Green Bay and Western Railroad: The Rise and Fall of a Wisconsin Railroad
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High-throughput sequencing assays such as RNA-Seq, ChIP-Seq or barcode counting provide quantitative readouts in the form of count data. To infer differential signal in such data correctly and with good statistical power, estimation of data variability throughout the dynamic range and a suitable error model are required. We propose a method based on the negative binomial distribution, with variance and mean linked by local regression and present an implementation, DESeq, as an R/Bioconductor package.
High-throughput sequencing of DNA fragments is used in a range of quantitative assays. A common feature between these assays is that they sequence large amounts of DNA fragments that reflect, for example, a biological system's repertoire of RNA molecules (RNA-Seq [1, 2]) or the DNA or RNA interaction regions of nucleotide binding molecules (ChIP-Seq , HITS-CLIP ). Typically, these reads are assigned to a class based on their mapping to a common region of the target genome, where each class represents a target transcript, in the case of RNA-Seq, or a binding region, in the case of ChIP-Seq. An important summary statistic is the number of reads in a class; for RNA-Seq, this read count has been found to be (to good approximation) linearly related to the abundance of the target transcript . Interest lies in comparing read counts between different biological conditions. In the simplest case, the comparison is done separately, class by class. We will use the term gene synonymously to class, even though a class may also refer to, for example, a transcription factor binding site, or even a barcode .
We would like to use statistical testing to decide whether, for a given gene, an observed difference in read counts is significant, that is, whether it is greater than what would be expected just due to natural random variation.
If reads were independently sampled from a population with given, fixed fractions of genes, the read counts would follow a multinomial distribution, which can be approximated by the Poisson distribution.
Consequently, the Poisson distribution has been used to test for differential expression [6, 7]. The Poisson distribution has a single parameter, which is uniquely determined by its mean; its variance and all other properties follow from it; in particular, the variance is equal to the mean. However, it has been noted [1, 8] that the assumption of Poisson distribution is too restrictive: it predicts smaller variations than what is seen in the data. Therefore, the resulting statistical test does not control type-I error (the probability of false discoveries) as advertised. We show instances for this later, in the Discussion.
To address this so-called overdispersion problem, it has been proposed to model count data with negative binomial (NB) distributions , and this approach is used in the edgeR package for analysis of SAGE and RNA-Seq [8, 10]. The NB distribution has parameters, which are uniquely determined by mean μ and variance σ2. However, the number of replicates in data sets of interest is often too small to estimate both parameters, mean and variance, reliably for each gene. For edgeR, Robinson and Smyth assumed  that mean and variance are related by σ2 = μ + αμ2, with a single proportionality constant α that is the same throughout the experiment and that can be estimated from the data. Hence, only one parameter needs to be estimated for each gene, allowing application to experiments with small numbers of replicates.
In this paper, we extend this model by allowing more general, data-driven relationships of variance and mean, provide an effective algorithm for fitting the model to data, and show that it provides better fits (Section Model). As a result, more balanced selection of differentially expressed genes throughout the dynamic range of the data can be obtained (Section Testing for differential expression). We demonstrate the method by applying it to four data sets (Section Applications) and discuss how it compares to alternative approaches (Section Conclusions).
which has two parameters, the mean μij and the variance σ i j 2 . The read counts Kij are non-negative integers. The probabilities of the distribution are given in Supplementary Note A. (All Supplementary Notes are in Additional file 1.) The NB distribution is commonly used to model count data when overdispersion is present .
In practice, we do not know the parameters μij and σ i j 2 , and we need to estimate them from the data. Typically, the number of replicates is small, and further modelling assumptions need to be made in order to obtain useful estimates. In this paper, we develop a method that is based on the following three assumptions.
First, the mean parameter μij , that is, the expectation value of the observed counts for gene i in sample j, is the product of a condition-dependent per-gene value qi, ρ(j)(where ρ(j) is the experimental condition of sample j) and a size factor sj,
qi,ρ(j)is proportional to the expectation value of the true (but unknown) concentration of fragments from gene i under condition ρ(j). The size factor sjrepresents the coverage, or sampling depth, of library j, and we will use the term common scale for quantities, such as qi, ρ(j), that are adjusted for coverage by dividing by sj.
This assumption is needed because the number of replicates is typically too low to get a precise estimate of the variance for gene i from just the data available for this gene. This assumption allows us to pool the data from genes with similar expression strength for the purpose of variance estimation.
The decomposition of the variance in Equation (3) is motivated by the following hierarchical model: We assume that the actual concentration of fragments from gene i in sample j is proportional to a random variable Rij , such that the rate that fragments from gene i are sequenced is sjrij . For each gene i and all samples j of condition ρ, the Rij are i.i.d. with mean qiρ and variance viρ . Thus, the count value Kij , conditioned on Rij = rij , is Poisson distributed with rate sjrij . The marginal distribution of Kij - when allowing for variation in Rij - has the mean μij and (according to the law of total variance) the variance given in Equation (3). Furthermore, if the higher moments of Rij are modeled according to a gamma distribution, the marginal distribution of Kij is NB (see, for example, , Section 4.2.2).
We now describe how the model can be fitted to data. The data are an n m table of counts, kij , where i = 1,..., n indexes the genes, and j = 1,..., m indexes the samples. The model has three sets of parameters: