Corrigendum: Differential expression analysis of RNA-seq data at single-base resolution

SUMMARY RNA-sequencing (RNA-seq) is a flexible technology for measuring genome-wide expression that is rapidly replacing microarrays as costs become comparable. Current differential expression analysis methods for RNA-seq data fall into two broad classes: (1) methods that quantify expression within the boundaries of genes previously published in databases and (2) methods that attempt to reconstruct full length RNA transcripts. The first class cannot discover differential expression outside of previously known genes. While the second approach does possess discovery capabilities, statistical analysis of differential expression is complicated by the ambiguity and variability incurred while assembling transcripts and estimating their abundances. Here, we propose a novel method that first identifies differentially expressed regions (DERs) of interest by assessing differential expression at each base of the genome. The method then segments the genome into regions comprised of bases showing similar differential expression signal, and then assigns a measure of statistical significance to each region. Optionally, DERs can be annotated using a reference database of genomic features. We compare our approach with leading competitors from both current classes of differential expression methods and highlight the strengths and weaknesses of each. A software implementation of our method is available on github (https://github.com/alyssafrazee/derfinder).

in experiments with small sample sizes. To be more specific, we present some of the details from Smyth and others (2004) here; further details can be found in that paper. For ease of notation, since we are in two-group case, we drop the "2" subscript from β 2 (l) in the following discussion. Following the framework from Smyth and others (2004), we assume a distribution on the estimated differential expression effect at base l: where σ 2 l represents the residual variance and v l represents the unscaled variance at base l. We also assume a distribution on the estimated residual variance for the model at base l, assuming d l is the residual degrees of freedom for that model: Then, a prior with parameters s 2 0 and d 0 is assumed on σ 2 l : The prior describes how variances are expected to vary across bases. A prior is also assumed on β(l) when β(l) = 0: This prior describes the distribution of differential expression parameters (here, log fold-changes) for differentially expressed bases. Under these priors, the posterior mean of σ −2 l given s 2 l iss −2 l , where:s Our test statistic s(l), here the moderated t-statistic at base l, is then defined by: This empirical Bayes approach, where the posterior variance is used in the t-statistic calulation instead of the sample variance, is implemented in the eBayes function in limma. Data-driven estimation of the values of d 0 and s 2 0 is built into the eBayes function, as described in Section 6 of Smyth and others 2004. This function has been incorporated into derfinder 's functions.
The Hidden Markov Model is then fit on the moderated t-statistics. In the general case, described in the main text, we assume a three-state Markov process D along genomic locations l, such that D(l) = 0 when base l is not expressed, D(l) = 1 when base l is equally expressed between conditions, and D(l) = 2 when base l is differentially expressed. However, in our implementation, we found it convenient to divide the differentialy expressed state into two separate states. So in derfinder, we define D(l) = 0 and D(l) = 1 the same way we do in the general case, but we assume here that D(l) = 2 corresponds to overexpression of base l in cases (compared to controls) and D(l) = 3 corresponds to underexpression of base l in cases.
The parameters left to estimate are π d = P r(D(l) = d), µ d , and σ 2 d for d = 0, 1, 2, 3. Recall that we assume s(l) | D(l) = d ∼ N (µ d , σ 2 d ). We estimate π 0 as the fraction of bases where average coverage is less than some threshold c, as described in the text. Estimates of π 1 , π 2 , and π 3 are obtained from the maximum likelihood approach of the two-groups model (Efron 2008).
Define n to be the total number of nonzero t-statistics that were generated from differential expression tests. Define the function n.above(x) as the observed number of nonzero t-statistics greater than x. Also define the function c(p) =σ e Φ −1 (p) + µ 1 , where Φ represents the cumulative distribution function of the standard normal distribution. Note that for p ∈ [0, 1], c(p) yields the 100p th percentile of the normal distribution for the equally expressed t-statistics. Using an iterative procedure, and using our estimate for π 0 , we find the value p ∈ [0, 1] such that The reason behind finding this p is as follows: note that 0.25(1 − π 0 )n is the estimate of half the number of nonzero t-statistics corresponding to bases with D(l) = 2: (1 − π 0 )n is the estimated number of differentially expressed bases (D(l) = 2 or 3), half of those have D(l) = 2, and we multiply by 0.5 again to get half that quantity. Also note that (1 − p)π 0 n gives the expected number of equally expressed t-statistics greater than c(p). Thus, the difference between the number of observed t-statistics greater than c(p) and (1 − p)π 0 n should yield the number of t-statistics with D(l) = 2 that are greater than c(p). When we find a p such that this difference equals half the estimated number of t-statistics with D(l) = 2, we can use c(p) as an estimate for the median of the distribution of overexpressed t-statistics. Since we assume this distribution is normal, c(p) also provides an estimate for its mean, µ 2 .
We can use µ 2 to estimate σ 2 2 : assume that p solves 1.2 above, and choose any value p ′ in (p, 1). Define the quantity: The numerator of the fraction in 1.3 gives the estimated number of overexpressed t-statistics greater than c(p ′ ), and the denominator gives the estimated total number of t-statistics with D(l) = 2. Therefore, q denotes what percentile of of the distribution of s(l) | D(l) = 2 is given by c(p ′ ). Then, since we know Φ −1 (q), c(p ′ ), and µ 2 , we can solve the equation for the unknown σ 2 . We then estimate µ 3 and σ 2 3 analagously.
Simulation studies comparing our data-driven method to an EM algorithm, implemented with the mclust package (Fraley and Raftery 2002), suggest that this algorithm is more conservative (i.e., distributions for s(l) | D(l) = 2 and s(l) | D(l) = 3 are estimated to be further from the distribution of s(l) | D(l) = 1) and more computationally efficient than the EM algorithm.
Using all these pre-set and estimated parameters, the HMM is fit in derFinder using a Viterbi algorithm (Forney Jr 1973). In derfinder, the dthmm and Viterbi functions from the Hidden-Markov package are utilized (Harte 2012). By default, a non-stationary, homogenous HMM is fit (non-stationarity is the default in dthmm), though the user may fit a stationary HMM for improved computational efficiency. The model outputs the most likely state for each base-pair in the genome given the observed t-statistics.
Runtime. We suggest running DER Finder's statistical analysis (beginning with the coverage matrix as input) for each chromosome separately, since this enables the pipeline to be parallelized.
The Y chromosome presented in this manuscript took about 1 hour to run. Larger chromosomes take longer: statistical analyses of chromosomes 1 and 12 took about 27 hours and 8 hours, respectively, when the HMMs were fit as non-stationary models. Assuming stationarity had very little effect on runtime. Efforts to make the derfinder software more efficient are ongoing -so future releases may have improved runtimes.
Correction to previous section (Supplement, Section 1) In the original supplement, section 1 (above), imprecise notation was used in the description of the method for estimating µ 2 , σ 2 2 , µ 3 , and σ 2 3 . Beginning at the paragraph immediately preceding equation (1.2), the supplementary material should read as follows: Define n to be the total number of nonzero t-statistics that were generated from differential expression tests. The two-groups model is only run on these n t-statistics, which means that it gives a direct estimate of what we will call π 0nz , i.e., the percentage of nonzero t-statistics with true state D(l) = 1. Then π 1 is estimated as π 0nz (1 −π 0 ), whereπ 0 is the empirical estimate of the percentage of t-statistics equal to 0).
Next, define the function n.above(x) as the observed number of nonzero t-statistics greater than x. Also define the function c(p) =σ 1 Φ −1 (p) + µ 1 , where Φ represents the cumulative distribution function of the standard normal distribution. Note that for p ∈ [0, 1], c(p) yields the 100p th percentile of the normal distribution for the equally expressed t-statistics, i..e, t-statistics emitted from bases with hidden state D(l) = 1. Using an iterative procedure, and using our estimate for π 0nz , we find the value p ∈ [0, 1] such that n.above[c(p)] − (1 − p)π 0nz n = 0.25(1 − π 0nz )n (1.5) The reason behind finding this p is as follows: note that 0.25(1 − π 0nz )n is the estimate of half the number of nonzero t-statistics corresponding to bases with D(l) = 2: (1 − π 0nz )n is the estimated number of differentially expressed bases (D(l) = 2 or 3), half of those have D(l) = 2, and we multiply by 0.5 again to get half that quantity. Also note that (1 − p)π 0nz n gives the expected number of equally expressed t-statistics (D(l) = 1) greater than c(p). Thus, the difference between the number of observed t-statistics greater than c(p) and (1 − p)π 0nz n should yield the number of t-statistics with D(l) = 2 that are greater than c(p). When we find a p such that this difference equals half the estimated number of t-statistics with D(l) = 2, we can use c(p) as an estimate for the median of the distribution of overexpressed t-statistics. Since we assume this distribution is normal, c(p) also provides an estimate for its mean, µ 2 .
We can use µ 2 to estimate σ 2 2 : assume that p solves 1.5 above, and choose any value p ′ in (p, 1). Define the quantity: The numerator of the fraction in 1.6 gives the estimated number of overexpressed t-statistics greater than c(p ′ ), and the denominator gives the estimated total number of t-statistics with D(l) = 2. Therefore, q denotes what percentile of of the distribution of s(l) | D(l) = 2 is given by c(p ′ ). Then, since we know Φ −1 (q), c(p ′ ), and µ 2 , we can solve the equation for the unknown σ 2 . We then estimate µ 3 and σ 2 3 analagously.
These changes do not affect the results presented in the manuscript, but they clarify the details of the methods implemented in the manuscripts analysis and in the beta version of the DER Finder R package.

HMM Assumptions
Here we provide explanations and empirical evidence regarding the assumptions made in the HMM step in the DER Finder pipeline. Markov model, so we believe the first-order model is sufficient for this analysis.

Stationarity and Homogeneity
By default, we assume that D(l) is a non-stationary, homogeneous Markov chain with hidden state probabilities π d = P r(D(l) = d). However, the user may assume stationarity (i.e., constant transition and probabilities across the genome) to improve computation time, i.e., he or she may assume the transition probabilities are the same across the genome. Assuming homogeneity means we assume the parameters of the mixture distribution generating the test statistics are the same across the genome. Both these assumptions seem reasonable: even though gene density differs across the genome, the probability of staying in the same state is quite high even in areas of high gene density (i.e., transitions between states are relatively rare), since genomic features like exons and introns tend to be hundreds of bases long, but only the two bases at the beginning and end of the feature will actually show the state changing. Therefore, fitting a stationary HMM is a reasonable way to speed up computation time (though the non-stationary model can also be fit for analyses where it is computationally feasible, such as the one presented in the manuscript).
Also, along the entire genome, a test statistic high in absolute value should indicate differential expression, while a test statistic low in absolute value indicates no differential expression, so using the same parameters for the test statistics' mixture distribution seems reasonable. If the user is particularly concerned about violations of these assumptions, separate HMMs (with different parameters and transition probabilities) can be fit on different sections of the genome. A nonstationary, non-homogenous HMM could also be implemented, but functions for this may not be available off the shelf, and computation time is likely to be greatly increased. Another option is to implement an alternative segmentation algorithm, such as circular binary segmentation (Olshen and others 2004). This segmentation option is available in the region-finding function (called getRegions) in the derfinder R package.

Test Statistic Distribution
We assume that the test statistic at base l, s(l) has latent state D(l) = 1, 2, or 3, and is a draw from a normal distribution, i.e., s(l) | D(l) = d ∼ N (µ d , σ 2 d ). We choose this normal distribution because the pre-built functions in the HiddenMarkov R package (Harte 2012) provided the computational framework for fitting this HMM, and because the observed distribution of test statistics seemed well-captured by a normal mixture distribution. As empirical evidence, we consider the test statistics obtained from the Y chromosome analysis presented in the paper (Supplementary Figure 4): the normal mixture distribution estimated using the process described in section 1 seems to fit the observed data quite well, though the distribution of the underexpressed statistics overlaps almost entirely with that of the equally expressed statistics, which is to be expected in Y chromosome data. We also investigated the effect of the prior estimate for π 1 , the proportion of base-pairs that are not differentially expressed, using the simulated data described in Section 7.1: there, we set 90% of transcripts to be differentially expressed, and DER Finder produced exactly the same results usingπ 1 = 0.8,π 1 = 0.9, andπ 1 = 0.98, the latter being the conservative estimate from the two-groups model. In general, we expect DER Finder to be quite robust to choice of parameters for the test statistic distribution: as long as large test statistics are classified as differentially expressed and test statistics close to zero are classified as not differentially expressed in a systematic manner, DER Finder will produce reasonable results.

Validity of p-value and FDR estimates
DER Finder assignes a measure of statistical significance to each candidate DER using a permutation p-value, as described in Section 3.3 in the main text. Each candidate DER is assigned a test statistic, defined as the mean base-level statistic over all bases contained in the region.
To estimate the null distribution of region-level test statistics, permutation is used, and the null distribution is created by pooling null statistics from the entire genome. Using permutation with pooled null statistics is standard practice, first introduced in Storey and Tibshirani (2003). It has been demonstrated that strong control of the FDR and FWER are guaranteed when a subset pivotality condition holds (see, e.g., Dudoit and Van Der Laan 2008). The subset pivotality requires that for any subset of the null hypotheses, the joint distribution of the p-values for the subset is identical to that under the complete null (Westfall and others 1993). This condition holds provided that the p-values under the null hypothesis are jointly uniform (see e.g. Leek and Storey 2011). Further justification for our approach is that this type of permutation procedure has been thoroughly studied both empirically and theoretically and is widely applied in the analysis of fMRI data: see for example, Genovese and others (2002) or Nichols and Holmes (2002).
We show empirically that our null p-values are uniformly distributed and that our estimated FDRs are conservative: using the simulated dataset described in Section 6.1, we analyzed all p-values from regions known to contain no differentially expressed bases and found that the distribution was approximately uniform (Supplementary Figure 5). Additionally, the true FDR in the simulation study at a q-value cutoff of 0.05 was 0, meaning our FDR estimate of 0.05 was indeed conservative. A false discovery in this case would be defined as calling a region differentially expressed when it did not overlap a transcript set to be differentially expressed.
We can also use the Y-chromosome experiment to show that the p-values and FDR adjustments used by DER Finder's permutation test are reasonable: for the Y-chromosome data, p-value histograms for each method were created (Supplementary Figure 6). For Cufflinks/Cuffdiff, we used Cufflinks version 2.0.2 for transcript assembly and Cuffdiff version 2.0.2 for differential expression analysis. Default parameters were used for both steps.
Detailed commands used are available upon request from the corresponding authors.
The main model for DER Finder (model 3.1) was fit as follows: g was defined as the function g(x) = log 2 (x + 32), X 2i = 1 if sample i was male and 0 if sample i was female (this is essentially the case/control scenario, so P = 2), and W i1 was defined as the median of nonzero coverage values for each sample. Using this model setup,β 2 (l j ) represents the estimated log (base 2) fold change in expression of base l j for males compared to females, when all coverage values are offset by 32 to ensure that zero counts would not cause problems in the log transformation. No other confounders were included in the analysis. The test statstic on the base level was limma's moderated t statistic (see section 1), and the HMM with derfinder 's default parameters was run on these t statistics to obtain candidate DERs (details are described in the supplement). To obtain p-values for the candidate DERs, a permutation test was run as described in section 3.3 of the main manuscript, using B = 10 permutations. All p-values (from all pipelines) were adjusted for multiple testing by controlling the false discovery rate, so the q-value (Storey and Tibshirani 2003) was used as a measure of statistical significance.
To connect the results from this experiment to annotated features, we labeled each DER with what type of genomic event it might indicate and which annotated features are involved (Supplementary Table 1). These labels aid in determining which exons and genes are showing differential expression signal and finding regions that may indicate phenomena like possible alternative splicing. Further exploration of these regions is possible using assemble-then-annotate methods to evaluate potential alternative or differential splicing events. Due to variance in read coverage across the genome, we observed some regions shorter than the length of an individual read. These small regions are particularly detrimental in the annotation and labeling step. We therefore choose to disregard regions shorter than the read length. Regions flanking very short transitions between states are merged. Figure 5 in the main text illustrates specific instances in the analysis of the human Y chromosome where DER Finder correctly identifies differential expression between sexes and EdgeR and DESeq do not, either because an exon was incorrectly annotated or because the differential expression did not occur within an exon at all. The instances shown in the text are not isolated: in fact, 280 non-exonic regions of the Y chromosome were identified by DER Finder as significantly differentially expressed (q < 0.05).

Additional figures illustrating problems with annotate-then-identify methods
Additionally, Supplementary Figure 1 demonstrates that differential expression does not always occur within exon boundaries, and as such, an identify-then-annotate method may be necessary to achieve high sensitivity. We examined differentially expressed Y-chromosome regions found by DER Finder that overlapped only part of an exon: for a fixed percentage x, we gathered the DERs (identified with DER Finder) that overlapped no more than x% of an exon. Then we calculated the fraction of the set exons overlapped by those DERs that EdgeR and DESeq called differentially expressed (q < 0.05). Supplementary Figure 1 plots different values of x against these fractions. The figure's message is that many exons showing a differential expression signal when analyzed with DER Finder are not called differentially expressed by EdgeR and DESeq, even in an easy analysis of differential expression between males and females on the Y chromosome. Figure 2 in the main text is a specific example of this problem, and Supplementary Figure   1 suggests that the issue is not confined to only one example.
Supplementary are also x% covered by a DER. As more overlap is required for a differential expression call, the percent agreement between DER Finder and the identify-then-annotate methods decreases, but overall, most of the exons identified by EdgeR and DESeq also show a signal in DER Finder.
Together, these two figures show that DER Finder identifies most of the differential expression found by EdgeR and DESeq, but the identify-then-annotate methods miss signals identified by DER Finder due to the heavy reliance on pre-specified exon annotation.

Additional Y-chromosome analysis: agreement between methods
To determine the extent to which the different pipelines discovered the same features to be differentially expressed, we quantified differential expression and overlap between findings at varying q-value cutoffs, comparing DER Finder to Tophat-Cufflinks-Cuffdiff (Supplementary Table 2) and DER Finder to EdgeR and DESeq (Supplementary Table 3). The new method produces better results than Cufflinks: we find differential expression between males and females on the Y chromosome, and find no differential expression between the males, while Cufflinks does not find differential expression between sexes unless the q-value cutoff is above 0.45. When the q-value cutoff is high (0.50), only 5.3% of the differentially expressed Cufflinks transcripts are also called differentially expressed by the new method: as expected, transcripts with high q-values are not overlapped by differentially expressed regions from the new method (regions that are equally expressed will not make it past the HMM segmentation step). On the other hand, 32.5% of the differentially expressed regions (q < 0.50) are overlapped by differentially expressed transcripts, which shows some agreement between the methods.
The comparison to EdgeR and DESeq shows the annotation-based results to be somewhat similar. The q-value cutoff did not seem to matter when assessing exon-specific results from DER Finder for the male-to-female comparison of the Y chromosome (Supplementary Table 3 in particular, the common counting methods do not allow reads to be counted toward more than one feature, so overlapping exons do not usually get any reads assigned to them at all.
In our experiment, this led to 345 exons having overlapping DERs assigned to them but not even being tested by EdgeR or DESeq. This issue explains some of the discrepancy between the exon-level findings for DER Finder and EdgeR/DESeq. Also, though DER Finder identifies more exons overall as being differentially expressed, 54 exons are identified only by EdgeR or DESeq.
Closer examination of the coverage patterns of these exons revealed that most of them were either (a) very lowly expressed overall, or (b) were less than 80% covered by DERs, so the exons themselves were not called differentially expressed because of the cutoffs defined in Table 1 of the main manuscript. Users can adjust DER Finder parameters if they are particularly interested in discovering differential expression of lowly-expressed features (e.g., the function g() chosen in model 3.1 could be g(x) = log 2 (x + 0.5) rather than log 2 (x + 32), which is what was was used the Y chromosome comparison). Also, DER Finder generally does show signal in the general area of the exons in question, even if that signal does not overlap the exon by 80%, so the results still give meaningful information. Overall, these findings confirm the result that EdgeR, DESeq, and DER Finder perform similarly when analyzing already-annotated features.

Experimental Design Concerns
Biologists who collect RNA-seq data must make several decisions when designing their experiments. Two important considerations are whether to use single-end or paired-end reads and how deeply to sequence the samples. We address these considerations and their impact on DER Finder's results in this section, using a small simulation study to support the conclusions drawn.

Simulation set-up
A small, 20-sample RNA-seq dataset with pre-defined differential expression was simulated using and coverage matrices were created from the Tophat alignment file.

Paired-end data in RNA-seq analysis
It is generally accepted that using paired-end data, i.e., data consisting of reads from both ends of the mRNA fragments instead of just one end, is better than using single-end data, even though paired RNA-seq experiments can cost up to twice as much as a single-end experiment (Katz and others 2010, Trapnell and others 2012). Mate-pair information is used during read alignment to more accurately determine the reads' best mappings. Furthermore, paired-end data is especially important in assemble-then-identify methods because it yields more reliable transcript assemblies and better per-transcript abundance estimates. Because annotate-then-identify and identify-thenannotate methods do not involve assembly or transcript-level quantification, paired-end data only improves these methods inasmuch as it improves the read alignment step. Therefore, since read alignment can be done with either single-end or paired-end reads, it is appropriate to use DER Finder with either type of data. The coverage matrix would be calculated the same way for paired data as it is for single-end data; each mate of a mate pair would contribute a coverage value of 1 to all the bases to which it aligns.
The Y-chromosome analysis in this manuscript was done using single-end data, which may put Cufflinks/Cuffdiff (the assemble-then-identify method) at a disadvantage when comparing it to the other tools. To determine whether the poor statistical results from Cufflinks/Cuffdiff may have been due to using single-end data, we ran Cufflinks and Cuffdiff (version 2.0.2, with default parameters) on the simulated dataset. Even though this dataset was paired-end and contained transcripts known to be highly differentially expressed, the statistical results from Cufflinks/Cuffdiff were unreasonable: they did not reflect any differential expression (Supplementary Figure 7). Therefore, we contend that while paired-end data may improve assembly methods, it is not the deciding factor in whether the Cufflinks/Cuffdiff pipeline produces reasonable statistical results.

Effect of sequencing depth
Sequencing depth (or read coverage) refers to how many times each mRNA nucleotide in the sample is read by the sequencing machine. Experiments with greater sequencing depth are better able to detect expression differences for features that are lowly expressed overall. This property holds for most existing differential expression analysis methods, including DER Finder. Therefore, experimenters wishing to use DER Finder and detect differential expression in for lowly expressed feature should deeply sequence their samples. One specific consideration in DER Finder is the choice of g() in model (3.1). In general, we recommend using g(x) = log 2 (x + k), where k is a constant that allows the method to avoid taking the log of 0. In our experiment, we set k = 32 because we were not particularly interested in differential expression in areas with low coverage, and offsetting all counts by 32 attenuates the fold changes observed in very low-coverage regions.
However, if the sequencing depth is high, the user may want to increase k (if lowly-expressed features are not of interest), since the method will be more sensitive to differential expression of lowly-expressed features with deep sequencing. Similarly, if the samples are not sequenced very deeply, the user may want to decrease k, since true differential expression may not be detected if the samples' coverage values are offset too much.
We also investigated the effect of sequencing depth using the simulated dataset described above in addition to two more simulated datasets. These additional datasets were generated in the same manner as the first dataset except for read coverage: the first additional dataset had half as many reads as the original dataset, and the second had 1/4 as many reads as the original dataset. Based on the median length of the transcripts included in these experiments, the coverages for these datasets were approximately 24x, 12x, and 6x, respectively. Coverage matrices were created using Tophat alignments, and DER Finder was run on the chromosome 22 coverage matrix for each dataset, with model 3.1 defined as follows: g(x) = log 2 (x + 32), X i = 1 for samples in group A and 0 for samples in group B, and W i1 was set as the median nonzero coverage value for each sample.
In this simulated dataset, DER Finder using the 24x and 12x datasets found 435 and 433 differentially expressed regions, respectively (q < 0.05), while the 6x dataset did not find any differential expression (minimum q-value 0.18). This is consistent with what we expect: the same offset (k = 32) was used for all three datasets, and this appears to be too much of an offset for the low-coverage (6x) dataset. To further investigate these findings, we used varying q-value cutoffs to create ROC curves for the different coverage levels (Supplementary Figure 8). DER Finder appears to be performing well in terms of sensitivity and specificy for the 12x and 24x experiments: for example, in the 24x experiment, 97 out of the 100 pre-set differentially expressed transcripts were overlapped by a significant (q < 0.05) DER, while 93% of the transcript features (exons, etc.) that were not simulated as differentially expressed were overlapped by regions in the equally expressed state or with q 0.05. In general, there was very little difference between 12x and 24x coverage in this simulation, but 6x read coverage appears to be too shallow when the offset is set at 32.

Code
Code -both general R functions and code used to do this particuar analysis -is available on github: (https://github.com/alyssafrazee/derfinder).    Fig. 6. P-value histograms for tests of differential expression on the Y chromosome between males and females. For all methods except Cufflinks, substantial differential expression is evident in the comparisons between sexes, as expected. The Cufflinks p-value distribution is quite unusual and indicates that using p-values adjusted for multiple testing to assess significance may be problematic.   Table 3. Comparison of results from DER Finder to EdgeR and DESeq, analyzing differential expression at the exon level on the Y chromosome between males and females. The first column is the number of differentially expressed regions found by DER Finder, and the second, third, and fourth columns are the number of differentially expressed exons found by each method at the specified q-value cutoff. Differentially expressed exons for DER Finder were defined as exons that were more than 80% covered by regions of state D = 2; the q-value for each exon was taken to be the q-value of the region most overlapping it. The last four colums show the number of exons found by two or all three methods. . ROC curves from DER Finder, created based on the simulation study with known differential expression. Sequencing depth is noted by color, while line type denotes different ways of determining differential expression calls: the dashed lines were created at the feature level, i.e., the true positive rate was the percentage of differentially expressed transcript f eatures (exons, etc.) that were overlapped by a significant DER. The solid lines were created at the transcript level, i.e., the true positive rate was the percentage of transcripts with at least one feature overlapped by a significant DER. DER Finder is performing well in terms of sensitivity and specificity when the sequencing depth is sufficient.