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The resulting MAP LFCs are biased toward zero in a manner that removes the problem of exaggerated LFCs for low counts. As Figure 2B shows, the strongest LFCs are no longer exhibited by genes with weakest expression. Rather, the estimates are more evenly spread around zero, and for very weakly expressed genes (with less than one read per sample on average), LFCs hardly deviate from zero, reflecting that accurate LFC estimates are not possible here.

The strength of shrinkage does not depend simply on the mean count, but rather on the amount of information available for the fold change estimation (as indicated by the observed Fisher information; see Materials and methods). Two genes with equal expression strength but different dispersions will experience a different amount of shrinkage (Figure 2C,D). The shrinkage of LFC estimates can be described as a bias-variance trade-off [18]: for genes with little information for LFC estimation, a reduction of the strong variance is bought at the cost of accepting a bias toward zero, and this can result in an overall reduction in mean squared error, e.g., when comparing to LFC estimates from a new dataset. Genes with high information for LFC estimation will have, in our approach, LFCs with both low bias and low variance. Furthermore, as the degrees of freedom increase, and the experiment provides more information for LFC estimation, the shrunken estimates will converge to the unshrunken estimates. We note that other Bayesian efforts toward moderating fold changes for RNA-seq include hierarchical models [8],[19] and the GFOLD (or generalized fold change) tool [20], which uses a posterior distribution of LFCs.

The shrunken MAP LFCs offer a more reproducible quantification of transcriptional differences than standard MLE LFCs. To demonstrate this, we split the Bottomly et al. samples equally into two groups, I and II, such that each group contained a balanced split of the strains, simulating a scenario where an experiment (samples in group I) is performed, analyzed and reported, and then independently replicated (samples in group II). Within each group, we estimated LFCs between the strains and compared between groups I and II, using the MLE LFCs (Figure 3A) and using the MAP LFCs (Figure 3B). Because the shrinkage moves large LFCs that are not well supported by the data toward zero, the agreement between the two independent sample groups increases considerably. Therefore, shrunken fold-change estimates offer a more reliable basis for quantitative conclusions than normal MLEs.

We demonstrate the use of the rlog transformation on the RNA-seq dataset of Hammer et al. [26], wherein RNA was sequenced from the dorsal root ganglion of rats that had undergone spinal nerve ligation and controls, at 2 weeks and at 2 months after the ligation. The count matrix for this dataset was downloaded from the ReCount online resource [27]. This dataset offers more subtle differences between conditions than the Bottomly et al. [16] dataset. Figure 5 provides diagnostic plots of the normalized counts under the ordinary logarithm with a pseudocount of 1 and the rlog transformation, showing that the rlog both stabilizes the variance through the range of the mean of counts and helps to find meaningful patterns in the data.

False positive rate To evaluate the false positive rate of the algorithms, we considered mock comparisons from a dataset with many samples and no known condition dividing the samples into distinct groups. We used the RNA-seq data of Pickrell et al. [17] for lymphoblastoid cell lines derived from unrelated Nigerian individuals. We chose a set of 26 RNA-seq samples of the same read length (46 base pairs) from male individuals. We randomly drew without replacement ten samples from the set to compare five against five, and this process was repeated 30 times. We estimated the false positive rate associated with a critical value of 0.01 by dividing the number of P values less than 0.01 by the total number of tests; genes with zero sum of read counts across samples were excluded. The results over the 30 replications, summarized in Figure 7, indicated that all algorithms generally controlled the number of false positives. DESeq (old) and Cuffdiff 2 appeared overly conservative in this analysis, not using up their type-I error budget.

To recover the desirable symmetry between all levels, DESeq2 uses expanded design matrices, which include an indicator variable for each level of each factor, in addition to an intercept column (i.e., none of the levels is absorbed into the intercept). While such a design matrix no longer has full rank, a unique solution exists because the zero-centered prior distribution (see below) provides regularization. For dispersion estimation and for estimating the width of the LFC prior, standard design matrices are used.

To incorporate empirical Bayes shrinkage of LFCs, we postulate a zero-centered normal prior for the coefficients β ir of model (2) that represent LFCs (i.e., typically, all coefficients except for the intercept β i0):

Fisher information. The effect of the zero-centered normal prior can be understood as shrinking the MAP LFC estimates based on the amount of information the experiment provides for this coefficient, and we briefly elaborate on this here. Specifically, for a given gene i, the shrinkage for an LFC β ir depends on the observed Fisher information, given by

If the size factors are not equal across samples, but not correlated with condition, conditioning on the mean of normalized counts should also provide uniformly distributed p as with conditioning on the mean of counts, K ̄ i . We may consider a pathological case where the size factors are perfectly confounded with condition, in which case, even under the null hypothesis, genes with low mean count would have non-uniform distribution of p, as one condition could have positive counts and the other condition often zero counts. This could lead to non-uniformity of p under the null hypothesis; however, such a pathological case would pose problems for many statistical tests of differences in mean.

Note that while a zero-centered prior on LFCs is consistent with testing the null hypothesis of small LFCs, it should not be used when testing the null hypothesis of large LFCs, because the prior would then favor the alternative hypothesis. DESeq2 requires that no prior has been used when testing the null hypothesis of large LFCs, so that the data alone must provide evidence against the null hypothesis.

The rlog transformation is calculated as follows. The experimental design matrix X is substituted with a design matrix with an indicator variable for every sample in addition to an intercept column. A model as described in Equations (1) and (2) is fit with a zero-centered normal prior on the non-intercept terms and using the fitted dispersion values α tr ( μ ̄ ), which capture the overall variance-mean dependence of the dataset. The true experimental design matrix X is then only used in estimating the variance-mean trend over all genes. For unsupervised analyses, for instance sample quality assessment, it is desirable that the experimental design has no influence on the transformation, and hence DESeq2 by default ignores the design matrix and re-estimates the dispersions treating all samples as replicates, i.e., it uses blind dispersion estimation. The rlog-transformed values are the fitted values,

where β ij is the shrunken LFC on the base 2 scale for the jth sample. The variance of the prior is set using a similar approach as taken with differential expression, by matching a zero-centered normal distribution to observed LFCs. First a matrix of LFCs is calculated by taking the logarithm (base 2) of the normalized counts plus a pseudocount of 1 2 for each sample divided by the mean of normalized counts plus a pseudocount of 1 2 . The pseudocount of 1 2 allows for calculation of the logarithmic ratio for all genes, and has little effect on the estimate of the variance of the prior or the final rlog transformation. This matrix of LFCs then represents the common-scale logarithmic ratio of each sample to the fitted value using only an intercept. The prior variance is found by matching the 97.5% quantile of a zero-centered normal distribution to the 95% quantile of the absolute values in the LFC matrix.

Reads were aligned using the TopHat2 aligner [65], and assigned to genes using the summarizeOverlaps function of the GenomicRanges package [60]. The sequence read archive fastq files of the Pickrell et al. [17] dataset (accession number [SRA:SRP001540]) were aligned to the Homo sapiens reference sequence GRCh37 downloaded in March 2013 from Illumina iGenomes. Reads were counted in the genes defined by the Ensembl GTF file, release 70, contained in the Illumina iGenome. The sequence read archive fastq files of the Bottomly et al. [16] dataset (accession number [SRA:SRP004777]) were aligned to the Mus musculus reference sequence NCBIM37 downloaded in March 2013 from Illumina iGenomes. Reads were counted in the genes defined by the Ensembl GTF file, release 66, contained in the Illumina iGenome.

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