- Open Access
Evaluation of normalization procedures for oligonucleotide array data based on spiked cRNA controls
© Hill et al., licensee BioMed Central Ltd 2001
- Received: 27 July 2001
- Accepted: 17 October 2001
- Published: 21 November 2001
Affymetrix oligonucleotide arrays simultaneously measure the abundances of thousands of mRNAs in biological samples. Comparability of array results is necessary for the creation of large-scale gene expression databases. The standard strategy for normalizing oligonucleotide array readouts has practical drawbacks. We describe alternative normalization procedures for oligonucleotide arrays based on a common pool of known biotin-labeled cRNAs spiked into each hybridization.
We first explore the conditions for validity of the 'constant mean assumption', the key assumption underlying current normalization methods. We introduce 'frequency normalization', a 'spike-in'-based normalization method which estimates array sensitivity, reduces background noise and allows comparison between array designs. This approach does not rely on the constant mean assumption and so can be effective in conditions where standard procedures fail. We also define 'scaled frequency', a hybrid normalization method relying on both spiked transcripts and the constant mean assumption while maintaining all other advantages of frequency normalization. We compare these two procedures to a standard global normalization method using experimental data. We also use simulated data to estimate accuracy and investigate the effects of noise. We find that scaled frequency is as reproducible and accurate as global normalization while offering several practical advantages.
Scaled frequency quantitation is a convenient, reproducible technique that performs as well as global normalization on serial experiments with the same array design, while offering several additional features. Specifically, the scaled-frequency method enables the comparison of expression measurements across different array designs, yields estimates of absolute message abundance in cRNA and determines the sensitivity of individual arrays.
- Additional Data File
- Array Design
- Absolute Decision
- Array Sensitivity
- cRNA Preparation
Affymetrix oligonucleotide arrays (referred to here as oligonucleotide arrays) are widely used to measure the abundance of mRNA molecules in biological samples . The investigator isolates total and/or polyadenylated RNA from cells or tissues, generates the corresponding complementary DNA (cDNA), transcribes complementary RNA (cRNA) from the cDNA template, and then hybridizes the cRNA to the array . There is a significant amount of assay noise associated with readouts from oligonucleotide arrays (for example [3,4]). For these arrays we have found additive and multiplicative noise affecting individual gene readouts (typically 5-20%), as well as multiplicative noise affecting entire arrays (often above 20%). As defined here, normalization attempts to correct for only the latter type of noise. The primary sources of this array-level noise are between-array variation in overall performance (due to inconsistencies in array fabrication, staining and scanning), and between-cRNA variation (as independently prepared cRNAs have variable purity and/or fluorescently-labeled mass fractions). Because these sources of variation contribute so significantly to array readouts, normalization is a critical first step in any analysis of gene expression data.
Most current normalization procedures for oligonucleotide arrays are global approaches, based on normalization of the overall mean or median array intensity to a common standard (for example [5,6,7]). Spiked standards have also been used to normalize cDNA  and oligonucleotide [9,10,11] arrays. All these techniques are inherently linear; there have been recent reports of nonlinear normalizations for cDNA , oligonucleotide [13,14] and other  arrays. Few detailed comparisons of oligonucleotide-array normalization procedures have been reported, however .
For oligonucleotide arrays, the normalization implemented in the Affymetrix GeneChip™ software (Affymetrix, Santa Clara, CA) is by far the most commonly used (for example [1,16]). In this approach, the mean hybridization intensities (the 'average differences' (AD)) of all probe sets on each array are scaled to an arbitrary, fixed level . In the rest of this paper, we refer to this procedure as 'global normalization' or scaled average difference (ADs). In practice, there are at least three limitations to this method. Of these, the first two relate to the normalization itself, and the last relates to the practical utility of the normalized readouts.
First, global normalization makes no attempt to absolutely quantify mRNA abundances. Readouts are normalized to an arbitrary scale, which may vary from one operator to another or between experiments. In contrast, previous experiments with spiked controls  and comparisons with serial analysis of gene expression (SAGE)  have shown that array response can be proportional to true transcript abundance, suggesting that absolute quantitation of transcripts is feasible. If sufficiently accurate, such an absolute scale for all array readouts could facilitate comparisons across large, diverse gene expression databases.
Second, global normalization implicitly assumes that the mean expression level of all monitored mRNAs is constant. The validity of this assumption depends on the number and biological characteristics of genes monitored by an array. For smaller arrays that monitor a limited set of mRNAs, this assumption is invalid and may result in erroneous normalization. Ideally, a quantitation method for arrays would be effective even in cases where this 'constant mean' hypothesis does not hold.
Third, as typically applied, global normalization does not deal well with transcripts expressed at low copy numbers. In a typical Affymetrix GeneChip assay, many low-abundance transcripts are present at levels below the sensitivity of detection of the array (typically about 1:100,000 mRNAs). Measurements for such mRNAs are not only noisy but are sometimes negative, due to cross-hybridization to mismatch probes . Negative intensity values are meaningless and problematic because they cannot be log-transformed, a manipulation that is a common prelude to downstream analysis of array data. Simply discarding negative values is objectionable as it can lead to missed observations of biologically significant upregulation. An automated normalization method that handles noisy and negative measurements and responds to variable array sensitivity is desirable, especially in a high-throughput setting.
The primary criterion for any alternative to global normalization is that it should expand the investigator's ability to compare diverse array experiments done at different times in different laboratories. In this paper, we describe alternative procedures that seek to quantitate array results in terms of transcripts per unit cRNA. We chose cRNA quantitation because it meets the primary criterion, and for several additional reasons.
First, cRNA quantitation is easily applied to array experiments using small amounts of starting total RNA that are difficult to quantitate accurately. Second, the spike reagents described here for cRNA quantitation can be used to specifically monitor the performance of individual arrays. Third, in our experience, the reproducibility, accuracy and scientific value of cRNA quantitation are at least as good as those of alternative techniques, such as procedures to quantitate transcripts per cell, transcripts per mass of input material, transcripts per total RNA or transcripts per polyadenylated RNA.
Spike-in transcript pool
ATCC accession number
Affymetrix gene qualifier
Final concentration (pmol)
Final concentration (ppm)
The constant-mean assumption
A key assumption underlying global normalization is that the mean expression level on an array should be the same for all samples and all arrays. This assumption is distinct from the additional implicit assumption that the fraction of polyadenylated mRNA per total RNA is constant. One can certainly construct special cases where the constant-mean assumption is invalid. One example would be using a small array containing only genes from a single pathway in an experiment that studies variable induction of that pathway. However, it is unclear how well even more general array experiments satisfy this assumption.
These results only apply when genes monitored by an array are randomly selected with respect to their expression characteristics. The example noted above (all genes on an array from a single pathway) is an extreme case of nonrandom selection. Other common ways of selecting genes for arrays may also violate this assumption, including selection based on matches in specific cDNA libraries.
Nonrandom selection of even large mRNA sets for individual arrays can also lead to between-array inconsistencies in mean expression level. For example, consider the case of two arrays, each monitoring a large, equal percentage (> 20%) of a transcriptome, where the first array monitors mRNAs with confirmed cDNA library matches, and the other array monitors mRNAs whose sequences are based on lower-quality expressed sequence tag (EST) sequence matches or computational gene predictions. While the constant mean assumption is justified for each array in isolation, comparison of globally normalized expression levels between the two arrays will give erroneous results because the mean expression level of transcripts on the first array is higher than that on the second.
Spike-in based normalization
The limitations of global normalization suggest the use of spiked transcripts to normalize array data. Our 'spike-in' normalization method, which we call 'frequency normalization', uses spiked transcripts for two purposes. First, they allow us to calibrate the arrays, transforming AD to cRNA frequency (F) estimates quoted in transcripts per million. Second, the spiked transcripts enable us to estimate the minimum detectable frequency on the array (the 'array sensitivity' value). The array sensitivity is useful as a quality-control metric for individual hybridizations and is also used to adjust signals from low-level transcripts. Specifically, frequency values below the array sensitivity are averaged with the sensitivity estimate to generate 'damped' frequencies that lie between 50% and 100% of the array sensitivity. This adjustment introduces a small systematic error into the damped data, but in return it eliminates problematic negative values and retains low-level readings that can be biologically informative in the context of additional experiments.
Fitting a power law model (AD = kF n ) to the data in Figure 2 yields the exponent n = 0.93. This indicates mild curvature in the response, consistent with progressive saturation of array readout for the highest abundance mRNAs. Experiments using 0.1 to 10 μg cRNA per hybridization with corresponding amounts of spike-in transcripts, as well as high and low gain settings on the scanner, indicated that readout saturation (not hybridization saturation) accounted for most of the observed curvature in the spike-in response. The use of approximately 1 μg cRNA in each hybridization, or reduced scanner gain, largely eliminated saturation with no penalty in sensitivity.
Scaled frequency normalization
Frequency normalization is appealing theoretically and effective even when the constant-mean assumption is known to be invalid. However, our experience suggests that frequency estimates might be biased by experimental limitations on the accuracy with which control transcripts can be spiked into cRNA. Specifically, because of the combination of small fluid-handling uncertainties and potentially larger variation in the purity of cRNA preparations, the actual ratio of the spiked transcripts to cDNA-template-derived cRNAs might be significantly skewed from one array to another. One source of variable impurities in cRNA preparations could be oligo(dT)-primer-dependent cRNA product . Such cRNA impurities would result in erroneous normalization in which all readouts from one array would be systematically higher or lower than those from another array. We use the term 'spike-skew' to denote this multiplicative skew in frequency values among multiple hybridizations. One expected symptom of spike-skew would be replicate hybridization readouts that are highly correlated but have widely divergent mean expression levels.
We developed the hybrid scaled frequency (Fs) normalization method to mitigate the effects of spike-skew. Fs normalization is based on the principle of removing technical variation in the ratio of spiked transcripts to cDNA-template-derived cRNAs, by averaging the response to spiked cRNAs over multiple hybridizations. To compute Fs values, globally scaled average differences are first computed for all arrays in a set. This initial step implicitly makes the constant-mean assumption. A calibration function is then computed by fitting a single linear model to the scaled average differences of all spiked cRNAs on all the arrays in the dataset, pooled together. Individual array sensitivities are still computed as described above, and the same damping of low-end frequencies is carried out using the sensitivity values for each array.
To compare F and Fs metrics, consider an experimental set of ten arrays. To compute F values, ten linear models are fitted to the ten distinct, unscaled AD responses to the spiked cRNAs, yielding ten different calibration factors, one for each array. In contrast, when computing Fs values, a single linear model is fitted to the pooled spike response curve consisting of 10 × 11= 121 globally scaled AD values, and a single calibration factor generated for all ten arrays. If there was no technical variation in the ratio of spiked transcripts to cDNA-template-derived cRNAs in the ten experiments, both approaches would give the same quantitation, up to a random error term arising from the difference between fitting ten 11-point responses versus a single 121-point response. If, in one of the ten arrays, the ratio of spiked transcripts to cDNA-template-derived cRNAs is different for technical reasons, then spike response for that array will be skewed, and the F-metric readout for that array will be skewed relative to the other nine arrays. In contrast, such a skewed array will only affect the Fs metric to the extent that the single skewed response shifts the fit to the pooled spike response. The skew for the single problematic array will be removed because all arrays in the set will be scaled and calibrated with a single factor. In other words, Fs values are estimates of transcript abundance in cRNA, based on the average response to the spiked cRNAs over multiple hybridizations and on the sensitivity of each individual array. F values provide the same estimate, but based solely on the response to spiked cRNAs in a single array hybridization.
Comparison of normalization methods: reproducibility
We tested if spike-skew could account for the relatively high CV of frequency in three of the four replicate sets (Figure 3) by comparing experimental data to simulated data with known levels of spike-skew. To approximate spike-skew, the concentration of the spike-in transcripts in simulations was multiplied by a random noise term. Over a series of simulations, we varied the standard deviation of the noise term from 0 to 40% to model the effect of increasing spike-skew. MEDACV values were then computed from the simulation results in the same way as for the experimental data in Figure 3.
Taken together, the experimental data and the simulations suggest that spike-skews of roughly 20% can explain the sometimes inferior MEDACV (but consistently high inter-replicate correlation coefficients) of the frequency metric.
Comparisons across array designs
Ratios of across- to within-array coefficients of variation (CVs)
The reason for the poor agreement of ADs readouts across distinct designs was that the mRNAs monitored by the A array are, on average, expressed at higher levels than those on the B or C array, as confirmed by two independent lines of evidence. First, the mRNAs on the A array were intentionally selected because they were represented in C. elegans cDNA libraries, whereas the B and C array genes (many of them computational predictions) were generally not represented in cDNA libraries. Second, A array mRNAs were more likely than B or C array mRNAs to be detected in the developmental time course by the Affymetrix Absolute Decision metric . Because of this systematic difference between gene sets, the mean AD of all A array genes was substantially higher than that of the genes on the B or C arrays. The ADs metric scales data under the assumption that mean expression levels for all arrays should be equal. Therefore, ADs values for genes on the B and C arrays were inappropriately inflated relative to ADs values from the A array.
Comparison of normalization methods: accuracy
Normalization methods should accurately measure true biological variation. We tested the accuracy of the four methods using simulated data. As a baseline we chose the experimental data from one of the 0-hour replicates on the A array. We generated 19 simulated experimental conditions to produce 20 raw average difference values for each of 6,617 genes. For each of the four metrics, computed fold-changes between the modulated condition and the baseline (considering only messages called Present) were compared to the true fold-changes. Accuracy was defined as the fraction of computed fold-changes that were accurate within twofold, and determined for assumed levels of spike-skew from 10-40% (percentage is the ratio of standard deviation (SD) to mean of the random spike-skew term in the simulation). Three simulations were carried out at each level of assumed spike-skew. ADs and Fs metrics performed equally well and best overall, with accuracies above 99% regardless of spike-skew. As expected, frequency was the only metric with a significant dependence on the level of spike-skew. At 10% spike-skew, frequency accuracy was (mean ± SD) 0.9951 ± 0.0006, at 20%, 0.96 ± 0.02, and at 40%, 0.82 ± 0.06. For comparison, the accuracy of AD was 0.88 ± 0.07 at 10% spike-skew, and did not change significantly at higher spike-skew levels.
We stress that the overall accuracy levels reported here are highly dependent on adjustable parameters in our simulation model (see Materials and methods). Nevertheless, the simulations demonstrate that at levels of spike-skew consistent with our experience, scaled frequency is as accurate as globally normalized ADs; this observation is robust to changes in the model parameters.
Absolute quantitation of cRNA and cellular RNA
There are several potential sources of inaccuracy in the cRNA quantitiation given by the scaled frequency metric.
Our results suggest that there is significant uncertainty in the molar ratio of spike-in mRNAs to template-derived cRNAs in any hybridization (the spike-skew effect). The MEDACV for the F metric in Figure 3 is likely one measure of this uncertainty, as it probably arises primarily from cRNA purity variation. This uncertainty leads to proportional differences between frequency metric readouts and true cRNA transcript abundances. However, in the scaled frequency method, the simultaneous normalization of larger datasets reduces these differences through averaging. We anticipate that inaccuracies of cRNA quantitation arising from this effect will be reduced by improved methods for quantitation of cRNA preparations.
For F and Fs, heterogeneity in probe response will lead to gene-specific biases in quantitation. Our data contains two observations that allow us to estimate the degree of heterogeneity among spiked probe sets. Cursory examination of the calibration curve (Figure 2) suggests relative responses of the 11 distinct probe sets shown do not vary more than two-fold: no observations fall more than about a factor of two from the fitted line. A more rigorous evaluation of probe set heterogeneity can be done by comparing the ratio of AD values from two distinct probe sets that monitor the same transcript in a single hybridization. This ratio estimates the difference in readout that would be observed for a single transcript if a different probe set were selected. This comparison was made for the 11 spiked transcripts (each array contained two probe sets for each of these mRNAs). On the basis of 138 ratio measurements from the C. elegans arrays, the 10th-90th percentile range for the ratio was 0.39-1.44 , indicating that for the set of control transcripts, the uncertainty in cRNA quantitation due to heterogeneity in probe set responses for 80% of transcripts was less than threefold.
In addition to these factors leading to inaccuracies in cRNA quantitation, there are at least two important factors leading to differences between cRNA abundances and cellular RNA abundances in the starting biological material.
First, because cRNA is generated from the polyadenylated fraction of total cellular RNA by a linear amplification process, frequency estimates will not reflect sample-specific changes in the fraction of polyadenylated RNA in total cellular RNA. This may be a desirable feature of frequency estimates, in cases where per-total-RNA abundance is less relevant than per-polyadenylated-RNA abundance.
Second, any gene-specific biases in the cRNA amplification procedure will lead to gene-specific differences between cRNA and per-total-RNA quantifications. Evidence to date suggests that these biases are small  and reproducible .
Taken together, the above-noted sources of inaccuracy suggest that there can typically be around two- to threefold differences between scaled frequency per-cRNA estimates and per-polyadenylated-RNA abundances in the starting material. These differences could be reduced by improved cRNA process control and quantitation, and by improved probe selection algorithms.
We have shown that cRNAs spiked into hybridization solutions at known concentrations covering two to three orders of magnitude can be used to normalize array data and to estimate array sensitivity. However, frequency normalization based solely on these control transcripts can be adversely affected by variations in the 'purity' of cRNA preparations. These observations underline the need for meticulous quality control during the production of cRNA samples and accurate quantitation of the resulting material. With better control of these processes, the frequency metric may provide a robust spike-based normalization that, unlike all the other metrics described here, does not rely on the constant-mean assumption.
In the presence of variation in cRNA purity, the Fs metric provides a compromise between the robustness of the ADs metric and the more absolute quantitation scale of the frequency metric, in cases where the constant-mean assumption is valid. In addition, the Fs metric provides a common scale for comparing data from distinct array designs. This is an important advantage over other metrics. For example, the Fs metric allows comparison of the expression levels of all worm mRNAs on all three of our array designs with comparable confidence to within-array-design comparisons. This is not possible with globally normalized average differences. We believe that cRNA quantitation and the damping of low-amplitude signals provided by the Fs normalization make this metric a valuable format for reporting diverse gene expression array results.
Experiments and arrays
Array experiments used the Genetics Institute C. elegans Affymetrix GeneChip™ oligonucleotide arrays, a set of three arrays (denoted A, B and C) which in aggregate monitor approximately 98% of the 19,099 predicted worm mRNAs in the October 1998 worm genome sequence release . The total number of probe sets on each array was 6,617 (A array), 5,768 (B), and 6,646 (C). Each probe set consists of 20 distinct probe pairs (each probe is a 25mer) designed to monitor a single transcript. On the C. elegans arrays described here, probe sets monitoring the spiked transcripts were each tiled twice with a different set of oligonucleotide probes. On arrays that are commercially available from Affymetrix, one probe set is tiled to monitor each of the spike-in transcripts. The probe sets are not fully randomly distributed across the arrays, although on the C. elegans arrays the different probe sets are tiled in widely different regions of the arrays. Experimental array data described here were taken from the developmental time course dataset reported in . Specifically, we examined individual replicate hybridizations of the A array from the worm developmental time course at each of 0 (n = 4), 36 (n = 3), 48 (n = 4) and 60 (n = 4) hours after synchronization of worm eggs by bleach, as well as a larger set of 13 hybridizations of all three arrays to samples ranging from oocytes to 2-week-old worms. Replicate hybridizations in the datasets included independently generated complementary RNA (cRNA) preparations from the same starting total RNA. Primary data for all transcripts on all arrays (including all replicates of all three array designs) is contained in the supplementary Excel spreadsheet (see Additional data files).
Spike-in transcript pool
A pool of biotin-labeled spike-in control transcripts was derived by in vitro transcription of 11 cloned Bacillus subtilis genes, using the methods described in . The spike-in pool was added into hybridization cocktails in proportion to the UV-quantitated cRNA mass in the hybridization, so as to achieve the desired final concentration of spike-ins. The spiked transcripts and their final concentrations in the hybridization cocktails are listed in Table 1. Final concentrations in pmol and parts per million (ppm) for each spiked transcript were computed from the known length of each spike-in, assuming a total mass of 2 μg worm cRNA in a 200 μl hybridization volume, and an average length of 1,000 bases for in vitro transcribed worm cRNAs.
Metrics for transcript abundance
Average difference (AD) is the basic measure of transcript abundance that is calculated by the Affymetrix GeneChip 3.1 software. The calculation of AD is described in detail in the Affymetrix GeneChip User Guide . Briefly, a background intensity is computed for each of 16 rectangular sectors on the array. This local background is subtracted from the intensity values of each probe cell in all sectors. After background subtraction, the difference between perfect match (PM) and mismatch (MM) feature intensity is calculated for all probe pairs in each probe set (in our case, 20 probe pairs in total). The AD for each probe set is the average of the PM - MM differences, after outlying values are removed.
A second important metric generated by the GeneChip software is the Absolute Decision. The Absolute Decision is a categorical call for each transcript: either Present, Absent, or Marginal. The Absolute Decision is a heuristic metric based on the number of probe pairs for a given transcript that show strong specific hybridization signals. See the Affymetrix GeneChip User Guide  for a detailed description of this metric.
Because of array-to-array variation in overall signal strength, AD values from different arrays are usually normalized to a common scale. We reproduced the scaled AD normalization of the Affymetrix GeneChip 3.1 software. The calculation is described in detail in the Affymetrix GeneChip User Guide . Scaling is done by equalizing the average intensity of all arrays in a given dataset, where the average intensity is defined as the trimmed average of the AD values of every probe set on the array, excluding the highest 2% and lowest 2% of the values. This normalization works on the assumption that the summed expression level of all genes on the array is constant across experiments, and that differences in expression levels between arrays can be corrected by array-specific scaling factors. We denote the normalized AD values as scaled average difference (ADs).
The calculation of frequency (F) values involved two steps: first, conversion of AD values to frequencies by use of the calibration curve, and second, estimation of the chip sensitivity of detection and 'damping' of frequency values below this sensitivity.
The calibration curve for each hybridization was constructed from the AD values for each of the 11 control transcripts and their known frequencies (Table 1). AD values that were negative, or associated with Absent or Marginal Absolute Decisions, were removed from the curve in order to improve the robustness of the fit. This calibration curve was fitted by a linear function with zero intercept, using a generalized linear model  fitting procedure in the statistical software S-PLUS (Insightful Corp., Seattle, WA). The fitting procedure assumed a gamma error structure, appropriate for data with constant coefficient of variation, and utilized iterative reweighting of errors. The single coefficient of this linear fit was multiplied with the average difference values for each gene on the array to yield initial frequency estimates. Calibration curves for the hybridizations described here were examined visually to rule out poor curve fits.
Chip sensitivity of detection was estimated from the Absolute Decisions (Present, Marginal, or Absent) for the 11 spike-in transcripts in one of two ways. In the general case, Absolute Decisions were considered as a binary response: Absent = 0, Present = 1, with Marginal calls treated as Absent to be conservative. This response was regressed against the log-transformed known frequencies, using a generalized linear model with a logit link function. The chip sensitivity was then defined as the frequency at which the predicted odds of a Present call were 70%. In the special case where all spike-in mRNAs called Absent were lower-abundance messages than all spike-ins called Present, the sensitivity was defined by linear interpolation as the frequency 70% of the distance between the highest Absent call frequency and the lowest Present call frequency.
Frequency values for all genes on the array that fell below the sensitivity were damped as follows. Negative frequencies (corresponding to negative AD values) were adjusted to one-half of the chip sensitivity. Frequencies between zero and the chip sensitivity were adjusted to the average of the frequency and the chip sensitivity. The rationale for this adjustment was threefold. First, one-half the chip sensitivity was a reasonable a priori estimate of abundance for many genes that were not reliably detected. Second, the adjusted frequencies were guaranteed to be positive-valued, making downstream analyses of frequency values (for example, log transformation) significantly easier. Third, retaining the adjusted low-level frequency estimates was preferable to discarding them, because discarding the values would make it impossible to detect potentially important regulation of these genes in future experiments.
Frequency normalization could be adversely affected by technical uncertainties in cRNA preparation (see Results and discussion). To attenuate these effects, an additional frequency variant termed scaled frequency (Fs) was introduced. Fs was a hybrid of ADs and frequency, and was computed as follows. ADs was first computed for a set of two or more arrays exactly as described above. Then a linear model (with zero intercept) was fitted to the pooled ADs values for the 11 spike-in transcripts from all arrays, ignoring negative ADs values or those associated with Marginal/Absent Absolute Decisions. The slope of this linear model was the single calibration factor for the entire dataset. This slope was multiplied with the ADs values from all arrays to yield Fs values. Per-array sensitivity values were computed exactly as described for F, and Fs values on any array that were below the array sensitivity were adjusted as described above.
Array data was simulated as follows. First, a single experimental dataset, one of the 0-hour replicates, was chosen as a baseline for generation of all simulated data. To this baseline dataset, several random noise sources were added to reproduce key sources of variability in array data. The relation describing the simulated data was:
AD ij = b ij + ADB i (a j m ij s ij r ij )
AD ij = simulated AD for the ith mRNA on the jth array
ADB i = baseline gene expression data for the ith mRNA
b ij = background noise for the ith mRNA on the jth array
a i = array intensity offset for the jth array
m ij = multiplicative noise for the ith mRNA on the jth array
s ij = spike-skew factor for the ith mRNA on the jth array (unity for all nonspiked mRNAs)
r ij = regulation factor for the ith gene on the jth array (unity for all spiked mRNAs)
Background b ij was Gaussian with a standard deviation (SD) that varied randomly from one array to another. The background noise SD had a mean of 20 AD units, and a standard deviation of 5 AD units. Array intensity offsets a j were Gaussian with a mean of one and SD of 0.3. Multiplicative noise, m ij , was drawn from a normally distributed zero-mean noise source with a constant CV of 0.1. Spike-skew factor s ij was a single random factor for all spiked cRNAs on a given array, and unity for all other messages. The spike-skew factor for the spiked cRNAs was Gaussian with mean 1 and a SD that was adjusted from 0.1 to 0.4 (in percentage terms, 10-40%). Regulation factors r ij were generated by a procedure in which the base-10 log (fold-change) for each gene was selected from a normal distribution with mean 0 and SD 0.5. Extreme random regulation factors were limited so that the regulated gene expression values had the same range as baseline data. After multiplication of each gene by its regulation factor, the mean expression level of all genes was adjusted so that the overall mean expression level was unchanged by regulation.
Primary data for all experimental hybridizations are available as an Excel spreadsheet (20,325 Kb).
We thank Yizheng Li, Bill Mounts and Scott Jelinsky for thought-provoking conversations about normalization approaches, Steve Rozen and Ken Griffiths for related software and database implementations, and Michael Byrne for contributions to initial normalization concepts.
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