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1 Department of Statistics, University of Florida
2 Department of Pathology, Immunology, and Laboratory Medicine, Center for Mammalian Genetics and Diabetes Center of Excellence, University of Florida, Gainesville, Florida, 32610-0275
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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ij) of background differences between a spot and the neighboring spots, i.e., a spot is considered as too weak if the signal is weaker than cij. Our studies suggest that normalization and ratio estimates were unacceptable when this threshold (c) is small. We further showed that when a reasonable compromise of c (c = 6) is applied, normalization using trimmed mean of log ratios performed slightly better than global intensity and mean of ratios. These studies suggest that decreasing the background noise is critical to improve the quality of microarray experiments. microarray; gene expression; statistics; normalization; functional genomics
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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For this study, we concentrated on microarray-based studies monitoring RNA expression levels using cDNA microarrays printed on glass microscope slides. Brown and coworkers (2, 3, 5, 11, 12) developed the protocols widely used to do this type of assay. The basic strategy for this type of analysis is to isolate RNA from two sources, a reference and an experimental sample. The RNA samples are converted to cDNA and labeled with a fluorophore, typically Cy3 or Cy5. The two probes are combined and hybridized to the microarray. Following the hybridization and washes, the array is scanned at both 532 nm (Cy3) and 635 nm (Cy5) to detect the labeled cDNA that has hybridized to the array. The two images produced from the scanner are combined, and the data for each spot (gene) is collected (along with background and error measurements). The data used for subsequent analysis is typically expressed in the form of a ratio of experimental expression to reference expression. The hybridizations are repeated multiple times to ensure reproducibility and confidence in the measurement. Once the data from several hybridizations are available, a variety of clustering and statistical methods can be used to determine which genes to study further (6, 7, 13).
One of the most critical steps in microarray experiments is to accurately assess the expression ratios between the sample and reference in a dual color experiment because most subsequent analyses of the data, such as cluster analysis (6), depend on the accuracy of these ratios. Even statistical decision rules such as shrinkage estimation (9) or hypothesis testing on yes/no types of expression differences depend on these original ratios. The ratio estimation consists of two major steps: 1) to eliminate elements with weak expression that are statistically too close to the background to avoid the detrimental effect in the ratio and 2) to adjust the expression for each gene by the overall expression. The ratio estimate follows by simply taking the ratio of the two adjusted expressions or the difference of the log transformation of the expressions. We used experimental replicates to determine the appropriate rules for each of the two steps.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Microarray Printing
The glass slides used in the production of the arrays were coated in-house with poly-L-lysine using a previously described protocol (10). The microarrays were printed using a GMS 417 arrayer (Affymetrix, Santa Clara, CA) and subsequently processed using previously described protocols (10). The details of these protocols can be found at http://www.genomics.ufl.edu/microarray. The microarrays can then be stored for several months in a dehumidified chamber.
RNA/Probe Preparation, Hybridization, and Data Capture
Total RNA was isolated from the spleen of NOD mice at 4 wk, 5 wk, and 20 wk of age using the Qiagen RNA extraction kits per the manufacturer’s instructions. The reference RNA was a pool of total RNA from 4-wk-old spleens from 10 NOD and 10 C57BL/6 animals. In our current assays, we can obtain excellent hybridization signals with 10–15 µg of total RNA. The hybridization probes are produced using an amino-allyl labeling strategy. This labeling strategy incorporates an amino-allyl modified dUTP [5-(3-aminoallyl)-2'-deoxyuridine 5'-triphosphate (aadUTP), from Sigma] during cDNA synthesis. Following the synthesis, a monofunctional NHS-ester Cye dye (either Cy5 or Cy3) is coupled to the modified dUTP in a sodium bicarbonate buffer. This labeling procedure circumvents the problem of a low incorporation rate with either the Cy3- or Cy5-coupled dUTP. The aadUTP will incorporate into the growing cDNA strands at rates similar to the unmodified nucleotide. After coupling the Cye dye to the aadUTP, the unincorporated dyes are removed using a Qia-Quick PCR purification kit (Qiagen). Once the reference and experimental probes are created, the hybridization is accomplished under a coverslip in a specially designed hybridization chamber (Array-It). The hybridization of the probes to the array is allowed to proceed for 16 h. Once the hybridizations were complete, detection of the hybridized probes to the array is accomplished using the GMS 418 Scanner (Affymetrix, Santa Clara, CA). The image data was captured by scanning the slide twice, the first time at 532 nm (Cy3-labeled probes) and the second time at 635 nm (Cy5-labeled probes). This process produces two 16-bit TIFF files, one for each wavelength. These images were then merged and analyzed using Scanalyze by Michael Eisen (6).
Four experiments (exp 1, 2, 3, and 4) were carried out for this study. The first experiment consists of three independent reference/reference hybridizations (RR1, RR2, and RR3) using the reference RNA (a pool of total spleen RNA samples from ten 4-wk-old NOD and ten 4-wk-old B6 mice) labeled independently with both Cy3 and Cy5. Since the same RNA was labeled with both fluorophores, the expected ratio is one for every target gene on the slide. Experiments 2, 3, and 4 were hybridizations using the same reference sample against spleen RNA from an NOD mouse at 4, 5, and 20 wk (diabetic) of age, respectively. Each experiment consists of three replicates.
Statistical Model, Estimation Theory, and Justification
Data output from a dual color microarray experiment contains the average fluorescence intensities for the target and background intensities for both the reference and experimental samples. On each microarray, let
 | (1) |
The observed fluorescent intensities for each spot j is determined primarily by two parameters: the amount of probes available for hybridization and the amount of target DNA on the microarray (dj for spot j). In dual color microarray experiments, the amount of target DNA for a given spot is identical for the reference and experimental sample. The amount of probe is proportional to the mRNA expression levels (the fraction of mRNA for gene j among the total mRNA for all expressed genes) in the reference and experimental samples (
1j and 2j for gene j) and the total amount of mRNA used for labeling in the corresponding samples (a1 and a2, respectively). The amount of probe available for hybridization to spot j should be a11j and a22j, and the fluorescence intensities at spot j should be a1dj1j and a2dj2j after hybridization. Both a1dj1j and a2dj2j are subject to random error due to variation in measuring mRNA concentration (measurement error and purity), efficiency of probe preparation (labeling and purification), and hybridization efficiency. It is reasonable to assume that the variation (noise) is proportional to the true values, i.e., the fluorescence after hybridization can be represented as aidjij x exp(
ij) where ij is a random variable with an ideal value of 0. A proportional constant may be added to the expression aidjij x exp(ij); however, it is not necessary, because we are only interested in the ratio of expressions. A proportional constant would disappear in the ratio. We let 
ij = aidjij as the ideal expression level for gene j in color i.
There is another type of noise from the fluorescence of the background due to the experimental conditions of the hybridization. The magnitude of this variation (noise) is variable across the glass slide. We assume that this noise is additive. Combined with the multiplicative variation (ij), the observed target intensity for sample i at spot j is
 | (2) |
The true sample to reference expression ratio (rj) and its log transformation (
j) are defined as
 | (3) |
ij, ij, and b'ij, we will make the following assumptions on the distribution of b'ij - bij and 1j - 2j.
Assumption 1.
The background noise is additive as defined in Eq. 2. Moreover, the differences of b'ij and bij (b'ij - bij) are independent random variables with 0 mean and variance ij2, where ij2 is almost a constant among neighboring spots. In other words, the large variation in background noise can be stabilized by taking the difference of adjacent spots.
Assumption 2.
The differences 1j - 2j are independent and identically distributed normal random variables for all spots j.
We will adjust the observed intensities by subtracting the background measurement as
 | (4) |
ij) defined in assumption 1 is estimated as 
ij, a spot is discarded if
 | (5) |
 | (6) |
ij = ln[1 + (b'ij - bij)/(aidjijeij)] 
(b'ij - bij)/(aidjijeij) is approximately a small zero mean random variable and becomes negligible if the signal is strong. Therefore, the observed log ratio (defined as yj) can be represented as
 | (7) |
= ln(a2/a1) and is the normalization constant, j = ln(rj), and 
j represents the last four random terms. The value (the normalization constant) is used to compensate for the fact that each color is scanned independently under a different laser power and gain for each wavelength in a two-color microarray experiment. The true ratio rj (or log ratio j) is not estimable without first determining this constant. However, to estimate the normalization constant , we face the confounding problem of and the mean value of j. Actually the mean of j is inseparable from because an increase in a2 is equivalent to the increase in all rj. Using mathematical notation, the confounding effect can be expressed as yj = ( + µ) + (j - µ) + j, i.e., we can claim the log ratio to be (j - µ) for any µ without affecting the validity of the model. If we are interested in the relative expression level of genes within the same array, then the constant µ plays no role. But, if we are interested in the expression of the same gene in different arrays, then the values of µ affect the level of their expressions when compared. Normalization could be used to put the reference mRNA expression in two different arrays on an equal footing. For example, we may adjust the expressions so that the total expression of the target and reference are the same. This is reasonable for normalization; however, it is not compatible with the commonly used logarithmic transformation for the ratio. In addition, because the majority of the genes have weak expression, they affect the accuracy of the total estimation whether or not they are eliminated. When the logarithmic transformation is taken for the ratio, we may use the model in Eq. 7 and assume that 
j = 0 as in Kerr et al. (8). This assumption is convenient for the usual analysis of variance model setting, Eq. 7, but not so easy to interpret from biological viewpoint. Another assumption would be that "most" of the genes (so-called housekeeping genes) have the same expression. In other words, "most" of the j are 0. We will define "most" later and will see that each of the assumptions will lead to a different method to estimate the . We will compare them. All the assumptions are equivalent to setting µ = 0 in the expression yj = ( + µ) + (j - µ) + j. Once can be estimated, the estimate of the log ratio (j) should be
 | (8) |
is the estimate of .
We have not found a method that can estimate the variance of j in a single experiment, because the variances of j and j are confounded. However, if the hybridization is repeated, then we can find the variance of j and consequently a confidence bound of j. Let the second experiment be represented in a similar notation to Eq. 7 as
 | (9) |
 | (10) |
j - 'j can be estimated by the sample variance of 
yj. We will denote this as s2. If we let the variances of j and 'j be 
2 and '2, then s2 estimates 2 and '2. The estimate for the log ratio (j) based on the two experiments should be
 | (11) |
'j is the j in Eq. 8 for the second experiment. Moreover, a 1 - 
confidence interval for the true log ratio j is
 | (12) |
/2 is the upper /2 quantile from a standard normal distribution. It can be easily shown that Eq. 12 does not need to assume that the variances of j and 'j are equal. Extension to more than two replicates can be worked out; however, we will omit the details here. |
RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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bij) is nearly normally distributed.bij = b2j - b2(j+1)) from the replicates in experiment 1. These histograms illustrate that bij tends to have a bell-shape histogram with a heavier concentration at the center than the normal distribution. Comparison of the standard deviations of bij and bij revealed a great reduction of variation for bij compared with the variation for bij (Table 2). The correlation between |bij| and the average background noise (bij + bi(j+1))/2 (last column in Table 2) assesses whether the local variation of bij (measured by |bij|) is affected by the magnitude of the background noise [measured by (bij + bi(j+1))/2]. The small correlation indicates that the variation of bij is not related to the uneven spread of the background.
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bij compared with the variation for bij is also illustrated by the distribution of bij (Fig. 2A) and bij (Fig. 2B) associated with each spot. As can be seen, the distribution of bij (Fig. 2B) is much more uniform than the original background (bij) (Fig. 2A). Although our data showed that assuming bij as independent and identically distributed was acceptable, a few outliers of bij still exist in microarrays containing a large number of spots (Fig. 2B). Therefore, we estimated the background noise standard deviation ij for each spot locally. We tried several spot matrix to define "local" for the ij estimation and found that using the eight nearest neighbors gives the best estimates ij.
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1j - 2j) is independent of the magnitude of the signal and normally distributed.j in Eq. 8 should be normally distributed with 0 mean for two identical RNA samples, in which all log ratios (j) are 0. This expectation is tested using the reference vs. reference hybridizations in experiment 1. As the assumption predicted, the histograms for the three replicates are all very close to a normal curve (Fig. 3). To evaluate that "(1j - 2j)" is independent of spot j as predicted in assumption 2, we used a linear regression analysis to assess the relationship between the absolute mean value of three log ratio estimates and their variance. We expect the variance to be independent of the mean. Indeed, the R2 values from the regression analysis are close to zero (R2 = 0.136, 0.014, 0.00, and 0.013 for experiments 1, 2, 3, and 4, respectively). The slightly positive value for experiment 1 (Ref vs. Ref) is not surprising given that all true log ratios for this experiment are zero. Any large value in the three ratios will produce both a large mean and a large variance and consequently a positive correlation. However, the R2 is not large enough to warrant a modification of the identical distribution assumption.
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Flagging of weak spots.
Intuitively, the quality of the spots on microarrays is an important parameter that determines the accuracy and reproducibility of microarray data. As discussed in validation 1, the quality of the spots can be evaluated by comparing the background adjusted signal intensity (t'ij) and the standard deviation (ij) of the local background difference (bij). If the background-adjusted intensity t'ij is smaller than a certain threshold (c) of ij, the spot is then flagged. We have found that the threshold (c) is the most important parameter for normalization and accuracy of the microarray data.
Normalization.
As we explained, after Eq. 7 that there are various ways to do the normalization. The method using the two-color total expressions to adjust each gene expression is equivalent to
 | (13) |
using a 25% trimmed mean method, i.e., to trim the top and bottom 25% of yj before taking the average. If we do not want to assume 50%, then we may use the mode to estimate . Obviously, under the assumption j = 0, should be estimated by the average of all the yj or the 0% trimmed mean. Another often-used method which uses the mean of the unadjusted ratios is also considered. The estimation of in this case is -ln[(t'2j/t'1j)/n], where n is the sample size, and this is referred to as the mean of ratio method. The mode estimation was based on an Splus subroutine with bandwidth equal to 1.57 x (q0.75 - q0.25)/n, where q is the th quantile of data with size n. We have evaluated the consistency of five different normalization methods using the average standard deviation of the log ratio estimates (Eq. 8) from the three replicates (Table 3). The smaller the standard deviation, the more consistent the method. Since the estimates in general depend on how weaker signals are eliminated, three thresholds, c = 0, c = 3, and c = 6 in Eq. 5 were evaluated. As shown in Table 3, little difference was observed between the different normalization methods. However, Table 3 demonstrates a flagging threshold of six (c = 6) performs better than a threshold of three (c = 3), and c = 0 is much worse than the other two thresholds in terms of consistency of ratio estimates. Evaluation of different thresholds (c) suggested that the larger the c, the more consistent the results. This is because a stronger signal has less chance of being blurred by the background noise. However, a large c will leave one with a smaller number of genes for analysis. Thus there is a trade-off between consistency and possibly missing a positive. We recommend the threshold of six as a reasonable compromise between the number of remaining genes and the quality of the data.
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j in Eq. 7.2 + '2. The two variances are inseparable. When there are three replicates, we have one s2 for each pair, and we can find 2 for each replica by solving three simultaneous equations. Table 4 shows obtained for all the replicas in each experiment. The data indicate that there is considerable variation between replicates of the experiments. Note the is the standard deviation of the noise ij in the exponent in Eq. 4. To recover the multiplicative noise, we list e and a 95% confidence interval for the ratio.
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j 95% of the time. Table 6 shows that the coverage is very accurate for those spots with a strong signal (1j > 101j). As expected, the confidence intervals for those spots with weaker signals (1j < 61j) are not as good. This implies that genes with strong true expression level (ij > 10ij) are likely to be picked, and the expression ratios can be reliably estimated by the confidence interval formula in Eq. 12.
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j) is greater than 2 or less that 0.5. Others have suggested a threefold threshold. To make this rule general, one could claim a difference in expression if
 | (14) |
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 | (15) |
) and the sensitivity or power (1 - ß, probability of accepting H1 when H1 is true). Determination of the specificity and sensitivity in Eq. 15 is more complicated than the usual hypothesis testing, because these depend on both the expression intensity ij and rj. A reasonable requirement for easily interpretable microarray results would be: 1) if r < R0, then the specificity is at least 1 - , or the false-positive rate is at most ; 2) if r > R1 and the expression level ij > 0, then the sensitivity is at least 1 - ß, where R0, R1, 0, , and ß are user-specified values. Based on 1,000 simulations, Table 7 presents the specificity and sensitivity data for a microarray experiment under various values for c and f when is set at 0.20 and 0.4, a reasonable error range from our experiments (see Table 4). Table 7 shows the trade-off between specificity and sensitivity for various thresholds of c and f as well as the multiplicative error . For example, if the is 0.2 and we want a specificity of 0.99 for r < 1.25 and a sensitivity > 0.85 when r > 3.5 for a strong signal (ij > 10ij), then we should let c = 6 and f = 2.5.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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ij) due to experimental variation from the probe preparation and hybridization efficiency. Similar models have been developed by other investigators. The assumption of multiplicative variation was first proposed by Chen et al. (1). In their model, the likelihood function was based on the null hypothesis that all genes have the same expression. Therefore, the application of their model is limited to known house keeping genes. Kerr et al. (8) proposed an analysis of variance model that is similar to our Eq. 6. However, they did not consider the background noise in their model. In addition, they did not extend Eq. 6 to Eq. 7, and consequently their model requires the independence of the multiplicative noise for sample (1j) and reference (2j) as well as equal variance between experiments. The former assumption is not easy to justify because both errors are from the same spot and the latter assumption was shown not to be a good approximation to the real situation (Table 4). We believe that our model has better assumptions reflecting the true experimental conditions.
Intuitively, the accuracy of microarray data depends on the signal/background ratio of the spots. It is a customized practice to eliminate weak spots on microarrays before subsequent data analyses. However, this practice was based on the user’s discretion, and a statistically sound procedure was lacking. Discovery of the distribution of additive variation allowed us to propose a model-based method to flag spots with weak expression levels, a critical procedure that can influence all subsequent analyses of microarray data such as normalization and ratio estimates. Since the background has a local distribution, the best way to flag weak spots is to identify those spots that have an adjusted signal (t'ij) less than a certain threshold of standard deviation of background intensity difference between the spot and its neighboring spots (cij). We found that the higher the threshold, the more reliable the results. However, increased threshold will decrease the number of genes that can be evaluated. A compromise would be c = 6. We have developed a computer program for this purpose, and it is available upon request. To provide flexibility to the users, we have used a 10-flag scheme with three different c values (c = 2, 4, or 6) for both the sample [channel 1 (CH1)] and reference (CH2): flag 0 (4ij > CH1 > 2ij and 4ij > CH2 > 2ij), flag 1 (CH1 < 2ij and 4ij > CH2 > 2ij); flag 2 (4ij > CH1 > 2ij and CH2 < 2ij); flag 3 (CH1 < 2ij and CH2 < 2ij); flag 4 (6ij > CH1 > 4ij and 6ij > CH2 > 4ij); flag 5 (CH1 < 4ij and 6ij > CH2 > 4ij); flag 6 (6ij > CH1 > 4ij and CH2 < 4ij); flag 7 (CH1 > 6ij and CH2 > 6ij); flag 8 (CH1 < 6ij and CH2 > 6ij); flag 9 (CH1 > 6ij and CH2 < 6ij).
Normalization is the next critical step for analyzing microarray data. Our experiments showed that the threshold for background noise (c) used for elimination of weak spots is the most critical parameter that influences the performance of normalization. Normalization with all weak spots (c = 0) gives high variation for ratio estimates, and it is not acceptable. When c = 6 is chosen, all normalization methods tested in this study performed well, although the trimmed mean of log ratios performed better than the global intensity and mean of ratio.
Since a large number of genes are simultaneously analyzed by microarray experiments, it is critical to assess the effects of multiple decisions. Using simulations based on parameters derived from experimental data, we have assessed the specificity and sensitivity of detecting expression differences depending on several variables including c, , f, and r (Table 7). However, the reproducibility is not solely controlled by these parameters. It also depends on the number of analyzed genes and the distributions of the expression ratios (rj) (or number of genes with different expression vs. number of genes with the same expression) and the true expression levels (ij). For example, assuming that the ratio is 1 for 99% of the genes (r = 1) and the remaining 1% of genes with strong signal expression (ij > 10ij) have a ratio of 3.0, we will observe a number of genes with different expression levels using an elimination threshold of six (c = 6) and cutoff of 2.0 (f = 2.0). Using the numbers in row 8 of Table 7, the chance that one of the chosen genes really has a different expression is, by the usual Bayes’ formula
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The false-positive rate can be dramatically reduced when the experiment is duplicated and only those genes that have different expressions in both hybridizations are considered. The p then becomes
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In summary, we have developed a statistical model that allows investigators to improve the normalization, accuracy of ratio estimates, and evaluation of the experimental variation, as well as sensitivity and specificity of their microarray experiment.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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Address for reprint requests and other correspondence: J.-X. She, Dept. of Pathology, Immunology and Laboratory Medicine, Box 100275, College of Medicine, Univ. of Florida, Gainesville, FL 32610-0275 (E-mail: she{at}ufl.edu).
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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j in Eq. 7 representing 
6