a model-based analysis of microarray experimental error and

A model-based analysis of microarray experimentalerror and normalisationYongxiang Fang1,2, Andrew Brass1,2, David C. Hoyle1, Andrew Hayes1,2,

Abdulla Bashein1, Stephen G. Oliver1, David Waddington3 and Magnus Rattray2,*

1School of Biological Sciences, University of Manchester, Manchester M13 9PT, UK, 2Department of ComputerScience, University of Manchester, Manchester M13 9PL, UK and 3Roslin Institute, Midlothian EH25 9PS, UK

Received April 4, 2003; Revised and Accepted June 25, 2003

ABSTRACT

A statistical model is proposed for the analysis oferrors in microarray experiments and is employed inthe analysis and development of a combinednormalisation regime. Through analysis of themodel and two-dye microarray data sets, this studyfound the following. The systematic error intro-duced by microarray experiments mainly involvesspot intensity-dependent, feature-speci®c and spotposition-dependent contributions. It is dif®cult toremove all these errors effectively without a suitablecombined normalisation operation. Adaptive nor-malisation using a suitable regression technique ismore effective in removing spot intensity-relateddye bias than self-normalisation, while regionalnormalisation (block normalisation) is an effectiveway to correct spot position-dependent errors.However, dye-¯ip replicates are necessary toremove feature-speci®c errors, and also allow theanalyst to identify the experimentally introduceddye bias contained in non-self-self data sets. In thiscase, the bias present in the data sets may includeboth experimentally introduced dye bias andthe biological difference between two samples.Self-normalisation is capable of removing dye biaswithout identifying the nature of that bias. The per-formance of adaptive normalisation, on the otherhand, depends on its ability to correctly identify thedye bias. If adaptive normalisation is combined withan effective dye bias identi®cation method thenthere is no systematic difference between the out-comes of the two methods.

INTRODUCTION

Microarrays are a technology for measuring the expressionlevels of genes simultaneously, allowing the completetranscriptome of a cell, tissue or organ to be de®ned.However, microarray experiments generate data that containerrors, which arise from various sources, such that the noisemay signi®cantly distort the real signal. Experimental error

can be classi®ed into two categories: systematic error andrandom error. The former re¯ects the accuracy of measure-ments, while the latter re¯ects the precision of measurements.Obtaining data with satisfactory precision and accuracy hasbeen one of the biggest challenges in the application ofmicroarray technology.

In a typical spotted slide microarray experiment, twomRNA samples are compared by reverse transcribing theminto cDNA, labelling using red and green ¯uorescent dyes,respectively, and then hybridising these labelled targetssimultaneously to denatured PCR product or cDNA probesspotted on a glass slide. The relative level of gene expressionin the two samples is then measured by determining the ratioof ¯uorescence intensity of the two dyes. This technique isvery effective in overcoming the weak point of microarrayexperiments, which is that their reproducibility in measure-ment of absolute (as opposed to relative) expression level ispoor. However, associating two samples with two different¯uorescent dyes introduces dye bias into the measurements.Dye bias is a systematic error that should be removed beforeany further analysis of microarray data is performed. Toremove systematic errors from microarray data one canemploy a normalisation method. A number of normalisationapproaches have been introduced for microarray data analysis,which include the housekeeping gene approach (1), total RNAapproach (2), global normalisation (3), ANOVA (4), thecentralisation method (5), self-normalisation (6) and adaptivenormalisation involving regression techniques such as LOESS(3).

The housekeeping gene approach builds on the assumptionthat there are sets of genes in any genome that areconstitutively expressed at a relatively constant level underany set of conditions. The total RNA approach is based on theassumption that each cell carries the same amount of totalRNA at different times. Unfortunately, there is substantialevidence that these assumptions are incorrect in many cases(7,8). Global normalisation is based on two assumptions.Firstly, that the centre (i.e. the mean or median) of thedistribution of gene expression ratios on a log scale is zero.Secondly, that the systematic error makes the central linemove vertically, but otherwise leaves the distribution invari-ant. Therefore, a global normalisation shifts the centre of thelog ratio distribution to zero. However, systematic error inmicroarray data is often not constant (3,9). In this study, we

*To whom correspondence should be addressed. Tel: +44 161 275 6187; Fax: +44 161 275 6204; Email: [email protected]

Nucleic Acids Research, 2003, Vol. 31, No. 16 e96DOI: 10.1093/nar/gng097

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will show that the log ratio distribution is not always centred tozero. Therefore, the performance of global normalisation isnot satisfactory. Kerr et al. (4) proposed an ANOVA approachfor microarray data analysis, but this is also global in nature.The dye bias contained in microarray data usually not onlyvaries from spot to spot over a slide, but also from replicate toreplicate for a given spot. Therefore, the interaction betweenslide and spot should be considered, but this cannot beachieved using the ANOVA approach. The centralisationmethod of Zien et al. (5) was shown to provide improvedresults; however, they discarded the data points whoseintensity was below 10 or above 1000 in their study. Thisoperation excluded 487 out of 1189 genes in their data sets. Inaddition, the number of replicates used was very large and sotheir approach may be dif®cult to apply in practice.

The self-normalisation method (6) assumes that experi-mentally introduced error is multiplicative and that, forcorresponding spots in replicated measurements, it is consist-ent. Based on this, the error on a log scale is additive and asubtract operation applied to the data sets from two replicateswill remove this systematic error. However, this approachrequires the association of a dye-¯ip technique, as otherwisethe biological difference between two samples being meas-ured will also be removed by the operation. The dye-¯ip (alsoknown as dye swap or reverse labelling) technique generatespaired slides where, on the ®rst slide, one mRNA sample islabelled by Cy5 and the other mRNA sample is labelled byCy3, while, on the second slide, the labels for the two samplesare exchanged. Based on self-normalisation, the normalisedresult (logged ratio of expression) for a measured spot is halfthe difference between logged ratios measured from a pair ofdye-¯ipped replicates for this spot (6). Therefore, self-normalisation has the property that it corrects feature-speci®c(i.e. probe- or spot-speci®c) differences so that, for a featuremeasuring the expression level of a given gene, the normalisedresult only measures the relative abundance of the gene itselfand is not in¯uenced by the measurement of any otherfeatures. Compared to global normalisation, in which thenormalised result for a given gene spot will depend on themeasurements of the whole set of genes, the self-normalisa-tion approach goes to the other extreme and ignores all othermeasurements. Self-normalisation typically performs muchbetter than global normalisation when dye-¯ip replicates areavailable (6).

Adaptive normalisation is an approach that falls somewherebetween global normalisation and self-normalisation. Theapproach employs the assumption that the bias introduced bythe experiment is dependent on a number of factors (spotintensity, print tips, spot position, etc.) and employs regressiontechniques to obtain a ®t of the speci®c relationship, and thenmakes the correction. Because it takes systematic error asneither a constant nor spot-speci®c, the method has theadvantages of both the global normalisation and the self-normalisation approaches, but without their disadvantages.Adaptive normalisation may perform differently for thedifferent regression techniques employed. Generally, theregression can be either global or local. For global regression,one can employ either a linear or a non-linear function for theregression. For local regression, the LOESS (LOWESS)regime (10) is currently the most popular (3), although somebasic knowledge of the method is required for the analyst to

choose appropriate values for the parameters involved in themethod.

Both self-normalisation and adaptive normalisation havebeen shown to provide advantages over global normalisation.Furthermore, adaptive normalisation may have advantagesover self-normalisation. Firstly, self-normalisation can only beapplied to dye-¯ip paired microarray slides. It cannot beapplied to single slides, in contrast to the adaptive approach.Secondly, for spots whose systematic errors are not consistenton a slide pair, adaptive normalisation may produce betterresults because the correction of any gene spot is dependent onthe bias of all gene spots that have the same value of spotintensity (i.e. the same value of the dependent variable used inthe regression). However, self-normalisation is much easier toapply since it can be used without identifying the nature of theexperiment-introduced bias or the genuine difference betweentwo samples. In contrast, one must know, or estimate, at leastone of these two components in order to apply an adaptivenormalisation. Furthermore, for non-self-self slides, both ofthese components are unknown.

Given the above complexity, researchers often ask whichcurrently used normalisation method performs better. Does thedye-¯ip technique play an indispensable role in obtainingbetter data quality? How does one improve the data qualitythrough more effective normalisation operations? This studytries to answer these questions by a comparison of data qualitybefore and after normalisation. This has permitted an inves-tigation of which type of replication (dye-¯ip or non-dye-¯ip)will lead to better data quality and what normalisationtechniques can generate data with higher accuracy. In orderto compare the data quality obtained by different experimentaldesigns and different normalisation regimes under a givennumber of replicates, we propose a method for assessing theaccuracy and precision of normalised data. The formerdepends on our ability to remove the systematic errorcontained in microarray data. For self-self hybridisations,the data points should centre to the zero line on an M±A plot(6) and this can be used for the assessment of the accuracy ofnormalised data. In general, for non-self-self data, it is notknown to what line the data spots should centre. However,self-normalisation has the intrinsic capacity to remove dyebias without detection of that bias. Hence, accuracy can beassessed by the systematic difference between normalised datafrom self-normalisation and from other normalisation tech-niques. The precision of normalised data can be assessed bythe data consistency, which is represented by the difference ofnormalised data from replicate experiments, since this differ-ence is only related to experimental noise and the performanceof the normalisation regime employed. Therefore, comparisonof this difference between a range of normalisation techniquesapplied to the same data sets can answer the question of whichnormalisation approach works better. Similarly, comparisonof data consistency corresponding to different experimentaldesigns shows which experimental design generates data ofhigher quality.

In this study, a statistical model for microarray experimentsis proposed and used to investigate how systematic error isremoved by different normalisation operations under certainsimplifying assumptions. Deductions derived from the modelare tested on real data from a set of microarray experiments.From analysis of the model and experiments, we make a

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number of useful conclusions. Firstly, we clarify the role ofdye-¯ip replicates for improving microarray data quality.Secondly, we introduce a high performance normalisationapproach that involves up to three stages in the regime. Detailsof the normalisation protocols and derivations of the resultsare contained in the Appendix.

ANALYTICAL METHODS

A statistical model is employed to investigate the main sourcesof systematic error and the effects of different normalisationapproaches on these terms. In the following discussion, weconsider log transformed data. Unless otherwise speci®ed, theratio (M) and spot intensity (A) are actually the ratio and meanspot intensity on a log scale, i.e. M = logR/G and A = log

��RGp

,where R and G are measurements from the red and greenchannels, respectively, and we use base 2 logarithms.

A statistical model of experimental error

Let n represent the number of replicates of an experiment, grepresent the number of the features (probes) on the slide, mj

( j = 1, 2, ¼, g) represent the true ratio of expression levels forthe gene measured by feature j, and Mjk ( j = 1, 2, ¼, g; k = 1,2, ¼, n) represent the measured ratio of expression levels forfeature j on replicate k. The measurement Mjk will be modelledas:

Mjk = mj + c + ck + e(Fj) + ek(Ajk) + ek(Pj) + ejk 1

withXg

j�1

e�Fj� � 0;Xg

j�1

ek�Ajk� � 0 "k;Xg

j�1

ek�Pj� � 0 "k;

where c represents the expected global measurement biasbetween two channels, ck represents the variation of globalmeasurement bias shown on replicate k, e(Fj) representsfeature-speci®c bias for feature j, ek(Ajk) represents spotintensity-dependent bias for feature j on replicate k, ek(Pj)represents spot location-related bias for feature j on replicate kand ejk represents zero mean random error introduced tofeature j on replicate k.

The global measurement bias c is a constant for the givendata sets of n replicates and ck is a constant which representsthe difference between the global bias of the kth replicate andmean global bias of n replicates. The function e(Fj) dependson the feature only, but the functions of spot intensity ek(Ajk)and the spot position ek(Pj) depend on both the feature andreplicate indices. The global measurement bias has beenincluded in c and ck and we therefore take e(Fj), ek(Ajk) andek(Pj) to be centred to zero. We make simplifying assumptionsthat the three systematic error terms are independent. This ismost likely sensible because their dependent factors areexpected to be unrelated or only very weakly related. The spotintensity is mainly dependent on the expression of genes. Thespot position-speci®c term is mainly related to the heterogen-eity of the experiment over a slide surface and the feature-speci®c term mainly re¯ects the different performance ofdifferent print tips and related factors. We also assume that thefunction of intensity and function of spot position are

continuous functions of their independent variables, but thatthe function of feature changes abruptly from feature tofeature.

The proposed model contains only multiplicative items andis therefore additive on a log scale. Such a multiplicativemodel appears to work very well on microarray data withoutbackground correction. The use of background correctionintroduces an additive error item into the data that can besigni®cant for low expression levels and may make this modelunsuitable. Data that contain a mixture of additive andmultiplicative terms can be modelled using, for example, themodel introduced by Rocke and Durbin (11). For themicroarray data studied here we believe it to be most likelythat naõÈve background subtraction has more negative effectsthan positive effects on microarray data quality and it istherefore not used here. This point is illustrated by examiningsome of our normalised self-self data (data measured fromself-self hybridisation chips). For self-self data, the genuinelog ratio of each data point should be zero and normalised self-self data re¯ects the data quality of the correspondingmicroarray experiment. A comparison of the outcomes fromapplying the normalisation to background-corrected self-selfdata and to the same data without background correction willshow the impact of background correction clearly. We havedone a few of these kinds of comparisons based on self-selfdata from different microarray experiments and all of themshow that background correction decreases the data quality.The plots presented in Figure 1 show normalised yeast datafrom self-self hybridisation microarray slides (for protocols,see 12,13). After normalisation, the background-correcteddata is much noisier than the data without backgroundcorrection.

Normalisation approaches and procedures

Our model contains three systematic error items which arefactor-speci®c [ek(Ajk), ek(Pj) and e(Fj)]. We employed anapproach that involves a number of normalisation stages inorder to remove these systematic effects wherever possible.We outline our normalisation approach below and providefurther details in the Appendix.

If the data comes from an experiment with dye-¯ipreplicates, then normalisation can be conducted in two orthree steps. The ®rst step is to correct the intensity-dependentdye bias. This task can be carried out by adaptive normalisa-tion or self-normalisation. Self-normalisation simply involvessubtracting the results of each pair of dye-¯ipped replicates.Adaptive normalisation involves the ®tting of a regressionfunction to the M±A plot for each slide and correctingaccordingly. In our experiments, we used a generalisation of astandard polynomial ®t that includes fractional powers(regression function M = b1 + å4

i = 1bi + 1A1/(i + 1) + bi + 5Ai

+ e). In some cases, it will be advantageous to identify the dyebias using a pooled set of dye-swap replicates. Theparameterised regression function from the pooled data setis then applied to each data set in the pool so as to correct itsdye bias. We call this technique `improved adaptive normal-isation' (iAN) (see Appendix for details).

The second step is to remove the spot location-speci®c errorusing regional normalisation. A two-dimensional regressiontechnique can be used to identify and correct the error in thiscase. The regression can be applied to each block of a replicate

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separately or can be applied to an entire replicate. Block-basedregression is used in this study and is recommended for thesituation where different blocks are printed by different printtips or are printed at different times. Figure 2 provides anexample of the outcome of applying a two-dimensionalquadratic polynomial regression [regression function f(x,y) =a1 + a2x + a3y + a4x2 + a5xy + a6y2 + e, where x, y are spotcoordinates in a block on a microarray slide] to each block of ayeast microarray data set (12,13). The plot shows that thedifference in print tips and printing time may cause an abruptchange in spot position-dependent error (see border area ofadjacent blocks). However, if a border of two adjacent blocksis in an area where the position-dependent error is dominatedby the heterogeneity of the experiment over a slide surface,then the impact of difference in print tips and printing timemay not be seen clearly. If a two-dimensional ®t is applied toan entire replicate, then the ®tted spot position-dependenterror will change smoothly across any borders of two adjacentblocks, and this is usually not the case.

If we use self-normalisation as the ®rst step, then thenormalisation process only requires the two steps describedabove. However, if we use adaptive normalisation then a thirdnormalisation step is required in order to remove anyremaining feature-speci®c bias. This can be done usingfeature normalisation, which is essentially identical to self-normalisation. We simply subtract the values (after adaptiveand regional normalisation) of each dye-¯ipped pair. The onlydifference between this and self-normalisation is that we carryout feature normalisation after the other normalisationmethods have been applied.

If dye-¯ip replicates are not available, then self-normalisa-tion and feature normalisation are not possible. In this case, wecan only carry out adaptive and regional normalisation. If the

intensity-dependent bias contains genuine biological signal,then there is no effective way to identify these two itemsseparately and it is dif®cult to achieve satisfactory normal-isation results, as we will demonstrate in the forthcomingsections.

To summarise, we will consider several normalisationmethods involving combinations of the following normalisa-tion steps: AN, adaptive normalisation; iAN, improvedadaptive normalisation; SN, self-normalisation; RN, regionalnormalisation; FN, feature normalisation.

Combinations of these abbreviations will be used to denotedifferent normalisation regimes. For data sets with dye-swap replicates we will use SN+RN, AN+RN, iAN+RN,AN+RN+FN and iAN+RN+FN, as outlined above. For datasets without dye swap replicates we will use AN+RN.

ANALYTICAL DEDUCTIONS

The analytical deductions given in this section are based onthe statistical model previously described (1). The analysis isapplied to four different situations: (i) data from self-selfreplicates (where they are treated as replicates without dye-¯ip); (ii) data from self-self replicates (where they are treatedas replicates with dye-¯ip); (iii) data from non-self-selfreplicates without dye-¯ip; (iv) data from non-self-selfreplicates with dye-¯ip. We should emphasise that treatingself-self replicates as dye-¯ipped replicates or as non-dye-¯ipped replicates is only a technique of data analysis and not areal experimental technique, because in a self-self microarrayexperiment the dye-¯ip is meaningless.

In the following analysis, we make the simplifyingassumption that the dependent terms can be identi®ed byregression without error. This is a fairly gross assumption, but

Figure 1. Plots of the estimated log ratio (base 2) after normalisation of four self-self replicates from a yeast microarray experiment. Red spots show resultsfrom data sets with background correction and green spots show results from data sets without background correction.


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we believe that the main differences between normalisationapproaches can be more easily illustrated in this way.Identifying error introduced by the particular regressiontechnique will be more dependent on details of the methodsused. We are assuming that this source of error is lesssigni®cant than the terms identi®ed below. In the remainder ofthis section, only the results and the conclusions based onthose results are provided, while details of the analyses arecontained in the Appendix.

Self-self replicates without dye-¯ip

For this situation, the model is:

Mjk = c + ck + e(Fj) + ek(Ajk) + ek(Pj) + ejk. 2

The genuine log ratio for each spot on each replicate is zero, sothat we have set mj = 0 in equation 1. The normalisationprocess for these data involves two steps: adaptive correctionof dye bias and regional normalisation (AN+RN). Let mj

represent the estimated log ratio for feature j after normalisa-tion, which is given by equation 3 below

mj � e�Fj� � ej where ej � 1

n

Xn

k�1

ejk: 3

Self-self replicates with dye-¯ip

For this situation, we can take the n replicated measurementsas s pairs (n = 2s). The statistical model can then be shownto be:

Mjk = c + ck + e(Fj) + ek(Ajk) + ek(Pj) + ejk

Mjk + s = c + ck + s + e(Fj) + ek + s(Ajk + s) + ek + s(Pj) + ejk + s

for k = 1, 2, ¼, s. 4

Normalisation of this data set can be carried out by twodifferent approaches. The ®rst approach includes the adaptivecorrection of dye bias, regional normalisation and thecorrection of feature-speci®c error (AN+RN+FN). The secondapproach involves self-normalisation and regional normalisa-tion (SN+RN). After AN+RN+FN the estimated log ratio mj

can be shown to be:

mj � eej � 1

n

Xs

k�1

ejk ÿXn

k�s�1

ejk

!; 5

where the random error term eej has equal variance to e and hasan identical distribution if the error is symmetrically distrib-uted. After SN+RN the estimated log ratio mj can be shown tobe:

mj � 1

n

Xs

k�1

�ek�Ajk� ÿ ek�s�Ajk�s�� eej: 6

Non-self-self replicates without dye-¯ip

For this situation the genuine log ratio mj may contain a spotintensity-dependent bias that will be removed by intensity-dependent regression. Therefore, we split mj into two parts: a

Figure 2. We show the spot position-dependent error contribution from one of the yeast microarray chips used in this study. The data set has 6400 (80 3 80)data points grouped as 16 (4 3 4) blocks. Graduated colours are used to show the spatially averaged log ratio (base 2) obtained by ®tting a 2D quadraticregression function to the log ratios from each block. Calibration of the colour scheme is shown by the bar on the right in log ratio units.


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spot intensity-independent term, mj*, and a spot intensity-dependent component, m(Aj), where Aj can be thought of asthe expectation value of the spot intensity for gene j over manyreplicates. We substitute mj = mj* + m(Aj) into equation 1 andget the appropriate statistical model below:

Mjk = mj* + m(Aj) + c + ck + e(Fj) + ek(Ajk) + ek(Pj) + ejk. 7

Self-normalisation is unusable since no dye-¯ipped measure-ment is available. Furthermore, when the dye bias is correctedby employing regression to the M±A plot, the ®t will containm(Aj) and the dye bias cannot be singled out from the ®t. Themost common current practice is to assume that m(Aj) isnegligible, take the whole ®t as dye bias and remove it fromthe data. In this case, we apply adaptive and regionalnormalisation (AN+RN), and mj can be shown to be:

mj = mj ± m(Aj) + e(Fj) + ej. 8

Here, we have made a simplifying approximation that m(Aj)would be completely removed by regression against Ajk. Themain point of equation 8 is that the expression ratio estimatewill be biased if the true expression ratio includes an intensity-dependent contribution, and this will hold in general.

Non-self-self replicates with dye-¯ip

In this situation, we can take the n replicated measurements ass pairs (n = 2s) and the statistical model is:

Mjk = mj* + m(Aj) + c + ck + e(Fj) + ek(Ajk) + ek(Pj) + ejk

Mjk + s = mj* ± m(Aj) + c + ck + s + e(Fj) + ek + s(Ajk + s)+ ek + s(Pj) + ejk + s, 9

where k = 1, 2, ¼, s. In this case, we consider threenormalisation approaches. After AN+RN+FN, the estimatedlog ratio mÅ j can be shown to be:

mj = mj ± m(Aj) + eej, 10

after iAN+RN+FN, mÅ j can be shown to be:

mj = mj + eej, 11

and after SN+RN, mÅ j can be shown to be:

mj � mj � 1

n

Xs

k�1

�ek�Ajk� ÿ ek�s�Ajk�s�� eej : 12

Conclusions based on analytical deductions

From the analytical deductions given by equations 3, 5, 6, 8and 10±12, we make the following conclusions. (i) Equations3 and 8 show that the normalised results from non-dye-¯ipreplicates contain the feature-speci®c error term, whilenormalised results from dye-¯ip replicates (equations 5, 6,11 and 12) do not. Therefore, dye-¯ip replication is aneffective way of removing feature-speci®c bias introduced bythe experiment and we can make use of it to achieve betterdata quality. (ii) Compared to the normalised results fromapproaches using adaptive normalisation (equations 5 and11), the results from approaches using self-normalization

(equations 6 and 12) contain a term which arises fromdifferences in spot intensity and dye response betweenreplicates. Hence adaptive normalisation is usually moreeffective in removing dye bias than self-normalisation. (iii) Inthe right-hand side of equations 8 and 10 there is a ±m(Aj) termwhich re¯ects the intensity-dependent biological bias. Thismeans that the dye bias will be overcorrected by a normal-isation approach that includes adaptive normalisation if thereis an intensity-dependent biological bias between twosamples. In this case, the traditional adaptive normalisationtechnique is not suitable and iAN is required. (iv) From thenormalised results (equations 10±12), we can see clearly thatthere will be an intensity-dependent systematic difference,m(Aj), between the results from the AN methods and theresults from the SN method. However, there is no such biasbetween the normalised results from the iAN method and theresults from the SN method. This provides a way to identifythe intensity-dependent biological difference between twomRNA samples.

EXPERIMENTAL METHODS

To test conclusions 1 and 2 (above), we will compare theoutcome of normalising self-self data sets using the differentmethods introduced. It is worth remembering that we can takeself-self replicates as replicates with or without dye-¯ip. Fortesting conclusions 3 and 4, we will use data from referencetreatment replicates with and without dye-¯ip. In order todetermine that the biological bias can be identi®ed andretained during normalisation, we compare the outcomes fromtraditional adaptive normalisation methods and from self-normalisation. Our model predicts that the systematic differ-ence between the outcomes of these two approaches is justequal to the spot intensity-dependent biological bias betweenthe two samples. The reason is that self-normalisationremoves the dye bias but not the biological bias, whiletraditional adaptive normalisation removes both the dye biasand the biological bias. The difference between them is thebiological bias between two measured samples. Finally, wecompare the outcomes from the improved adaptive method ofdye bias correction with self-normalisation so as to demon-strate which method produces results with higher accuracy.

Data and analysis

Data from a microarray study on yeast was employed for ouranalysis (12,13). The microarray study included four micro-array slides and on each slide there were two sections, whichwere actually two replicates. In each replicate, there are 16blocks with 400 spots per block. Among 6400 spots in areplicate, 6277 are spotted with gene products and theremaining 123 are empty spots. Of the four slides, two areself-self hybridisation slides, which provide four replicatedmeasurements. The remaining two are hybridisation slides ofreference versus heat shock samples. Each of the two slidesprovides two replicates without dye-¯ip and, at the same time,the two slides constitute two dye-¯ipped replicates. Thoughthe number of slides is small, they contain data from self-selfslides, reference heat shock slides, replicates in dye-¯ip formand replicates in non-dye-¯ip form. This means that allexperimental combinations required for our analysis areavailable from this set of slides.


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So far, we have assumed that a slide contains only one set ofmeasurements. However, the microarray experiments used forthis test have two duplicate sets of measurements on each slideand, effectively, we treat these sets as separate slides.Therefore, we have eight sets of measurements from theseslides. We denote the eight data sets as SS1A, SS1B, SS2A,SS2B, RT1A, RT1B, RT2A and RT2B, where SS stands forself-self slides, RT stands for reference treatment slides,1 stands for the ®rst of two replicated slides and 2 stands forthe second of two replicated slides, A stands for the ®rstsection on one slide and B stands for the second section on oneslide.

Analysis of SS1 and SS2

We ®rst take the two slides as non-dye-¯ip replicates andemploy AN+RN normalisation. We then take the two slides asdye-¯ip replicates and use the AN+RN+FN and SN+RNnormalisation approaches. Because the genuine log ratio iszero (mj = 0) for self-self microarray data, the normalisedresults provide a measure of the error contained in the data. If anormalisation method can remove all systematic errorscontained in the data, then the normalised data contains onlythe random error inherent in the corresponding microarrayexperiment. However, if a normalisation approach cannotremove all the systematic errors, then this will result inincreased errors, and if the systematic errors are not constant,then the normalised data will show greater variation.Therefore, a comparison of normalisation approaches can beconducted by comparing the variation (standard deviation andrange) of self-self microarray data after normalisation. Wealso measure the correlation between replicated data sets afterdifferent normalisation approaches, since this also provides anindication of any systematic variation. This correlation shouldbecome weaker if a better normalisation approach isemployed.

Analysis of RT1 and RT2

Four different normalisation approaches are employed tonormalise the four data sets. They are iAN+RN, SN+RN,AN+RN+FN and iAN+RN+FN. The outcomes are comparedto evaluate the performance of the approaches. It is necessaryto emphasise that we do not know the genuine log ratio (mj)and, therefore, we cannot assess the performance of anormalisation approach by observing the variation of anormalised data set. For this reason, we used a methoddifferent from that used in the analysis of SS1 and SS2. In thisexperiment, we have four replicated measurements and eachreplicate contains both random noise and systematic errors.The random error is related to uncontrollable factors in themicroarray experiment and systematic error may be removedby normalisation. If the systematic error can be removedcompletely by a normalisation approach, then the consistency(difference between two normalised results from two repli-cates) re¯ects the random noise of the experiments. However,if there is still a part of the systematic error in the normaliseddata, then the consistency of the data will deteriorate.Therefore, the performance of the normalisation approachescan be assessed by comparison of the consistency of replicateddata after being normalised by different normalisationapproaches. In practice, the consistency of replicated datashould be equivalent to the random error contained in the data

set. We measure consistency using the standard deviation andrange of the difference between normalised replicates. Wealso visually compare normalised results from differentnormalisation approaches in order to identify any systematicdifferences between normalisation approaches.

Additional data

The yeast data set contains all of the different types ofexperiment required to test our theoretical predictions;however, this data set is small and analysis of some additionaldata is therefore desirable in order to con®rm our results. A setof chips comparing tissue type in carp (mixed versus gilltissue), with dye-¯ip biological replicates, was obtained fromA. Cossins, A. Gracey and J. Fraser of the University ofLiverpool. The data consist of eight chips comprising fourdye-¯ip biological replicate pairs, which we refer to as A, B, Cand D. Each pair provides results from the same ®sh, while thefour pairs are also biological replicates in the sense that theyare comparing the same tissue types. Each slide contains14 112 spots, of which 13 440 are labelled with gene products.The spots are arranged in 32 blocks, each containing 441spots. The slides do not contain a large number of replicatedspots and therefore we cannot use within-chip consistency as ameasure of error for this data set. However, we would expect asuccessful normalisation scheme to reduce the differencebetween results derived from different replicate pairs.

EXPERIMENTAL RESULTS

The outcomes of normalisation from self-self slides SS1 andSS2 are plotted in Figure 3. The three subplots in Figure 3show the same results using different values on the x-axis: thelog intensity, the rank of intensity and the spot index. Spots areindexed block by block and row by row, left to right and thentop to bottom in each case, e.g. the top left spot of the secondblock from the left at the top of the slide would be indexed401. In each subplot, results from three different normalisationmethods are plotted (AN+RN, AN+RN+FN and SN+RN). Thelegend indicates which normalisation approach has been usedin each case. The data sets are measurements of self-selfdata, so that the true log ratio for every spot should be zeroand the three plots show the experimental error after datanormalisation.

The normalised results of slides RT1 and RT2 are plotted asFigures 4 and 5. Figure 4 shows the error contained in theresults from the three different normalisation approachesiAN+RN, SN+RN and iAN+RN+FN. The error here is de®nedas the difference in log ratio between the replicated spots oneach slide after normalisation. Figure 5 shows the mean andthe difference between the results (log ratios) from the threenormalisation approaches AN+RN+FN, iAN+RN+FN andSN+RN. Figure 6 shows the same normalisation methods as inFigure 4 applied to the carp data set. In this case the error isde®ned as the difference between the estimated log ratios fromdifferent dye-¯ip pairs.

Statistics of the normalised yeast data are presented inTables 1 and 2. Table 1 lists the range and standard deviationof the normalized self-self replicated data using AN+RN,SN+RN and AN+RN+FN. Table 2 lists the range and standarddeviation of errors after applying the AN+RN, SN+RN,AN+RN+FN and iAN+RN+FN approaches to reference


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Figure 3. We plot the estimated log ratios for every spot after applying different normalisation approaches to self-self replicated microarray data sets fromyeast experiments (AN+RN, red crosses; SN+RN, blue diamonds; AN+RN+FN, green dots). In the top subplot the estimated log ratios (base 2) are plottedagainst mean log intensity. In the middle and bottom subplots the estimated log ratios are plotted against spot index and rank, where the rank of spot intensityis obtained by sorting the data set by mean log intensity.

Figure 4. We plot the estimated error for every spot after applying different normalisation approaches to reference treatment replicated yeast microarray datasets (iAN+RN, red crosses; SN+RN, blue diamonds; iAN+RN+FN, green dots). The error is de®ned as the difference in estimated log ratio (base 2) betweenreplicated spots. In the top subplot the errors are plotted against mean log intensity. In the middle and bottom subplots the errors are plotted against spotindex and rank, where the rank of spot intensity is obtained by sorting the data set by mean log intensity.


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Figure 6. We plot the estimated error for every spot after applying different normalisation approaches to the carp microarray data sets (iAN+RN, red crosses;SN+RN, blue diamonds; iAN+RN+FN, green dots). The error is de®ned as the difference in estimated log ratio (base 2) from two distinct dye-¯ip pairs afternormalisation. Other details are as in Figure 4.

Figure 5. Comparison of the results (estimated log ratios, base 2) after using three different normalisation approaches on reference treatment replicated yeastmicroarray data sets (AN+RN+FN, res1; iAN+RN+FN, res2; SN+RN, res3). The mean of each pair of results are plotted as red crosses and the difference be-tween each pair of results are plotted as green dots.


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treatment samples, where the error is de®ned as the differencein log ratio between the replicated spots on each slide afternormalisation. In Table 3 we show results for these fournormalisation methods applied to all possible pairings of dye-¯ip replicates available in the carp data set.

We also calculated the correlation of the normalised self-self results. For the four self-self results (estimated log ratios)after using the AN+RN approach, we found that the meanvalue of the correlation coef®cients of replicated spots is 0.62.Taking the four replicates as two pairs and conducting featurenormalisation, we obtained two sets of results with a meancorrelation coef®cient of 0.22.

DISCUSSION

There are three important points revealed by Figures 3 and 4.The ®rst is that, through normalisation, we can get moreaccurate results from dye-¯ip replicated experiments thanfrom non-dye-¯ip replicates. This point can also be demon-strated by the summary statistics of the resulting data sets (seeTables 1 and 2). In addition, based on our statistical model ofexperimental errors and the corresponding analytical deduc-tions, it is easy to understand why this conclusion isreasonable, because the feature-speci®c bias introduced byexperiment cannot be removed if there is no dye-¯ip replicate

available. Furthermore, comparing the correlation of normal-ised results shows the existence of feature-speci®c bias. Forself-self replicated data sets, our model predicts that there maybe a strong correlation among replicates after the AN+RNnormalisation approach where the feature-speci®c error is notremoved. In contrast, a much smaller correlation will beshown after applying the AN+RN+FN approach to pairedreplicates.

The second point revealed by Figures 3 and 4 is thatadaptive normalisation can achieve higher precision than self-normalisation. This agrees with the inference based on ourmodel, and the statistics in Tables 1 and 2 also support thispoint.

The third point revealed by these two ®gures is that therandom errors introduced by microarray experiments may bedifferent from spot to spot, but these differences canreasonably be taken as independent of the spot intensity andspot position (on a log scale). This can be seen clearly from thesubplots showing error versus spot index and error versus therank of spot intensity. At ®rst glance, the error versus logintensity subplot seems to contradict this conclusion.However, in this subplot, the spots are not plotted evenlyover the horizontal axis. It is clear that the error plotted in aregion will appear to show a larger range of variation if thereare more spots. Plotting against spot rank provides a bettervisualisation of the range and variability of the data.

Figure 5 demonstrates an important point. Compared to thereference sample, the up-regulation and down-regulation ofgenes in the treatment sample may not be balanced.Furthermore, the direction and degree of this imbalance mayrelate to the spot intensity. If the AN method is employed inthis case, then the intensity-dependent imbalance between up-and down-regulation will be removed. Therefore, the system-atic difference between the outcome from the AN+RN+FNand SN+RN methods (shown by the middle subplot) re¯ectsthe intensity-dependent biological difference between twosamples. This demonstrates the necessity of identifying theexperimentally introduced bias and the intensity-dependentbiological bias [m(Aj) in our model]. Based on our analysis, ifwe conduct a ®t of M versus A of a replicate, then both thesebiases (as well as other global biases) will be contained in the®tting. The biological bias should be retained and the otheritems should be removed, but this ®t cannot single out thebiological bias. To achieve this, we proposed an improvedadaptive normalisation method. The top subplot in Figure 5shows that the iAN method behaves differently from AN. Thebottom subplot provides strong evidence that this method

Table 3. Statistics measuring the difference between estimated results from distinct dye-¯ip pairs of carp slides after four different normalisation methods

Pair A±Pair B Pair C±Pair D Pair A±Pair C Pair B±Pair D Pair A±Pair D Pair B±Pair C

AN+RN Range 1.4917 1.2257 1.5175 1.3812 1.3124 1.4804SD 0.0994 0.0891 0.0983 0.0864 0.0972 0.0942

SN+RN Range 1.3352 0.9798 1.4896 1.1591 1.1345 1.1549SD 0.0652 0.0475 0.0783 0.0618 0.0726 0.0549

AN+RN+FN Range 1.0460 0.8125 0.8994 0.9876 0.6851 1.0098SD 0.0524 0.0455 0.0534 0.0547 0.0502 0.0464

iAN+RN+FN Range 1.0526 0.8245 1.0571 1.0515 0.8896 1.0982SD 0.0549 0.0468 0.0556 0.0482 0.0521 0.0481

We show the range and standard deviation of the difference between log ratios (base 2) estimated from the same spot in different dye-¯ip pairs.

Table 1. Statistics measuring the experimental error of self-self yeastslides after three different normalisation methods

AN+RN SN+RN AN+RN+FN

Range 1.9200 1.1219 0.7805Standard deviation 0.2124 0.1046 0.0865

We show the range and standard deviation of estimated log ratios (base 2)of all spots.

Table 2. Statistics measuring the experimental error of reference heatshock yeast slides after four different normalisation methods

AN+RN SN+RN AN+RN+FN iAN+RN+FN

Range 2.0263 1.5939 0.9725 0.9964Standard deviation 0.1970 0.0838 0.0682 0.0758

We show the range and standard deviation of the difference between logratios (base 2) estimated from replicated spots.


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performs well as it shows no systematic difference from theSN+RN method. Therefore, we conclude that theiAN+RN+FN method removes systematic bias effectively,while retaining the improved dye-bias correction displayed bythe adaptive method.

Results from the additional carp data set are shown inTable 3 and Figure 6 and are consistent with our results usingthe yeast data set. The iAN+RN+FN approach provides agreater reduction in variation in log ratio after normalisationwhen compared to the AN+RN, iAN+RN and SN+RNapproaches. We have also con®rmed that there is no system-atic difference between results from the SN+RN andiAN+RN+FN methods, which shows that the improvedadaptive normalisation method is not introducing any biasinto the results.

From the above discussion, we can see that the experimentalresults strongly support all the analytical deductions based onour statistical model of experimental error. However, we havealso made a few observations from the experimental resultsthat cannot be obtained through model-based analyses. Firstly,from comparison of Tables 1 and 2, we can see that the dataquality of reference heat shock replicates may be as good asthe data quality of self-self replicates. In fact, the random errorof RT (reference treatment) replicates has a slightly smallerstandard deviation than that of SS (self-self) replicates. Incontrast, the former has a larger range of variation; this meansit has more notable outliers. Secondly, we can see fromFigure 5 that heat shock treatment makes a number of genesbecome differentially expressed. Furthermore, it appears thatmore highly expressed genes are more highly regulated sincethey tend to show a systematic increase in log ratio on averageas intensity increases. It would be interesting to see whetherthis is true of other organisms. Thirdly, the feature-speci®cerror is the dominant error item in normalised yeastmicroarray data from the AN+RN approach (see Tables 1and 2). This is clear evidence of the impact of feature-speci®cerror and it also reveals that using dye-¯ip replicates is crucialfor improving the performance of microarray data normalisa-tion. Finally, Figure 4 shows that the error contained in the®nal results from the iAN+RN+SN method is bounded in theinterval (±0.5, 0.5). This means that we can generatemicroarray data with such a level of quality comfortably(only two dye-¯ip pairs were used in the experiment) if dye-¯ip replicates and proper normalisation approaches areemployed. Therefore, the genes with 2-fold or larger changesin expression level can be identi®ed very con®dently (weemployed base 2 log transformations in this study so that a2-fold change corresponds to a log ratio of 1 or ±1).

SUMMARY AND CONCLUSIONS

Microarray data contains different sources of variation, whichcan be classi®ed as systematic and random errors. Removal ofsystematic error items is a key issue for the improvement ofmicroarray data quality. Though increasing the number ofreplicates can reduce random error in the data set, it can donothing to reduce the systematic error. From consideration ofthe statistical model proposed in this study, we demonstratethat removing systematic error is not only the business of thedata analyst but is also greatly in¯uenced by experimentaldesign. Employment of dye-¯ip replicates is critical for

obtaining better results through normalisation. In addition,dye-¯ips provide the precondition for the analyst to identifythe experiment-introduced systematic bias effectively. Incomparison to non-dye-¯ip replicates, the use of dye-¯ipreplicates associated with a suitable normalisation method cangenerate much improved results.

The statistical model of experimental errors introduced inthis study provides useful guidance for microarray dataanalysis. Based on the model, we know why better resultscan be obtained from dye-¯ip replicates and we know whyadaptive normalisation can achieve higher precision than self-normalisation. We also developed an improved method foridentifying the experiment-introduced bias correctly. Usingthe model, we adopted suitable methods for removal of erroritems from different sources and so achieved very promisingresults. In turn, all the inferences from the model have beencon®rmed experimentally by analysis of real data sets.

This study provides a sound basis for microarray dataanalysis. The approach proposed can be used to perform notonly a data normalisation, but also a data quality assessment.The data quality assessment method is potentially useful fordata ®ltering and experimental quality control. For data ®lterpurposes, we suggest that one could classify the replicatedmeasurements of a given gene spot into good measurementand poor measurement subsets, based on the error contained ineach of the measurements, and then discard the poormeasurements of the gene spot, while retaining the goodreplicates of the gene spot for use in further analysis. Todiscard any gene spot being measured is a method that hasbeen adopted by some researchers (5,9); however, there istypically no evidence that the gene spots being discarded arenot of biological importance. We leave the testing of thisapproach and the details of the method for microarray dataquality control for further study.

The main conclusions of our study appear to be robust withrespect to the particular regression method used, e.g. usinglocally weighted regression in our adaptive normalisationmethod gives very similar errors after normalisation (resultsnot shown). However, the analytical deductions presented inthis paper are based on the assumption that all the normal-isation processes involved can be carried out perfectly (therelevant error item can be identi®ed and removed exactly), andthis is clearly an idealisation. For example, in the adaptivenormalisation process, the dye bias may not be identi®ed andremoved exactly. One problem with doing standard regressionon an M±A plot is that both variables contain experimentalerror. In this case, it may be better to use a total least squaresmethod in which both variables are modelled as variables thatdepend on some latent independent variable. Since the noise inM and A is correlated, it may be better to carry out this analysisusing the original channel intensities. We leave the analysis ofimproved regression methods for future study.

ACKNOWLEDGEMENTS

We would like to thank Andrew Cossins, Andrew Gracey andJane Fraser for providing the carp microarray data. This workwas supported by a COGEME grant made to S.G.O. and A.B.under the `Investigating Gene Function' Initiative of theBBSRC, a contract to S.G.O. within the frame of theGARNISH network of FP5 of the European Commission


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(contract no. QLK3-2000-00174) and by grants to A.B. andS.G.O. from the North-West Science Initiative's Bioarrayprogramme. Y.F. was supported by an ARK-Genome grant toA.B. and M.R. under the `Investigating Gene Function'initiative of the BBSRC.

APPENDIX: AN ANALYSIS OF THE PERFORMANCEOF NORMALISATION APPROACHES

In adaptive normalisation (AN), we conduct a regression to anM±A plot of a replicate and then take the ®tting as the dye biasand remove it from the data. The ®tting is obtained by takingthe spot intensity A as the independent variable and log ratio Mas the dependent variable. The method for obtaining this ®tcan be either global linear regression or local regression. Inthis study, we used a non-linear regression method which is ageneralisation of a standard polynomial ®t. We followed themodel in equation 1 and used fk(Ajk) to represent the ®ttedvalue for spot j on slide k after carrying out regression, i.e. fk(.)was the ®tted function from the kth replicate and Ajk was theintensity of the jth spot on the kth replicate. The correction wasachieved by subtracting the ®tted value from Mjk,

Mjk ¬ Mjk ± fk(Ajk). A1

It is easy to see that AN performs a slide-speci®c correction,i.e. for different slides, the correction may be different. Themethod can be applied to any type of microarray data set.

Improved adaptive normalisation (iAN) is similar to AN.However, a general function of dye bias is used for all nreplicates. Although the function for n replicates staysunchanged, the correction for a given gene spot may bedifferent from replicate to replicate because the spot intensityon different replicates will be measured as a different value.We propose a technique for obtaining this general function forreplicates with dye-¯ip. The technique includes four steps:Firstly, we conduct a scale and location adjustment of spotintensity so that the intensity distribution of n replicated datasets has the same range and location. Secondly, we pool thedata from all replicates together into one big data set. Thirdly,we make an M±A plot of this pooled data. Finally, we carry outregression on the M±A plot and obtain the parameterisedregression function. The correction is then carried out for eachof the replicates and the amount of adjustment of each datapoint is computed by substituting the spot intensity into theregression function. Therefore, the dye bias correctionperformed by iAN is:

Mjk ¬ Mjk ± f(Ajk), A2

where f(.) is the regression function which has beenparameterised by the pooled data.

The pooled data set contains both measurements fromforward labelling slides and reverse labelling slides and theyare matched into pairs, hence the in¯uence of m(Aj) (inforward labelled measurements) and ±m(Aj) (in reverselabelled measurements) cancels out. Therefore, the ®tting ofthe iAN technique only picks up the systematic error itemsek(Ajk) and c and not the terms m(Aj) and ck. Another basicpoint of the iAN method is the commonly used assumptionthat dye bias is spot intensity-dependent. We further assumethat the dye bias function for a pool of replicated data sets andfor a single data set will behave similarly. Based on these

assumptions, the main reason for a given spot being measuredwith a different dye bias contribution is that the spot intensitywill differ between replicates. The in¯uence of the differencebetween dye bias functions (for pooled replicates and for eachof the replicates) will be much weaker. This point has beencon®rmed by analysis of the yeast data described in the maintext. The variation (measured by variance) of dye bias causedby spot intensity differences is about 100 times as large as thatcaused by differences due to changes in the form of dye biasfunction between replicates. Therefore, iAN can be expectedto perform well.

Self-normalisation (SN) can only be applied to data from adye-¯ipped pair of slides as otherwise the operation willremove any genuine expression ratio. We assume that thenumber of replicates is an even number n = 2s. Let slide k andk + s be a dye-¯ip pair. A single measurement is obtained froma pair of replicates:

Mjk ¬ (Mjk ± Mj,k + s)/2 for k = 1, 2, ¼, s. A3

Regional normalisation (RN) is applied to each block of areplicate. The position-dependent error is identi®ed by a 2Dregression that takes the spot position as the independentvariable and M as the dependent variable. Let fk(Pj) representthe ®t for feature j on slide k from the regional ®tting. Thecorrection is:

Mjk ¬ Mjk ± fk(Pj) for k = 1, 2, ¼, s. A4

Feature normalisation (FN) is identical to SN except that it iscarried out after AN and RN (or after iAN and RN). Thetransformation is identical to equation A3 except that Mjk inthe right hand side has been transferred by the othernormalisations before this transformation.

In the following sections we consider the process andperformance of different normalisation approaches in remov-ing the experimental error contained in microarray data.

Self-self replicates without dye-¯ip

For this situation, the statistical model is shown by equation 2in the main text; iAN, SN and FN are not usable, and so we useAN+RN (adaptive normalisation plus regional normalisation)to normalise the data.

In the AN process, the ®t is fk(.) = c + ck + ek(Ajk). Therefore,after transformation A1, the measured expression ratiobecomes:

Mjk = e(Fj) + ek(Pj) + ejk. A5

In the RN process, the ®t is fk(Pj) = ek(Pj) and, aftertransformation A4, we get:

Mjk = e(Fj) + ejk. A6

Finally, it is sensible to take the mean of the n replicates as the®nal result. Let mÅ j represent the mean of the log ratio for agene on spot j. After normalisation, mÅ j can be represented byequation 3 in the main text.

Self-self replicates with dye-¯ip

For this situation, we can take the n replicated measurement ass pairs (n = 2s). The model is then given by equation 4 in the


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main text. The two normalisation approaches AN+RN+FNand SN+RN are considered below.AN+RN+FN. The outcome of AN and RN can still berepresented by equation A6. Moreover, because dye-¯ipreplicates are available, we can apply FN to the outcome inequation A6, and using the transformation in equation A3 weget:

Mjk = [e(Fj) ± e(Fj) + ejk ± ej,k + s]/2 = (ejk ± ej,k + s)/2 A7

Finally, we take the mean of the s outcomes as the normalisedresult shown by equation 5 in the main text.SN+RN. The SN process performs the transform A3, afterwhich we have:

Mjk = (ck ± ck + s)/2 + [ek(Ajk) ± ek + s(Aj,k + s) + ek(Pj) ±ek + s(Pj) + ejk ± ej,k + s]/2 A8

after the FN transformation in A4 and, taking the mean of ssets of results as the normalised result, the ®nal result is shownby equation 6 in the main text.

Non-self-self slides without dye-¯ip

We split mj into an intensity-independent term mj* and anintensity-dependent term m(Aj) so that mj= mj* + m(Aj). Thestatistical model is then shown by equation 7 in the main text.We consider the AN+RN approach. For AN the ®t fk(.) willpick up the terms m(Aj), c, ck and ek(Ajk) and the expressionratio after transformation A1 becomes:

Mjk = mj* + e(Fj) + ek(Pj) + ejk. A9

For the RN process, the ®t fk(.) will pick up the term ek(Pj) andafter the correction based on equation A4 the outcome can beshown to be:

Mjk = mj* + e(Fj) + ejk. A10

Finally, we make use of the relationship mj* = mj ± m(Aj) andthe mean of the n replicated measurements is represented byequation 8 in the main text.

Non-self-self slides with dye-¯ip

The model of the data set is then given by equation 9 in themain text. We consider the three different normalisationapproaches AN+RN+FN, SN+RN and iAN+RN+FN.AN+RN+FN. It is easy to see that the outcome of AN is:

Mjk = mj* + e(Fj) + ek(Pj) + ejk A11Mj,k + s = ±mj* + e(Fj) + ek + s(Pj) + ej,k + s

RN will pick up ek(Pj) and the transformation A4 will removeit, so we obtain:

Mjk = mj* + e(Fj) + ejk A12Mj,k + s = ±mj* + e(Fj) + ej,k + s.

Finally, applying FN to each pair of slides and computing themean of the s measurements we obtain equation 10 in the maintext.

iAN+RN+FN. As before, iAN removes the error terms ek(Ajk)and c, but leaves m(Aj) and ck untouched, so that:

Mjk = mj + ck + e(Fj) + ek(Pj) + ejk A13Mj,k + s = ±mj + ck + s + e(Fj) + ek + s (Pj) + ej,k + s

RN removes the terms ck and ek(Pj) and we ®nd:

Mjk = mj + e(Fj) + ejk A14Mj,k + s = ±mj + e(Fj) + ej,k + s

Finally, use of the FN process removes the term e(Fj), andtaking the mean of the outcomes we obtain equation 11 in themain text.SN+RN. The result is similar to the self-self case and theresult is given by equation 12 in the main text.

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a model-based analysis of microarray experimental error and

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