A new non-linear normalization method for reducing variability in DNA microarray experiments
© Workman et al., licensee BioMed Central Ltd 2002
Received: 8 January 2002
Accepted: 10 June 2002
Published: 30 August 2002
Microarray data are subject to multiple sources of variation, of which biological sources are of interest whereas most others are only confounding. Recent work has identified systematic sources of variation that are intensity-dependent and non-linear in nature. Systematic sources of variation are not limited to the differing properties of the cyanine dyes Cy5 and Cy3 as observed in cDNA arrays, but are the general case for both oligonucleotide microarray (Affymetrix GeneChips) and cDNA microarray data. Current normalization techniques are most often linear and therefore not capable of fully correcting for these effects.
We present here a simple and robust non-linear method for normalization using array signal distribution analysis and cubic splines. These methods compared favorably to normalization using robust local-linear regression (lowess). The application of these methods to oligonucleotide arrays reduced the relative error between replicates by 5-10% compared with a standard global normalization method. Application to cDNA arrays showed improvements over the standard method and over Cy3-Cy5 normalization based on dye-swap replication. In addition, a set of known differentially regulated genes was ranked higher by the t-test. In either cDNA or Affymetrix technology, signal-dependent bias was more than ten times greater than the observed print-tip or spatial effects.
Intensity-dependent normalization is important for both high-density oligonucleotide array and cDNA array data. Both the regression and spline-based methods described here performed better than existing linear methods when assessed on the variability of replicate arrays. Dye-swap normalization was less effective at Cy3-Cy5 normalization than either regression or spline-based methods alone.
Microarray data is almost always described as containing large measurement noise or high variability. Some sources of variability are random but most are systematic and due to specific features of the particular microarray technology. Systematic effects resulting from the biological process under study are of interest whereas other systematic sources should be removed. Normalization methods are required for controlling uninteresting variability and it is our belief that much of the systematic variability in oligonucleotide array data is controllable through careful normalization. Improper normalization can lead to incorrect conclusions or unacceptably high false-positive or false-negative rates. Here we describe qspline, a non-linear method for controlling signal-dependent sources of variability in Affymetrix oligonucleotide array data, and show that the approach can also be applied to cDNA array data.
Few microarray studies quantify specific sources of variability and fewer still suggest methods for controlling them. Much of the literature addressing microarray normalization concerns cDNA array data, whereas only a few examples can be found for oligonucleotide arrays [1,2,3,4]. Here we review microarray normalization techniques, present the qspline method and show its application to Affymetrix oligonucleotide arrays of human T-cell cultures and cDNA arrays of Bacillus subtilis and Arabidopsis thaliana (H. Næsted, A. Holm, H.B. Nielsen, C.A. Harris, M.H. Beale, M. Andersen, O. Mattsson and J. Mundy, unpublished observations). The oligonucleotide array study compares human T-cell cultures infected with human immunodeficiency virus (HIV) to uninfected cultures and the B. subtilis arrays investigate mutant strains for the genes glnA and tnrA. The A. thaliana experiment compares wild-type to mutant strain and in this context is used to illustrate the effects of normalization on a dye-swap experiment.
Linear normalization methods
Linear methods are the predominant means of microarray normalization. Scaling, the simplest linear approach, assumes a linear relationship passing through the origin. Forcing array distributions to have the same central tendency (arithmetic mean, geometric mean, median) can be accomplished by a scaling factor and has been the method chosen by Affymetrix and others . Linear regression [6,7] and general linear modeling such as ANOVA [8,9] provide offsets, scaling factors and other parameters, but again assume linear relationships and properties of the data or log data distributions, such as normality, that are not always true. In all cases, linear approaches will fall short when the signal bias is not linear.
Approaches for two-channel cDNA arrays apply scaling in order to zero a central tendency of the log expression ratios [10,11,12]. These approaches may also be applied to pairs of single-channel oligonucleotide arrays. Applying an additive offset to the log data is equivalent to a multiplicative scaling of one channel before taking the log. Moving the median or mode of the log-ratios to zero makes the assumption that the majority of genes are not differentially regulated. Additional steps may scale the log-ratios by a measure of dispersion such as the standard deviation or median absolute deviation (MAD), possibly grouping spots according to the printing-tip. This is equivalent to raising both channels to a power before taking the log and is therefore a non-linear transformation of the original signals though giving rise to a linear transformation of the log-signals. Unfortunately, systematic biases are not always linear in log-space either.
Non-linear normalization methods
Non-linear normalization methods have been shown to control signal-dependent non-linear bias between Cy5 and Cy3 channels of cDNA arrays [11,13]. The promising approach of Yang et al.  uses lowess local regression  directly to paired data as a function of signal intensity. For the cDNA data used by Yang et al. and Tseng et al. [11,13], lowess was used for local linear regression of log(R)-log(G) versus 1/2(log(R) + log(G)) where R, and G are the intensities of the Cy5 and Cy3 channels respectively. Rather than regressing log(R) directly to log(G), these approaches correctly attribute uncertainty to both channels by regressing to the geometric mean of the intensity. Even so, standard regression techniques can be sensitive to outliers, which are likely to occur in microarray data. Robust regression techniques, as found in the R version of lowess , are relatively insensitive to outliers. Robust regression techniques incur significant computational costs and, in practice, a small random sample of data must be used for today's oligonucleotide arrays. The approach taken by Tseng et al.  and Schadt et al.  addresses the outlier issue by selecting non-regulated features based on a rank invariant criterion where all signals from both arrays are sorted and signals with ranks deviating by less than a threshold are included. Normalization of replicate arrays raises the question of which signals should be excluded when none of the genes are differentially expressed. In the method we will describe, no data need to be excluded for curve fitting and it is therefore global and unbiased by subset selection.
Signal-dependent non-linear normalization
The goal of signal-dependent normalization is to make signal distributions comparable across the intensity range. This suggests that the expectation for the difference of paired measurements, or log-ratios, should not significantly deviate from zero anywhere over the intensity range. Again, this is the motivation for centering Cy5 to Cy3 log-ratio distributions of cDNA arrays, but now we consider multiple arrays. If the log-ratio distributions between arrays or color channels are to be centered across the intensity range, then the resulting correlation will be linear with intercept zero and slope one. The resulting data correlation will retain stochastic noise and, more important, the biological variation. This linear data correlation presupposes that the distributions of signals be roughly the same. Our approach seeks to transform the distribution of one array to the distribution of a target array in order to achieve this goal. If the target array is chosen to be the geometric mean probe intensities over the arrays in an experiment, then each array distribution is fitted to an empirical estimate of the signal distribution for the experiment. In principle, any measure of central tendency may be used, although the geometric mean is a natural extension of existing cDNA methodologies. Alternatively, each array can be fitted to the theoretical quantiles of a known distribution (for example, the log-normal distribution). This approach was not chosen here as the observed distributions did not fit a theoretical model and contained well determined, systematic features. In this work we seek only to make the distributions similar to each other whatever those distributions may be.
For Affymetrix data, all the arrays of an experiment can be used to define a target array and thus a target distribution that all arrays are normalized to. As a result of the dye effects, data from two-channel cDNA arrays display significantly different Cy3 and Cy5 signal distributions, suggesting that means calculated over both signal types may not make sense for the definition of a target distribution. Instead, either the geometric mean of the Cy3 or Cy5 channels are used, and all channels from all arrays are normalized to a single Cy3 or Cy5 distribution. This accomplishes both Cy3 to Cy5 normalization within and between arrays. The approach described here can be used to normalize the Cy3 channel to the Cy5 channel within a single array or may also be applied to pairs of Affymetrix arrays.
Spatial heterogeneity of signal can be observed in microarray data and in particular in cDNA microarray data. Variability in Cy3/Cy5 ratios has been shown to be generated, in part, by the specific print-tip used during the spotting of the cDNA probes . In fact, an F-test found that at least one print-tip was very significantly correlated to log-ratio (log(p-value) less than -20) in all cDNA arrays used in this study. Spatial effects are not only caused by the printing device but may also be related to; temperature or humidity during the time of printing, the batch of cDNA represented by a specific microtiter plate, reagent flow during the washing procedure after hybridization, or from uneven or tilted glass surfaces during scanning. Microtiter plate effects can appear as vertical or horizontal bands across the array whereas other effects may generate smooth gradients of arbitrary orientation. Signal gradients can be normalized by subtracting local signal estimates (log intensities or log-ratios) and a preliminary approach for this was tested.
These effects, though not as significant as the Cy3 Cy5 bias, can be quite pronounced, as will be observed in the A. thaliana microarrays. Older-generation Affymetrix arrays like those used in this study were subject to local signal biases due to the adjacent placement of probes for each gene-probe set. This bias has since been corrected in more recent oligonucleotide array designs. Other spatial biases can be observed for Affymetrix arrays and are best described as smooth gradients.
Log-intensity scaling, lowess and invariant set normalization approaches were compared to qspline for the reduction of variation between replicate arrays in oligonucleotide and cDNA array experiments. Six microarrays were used for each of the HIV and B. subtilis studies. Three control replicate and three treatment replicate single-channel arrays were used in the oligonucleotide array study of HIV-infected T-cells. Six replicate two-channel cDNA arrays provided six control and six treatment channels for each B. subtilis study. In addition, a pair of dye-swapped replicate arrays from A. thaliana were also normalized to investigate the various normalizations methods versus dye-swap normalization.
The lowess normalization method for cDNA array data was taken from the Rarray module of the sma package (version 0.5.8)  and was used for the normalization of all cDNA microarray data sets. For oligonucleotide array data, a lowess method was adapted from Raffy module of the sma package. Instead of averaging all pairwise array regressions on a single random sample of feature pairs, iterative regressions between an array and the target array were averaged using a new random sample for each iteration. The former method tended to significantly under-normalize whereas the adapted method used essentially the same procedure but performed significantly better.
In addition to print-tip normalization for cDNA arrays, a spatial gradient normalization was devised using a two-dimensional Gaussian function. This function was used to estimate local background bias over a window of probes for the log-ratios of cDNA and oligonucleotide array data. Log-ratios for oligonucleotide array data were calculated versus the geometric mean of each probe across arrays, as will be seen throughout this work. Lastly, although it is only a matter of preference, we used log base 2 throughout.
Global assessment of normalization
Figure 2 shows that signal-distribution inconsistencies between replicate oligonucleotide arrays are comparable to those between treatments. For this example, distribution discrepancies appear minor though we have encountered Affymetrix experiments showing dramatically disparate distribution profiles. Scaling by the trimmed mean of PM-mismatch (MM), as done by the Affymetrix GeneChip software (MAS 4.0), was observed to separate PM intensity distributions more than is observed for unnormalized data and was not considered in this study. Newer versions of the Affymetrix software (MAS 5.0) incorporate additional log-intensity scaling routines that improve normalization performance but were not investigated in this study. Log-intensity scaling used in this comparison also did not make signal distributions more comparable than the unnormalized case. Both lowess and invariant set normalization methods resulted in similar array distributions, whereas our version of lowess normalization gave more comparable distributions than the Raffy version. The qspline normalization resulted in the most similar array distributions, which are indistinguishable from each other in Figure 2.
Normalized array distributions for the two cDNA experiments showed a similar trend, although cDNA array distributions strongly depended on the cyanine dye and in all cases, distributions were much less smooth. The dye bias can clearly be seen in the unnormalized Cy3 and Cy5 signal distributions shown in green and red respectively in Figure 2. Both global and scaled print-tip lowess and qspline normalizations showed similar signal-distribution comparisons. Lowess and qspline normalizations generated more comparable Cy3 and Cy5 distributions than scaling alone, whereas qspline methods generated the most similar signal distributions.
The effects of normalization on the relative signal distributions, log(x i /v), are shown in Figure 3 for both oligonucleotide and cDNA arrays for the same normalizations shown in Figure 2. These probe-deviation distributions show the variability from each v calculated over all the channels in an experiment and therefore show log-intensity deviations from the experiments' centroid. The original signal intensities are lost in these plots of relative information, but global biases can still be seen. Comparable decreases in global array bias were observed for lowess, qspline and invariant set normalized results. Systematic differences between cDNA Cy3 and Cy5 channel distributions can be seen before normalization but display similar distributions after signal-dependent normalization. By visual inspection, print-tip scaling followed by qspline normalization resulted in the greatest decrease of relative signal bias for the cDNA array data. Little if any difference was observed between global lowess and scaled print-tip lowess with regard to relative signal distributions. The same was observed for the two variants of qspline, suggesting that print-tip effects are much smaller than the global biases. The same trends were observed for the four channels of the dye-swap replicate cDNA arrays.
Variance of replicates
Signal-dependent assessment of normalization
Biological assessment of normalization
t-tests were applied to gene-expression estimates calculated from the Li and Wong  reduced model for the oligonucleotide array data. Unfortunately, the set of genes believed to be differentially regulated due to HIV infection was found to be randomly distributed throughout the two sample t-test rankings both before and after different normalizations. The overall effects of normalization on the p-values can be seen for the HIV and glnA experiments in Figure 12. For this comparison, only the relative characteristics between normalization methods are of interest. The actual p-values are not correct as a result of multiple testing. Log rank versus log p-value plots show differing trends for oligonucleotide and cDNA array data. Unnormalized data show lower p-values for genes in the lower half of the ranking in cDNA because of the dye bias and lack of dye swapping over the six replicates, and suggest poorer signal-to-noise characteristics. Lack of normalization in the HIV experiment showed less significant p-values over all genes. Log-signal scaling showed slightly higher p-values whereas lowess, invariant set and qspline showed comparably lower p-values.
Three important assumptions must hold for signal-dependent normalization methods. The first two were suggested by Zien et al.  in their linear centralization approach: the majority of genes are not differentially regulated (assumption 1); and the number of upregulated genes roughly equals the number downregulated (assumption 2). The third assumption is that these two assumptions hold across the signal-intensity range. Assumption 1 was used as a justification for centering log-ratio distributions and is likely to hold when an unbiased selection of thousands of genes is measured. If assumption 1 is true, then assumption 2 can be relaxed as long as random effects on non-differentially expressed genes are equally positive and negative. The third assumption is important for signal-dependent normalizations and is supported by the intuition that genes with 100 to 1,000 mRNA copies per cell will be no more or less biased toward up- or downregulation than genes with 10,000 to 100,000 copies per cell. Microarrays with a small number of genes may be biased to over- or underexpression and could generate data that do not conform to any of the listed assumptions. We believe that future microarrays will only include more features and genes, making these assumptions even more valid. In coming years, oligonucleotide arrays for humans will contain all 35,000 genes (or however many there may be) and therefore will not contain a biased selection of genes.
A potential fourth assumption motivates spatial- or pin-specific normalization and would state that assumption 2 should hold spatially across the surface of the array. These effects were observed to be more significant for cDNA microarrays, whereas global signal-dependent biases dominated over spatial effects for oligonucleotide array data. The MAD for the difference of log(PM) before and after global normalization was 30 times larger than the MAD of the difference between globally normalized and spatially normalized (0.3 versus 0.01). For the glnA experiment, the spatial effect as measured by a similar MAD was ten times larger than that of the HIV experiment but still roughly ten times smaller than the global cDNA effects (1.3 versus 0.1). Although spatial effects may not always be significant, probe-specific effects are a significant issue for oligonucleotide arrays, and must be addressed by other techniques such as the method of Li and Wong . The preliminary spatial normalization method presented in this work will require further validation and development as the current method, by visual inspection, appears to be overfitting the cDNA data. Probe placement and microarray designs do not always correctly randomize strongly or weakly expressed genes, and may contain spatial biases that should not be removed. That said, the ranking of genes known to be regulated did not change significantly as a result of this overfitting.
The process of comparing and validating these methods should depend most heavily on the biological assessment. Unfortunately, this evidence was found to be the most unreliable. Genes from the list that were known to be differentially regulated were found throughout the t-test ranking in both the HIV and the glnA experiments. Statistical measures, such as reproducibility, were less ambiguous, but showed similar performance for global and print-tip, although clear print-tip biases were present. Each microarray system presented different sources of bias and in the end, no one criterion was sufficient for assessing whether microarray data was properly normalized. Observing global, signal-dependent and spatially dependent distributions is recommended in all cases.
Assessments of normalizations were shown globally for array signal and relative signal distributions as well as for signal-dependent MA-style plots and medians of relative signals by quartile. In all cases, these analyses showed favorable results for lowess, invariant set and qspline methods. Certain aspects of the spline-based methods make it preferable to the other methods. First, the qspline method is computationally more efficient than the lowess methods; second, owing to the random sampling procedure, lowess-based methods will give slightly different results for each data fitting whereas the spline methods are deterministic; and third, the quantile-based approach of qspline can normalize arrays with different numbers of features. Only through population-based methods, such as qspline, can arrays with different features be normalized for signal dependent bias.
Materials and methods
T-cell cultures infected with HIV
Ten million MT-4 cells were incubated with either 1 ml virus stock (2 multiplicity of infection (MOI) units of strain IIIB) or 1 ml mock virus for 3 h at 37°C. After extensive washing the cells were transferred to 80 cm3 Nunclon bottles in 20 ml culture medium (RPMI 1640 with 10% fetal calf serum and antibiotics) and cultured at 37°C, 5% CO2 for 7 days. Messenger RNA was extracted using QIAGEN RNeasy Mini kit and prepared for hybridization to Affymetrix HuGeneFL chips according to protocols provided by Affymetrix.
The Affymetrix HuGeneFL arrays contain about 1.4 × 105 PM, MM probe pairs. Probe-based normalization was found to work on both PM and pooled PM, MM values, although the MM values were ignored in this analysis (We believe that MM values are unreliable indicators of cross-hybridization and can be shown to have a confounding effect on all but the top 20% of the signal intensity range.) Control features designed for grid alignment, spiked controls and ALU controls ("AFFX", "hum_alu") were removed.
Bacilluscultures on cDNA microarrays
Cells were grown at 37°C in a modified Spizizen salt-buffered minimal medium as described previously  supplemented with 100 μg/ml L-tryptophan. Na2SO4 was used instead of (NH4)2SO4 and glutamine (0.2%) was added as the sole nitrogen source. At an OD = 1 (450 nm), approximately 100 ml of culture was harvested by centrifugation at 7,000 rpm for 5 min. The pellet was resuspended in 400 μl water and RNA was isolated from this by the use of four tubes from the fastRNA kit blue (BIO 101, Carlsbad, CA) and as recommended by the supplier. cDNA synthesis, dye coupling of probe, array, hybridization and scanning were carried out as described by DeRisi and co-workers (protocol available at ). RNA preparations were run in duplicate and each sample was hybridized to triplicate arrays. In all cases the control RNA was labeled with Cy5 and treatment RNA with Cy3. Each of the 4,100 genes was spotted on the array twice, which gave 12 measurements per gene per experiment.
The variance within replicate group was estimated for each normalization described below,
where i is an index over n probe signals, and j is an index over m replicates.
Global scaling of log intensities
As a comparison with the signal-dependent qspline and lowess approaches, a scaling of log intensities was carried out as follows. Probe intensities were log-transformed (in all cases, we used log base 2 on signal-intensity data) and a measure of central tendency, c j , was calculated for each array and for the entire matrix (c). The mean of the log values was used but the median was found to work just as well. Each array was then scaled in the log-space to the global mean, log(x j )(c/c j ). This simple transformation often accounted for much of the variability between oligonucleotide arrays. The approach used on cDNA array data scaled each Cy3 and Cy5 channel to the same central tendency with the additional constraint that each print-tip group also had the same Cy3 and Cy5 central tendency.
The Affymetrix normalization method found in the Raffy module of the sma package uses the lowess function ('loess') with a robust local linear regression mode ('symmetric'). The Raffy method performed all pairwise normalizations between the six HIV arrays and averaged the five results (m-1) for each array. The process was then repeated on the new data, which resulted in 30 total fits for the six arrays. Instead of averaging all pairwise array regressions on a single random sample of feature pairs, the modified version averaged iterative regressions between each array and the target array using a new random sample on each iteration. This resulted in 18 total fits for the six arrays. The former method tended to undernormalize and did not perform as well as simple log-intensity scaling. The adapted method used essentially the same procedure and code but performed significantly better when used in this scheme.
The six replicates for each glnA and tnrA experiment were normalized to a prototype, v, denned by the median or geometric mean of the six Cy3 channels used to measure the control mRNA (wild type). By this method, all 12 channels from the six arrays were normalized to each other and at the same time the Cy5 channels were normalized to the Cy3 channels. The cDNA normalization method found in the Rarray module of the sma package uses the lowess function ('lowess') with analogous settings as described for oligonucleotide array normalization. Both global and scaled print-tip group lowess normalization were performed (norm = 'l' and norm = 's', respectively). The scaled print-tip mode performed lowess normalization for within each print-tip group and then scaled the log-ratios of each group by its respective MAD .
Cubic spline normalization using quantiles
For this approach, all signal channels are normalized to a target array. The same method was used for cDNA and oligonucleotide array data.
The geometric mean of each probe was calculated over all arrays, x j , in the experiment,
where j indexed over the m = 6 arrays and i indexed over the n probes.
From each array and the vector v, 100 quantiles were taken, q j andq v (percentiles). The size of the quantile sample represented < 0.1% of n. Figure 1 shows an example comparison of array signal distributions x j and v along with a quantile-quantile plot showing the correlation between the corresponding q j and q v . Each q v ,q j pair was used to fit a cubic spline function, s j = f(q v ,q j ), where f was a spline function generator that fits the parameters of a natural cubic spline (B-spline). The splinefun function in the R base package was used for this purpose. Spline parameters were fit for each interval between consecutive quantiles. The interpolating spline function defined over the kth interval, is defined for the parameters a jk and y jk in Equation 3. For an unsmoothed cubic spline, y jk = q vk ,
s j (x) = ajk1 + ajk2 (x-y jk ) + ajk3 (x-y jk )2 + ajk4 (x-y jk )3 (3)
In an iterative approach, quantiles were resampled from percentiles shifted by a small offset np0/k where p0 was the first percentile and k the number of iterations (defines the difference in rank between consecutive quantiles). This provided a different set of evenly spaced quantiles for each curve fitting. For the normalizations performed here, the results from five interpolations were averaged. The qspline method was implemented in R and included in the Bioconductor package, which will be described elsewhere and is available from the web .
Data for cDNA experiments were qspline normalized with percentile samples and fitted in an iterative approach as described for the oligonucleotide arrays. The signal distributions were found to contain many non-smooth features not common for both channels. To account for the print-tip effect, an additional normalization was performed where Cy3 and Cy5 signals were scaled within each print-tip group, as described in global scaling of log intensities, before qspline normalization as just described.
Local signal (log-ratio) was estimated for each probe using a weighted mean of neighboring probe signals. A sliding square window centered on the each probe (50 × 50 for oligonucleotide arrays, and 10 × 10 for cDNA) was used to define the local neighborhood. Weights were defined by their Euclidean distance to the center probe using a Gaussian function (standard deviation 19 for 50 × 50 neighborhood and 3 for the 10 × 10 neighborhood). For both oligonucleotide and cDNA array data, this adjustment was made after global qspline normalization.
Additional data files
C.W., S.B. and S.K. acknowledge the Danish National Research Foundation for financial support. L.G. was supported by a grant from the Danish Biotechnology Instrument Center. H.J. would like to thank Maria Tang and Alan Sloma at Novozymes Biotechnology, Davis.
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