally small for moderae correlaion. In our modeling of he sandardized gene expression levels u gj across boh genes and samples, he proporion of correlaed u gj s is expeced o be relaively small because of independence across samples, bu furher invesigaion is needed. As o he exising mehods of cancer oulier analysis, our simulaions suggesed superioriy of he sandardizaion based on he reference, normal sample daa, no he pooled daa from boh cancer and normal samples. The poor performance of he mehod for greaer proporions of cancer ouliers, such as φ =.5, can be explained by he useofheiqrbasedonhepooleddaa.insuchsiuaions wih relaively large numbers of cancer ouliers, he IQR may cover some of cancer ouliers, resuling in a very large oulier hreshold, so ha a subsanial fracion of cancer ouliers migh be missed by using he saisic. In conras, he performance of he mehod, which is based on he IQR based only on he normal sample daa, was no deerioraed as φ increased in our simulaions. Afer screening cancer-relaed genes wih cancer oulier profiles, researchers will need clusering of genes o idenify coregulaedgeneshabelongohesamemolecularpahway relaed o disease biology and aggressiveness. A he same ime, clusering of cancer samples based on he idenified gene clusers can help discovering new axonomy of cancer based on gene expression profiles of cancer ouliers, possibly relaed o paiens clinical courses such as prognosis and response o herapeuics. A wo-way model-based clusering of genes and samples in he conex of cancer oulier analysis, as an exension of he proposed model-based mehod in hispaper,wouldbeanimporanopic,andoneofsuch clusering mehods will be repored elsewhere.

7 Compuaional and Mahemaical Mehods in Medicine 7 Normal sample Cancer sample 5 Sandardized expression value 5 Sandardized expression value (a) No. 2592: 2347 a Normal sample Cancer sample 5 Sandardized expression value 5 Sandardized expression value (b) No. 37: a Normal sample Cancer sample Sandardized expression value Sandardized expression value (c) No. 2595: 235 a Figure 3: Hisograms of he sandardized expression values of hree genes seleced by our mehod, bu no by he oher mehods.

8 8 Compuaional and Mahemaical Mehods in Medicine Acknowledgmen This research was parly suppored by a Gran-in-Aid for Scienific Research (242442) from he Minisry of Educaion, Culure, Spors, Science and Technology of Japan. References [] M. Soda, Y. L. Choi, M. Enomoo e al., Idenificaion of he ransforming EML4-ALK fusion gene in non-small-cell lung cancer, Naure, vol. 448, no. 753, pp , 27. [2] C. Kumar-Sinha, S. A. Tomlins, and A. M. Chinnaiyan, Recurren gene fusions in prosae cancer, Naure Reviews Cancer, vol.8,no.7,pp.497 5,28. [3] S.A.Tomlins,D.R.Rhodes,S.Pernereal., Recurrenfusion of TMPRSS2 and ETS ranscripion facor genes in prosae cancer, Science,vol. 3, no. 5748, pp , 25. [4] R. Tibshirani and T. Hasie, Oulier sums for differenial gene expression analysis, Biosaisics,vol.8,no.,pp.2 8,27. [5] B. Wu, Cancer oulier differenial gene expression deecion, Biosaisics,vol.8,no.3,pp ,27. [6] H. Lian, : deecing cancer differenial gene expression, Biosaisics,vol.9,no.3,pp.4 48,28. [7] G. McLachlan and D. Peel, Finie Mixure Models,WileySeries in Probabiliy and Saisics, 2. [8] S. H. Jung, Sample size for -conrol in microarray daa analysis, Bioinformaics,vol.2,no.4, pp ,25. [9] Y. Shao and C. H. Tseng, Sample size calculaion wih dependence adjusmen for -conrol in microarray sudies, Saisics in Medicine,vol.26,no.23,pp ,27. [] K. I. Mills, A. Kohlmann, P. M. Williams e al., Microarraybased classifiers and prognosis models idenify subgroups wih disinc clinical oucomes and high risk of AML ransformaion of myelodysplasic syndrome, Blood, vol. 4, no. 5, pp , 29. [] B. M. Bolsad, R. A. Irizarry, M. Åsrand, and T. P. Speed, A comparison of normalizaion mehods for high densiy oligonucleoide array daa based on variance and bias, Bioinformaics, vol.9,no.2,pp.85 93,23. [2] G. J. McLachlan, R. W. Bean, and L. B. T. Jones, A simple implemenaion of a normal mixure approach o differenial gene expression in muliclass microarrays, Bioinformaics,vol. 22,no.3,pp.68 65,26.

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