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496 nature genetics supplement • volume 32 • december 2002

Microarray data normalization andtransformationJohn Quackenbush

doi:10.1038/ng1032

Underlying every microarray experiment is an experimental question that one would like to address. Finding a

useful and satisfactory answer relies on careful experimental design and the use of a variety of data-mining tools

to explore the relationships between genes or reveal patterns of expression. While other sections of this issue

deal with these lofty issues, this review focuses on the much more mundane but indispensable tasks of ‘normaliz-

ing’ data from individual hybridizations to make meaningful comparisons of expression levels, and of ‘transform-

ing’ them to select genes for further analysis and data mining.

The Institute for Genomic Research, 9712 Medical Center Drive, Rockville, Maryland 20850, USA (e-mail: johnq@tigr.org).

The goal of most microarray experiments is to survey patternsof gene expression by assaying the expression levels of thou-sands to tens of thousands of genes in a single assay. Typically,RNA is first isolated from different tissues, developmentalstages, disease states or samples subjected to appropriate treat-ments. The RNA is then labeled and hybridized to the arraysusing an experimental strategy that allows expression to beassayed and compared between appropriate sample pairs. Com-mon strategies include the use of a single label and independentarrays for each sample, or a single array with distinguishablefluorescent dye labels for the individual RNAs. Regardless of theapproach chosen, the arrays are scanned after hybridizationand independent grayscale images, typically 16-bit TIFF(Tagged Information File Format) images, are generated foreach pair of samples to be compared. These images must thenbe analyzed to identify the arrayed spots and to measure the rel-ative fluorescence intensities for each element. There are manycommercial and freely available software packages for imagequantitation. Although there are minor differences betweenthem, most give high-quality, reproducible measures ofhybridization intensities.

For the purpose of the discussion here, we will ignore the par-ticular microarray platform used, the type of measurementreported (mean, median or integrated intensity, or the averagedifference for Affymetrix GeneChips™), the background correc-tion performed, or spot-quality assessment and trimming used.As our starting point, we will assume that for each biologicalsample we assay, we have a high-quality measurement of theintensity of hybridization for each gene element on the array.

The hypothesis underlying microarray analysis is that the mea-sured intensities for each arrayed gene represent its relativeexpression level. Biologically relevant patterns of expression aretypically identified by comparing measured expression levelsbetween different states on a gene-by-gene basis. But before thelevels can be compared appropriately, a number of transforma-tions must be carried out on the data to eliminate questionableor low-quality measurements, to adjust the measured intensitiesto facilitate comparisons, and to select genes that are significantlydifferentially expressed between classes of samples.

Expression ratios: the primary comparisonMost microarray experiments investigate relationships betweenrelated biological samples based on patterns of expression, andthe simplest approach looks for genes that are differentiallyexpressed. If we have an array that has Narray distinct elements,and compare a query and a reference sample, which for conve-nience we will call R and G, respectively (for the red and greencolors commonly used to represent array data), then the ratio (T)for the ith gene (where i is an index running over all the arrayedgenes from 1 to Narray) can be written as

(Note that this definition does not limit us to any particular arraytechnology: the measures Ri and Gi can be made on either a sin-gle array or on two replicate arrays. Furthermore, all the trans-formations described below can be applied to data from anymicroarray platform.)

Although ratios provide an intuitive measure of expressionchanges, they have the disadvantage of treating up- and down-regulated genes differently. Genes upregulated by a factor of 2have an expression ratio of 2, whereas those downregulated bythe same factor have an expression ratio of (–0.5). The mostwidely used alternative transformation of the ratio is the loga-rithm base 2, which has the advantage of producing a continuousspectrum of values and treating up- and downregulated genes ina similar fashion. Recall that logarithms treat numbers and theirreciprocals symmetrically: log2(1) = 0, log2(2) = 1, log2(1⁄2) = −1,log2(4) = 2, log2(1⁄4) = −2, and so on. The logarithms of theexpression ratios are also treated symmetrically, so that a geneupregulated by a factor of 2 has a log2(ratio) of 1, a gene down-regulated by a factor of 2 has a log2(ratio) of −1, and a geneexpressed at a constant level (with a ratio of 1) has a log2(ratio)equal to zero. For the remainder of this discussion, log2(ratio)will be used to represent expression levels.

NormalizationTypically, the first transformation applied to expression data,referred to as normalization, adjusts the individual hybridiza-

Ti =

Ri

Gi

.

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nature genetics supplement • volume 32 • december 2002 497

tion intensities to balance them appropriately so that mean-ingful biological comparisons can be made. There are a num-ber of reasons why data must be normalized, includingunequal quantities of starting RNA, differences in labeling ordetection efficiencies between the fluorescent dyes used, andsystematic biases in the measured expression levels. Conceptu-ally, normalization is similar to adjusting expression levelsmeasured by northern analysis or quantitative reverse tran-scription PCR (RT–PCR) relative to the expression of one ormore reference genes whose levels are assumed to be constantbetween samples.

There are many approaches to normalizing expression levels.Some, such as total intensity normalization, are based on simpleassumptions. Here, let us assume that we are starting with equalquantities of RNA for the two samples we are going to compare.Given that there are millions of individual RNA molecules ineach sample, we will assume that the average mass of each mole-cule is approximately the same, and that, consequently, thenumber of molecules in each sample is also the same. Second, letus assume that the arrayed elements represent a random sam-pling of the genes in the organism. This point is importantbecause we will also assume that the arrayed elements randomlyinterrogate the two RNA samples. If the arrayed genes areselected to represent only those we know will change, then wewill likely over- or under-sample the genes in one of the biologi-cal samples being compared. If the array contains a large enoughassortment of random genes, we do not expect to see such bias.This is because for a finite RNA sample, when representation ofone RNA species increases, representation of other species mustdecrease. Consequently, approximately the same number oflabeled molecules from each sample should hybridize to thearrays and, therefore, the total hybridization intensities summedover all elements in the arrays should be the same for each sam-ple.

Using this approach, a normalization factor is calculated bysumming the measured intensities in both channels

where Gi and Ri are the mea-sured intensities for the itharray element (for example, thegreen and red intensities in atwo-color microarray assay)and Narray is the total numberof elements represented in themicroarray. One or both inten-sities are appropriately scaled,for example,

so that the normalized expres-sion ratio for each elementbecomes

which adjusts each ratio such that the mean ratio is equal to 1.This process is equivalent to subtracting a constant from the log-arithm of the expression ratio,

which results in a mean log2(ratio) equal to zero.There are many variations on this type of normalization,

including scaling the individual intensities so that the mean ormedian intensities are the same within a single array or across allarrays, or using a selected subset of the arrayed genes rather thanthe entire collection.

Lowess normalizationIn addition to total intensity normalization described above,there are a number of alternative approaches to normalizingexpression ratios, including linear regression analysis1, log cen-tering, rank invariant methods2 and Chen’s ratio statistics3,among others. However, none of these approaches takes intoaccount systematic biases that may appear in the data. Severalreports have indicated that the log2(ratio) values can have a sys-tematic dependence on intensity4,5, which most commonlyappears as a deviation from zero for low-intensity spots. Locallyweighted linear regression (lowess)6 analysis has beenproposed4,5 as a normalization method that can remove suchintensity-dependent effects in the log2(ratio) values.

The easiest way to visualize intensity-dependent effects, andthe starting point for the lowess analysis described here, is to plotthe measured log2(Ri/Gi) for each element on the array as a func-tion of the log10(Ri*Gi) product intensities. This ‘R-I’ (for ratio-intensity) plot can reveal intensity-specific artifacts in thelog2(ratio) measurements (Fig. 1).

Lowess detects systematic deviations in the R-I plot and cor-rects them by carrying out a local weighted linear regression as afunction of the log10(intensity) and subtracting the calculatedbest-fit average log2(ratio) from the experimentally observedratio for each data point. Lowess uses a weight function that de-emphasies the contributions of data from array elements that arefar (on the R-I plot) from each point.

log2(Ti)=log2(Ti)–log2(Ntotal),′

Ti =

Ri

Ri

Gi

Gi

=1

Ntotal,

Gk = NtotalGk and Rk = Rk , ′ ′

Ntotal

=

Ri

Gi

Σ

Σ

Narray

Narrayi=1

i=1

,

R-I plot raw data

–4

–3

–2

–1

0

1

2

3

7 8 9 10 11 12 13

log10(R*G)

log

2(R

/G)

Fig. 1 An R-I plot displays thelog2(Ri/Gi) ratio for each element onthe array as a function of thelog10(Ri*Gi) product intensities andcan reveal systematic intensity-dependent effects in the measuredlog2(ratio) values. Data shown hereare for a 27,648-element spottedmouse cDNA array.

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498 nature genetics supplement • volume 32 • december 2002

If we set xi = log10(Ri*Gi) and yi = log2(Ri/Gi), lowess first esti-mates y(xk), the dependence of the log2(ratio) on the log10(inten-sity), and then uses this function, point by point, to correct themeasured log2(ratio) values so that

or equivalently,

As with the other normalization methods, we can make thisequation equivalent to a transformation on the intensities, where

The results of applying such a lowess correction can be seen inFig. 2.Global versus local normalization. Most normalization algo-rithms, including lowess, can be applied either globally (to theentire data set) or locally (to some physical subset of the data).For spotted arrays, local normalization is often applied to eachgroup of array elements deposited by a single spotting pen(sometimes referred to as a ‘pen group’ or ‘subgrid’). Local nor-malization has the advantage that it can help correct for system-atic spatial variation in the array, including inconsistenciesamong the spotting pens used to make the array, variability in theslide surface, and slight local differences in hybridization condi-tions across the array. When a particular normalization algo-rithm is applied locally, all the conditions and assumptions thatunderlie the validity of the approach must be satisfied. For exam-ple, the elements in any pen group should not be preferentiallyselected to represent differentially expressed genes, and a suffi-ciently large number of elements should be included in each pengroup for the approach to be valid.Variance regularization. Whereas normalization adjusts the meanof the log2(ratio) measurements, stochastic processes can cause thevariance of the measured log2(ratio) values to differ from oneregion of an array to another or between arrays. One approach todealing with this problem is to adjust the log2(ratio) measures sothat the variance is the same4,7. If we consider a single array withdistinct subgrids for which we have carried out local normaliza-tion, then what we are seeking is a factor for each subgrid that we

can use to scale all of the mea-surements within that subgrid.

An appropriate scaling factoris the variance for a particularsubgrid divided by the geomet-ric mean of the variances for allsubgrids. If we assume thateach subgrid has M elements,because we have alreadyadjusted the mean of thelog2(ratio) values in each sub-grid to be zero, their variancein the nth subgrid is

where the summation runs over all the elements in that subgrid.If the number of subgrids in the array is Ngrids, then the appro-priate scaling factor for the elements of the kth subgrid on thearray is

We then scale all of the elements within the kth subgrid by divid-ing by the same value ak computed for that subgrid,

This is equivalent to taking the akth root of the individualintensities in the kth subgrid,

It should be noted that other variance regularization factors havebeen suggested4 and that, obviously, a similar process can be usedto regularize variances between normalized arrays.

Intensity-based filtering of array elementsIf one examines several representative R-I plots, it becomes obvi-ous that the variability in the measured log2(ratio) valuesincreases as the measured hybridization intensity decreases. Thisis not surprising, as relative error increases at lower intensities,where the signal approaches background. A commonly usedapproach to address this problem is to use only array elementswith intensities that are statistically significantly different frombackground. If we measure the average local background neareach array element and its standard deviation, we would expectat 95.5% confidence that good elements would have intensitiesmore than two standard deviations above background. By keep-ing only array elements that are confidently above background,

we can increase the reliability of measurements. Other approachesinclude the use of absolute lower thresholds for acceptable array

Gspot >2 ∗ σ (Gbackground) and Rspot > 2 ∗ σ(Rbackground),i i i i

Gi = [Gi] and Ri = [Ri] .1⁄ak1⁄ak′ ′

log2(Ti)=log2(Ti)

ak

.

ak = σ2

k

σ2n

Ngrids

n=1

1Ngrids⁄Π

.

σ2n

=ΣM

j=1

[log2(T

j)]2,

Gi = Gi ∗ 2y(xi) and Ri = Ri .′ ′

log2(T

i)=log

2=log

2T

i

Ri

Gi

∗ ∗1

2y(xi)

1

2y(xi).

log2(Ti)=log2(Ti)–y(xi)=log2(Ti)–log2(2y(xi)),′

R-I plot following lowess

–3

–2

–1

0

1

2

3

7 8 9 10 11 12 13

log10(R*G)

log

2(R

/G)

Fig. 2 Application of local (pengroup) lowess can correct for bothsystematic variation as a function ofintensity and spatial variationbetween spotting pens on a DNAmicroarray. Here, the data from Fig. 1has been adjusted using a lowess inthe TIGR MIDAS software (with a tri-cube weight function and a 0.33smooth parameter) available fromhttp://www.tigr.org/software.

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nature genetics supplement • volume 32 • december 2002 499

elements (sometimes referredto as ‘floors’) or percentage-based cut-offs in which somefixed fraction of elements is discarded.

A different problem canoccur at the high end of theintensity spectrum, where thearray elements saturate thedetector used to measure fluo-rescence intensity. Once theintensity approaches its maxi-mum value (typically 216–1=65,535 per pixel for a 16-bit scan-ner), comparisons are no longer meaningful, as the arrayelements become saturated, and intensity measurements can-not go higher. Again, there are a variety of approaches to deal-ing with this problem as well, including eliminating saturatedpixels in the image-processing step or setting a maximumacceptable value (often referred to as a ‘ceiling’) for each arrayelement.

Replicate filteringReplication is essential for identifying and reducing the variationin any experimental assay, and microarrays are no exception.Biological replicates use RNA independently derived from dis-tinct biological sources and provide both a measure of the nat-ural biological variability in the system under study, as well asany random variation used in sample preparation. Technicalreplicates provide information on the natural and systematicvariability that occurs in performing the assay. Technical repli-cates include replicated elements within a single array, multipleindependent elements for a particular gene within an array (suchas independent cDNAs or oligos for a particular gene), or repli-cated hybridizations for a particular sample. The particularapproach used will depend on the experimental design and theparticular study underway (see also the review by G. Churchill,pages 490–495, this issue)8.

To illustrate the usefulness of technical replicates, considertheir use in identifying and eliminating low-quality or question-able array elements. One widely used technical replication intwo-color spotted array analysis is dye-reversal or flip-dye analy-sis8, which consists of duplicating labeling and hybridization byswapping the fluorescent dyes used for each RNA sample. Thisprocess may help to compensate for any biases that may occurduring labeling or hybridization; for example, if some genes pref-erentially label with the red or green dye. Let us assume we havetwo samples, A and B. In the first hybridization, we label A withour red dye and B with our green dye and reverse the dye labelingin the second, so that the ratios for our measurements can bedefined, respectively, as

As we are making two comparisons between identical samples,we expect

or equivalently,

However, we know that experimental variation will lead to a dis-tribution of the measured values for the log of the product ratios, log2(T1i*T2i). For this distribution, we can calculate the mean andstandard deviation. One would expect the consistent array ele-ments to have a value for log2(T1i*T2i) ‘close’ to zero and inconsis-tent measurements to have a value ‘far’ from zero. Depending onhow stringent we want to be, we can choose to keep and use arrayelement data for which log2(T1i*T2i) is within a certain number ofstandard deviations of the mean. Although one cannot determine a priori which of the replicates is likely to be in error, visual inspec-tion may allow the ‘bad’ element to be identified and removedbefore further analysis. Alternatively, and more practically for largeexperiments, one can simply eliminate questionable elements fromfurther consideration. The application of such a replicate trim isshown in Fig. 3. Obviously, similar approaches can be used to filterdata from other replication strategies, including replicates within asingle array or measurements from replicated single-color arrays.

Averaging over replicatesTo reduce the complexity of the data set, we may average over thereplicate measures. If, as before, we consider replicate measuresof two samples, A and B, we would want to adjust the log2(ratio)measures for each gene so that the transformed values are equal,or, for the ith array element,

This equation can be easily solved to yield a value for the con-stant ci that is used to correct each array element,

log2 A2i

A1iB2i

B1i = log2

A2i

A1iB2i

B1i .ci =12

+ ci = log2 A1i A2i

B1iB2i

log2 – ci .

log2(T1i ∗ T2i) = log2 A1i

A2iB1i

B2i = 0.∗

(T1i ∗ T2i) = A1i

A2iB1i

B2i = 1,∗

T1i =R1i

G1i

A1i

B1i

R2i

G2i A2i

B2iand T2i

= = = .

Replicate comparison and trim

–6

–4

–2

0

2

4

6

–6 –4 –2 0 2 4 6

log2(A/B)1

log

2(A

/B) 2

Fig. 3 The use of replicates can helpeliminate questionable or inconsis-tent data from further analysis. Here,the lowess-adjusted log2(Ai/Bi) valuesfor two independent replicates areplotted against each other elementby element for hybridizations to a32,448-element human array. Outliersin the original data (in red) areexcluded from the remainder of thedata (blue) selected on the basis of atwo-standard-deviation cut on thereplicates.

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500 nature genetics supplement • volume 32 • december 2002

If we use this equation to aver-age our measurements, theresult is equivalent to takingthe geometric mean, or

where the average measure-ments for expression in eachsample is given by

The adjusted average measures

and for each gene can then be used to carry out further analyses.For example, one can create an R-I plot, with

plotted as a function of the

product intensities for each arrayed element, or for any otherapplication. This procedure can obviously be extended to averag-ing over n replicates by taking the nth root.

Propagation of errorsOne advantage of having replicates is that they can be used toestimate uncertainty in derived quantities based on the measuredor estimated uncertainties in the measured quantities. The the-ory of error propagation9 tells us that, in general, if we have afunction f(A,B,C,…), with error estimates σ for each of the inputvariables, A,B,C,…, then the squared error in f is

where

is the partial derivative of f with respect to A. For microarrays, thefunction with which we are most often concerned is the log2(ratio),

Here, we can take advantage of the fact that logarithms caneasily be changed between bases and that we know the derivativeof the natural logarithm,

to derive the following equation

It then follows that the squared-error in our average log2(ratio) is

so that the errors in the expression measures for A and B can beused to estimate the error in the log2(ratio).

Identifying differentially expressed genesRegardless of the experiment performed, one outcome that isinvariably of interest is the identification of genes that are differ-entially expressed between one or more pairs of samples in thedata set. Even if data-mining analysis is going to be done using,for example, one or more of the widely used clustering meth-ods10–12, it is still extremely useful to reduce the data set to thosegenes that are most variable between samples. In many earlymicroarray analyses, a fixed fold-change cut-off (generally two-fold) was used to identify the genes exhibiting the most signifi-cant variation. A slightly more sophisticated approach involvescalculating the mean and standard deviation of the distributionof log2(ratio) values and defining a global fold-change differenceand confidence; this is essentially equivalent to using a Z-scorefor the data set. In an R-I plot, such criteria would be representedas parallel horizontal lines; genes outside of those lines would becalled differentially expressed.

Analysis of the R-I plot suggests, however, that this approachmay not accurately reflect the inherent structure in the data. Atlow intensities, where the data are much more variable, onemight misidentify genes as being differentially expressed, while athigher intensities, genes that are significantly expressed mightnot be identified. An alternative approach would be to use thelocal structure of the data set to define differential expression.Using a sliding window, one can calculate the mean and standarddeviation within a window surrounding each data point, anddefine an intensity-dependent Z-score threshold to identify dif-ferential expression5, where Z simply measures the number ofstandard deviations a particular data point is from the mean. If

σlocallog

2(T)

i

σ2 = σ2f A + σ2

B + σ2B= σ2

A∂f∂A

2 ∂f∂B

2 2

A ln(2)1

2

B ln(2)1

[log2(x)] = ln(x)ln(2)

lln(2)

lln(2)

lx ln(2)

∂∂x

∂∂x

∂∂x

1x

[ln(x)]== = .

log2(x) = ln(x)ln(2)

and∂∂x

1x

[ln(x)]= ,

= log2 (Ai)–log2(Bi). A1

B1

f = log2

∂f

∂A

σ2f σ2

A + σ2B=

∂f

∂A

2 ∂f

∂B

2

+... ,

log10(Ai ∗ Bi)− −

log2(Ai / Bi)− −

Ai and Bi , − −

AiA1i A2i B1i B2i .Bi

− −= =and

log2 Ai

Bi

= log2

A2i

B1iB2i

A1i ,−−

Intensity-dependent Z -scores for identifying differential expression

–3

–2

–1

0

1

2

3

7 8 9 10 11 12 13

log10(R*G)

log

2(R

/G)

Z<11<Z<2Z>2

Fig. 4 Local variation as a function ofintensity can be used to identify dif-ferentially expressed genes by calcu-lating an intensity-dependentZ-score. In this R-I plot, array elementsare color-coded depending onwhether they are less than one stan-dard deviation from the mean (blue),between one and two standard devi-ations (red), or more than two stan-dard deviations from the mean(green).

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nature genetics supplement • volume 32 • december 2002 501

is the calculated standard deviation in a region of the R-I plotsurrounding the log2(ratio) for a particular array element i, then

With this definition, differentially expressed genes at the 95%confidence level would be those with a value of

or, equivalently, those more than 1.96 standard deviations fromthe local mean. At higher intensities, this allows smaller, yet stillsignificant, changes to be identified, while applying more strin-gent criteria at intensities where the data are naturally more vari-able. An example of the application of a Z-score selection isshown in Fig. 4.

Having identified differentially expressed genes for each pairof hybridized samples, one can then further filter the entireexperimental data set to select for further analysis only thosegenes that are differentially expressed in a subset of the experi-mental samples, such as disease or normal tissues, or using anyother criteria that make sense for the experimental design used.Another alternative would be to use the ANOVA techniquessummarized by Churchill on pages 490–495 of this supplement8

to select significantly differentially expressed genes. In eithercase, the resulting reduced data set can then be used for furtherdata mining and analysis.

Looking aheadThere is a wide range of additional transformations that can beapplied to expression data, and we have presented only a smallsample of the available techniques that can be used. For example,there are specific transformations that have been developed foranalysis of data from particular platforms13,14, and sophisticatedmethods have been proposed for development of error modelsbased on analysis of repeated hybridizations14,15. Regardless ofthe sophistication of the analysis, nothing can compensate forpoor-quality data. The single most important data-analysis tech-nique is the collection of the highest-quality data possible. Thestarting point for effective data collection and analysis is a goodexperimental design with sufficient replication to ensure thatboth the experimental and biological variation can be identifiedand estimated. A second important element in generatingexpression data is optimization and standardization of the exper-imental protocol. Like northern blots and RT–PCR, microarraysdirectly assay relative RNA levels and use these to infer geneexpression. Therefore, at every step in the process, from samplecollection through RNA isolation, array preparation, samplelabeling, hybridization, data collection and analysis, every possi-ble effort must be made to minimize variation.

Defining objective criteria for the quality of a DNA microarrayassay remains an open problem, but one that clearly needs to beaddressed as microarray assays become more widespread. Onecould easily write another review article on elements that con-tribute to good-quality arrays and which could be used, in part,to define such a quality standard. The challenge is to use these

often qualitative objectives to define one or more appropriatequantitative measure. The Normalization Working Group of theMicroarray Gene Expression Data (MGED) organization(http://www.mged.org) is attempting to define such a standard,and community input and participation is welcome (see com-mentary by C. Stoeckert, pages 469–473, this issue)16.

Finally, as standards for publication of microarray experi-ments evolve, reporting data transformations is becoming asimportant as disclosing laboratory methods. Without an accu-rate description of either of these, it would be difficult, if not impossible, for the results derived from any study to bereplicated.

AcknowledgmentsThe work presented here evolved from looking at a large body of data andwould have been much less useful without the contributions of NormanH. Lee, Renae L. Malek, Priti Hegde, Ivana Yang, Shuibang Wang, YonghongWang, Simon Kwong, Heenam Kim, Wei Liang, Vasily Sharov, John Braisted,Alex Saeed, Joseph White, Jerry Li, Renee Gaspard, Erik Snesrud, Yan Yu,Emily Chen, Jeremy Hasseman, Bryan Frank, Lara Linford, Linda Moy, TaraVantoai, Gary Churchill and Roger Bumgarner. J.Q. is supported by grantsfrom the US National Science Foundation, the National Heart, Lung, andBlood Institute, and the National Cancer Institute. The MIDAS softwaresystem used for the normalization and data filtering presented here is freelyavailable as either executable or source code from http://www.tigr.org/software, along with the MADAM data-management system, the Spotfinderimage-processing software, and the MeV clustering and data-mining tool.

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