Bioinformatics Advance Access originally published online on September 19, 2007
Bioinformatics 2007 23(21):2881-2887; doi:10.1093/bioinformatics/btm453
Moderated statistical tests for assessing differences in tag abundance
1Department of Medical Biology, The University of Melbourne, Parkville, Victoria 3010 and 2Bioinformatics Division, Walter and Eliza Hall Institute of Medical Research, Parkville, Victoria, Australia
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 PRELIMINARIES3 MODERATED DISPERSION...4 RESULTS5 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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Motivation: Digital gene expression (DGE) technologies measure gene expression by counting sequence tags. They are sensitive technologies for measuring gene expression on a genomic scale, without the need for prior knowledge of the genome sequence. As the cost of sequencing DNA decreases, the number of DGE datasets is expected to grow dramatically.
Various tests of differential expression have been proposed for replicated DGE data using binomial, Poisson, negative binomial or pseudo-likelihood (PL) models for the counts, but none of the these are usable when the number of replicates is very small.
Results: We develop tests using the negative binomial distribution to model overdispersion relative to the Poisson, and use conditional weighted likelihood to moderate the level of overdispersion across genes. Not only is our strategy applicable even with the smallest number of libraries, but it also proves to be more powerful than previous strategies when more libraries are available. The methodology is equally applicable to other counting technologies, such as proteomic spectral counts.
Availability: An R package can be accessed from http://bioinf.wehi.edu.au/resources/
Contact: smyth{at}wehi.edu.au
Supplementary information: http://bioinf.wehi.edu.au/resources/
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 PRELIMINARIES3 MODERATED DISPERSION...4 RESULTS5 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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1.1 DGE technology
DGE technologies measure gene expression by generating sequence tags. A number of DGE technologies are now available, including serial analysis of gene expression (SAGE; Velculescu et al., 1995), massively parallel signature sequencing (MPSS; Brenner et al., 2000), sequencing by synthesis (SBS, Margulies et al., 2005) and polony multiplex analysis of gene expression (PMAGE, Kim et al., 2007). The affordability of DGE seems set for a breakthrough in the next few years. The same technologies that aim to produce a $1000 genome may also be applied to expression profiling studies, since the cost-limiting step until recently has been sequencing. In addition, there are a number of promising sequencing-based approaches that are commercially available for gene expression profiling or will be in the near future (Shaffer, 2007).
There are now several variations on the technique (Wang, 2007), but essentially a DGE system can quantify a snapshot of gene expression, without the necessity of either knowing the gene sequence or designing probe sequences, as is needed for microarrays. In the case of SAGE, messenger RNA (mRNA) is extracted from a sample of interest (e.g. a cancer tumour), reverse transcribed into cDNA, fragmented by an enzyme into small tags from a fixed location within the transcript. These tags are 10–20 bp in length, depending on the protocol. The tags are then sequenced, either by concatenating them and sequencing a stretch of them, or are sequenced in parallel. For each tag, a count of the number of times it was observed is recorded in a library and a larger count is indicative of higher expression. Where an mRNA database or a genome exists, tags can be mapped to a particular mRNA or location in the genome.
SAGE was initially used for determining transcripts expressed in pancreas (Velculescu et al., 1995). Since then, SAGE and its variants have been successful in a number of applications including creating a database of gene expression in human cancers (Lal et al., 1999), discovering prognostic factors in cancer (Aung et al., 2006) and the creation of an atlas of mouse tissue expression (Siddiqui et al., 2005).
Like microarrays, many sequencing-based techniques have applications beyond transcript profiling and we expect the approach developed here to have applications elsewhere. Examples include quantifying micro RNAs (known as miRAGE) (Cummins et al., 2006), copy number analysis (Chen et al., 2002), genome-wide DNA methylation analysis (Hu et al., 2005) and serial analysis of chromatin occupancy (SACO) (Impey et al., 2004).
We focus our attention on the problem of inferring differential expression between two sets of libraries (e.g. cancer versus normal), assuming minimal replication (at least one class has more than one sample). Traditional SAGE is laborious and expensive due to the cost of sequencing. Even with recent developments in high-throughput sequencing, typically most of the ‘real estate’ is given to sequencing more tags, as opposed to, sequencing more libraries (samples or replicates). So, there are rarely large numbers of libraries to compare. For this reason, it is essential that a statistical analysis method be stable in small samples.
Throughout this article, we refer to DGE data. However, our approach should be equally applicable to other count data of this type, such as peptide counts from mass spectrometry data Lu et al. (2007). We have optimized our calculations to the two class comparison problem, but the extension to many classes or accounting for covariates is very straightforward.
Many genome-wide statistical inference methods that share information over all genes, be it in an ad hoc way (Tusher et al., 2001) or via hierarchical models (Smyth, 2004), have proven much more sensitive than standard methods. To the best of our knowledge, this is the first exploration of a moderated test statistic applied to the differential expression analysis of tag count data. The novelty in our method is that we share information over all tags in order to stabilize dispersion estimation in small samples.
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2 PRELIMINARIES |
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TOPABSTRACT1 INTRODUCTION2 PRELIMINARIES3 MODERATED DISPERSION...4 RESULTS5 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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2.1 Differential expression between multiple DGE libraries
Early methods for differential expression between multiple libraries involved pooling the libraries in each class and using the standard two-sample difference in proportions test or the Fisher exact test. As mentioned previously (Baggerly et al., 2003, 2004, Lu et al., 2005), this pooling inadequately deals with the within-class variability and more flexible models have been proposed. A later method (Ryu et al., 2002) computed two-sample t-statistics on the proportions, thereby taking into account the library-to-library variability. However, t-statistics for very small samples when the data are genuinely non-normal can be problematic.
Natural choices for a statistical model for tag counts may be Poisson or Binomial. In practice, the mean-variance relationship of either the Poisson or Binomial distribution may not provide enough flexibility. More variation is typically observed than the model allows, known as overdispersion. Hence, more recent methods have explored the use of beta-binomial (Baggerly et al., 2003) [and more generally, overdispersed logistic (Baggerly et al., 2004)] and overdispersed log-linear (i.e. gamma-Poisson or negative binomial) models (Lu et al., 2005). The simulation studies of (Lu et al.) suggest the negative binomial (NB) assumption can be reliable even with non-NB sampling situations and thus should provide a more flexible framework for real data. For this reason, we make comparisons of our model against that of Lu et al. (2005).
2.2 Statistical framework: negative binomial model
For ease of notation, we first consider a single tag. Let Yij denote the observed count for class i and library j for a particular tag. Here j = 1, ..., ni and for now, we assume just a two-group comparison so that i = 1, 2. A special feature of our analysis is that we require only one of n1 or n2 to be greater than 1. Strictly speaking, previous methods (Baggerly et al., 2003, 2004; Lu et al., 2005) may be able to operate in this setting. However, in the extreme case of 2 libraries versus 1, one-tag-at-a-time inference would require estimation of three parameters from three observations, which is a rather futile exercise.
Assuming an NB distribution for the tag counts Yij, we have:
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is the dispersion. We choose the parameterization such that E(Yij) = µij and Var(Yij) =µij (1 +µ ij ), making = 0 the Poisson distribution.
Let 
i be the true relative abundance of this tag in RNA of class i. Then µij = mij i where mij is the library size for sample j. To assess differences in relative abundance, the null hypothesis H0: 1 = 2 is tested against the two-sided alternative, and this is repeated for each tag.
2.3 Dispersion estimation
Robinson and Smyth (2007) discuss a common dispersion model for SAGE data, which uses all tags to estimate a common dispersion (). The conditional likelihood for a single tag is formed by conditioning on the sum of counts for each class, a straightforward calculation since the sum of identically distributed NB random variables is also NB. The conditioning has the effect of removing the ‘nuisance’ parameter, and is a generalization of restricted maximum likelihood (REML). If the library sizes mij are equal within each class, the single-tag conditional log-likelihood for given 
is:
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| (1) |
where G is the number of tags.
In the real situation of unequal library sizes, the counts are not identically distributed, and the conditioning argument does not hold exactly. Robinson and Smyth (2007) use a quantile adjustment to adjust the observed counts up or down depending on whether the corresponding library sizes are below or above the geometric mean (called qCML for quantile adjusted conditional maximum likelihood). This creates approximately identically distributed pseudodata that can be inserted into Equation (1), summed over all tags and maximized with respect to , resulting in a common estimate. With even as few as 100 tags, the qCML estimate is the least biased over a broad range of conditions among a panel of commonly used estimators (Robinson and Smyth, 2007).
2.4 Statistical testing
For testing the difference in expression between two conditions, we compare two statistical tests in what follows. As a default, we use the Wald test that was used in Lu et al. (2005). The Wald test simply divides
by its estimated standard error. Secondly, we use our previously developed exact test (Robinson and Smyth, 2007).
Briefly, the exact test works as follows. The same quantile adjustment that is used to adjust the tag counts to a common library size for estimation is used to construct the exact test. Using this pseudodata, we again use the fact that a sum of independent and identically distributed NB random variables is also NB. By conditioning on the total pseudosum (an NB random variable), we can calculate the probability of observing counts as or more extreme than what we observed, resulting in an exact P-value.
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3 MODERATED DISPERSION ESTIMATION VIA WEIGHTED LIKELIHOOD |
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TOPABSTRACT1 INTRODUCTION2 PRELIMINARIES3 MODERATED DISPERSION...4 RESULTS5 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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3.1 Weighted conditional likelihood framework
The assumption of common dispersion, as in Robinson and Smyth (2007), offers a significant stabilization, compared with tag-wise estimation, especially in very small samples. However, it is not generally true that each tag has the same dispersion, suggesting that inference can be improved by more sophisticated and less drastic stabilization techniques. For microarray data, empirical Bayesian (EB) hierarchical models have been used to stabilize the variance estimates by sharing structure over all genes (Smyth, 2004). Such strategies are adaptive. If the variances are not very different, the EB model arrives at essentially a pooled estimate. However, if the variances are very different, the EB model shrinks a lesser amount. For our NB model, an EB solution is hampered by the fact that the NB falls outside the exponential family and no conjugate prior for
exists. Bradlow et al. (2002) suggest a polynomial approximation in order to avoid the computational overhead of stochastic Markov Chain inference methods. Instead of enforcing a common dispersion on all tags, we propose instead to squeeze each tag-wise dispersion (denoted as g, with an extra subscript to denote the tag) towards the common value (). We employ weighted likelihood and choose likelihood weights so as to approximate an EB solution.
We define the weighted conditional log-likelihood (WL) for g to be a weighted combination of the individual and common likelihoods:
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| (2) |
is the weight given to the common likelihood. This is a special case of weighted likelihood defined by Wang (2006).
The common likelihood plays the same role in the WL as the prior for g would play in a Bayesian hierarchical model, with the prior precision. If = 0 in (2), then we get tag-wise qCML estimates. At the other extreme, if is chosen sufficiently large, the contributions from any individual log-likelihood is outweighed by the common likelihood and the result is a common dispersion. In between these two extremes lies an estimation scheme where the tag-wise estimates are somewhere between the individual and common estimates.
3.2 Selecting as an approximate EB rule
We wish to select an appropriate that will make the estimation adaptive. If evidence suggests that dispersions are not very different, should be chosen high enough to encourage all tags to shrink strongly towards the common estimate. However, if there is evidence for variable dispersions, should be selected to shrink a lesser amount.
To understand our strategy for selecting , suppose that the qCML individual estimators 
were normally distributed with means g and known variances 
, and assume the hierarchical model:
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g would be: |
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0 and 
are unknown but can be estimated from the marginal distribution of 
to obtain an EB rule. Our strategy is to choose so that WL coincides with this EB rule. Under this idealistic normal model, the maximum WL estimator is: |
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if 0 equals the common dispersion estimator |
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It only remains to have an estimator for 
. Under the normal model, 
, so a consistent estimator of 
0 is obtained by solving
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| (4) |
This rule for choosing is not available to us directly, because the qCML estimators 
are far from normally distributed, do not have known variances, and in fact can take values on the boundary of the sample space at zero or infinity with positive probability. To evade these difficulties, we take advantage of the fact that score statistics (log-likelihood derivatives) converge to normality more rapidly than do maximum-likelihood estimators. We also note that the estimating Equation (4) can be written in terms of the likelihood score Sg() = 
lg()/ and expected information Ig() = E(Jg), Jg = – 2lg()/ 2, functions for g. This allows us to state our estimation algorithm as follows.
- Find the common dispersion estimator
which maximizes lC.
- Evaluate
and 
for each tag.
- Estimate 0 by solving
If
then 0 = 0.
- Set
- Obtain weighted likelihood estimators
by maximizing WL(g).
to estimate 0.
The expected informations Ig are difficult to compute directly, but can be well approximated using the observed informations Jg. For any given value of g, Ig should be very nearly directly proportional to the total count z1 + z2. Hence, we compute the linear regression with intercept zero of Jg on the total pseudo-count (see Fig. 1), and use the fitted values to represent Ig.
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The algorithm can actually be applied to any transformation of
. We have found it convenient to implement the algorithm on the 
= /( + 1) scale because takes strictly bounded values.
3.3 Interpretation of the approach
The above algorithm has a nice statistical interpretation. If the dispersions are truly equal (all g = 0), then 
so that 
will be estimated close to zero and hence will be large. If, however, the dispersions are truly different, then E(Sg) will be non-zero and 
will be greater than Ig on average, forcing 
to be greater than zero and less weight is given to the common likelihood. The more dissimilar the dispersions are, the greater 
will be estimated and the less shrinkage is done. The fact that 
under the null hypothesis is an exact result which does not rely on asymptotic normality. This ensures that our algorithm has good qualitative behaviour even when the number of libraries is small.
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4 RESULTS |
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TOPABSTRACT1 INTRODUCTION2 PRELIMINARIES3 MODERATED DISPERSION...4 RESULTS5 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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4.1 Squeezing improves estimation of dispersion in the negative binomial model
For the NB model of tag counts, estimation of
is a crucial step and can have an impact on the determined significance of differential expression. Note that does not have as direct an influence on statistical tests as the variance does in microarrays, since the variance in our data is also a function of the mean. Lu et al. (2005) use a PL model for the estimation of and estimate separately for each tag. Here, we calculate a common dispersion over all tags and shrink the tag-wise dispersions towards it, in a novel approximate EB strategy.
We first show that our approximate EB approach improves overall estimation of the dispersions in terms of mean squared error (MSE). We compare four estimation strategies: tag-wise qCML, WL using the approximate EB rule, common qCML (Robinson and Smyth, 2007) and tag-wise PL (Lu et al., 2005), over three true situations. For all comparisons here, we fixed the library size at 50 000 and means at 10 ( = 0.0002), sampled 1000 tags from NB and repeated the simulation 50 times. Overall MSE was calculated for each simulation. MSEs are calculated on the 
scale, since there is a non-zero probability of an infinite tag-wise qCML estimate. The first situation considered is a medium number of libraries (n = 4) and a fixed dispersion, shown in Figure 2A. This situation obviously favours a common dispersion estimate, and the common dispersion estimate has the lowest MSE. But, note that the approximate EB strategy does well here also, giving enough weight to the common model so as to squeeze the tag-wise dispersions almost entirely to the common value. The remaining two situations involve random dispersions, here taken from a gamma distribution. We picked gamma parameters for the simulation to match the empirical dispersion estimate distribution (using approximate EB estimates) on the Zhang dataset (Zhang et al., 1997), and used a medium (n = 4) and large number of libraries (n = 10) (Fig. 2B and C, respectively). We can see that the approximate EB solution provides a significant advantage for estimating dispersions and adapts to the situation, showing the relevance of the evidence contained in the scores and informations.
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The weights adapt well to the situation. In the case of fixed dispersions, the weights are large enough to shrink estimates almost to the common value. In the face of more dispersed true dispersions, the weights decrease, as expected. In the case of random dispersion and a larger sample, again the weights decrease to automatically adjust for having more information contained in the tag-wise estimates, showing the approximate EB system seems to be achieving what is expected and therefore provides a suitable rule in practice.
4.2 Comparison of methods: simulated data
Improvements in dispersion estimation can improve our ability to separate the differentially expressed (DE) tags from non-DE. We repeat a subset of the simulation study of Lu et al. (2005) and consider an extended, more realistic study.
The simulation in Lu et al. (2005) considered sampling 10 000 tags under two conditions, with a fixed 1, libraries sizes sampled uniformly between 30 000 and 90 000, comparing 5 libraries of 1–5 libraries of the other. For 5000 tags, an implanted difference of 2 = b.1 is used and the remaining tags have no difference (2 = 1). When sampling from NB, they choose fixed dispersions at 0.17, 0.42 and 0.95. We repeated their performance analysis for 1 = 0.0002 and b = 4, which is directly comparable to Figure 2 of Lu et al. (2005). In Figure 3 in this article, we compare receiver-operating-characteristic (ROC) curves using the Wald test statistic they used, both for their PL estimate and our shrunken estimate. That is, we fit the generalized linear model of Lu et al. (2005) with our estimates to show that an improved estimator is beneficial. In all cases, an improved estimator for the dispersions improves the ability to separate the truly DE and non-DE tags. Here, there is very little difference between the exact and Wald tests.
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Next, we extend their simulation study in a number of simple yet important ways. First, we make the problem more realistic by having non-fixed dispersions. Again, we set the random dispersions to the gamma-approximated empirical estimate distribution (shape = 0.85, scale = 0.5). Instead of a fixed
, we use the empirical distribution of estimates for the Zhang dataset and assign the gamma-sampled dispersions to the s at random. A subtle change we make to our simulation is that the multiplier of the implanted differences, b, does not always increase the true means, since larger counts lead to easier estimation problems. Instead, for the tags sampled with true differences, we use 
and 
as the true proportions. Finaly, we consider a more realistic 10% differentially expressed tags and compare small (n1 = n2 = 2) and moderate (n1 = n2 = 5) numbers of libraries. Instead of ROC curves, we prefer false discovery (FD) plots since they highlight the performance at the top ranked tags. Figure 4 shows FD plots for four situations: small and medium true difference (b = 4, 8) and small and medium number of libraries. Tags are ranked according to their test-statistic (Wald) or P-value (approximate exact test). Since there are 1000 truly DE tags, the top 1000 are selected (x-axis) and the number of false detections is plotted (y-axis) at each point. Fewer FDs is obviously preferred. We see that using the approximate EB estimate for dispersion estimation results in fewer FDs on average in all cases, regardless of the statistical test being used. If we were to validate the top 200 tags, e.g. the ratio of ‘Exact.EB’ to ‘Wald.PL’ FDs is 80/95, 42/59, 31/35 and 3/7 for the four cases presented in Figure 4, thus showing a consistent and practically meaningful improvement.
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Of course, this only suggests the statistics are in a desirable order. Since the exact test does not rely on asymptotics for its distributional assumptions, it is best able to achieve a set false positive rate, allowing one to set reasonable cutoffs, presumably after adjusting for multiple testing. In a study of false positive rates for small NB samples, Robinson and Smyth (2007) demonstrate that the exact test is best able to achieve the nominal false positive rate and in fact, the Wald test has the highest false positive rate of all the asymptotic tests. Permutation tests (Tusher et al., 2001) with such few libraries are unlikely to create a reasonable null distribution.
4.3 Application to SAGE data
We apply the method to the SAGE data from Zhang et al. (1997), since true biological replicates are available. Our comparison is just between two libraries of normal colon to two libraries of colon tumour. Figure 5A shows the both the dispersion and proportion () estimates, assuming no difference between the sets of libraries. Here, we see that at low abundance, there are a small number of tags with large dispersion. Most of these are cases where one of two replicate counts is zero and the other is non-zero, and in some cases, the tags counts are all zero in one condition and have only one non-zero in the other. In these cases, the one-tag-at-a-time qCML estimate is = 
( = 1), so the squeezing towards the common estimator is essential. Note that it is not surprising that there is a decreasing trend of dispersions as the abundance increases, since the variance of the observations is a function of the mean.
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Applying the exact test to the comparison of normal colon samples to colon tumours, we find that 49 genes are up-regulated and 115 genes are down-regulated in tumours, at a 5% false discovery rate (using a Benjamini–Hochberg correction). Figure 5B shows the analogous plot to an ‘MA’ plot for microarray data, where the x-axis is indicative of abundance and the y-axis show the magnitude of the change between the two conditions.
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5 CONCLUSION |
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TOPABSTRACT1 INTRODUCTION2 PRELIMINARIES3 MODERATED DISPERSION...4 RESULTS5 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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Estimation of dispersion for NB data is critical for assessing the significance of changes in the mean. For tag count data with even the most minimal amount of replication, we have introduced a weighted conditional likelihood estimator that squeezes individual tag-wise dispersions towards the common dispersion. The procedure can be thought of as using a data-dependent prior and finding the maximum a posteriori estimate, or simply as weighted likelihood. We choose the amount of shrinkage according to an approximate empirical Bayes rule. The EB rule works well in practice, adapting to the true similarity of the dispersions, according to sample size and the evidence contained in the first and second derivative of the conditional log-likelihood (scores and informations). The increased precision in estimating dispersion gives an increase in power for testing between experimental conditions. The exact test performs as well or better than the Wald test for testing differences between two experimental conditions. The exact test has the added advantage that it achieves close to its nominal error rates.
Our weighted likelihood shrinkage algorithm is of very general application, requiring only the log-likelihood function and its first two derivatives evaluated at a common parameter estimate. This approach may prove useful in a number of other genome-scale estimation and inference problems.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 PRELIMINARIES3 MODERATED DISPERSION...4 RESULTS5 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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The authors wish to thank Terry Speed for valuable discussions. This research was supported by NHMRC Program Grant 406657.
Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: David Rocke
Received on June 18, 2007; revised on July 16, 2007; accepted on August 27, 2007
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