Gene Selection and Ranking with Microarray Data

Alfred O. HeroDept. of Electrical Engineering & Computer Science

University of Michigan1301 Beal AveAnn Arbor, MI

Abstract

Over the past decade there has been an explosion in theamount of genomic data available to biomedical researchersdue to advances in biotechnology. For example, using genemicroarrays, it is now possible to probe a person’s gene ex-pression profile over the more than 30,000 genes of the hu-man genome. Signals extracted from gene microarray exper-iments can be linked to genetic factors underlying disease,development. and aging in a population. This has greatly ac-celerated the pace of gene discovery. However, the massivescale and experimental variability of genomic data makes ex-traction of biologically significant genetic information verychallenging. One of the most important problems is to se-lect a ranked list of genes which are both biologically andstatistically significant based on a gene microarray experi-ment. We will describe multicriterion methods that we havedeveloped for this gene selection and ranking problem.

1. INTRODUCTION

Since Watson and Crick discovered DNA more than fiftyyears ago, the field of genomics has progressed from a spec-ulative science starved for data and computation cycles toone of the most thriving areas of current research and de-velopment. It was not until almost 45 years after Watsonand Crick’s discovery that the first entire genome was se-quenced, the E Coli bacterium containing over 4000 genes,after several years of effort. In 2001 the first draft of the hu-man genome, containing more than 30,000 genes, was ob-tained. In spring 2003 the genome for the SARS coronavirus (SARS-CoV) was sequenced and authenticated in lessthan 2 months time.1,2 These recent leaps in progress wouldnot have been possible without significant advances in genesequencing technology. One such technology, which is themain focus of this paper, are gene microarrays and their as-sociated signal extraction and processing algorithms.

Gene microarrays provide a high throughput method tosimultaneously probe a large number gene expression lev-els in a biological sample. Current state-of-the-art microar-rays contain up to 50,000 gene probes that interact with thesample producing probe responses that can be measured asa multichannel signal. When the probes are suitably repre-sentative of the range of genetic variation of the organism,

this signal specifies a unique gene expression signature ofthe sample. Gene microarrays are a very powerful tool whichcan be used to perform gene sequencing, gene mapping andgene expression profiling. They will be critical in determin-ing the genetic circuits that regulate expression levels overtime and genetic pathways that lead to specific biologicalfunction or dysfunction of an organism.

In this paper we will describe some signal processingchallenges in gene microarray analysis and present a fewapproaches we have developed in interacting with our col-laborators in molecular biology. The focus application ofthe paper is the analysis of temporal gene expression profilesand their role in exploring genetic factors underlying disease,regulatory pathways controlling cell function, organogenesisand development. In particular we and our collaborators inthe Dept. of Human Genetics at the University of Michi-gan are interested in analyzing retinal data to determine ge-netic factors underlying dysfunction of the eye due to aging,glaucoma, macular degeneration, and diabetes. Our exam-ples will be primarily drawn from these areas and we willfocus on the problem of selection and ranking of genes thatare both biologically and statistically significant from exper-imentally replicated microarray data.

In our past work on signal processing for gene microar-rays3{7 our primary goal has been be to develop statisticallyreliable methods for selecting and ranking temporal gene ex-pression profiles. The work most closely related to this paperis our multi-criterion optimization approach togene rankingusing a statistical version of Pareto front analysis.5,7 In thiswork two methods for ranking data from multiple microar-ray experiments were introduced: cross-validation leadingto resistant Pareto front (RPF) analysis, and Bayes smooth-ing, leading to posterior Pareto front (PPF) analysis. In thispaper we focus on thegene selectionproblem and adopt astatistical multiple criteria approach similar to our previouswork. We then illustrate these methods for two AffymetrixGeneChip experiments for probing the genes of the retina. Inthese experiments we adopt pairs of criteria which trade-offhigh selectively for robustness. Specifically, one selectioncriterion is a (multivariate) paired t-test statistic for selectinggene profiles. This criterion has optimal gene selection prop-erties under a Gaussian microarray probe response model.The other criterion is based on distribution-free rank order

statistics. This criterion is robust to violations of distribu-tional assumptions on the data.

The outline of the paper is as follows. In Sec. 2 we givesome background on genomics and review gene microarraysin the context of temporal profile analysis. In Sec. 3 we mo-tivate and describe the multicriterion selection and rankingapproach. In Sec. 4 we apply false discovery rate (FDR) tomulticriterion gene selection. Finally, in Sec. 5 we illustratethese techniques for experimental data.

2. GENOMICS BACKGROUND

We start with some definitions and a brief review of molecu-lar biology and genetics. The genome refers to the geneticoperating system which controls structure and function ofcells in an organism. This genome consists of genes thatlie on segments, called exons, of the double stranded DNAhelix which lie on a number of chromosomes in the nucleusof every cell in the organism. The number of genes in theDNA of a given organism can range from a few thousand forsimple organisms to tens of thousands for more sophisticatedorganisms. Each exon contains a gene which is encoded asa nucleotide sequence of symbols A,C,G,T forming a 4-aryalphabet.

Gene expression occurs when the DNA sheds certain ofits genes in the cell nucleus in order to stimulate or inhibitvarious functions, e.g., cell growth or metabolism. This stim-ulation occurs through production of derivatives of DNA, themRNA and tRNA, produced by a process called transcrip-tion and translation. Stimulated by mRNA and tRNA theribosome of a cell produces specific amino acids in polype-tide chains. These chains form proteins that carry out the in-tended function expressed by the DNA. While the DNA doesnot change, the specific genes expressed in this fashion canchange over time, environmental conditions, and treatments.The objective of genomics is to identify the very large num-bers of genes that are expressed by the organism.

Biotechnology, such as gene microarray hybridization,Northern hybridization, and gell electrophoresis, is essentialto reliably probe the gene expression of a biological sam-ple. Bioinformatics provides tools for computational extrac-tion and analysis of the vast amounts of information in proberesponse data. As scientists and genetic engineers becomeincreasingly interested in studies of gene expression profilesover time, signal processing will become a major bioinfor-matics tool. We next briefly describe the signals generatedby gene microarrays.

A gene microarray consists of a large numberN of knownDNA probe sequences that are put in distinct locations on aslide. See one of the references8,9 for more details. After

hybridization of an unknown tissue sample to the gene mi-croarray, the abundance of each probe present in the sam-ple can be estimated from the measured levels of hybridiza-tion. Two main types of gene microarrays are in wide use:photo-lithographic gene chips and fluorescent spotted cDNAarrays. An example of the former is the Affymetrix10 prod-uct line. An example of the later is the cDNA microarrayprotocol of the National Human Genome Research Institute(NHGRI).11 A suite of software tools are available fromAffymetrix and elsewhere for extracting accurate estimatesof abundance, called probe responses. When probe responsesare to be compared across different microarray experimentsthey must also be normalized. Extraction and normalizationmethods can range from simple unweighted sample averag-ing, as in the Affymetrix MAS4 software, to more sophisti-cated model-based analyses, such as MAS5,10 the Li-Wongmethod12,13 and RMA.14,15 Many of the more sophisticatedpackages are available as freeware, e.g., see Strimmer’s web-site16 for links to relevant software written in the R soft-ware language. When several microarray experiments are

Figure 1. Probing gene expression at several time pointsleads to a temporal sequence of gene microarrays (left). Afew of the sequences can be extracted at specific probe loca-tions on the microarrays and plotted as time signals (right).

performed over time they can be combined in order to findgenes with interesting temporal expression profiles (see Fig.1). This is a data mining problem known variously as ”geneselection” and ”gene filtering” for which many methods havebeen proposed.17{19 Crucial for gene ranking is the speci-fication of a preference ordering for the ranking. A populargene selection and ranking method is based on optimizingsome single fitness criterion such as: the ratio of between-population-variation to within-population-variation; or the

temporal correlation between a measured profile and a pro-file template. A problem with this single criterion rankingmethod is that it is often difficult for the molecular biolo-gist to articulate what he is looking for in terms of a singlequantitative criterion. It is for this reason that our group hasproposed multiple criteria methods for selecting and rankinggene profiles.3,5,7

3. MULTICRITERIA SELECTION ANDRANKING

As contrasted to maximizingscalarcriteria, multiple objec-tive gene filtering seeks gene profiles that strike an optimalcompromise between maximizing several criteria. It is of-ten easier for a molecular biologist to specify several criteriathan a single criterion. For example the biologist might be in-terested in aging genes, which he might define as those geneshaving expression profiles that are increasing over time, havelow curvature over time, and whose total increase from initialtime to final time is large. As another example, one may haveto deal with two biologists who each have different criteriafor what features constitute an interesting aging gene.

Multicriterion Gene Selection: To illustrate, let fitness cri-teria �1(g); : : : ; �p(g) be defined for each geneg in the mi-croarray. A reasonable gene selection criterion would bethat the fitness for each selected geneg lies in the quadrant�1(g) > u1; : : : ; �p(g) > up. Hereu1; : : : ; up are thresh-olds which are selected by the experimenter to reflect the bi-ological significance of a particular level of measured genefitness�k(g). This is illustrated in Fig. 2 where the selectedsector for two aging criteria (the orthogonalized criteria de-scribed in Sec. 5.1) is superimposed over the scatter plot offitness levels extracted for all the genes probe in the microar-ray. This scatter plot is called the multicriteria scattergram ofthe fitness responses.

Multicriterion Gene Ranking : In a well designed gene mi-croarray experiment, multicriterion (or other) methods of se-lection will generally result in a large number of genes andthe biologist must next face the problem of selecting a fewof most “promising genes” to investigate further. Resolutionof this problem is of importance since validation of gene re-sponse requires more sensitive techniques, such as RT-PCR,which are much more time consuming and expensive. Somesort of rank ordering of the selected genes would help guidethe biologist to a solution. As a linear ordering of set ofvector quantities such asf[�1(g); : : : ; �p(g)]gg does not gen-erally exist, an absolute ranking of the selected genes is ofcourse generally impossible. However a partial ordering ofthese vectors is possible and such a ”partial ranking” canbe formulated as a multiple objective optimization problem.Multiple objective optimization captures the intrinsic com-

Figure 2. Multicriteria scattergram of gene fitness responseswith overlaid gene selection sector. The choice of position[u1; u2] of the sector depends on the experimenter’s chosenbiological significance levels for gene discovery.

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Figure 3. A hypothetical multicriterion scattergram for genesA,B,C,D,E plotted as vectors in the plane described by a pair offitness criteria�1 and �2. A, B, C are non-dominated genes andform the (first) Pareto front. A second Pareto front is formed bygenes D,E.

promises among possibly conflicting objectives in a naturalway. Consider the multicriterion scattergram in Fig. 3 andsuppose that fitness criteria�1 and�2 are to be maximized.Gene D isdominatedby both gene A and gene B since geneD has lower fitness in both criteria�1 and�2. Likewise geneE is dominated by gene B and gene C. On the other handgenes A, B and C are not dominated by any other gene andare therefore preferable to genes D and E. Multi-objectivefiltering uses this non-dominated property as a way to estab-lish a preference relation among genes given a set of criteriaf�qgq. More formally, genei is said to be dominated if thereexists some other geneg 6= i such that for at least oneq

�q(i) < �q(g) and �p(i) � �p(g); p 6= q:

The set of non-dominated genes are defined as those genes

that are not dominated. All the genes which are non-dominated constitute a set of points called the (first) Paretofront. A second Pareto front can be obtained by strippingoff the points on the first front and computing the Paretofront on the remaining points. For the example in Fig. 3 thefirst Pareto front isfA;B;Cg and the second Pareto front isfD;Eg.

The above multiple criterion selection and ranking meth-ods are applicable to any set of criteria�1; : : : ; �p. How-ever, these method do not account for any statistical uncer-tainty. The study of gene expression almost always requireshybridizing several microarrays from a population to captureand reduce response variability. This variability can be due totwo factors: biological variability of the population and ex-perimental variability. It is difficult to separate these two fac-tors and most analysis is performed with a statistical modelwhich lumps them together.

4. FDR FOR MULTIPLE CRITERIA

For comparing experiments in a way that accounts for statis-tical variations it is essential to report a figure of statisticalsignificance of the each of the findings. Two important quan-tities indicative of statistical significance are the p-value, as-sociated with testing a single gene response, and the falsediscovery rate (FDR), associated with testing all the geneprobes simultaneously (multiple comparisons). In gene mi-croarray experiments the biologist is always making multi-ple comparisons so FDR is the more appropriate quantity.Let each gene on the microarray have measured aggregatefitness�1(g) = u1(g); : : : ; �p(g) = up(g), e.g., a statisticcomputed as the average fitness ofg over all of the microar-ray replicates. For ease of presentation, we assume that thestatistical distributionP of �1(g); : : : ; �p(g) is known whenthe probe responses are spatially independent and identicallydistributed (i.i.d.) random variables over the microarray. Inother words the aggregate fitness statistic is distribution freeunder the null hypothesis that all probe responses are i.i.d.The p-value is computed for a single gene probe, say genego, and is the probability that purely random effects, i.e., i.i.d.probe responses, would have causedgo to be selected. Moreprecisely the p-value forgo is defined as:

pv(go) = P (�1 > u1(go); : : : ; �p > up(go))

where�1; : : : ; �p are random variables are computed fitnesslevels of an i.i.d. random sample. If an experimenter wereonly interested in deciding on the biological significance ofa single genego based only on observing that gene, then re-portingp(go) would be sufficient for another biologist to as-sess the statistical significance of the experimenter’s finding.In contrast to the p-value, FDR communicates statistical sig-nificance of an experimenters decisions made on the basis of

Figure 4. The maximum p-value for multiple criteria geneselection in the aging gene mouse retina microarray exper-iment (left). The FDR, computed from the p-value using awell known formula,20 for the same experiment (right). Thegenes are rank ordered in terms of their p-value and FDRprobabilities, respectively.

many gene probes. The FDR is the probability that purelyrandom effects would have caused specific genes to be se-lected among all probes on the microarray.

When the null distributionP is unknown, the p-value andthe FDR can be computed empirically by simulation or re-sampling. More information on FDR can be found in thereferences.20{22 In general an experimenter would like themaximum p-value and the FDR for his selected genes to beas low as possible to ensure a high level of statistical signifi-cance. However, as compared to the more conservative FDR,use of the maximum p-value gives an overoptimistic measureof significance. This is illustrated in Fig. 4 for the aging genemicroarray study described in the next section. In terms ofFig. 7 the FDR is related to the probability that at least one ofthe many gene responses would fall into the selected sector.

5. APPLICATIONS

Here we illustrate statistical multi-criterion selection andranking techniques for data from two gene microarray exper-iments. The biological significance of the experiment andthe list of statistically significant genes found will be re-ported elsewhere. Our purpose here is simply to illustratethe application of our gene selection and ranking techniques.Both experiments used oligonucleotide-arrays, specificallythe Affymetrix U74 mouse chips, and probe responses wereextracted using the Affymetrix MAS5 data analysis pack-age.10

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5.1. Strongly Increasing Profiles

The experiment consists of 24 retinal tissue samples takenfrom a population of age-sorted mice at 6 ages (time points)with 4 replicates per time point. These 6 time points con-sisted of 2 early development (Pn2-Pn10) and 4 late develop-ment (M2-M21) time points. DNA from each sample of reti-nal tissue was amplified and hybridized to the 12,422 probeson one of 24 Affymetrix U74 GeneChips. The data arraysfrom the GeneChips were processed by Affymetrix MAS5software to yield probe response data. We eliminated fromanalysis all genes that MAS5 called out as “absent” from allchips, leaving 8826 genes for analysis. Figure 5 shows the24 data points for a particular gene among the 8826 genesstudied. Define the gene response datum extracted from them-th microarray replicate at timet for the g-th gene probelocation:

xt;m(g); g = 1; : : : ; G; m = 1; : : : ;M; t = 1; : : : ; T: (1)

whereG = 8826, M = 4, T = 6. Figure 5 shows theresponse datafxt;m(g)gt;m for one of the genes extractedfrom the GeneChip. The scientific objective of the exper-iment is to find genes which are strongly associated withaging and development, i.e. those that are strongly mono-tonic over time. Template matching methods are not effec-tive here since they require specification of a profile patternand, due to variability in the experiment, this can miss genesthat have the desirable monotonicity characteristics but donot agree with the specified pattern. Thus we adopted thefollowing multi-criteria approach. We designed criteria tokey onto three types of profiles: 1) those that are monotoni-cally increasing; 2) those that are monotonically decreasing;3) those that display a large end-to-end change. We only de-scribe the gene selection method for monotonic increasingcase as the treatment of the decreasing case is analogous. Inorder to tease out the monotonic increasing profiles we previ-ously proposed a naturalvirtual profile criterion that counts

Figure 6. 3 of the64 = 4096 virtual profiles that can bedrawn through the 24 gene responses in mouse retinal ag-ing study. None of these 3 are monotonic. Label at top leftdenotes the gene’s Unigene number.

the number of monotonic increasing trajectories among the64 = 4096 possible trajectories that could pass through the24 data points7 (See Fig. 6). However, even though it isarguably a more compelling monotonicity statistic, it hasexponential computational complexityO(MT ) and, to ourknowledge, its p-values are not tabulated. For these rea-sons, for the gene selection application we prefered to use thewell known Jonckheere-Terpstra (JT) test statistic23 as crite-rion �1. For end-to-end change we adopted the one sidedpaired t-test statistic24 as criterion�2. The JT statistic es-sentially counts the number of times that a sample at a fu-ture time point is larger than a sample at a previous timepoint and its computation is only of polynomial complexity(O((T + 1)T=2M2)). The paired t-test statistic is an opti-mal end-to-end selection criterion when the extracted proberesponses are Gaussian random variables with identical vari-ances. An implicit assumption underlying the use of the JTand student-t test statistics is that the probe responses haveidentical distributions except for a possible shift in location,as measured by the mean or median. This assumption is rea-sonable after normalization of the gene microarrays, e.g. af-ter using the RMA procedure.14 As our collaborators areprimarily interested in the genes that are implicated in latedevelopment or aging, we dropped the first two time pointsin the data set for the analysis described below.

While the sampling distribution of the JT statistic isknown exactly under an assumption of spatially i.i.d. proberesponses, the sampling distribution for the paired t-test isnot known exactly unless the responses are Gaussian dis-tributed. Therefore we chose to generate the FDR contoursempirically using a resampling method. In this method we

Figure 7. The multicriterion scattergram of pairsf�1(g); �2(g)g

Gi=1 for i.i.d. resampled GeneChip probe re-

sponses appears approximately Gaussian distributed with re-gression line as indicated. Here�1 is equal to the JT statisticand�2 is equal to T2 which denotes the end-to-end paired ttest statistic.

simulated 500 sets of i.i.d. probe responsesfxt;m(g)gt;m;g

for which the marginal distribution matches the empiricalmargin distribution offxt;m(g)gm;g at each time pointt. Us-ing these 500 simulated GeneChip data sets we determinedFDR by computing the relative frequency that any gene fit-ness statistic[�1(g); �2(g)] falls in a given sector. By varyingthe position[u1; u2] of these sectors over the plane constantFDR contours were determined. To obtain the most dis-criminating multicriterion test we made an orthogonalizingtransformation to data in the multicriterion plane. This trans-formation was motivated by the observation that the scatter-grams of the resampled data (see Fig. 7) appeared to be acorrelated approximately bivariate Gaussian sample. Usinga regression of�2 on�1 we determined a monotonic transfor-mation that converted these resampled scattergrams into ap-proximately orthogonal bivariate Gaussian scatter plots. Thistransformation was then applied to the original data set to de-termine a set of monotonic increasing genes at a FDR level of0.1 (see Fig. 8). Shown in Fig. 9 are the 9 top ranked mono-tone increasing gene profiles among the 16 genes selected.

5.2. Differentially Expressed Profiles

The second experiment we describe is concerned with find-ing genes whose expression profiles change significantly af-ter a treatment. Such genes are called ”differentially ex-pressed” after treatment. One variant of this experiment is

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called a wildtype vs knockout experiment. In this experi-ment one has a control population (wildtype) of subjects anda treated population (knockout) of subjects whose DNA hasbeen altered in some way. One then collects cell samplesfrom both populations at different times and generates mi-croarray data sets to find any genes that are differentially ex-pressed. Figure 10 shows gene probe responses from sucha wildtype and knockout experiment. We label the wildtypeand knockout responsesWt;m(g) andKt;m(g) in a similarmanner to (1) where hereM = 4, T = 3.

The dual criteria chosen were: 1) a Mack-Skillings (MS)statistic for testing for parallel W vs. K responses (profiles)in a two way layout23; and 2) a multivariate paired t (MVPT)test statistic for quantifying the amount of difference in theW vs. K responses.24 Similarly to the previous experimentthese two criteria are complementary: the MS test is a dis-tribution free rank-order statistical test while the MVPT isoptimal under the Gaussian assumption. We applied non-linear transformations to these two criteria to stabilize theirvariances. Similarly to before we used a resampling methodto empirically compute FDR contours in the dual criteriaplane. These contours were superimposed on the multicrite-rion scattergram (see Fig. 11) to find the set of genes that aredifferentially expressed at a FDR of prescribed level. Fig-ure 12 shows the 9 top ranked differentially expresse geneprofiles among the 142 genes selected.

6. CONCLUSION

Signal processing for analysis of gene microarray and othergene experiments is a growing area and there are enoughchallenges to keep the community busy for years. In our col-laborations we have found it crucial to interact closely withour biology colleagues to ensure that our signal processingmethods are relevant and capture the biological aims of theexperimenter. To illustrate this point, in this paper we havedescribed one of our projects involving gene selection andranking. To respond to the needs of our collaborators we hadto develop a flexible multi-criterion approach to gene selec-tion and ranking. A single criterion would have much greater

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difficulty in capturing the variety of properties that our col-laborators considered biologically significant. To account forstatistical variation, we had to extend multi-criterion opti-mization to a stochastic setting. We continue to refine ourmethods to meet the changing requirements of interactingwith a very rapidly changing field.

Acknowledgement

The author would like to thank Prof. G. Fleury at EcoleSuperieure d’Electricite for the collaboration that has influ-enced work reported here. I would also like to thank Prof.A. Swaroop, Dr. S. Yosida, R. Farjo, and A. Mears in theDept. of Human Genetics at University of Michigan for theirguidance and support.

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