Candidate statistical measures

Pearson’s correlation coefficient r was the main competitor because of the ubiquity of its use in the current RNA-Seq literature. The Kappa statistic was recently proposed as an alternative[5]. This requires the counts to be binned. Here, we used the same binnig as suggested in that paper. The SERE statistic is a ratio of the observed standard deviation between replicates divided by the value that would be expected from an ideal experiment.

Sensitivity experiment

Figure1 shows results from constructed datasets representing two lanes from an ideal replication experiment (“perfect” in silico replicates affected only by the random sampling effect), and pairs of lanes where one of the two has various amounts of contamination. The contaminating sample in this experiment was from a litter mate under the same experimental condition (biological replicate) in order to make detection purposefully difficult. Each point of the figure represents the average of 200 independent realizations of the simulation experiment. The last point for each method represents a duplication, i.e., comparing a lane with itself (This could be the result of a copy/paste error during analysis, for instance).

For the correlation and concordance measures the value 1 is usually viewed as the “ideal”. This is only achieved for the duplication, a situation where the randomness inherent to the process of read sampling is not allowed and instead a greater than expected congruence between two sample pairs is forced, resulting in an extreme case of “underdispersion”. Data from an actual ideal experiment (0% contamination) had on average correlation values of 0.89 and concordance values of 0.41. SERE on the contrary yielded the expected baseline value of 1 for perfect in silico replicates (0% contamination) and detected contamination as early as 25%. Marked differences appeared when contamination reached 50%. The SERE measure also clearly marks the duplication comparison as unusual (SERE = 0). The sensitivity of both the correlation and concordance measures is much lower, making it difficult to distinguish contaminated samples from the ideal experiment.

Stability experiment

Another characteristic, stability, interrogates whether the behavior of the underlying statistic is independent of ancillary aspects of the experiment; the obvious such factor in RNA-Seq is the sequencing depth. Therefore, RNA-Seq perfect replicate datasets of different sizes were generated by drawing random reads from the universal read pool. We simulated two types of scenarios: In our first experiment (Figure2) we decreased the number of reads in both lanes from 10 to 0.5 million. Pearson’s r fell markedly from 0.93 to 0.71 when the read counts of both datasets in a pair were reduced (Figure2). Kappa was equally sensitive to the total read count, decreasing from 0.54 for perfect replicate pairs with 10⁷ reads per sample down to 0.09 for pairs with only 0.5x10⁶ reads. All datasets represented perfect replicates by definition as they were generated in silico by sampling from a common pool. Therefore, low values of Pearson’s r such as <0.8 and Kappa <0.3 are not in all cases indicative of poor RNA-Seq experimental replication. SERE was unaffected by the total count of RNA-Seq reads, remaining stable at 1.0.

In our 2nd experiment (Figure3) we kept the total read count of 200 in silico replicates constant, but varied the relative size of both samples, simulating multiplexed RNA-samples, where both samples will not always yield the same number of reads. Pearson’s r and the Kappa statistic performed continuously worse, as the relative difference between the two perfect replicates became bigger, reaching values of approximately 0.82 and 0.19 respectively at the extremes (1 million versus 9 million uniquely mapped reads per sample). SERE on the contrary stayed at a stable value of 1.0 through all the scenarios. A minor increase of the confidence intervals could be observed as the relative sample size tended to the extremes, yet remaining between ± 0.01.

Performance of the statistics on empirical data

To put the above findings into perspective, we studied the candidate statistics on an empirical dataset which included technical and biological replicates, as well as samples from different experimental conditions (“control” vs. “SNL”, see Methods). Figure4 shows the result for 14 lanes of data, consisting of 3 replicate lanes for each of the 2 “control” rats and 4 replicate lanes for each of the 2 “SNL” rats. First the replicate lanes of each rat were compared (technical replicates), second the biological replicates (the 2 “controls” and the 2 “SNL” respectively) and third the animals belonging to different experimental groups (“SNL” vs. “control” rats). Note that SERE results in a single value for a set of lanes that are being compared, while the correlation and concordance measures apply only to pairs of lanes.

R and Kappa were slightly lower for the biological replicates as compared to the technical replicates and further decreased when comparing the two experimental conditions. However, differences were small compared with those caused by total read counts (Figures2 and3) suggesting that both measures are poor candidates for detection of global alterations in practice. SERE was highly sensitive to global differences, with scores of approximately 1.15 for biological replicates and 1.7 to 2.0 for comparisons between different experimental conditions.

The SERE statistic can also be computed pairwise. For the 3 technical replicates of “control 1” for instance, the overall ratio for the three lanes is 1.005, with pairwise values of 1.003, 1.002, and 1.008. When the overall SERE statistic for a set of lanes is large we can use these individual comparisons to further sort out which lane(s) is the source of concern. A simple way to display this is to use SERE to create a cluster map. Figure5 shows the resulting dendrogram for the 14 lanes of data used in Figure4. The dendrogram clearly reflects to the experimental design by first distinguishing between the two conditions, then separating the biological replicates within an experimental configuration and finally grouping the technical replicates. The vertical axis of dendrograms is often left unlabeled since the values are on an arbitrary scale, but in this case they have a direct interpretation as “excess dispersion”. A more interesting example is shown in Additional file1: Figure S1 where we applied SERE on a drosophila melanogaster dataset (SRA id GSE17107) that was employed by McIntyre et al. Interestingly, it revealed 2 distinct groups (no sign of overdispersion within the groups) although the 5 samples originated from the same RNA-Seq library suggesting a possible batch effect.

The drawbacks of r and Kappa

This study was focused on a global approach that is useful in both quality control and early analysis of RNA-Seq experiments. Therefore, an ideal measure for this task was defined to be easy to compute and have three features of sensitivity, calibration and stability. The SERE measure does well, but the correlation and concordance have serious flaws. Why?

Deficiencies in the correlation coefficient have long been known. Chambers et al.[13] for instance showed a panel of 8 graphs with very different patterns all with the same value of r. Of most relevance here is that r can be dominated by values at the extremes of the data. Count data for RNA-Seq is very skewed. For example, in the first lane of “control 1”, 55.4% of the observed exons had a count of 10 or less out of 5,512,030 million total uniquely mapped reads, while the highest had a count of 95660. Even under a log-transformation, the largest few counts have in inordinate influence. Further examination of the results underlying Figures2 and3 shows that the value of r for the ideal samples is essentially determined by the range of the counts, which in turn is closely related to the total sequencing depth (Additional file1: Figure S2). This causes r to have both a varying target value (the expected result for a perfect replicate) and high variability. Even a low value (e.g <0.8) might result from an “ideal” experiment. At the same time, markedly different samples can yield correlation coefficients of >>0.8 if the total read count is high. Moreover, Pearson’s r can be computed in several ways: on square root of the raw counts, on count data after addition of a pseudo-count and log-transformation or on RPKM data after adding a pseudo-count and log-transformation. The latter is the most commonly used normalization and therefore employed here. These different approaches can substantially change the outcome since they influence the distance between the extremes (data not shown).

By categorizing the data into bins, as performed by the Kappa statistic, one avoids the susceptibility to values on the extreme of the scale. However the choice of the bin sizes becomes the driving factor for this statistic. Additional file1: Figure S3 demonstrates how the binning influences the result of Kappa for one and the same dataset. If the bins are chosen as reported previously by McIntyre et al. (Additional file1: Figure S3A) Kappa is 0.41. When the bins are chosen wider (Additional file1: Figure S3D), the value for Kappa raises to approximately 0.73. By choosing very small bin sizes (Additional file1: Figure S3C), the Kappa value decreases to approximately 0.3. We also computed a weighted Kappa[15] employing the most common definition where disagreement is proportional to the distance from the diagonal. The result shown in Additional file1: Figure S4 demonstrated the same characteristics of Kappa shown for the unweighted procedure.

For the simulation study, we chose the unweighted Kappa. We took the same bin sizes as proposed by McIntyre et al., which used 0 counts as the smallest bin. Therefore, whenever the expression of an exon is so sparse that only a single read is detected among two or more samples (the exon is a singleton) the exon will be scored as “off the diagonal” since it will fall into the bin “0” for one sample and in “1-10” for the other sample. The fraction of singletons in our in silico samples with 5 million UMRs is 11.56-11.96%, which alone limits the Kappa to a maximum of about 0.89. The total fraction of singletons tended to decrease by increasing the total read count and the calculated Kappa value rises as seen in Figure2.

Computational simulation can be helpful in estimating the expected values for Pearson’s r or Kappa but need to take the specific experimental condition into account and cannot be generalized easily. Therefore the use of r and Kappa to investigate whether two RNA-Seq datasets are faithful replicates or subject to systematic differences or bias leads to ambiguity in most cases.

The Simple Error Ratio Estimate (SERE)

The third candidate statistic appears to be a useful measurement to identify global differences between RNA-Seq data by fulfilling the set criteria of a good measure. A primary reason is that it compares the observed variation to an expected value, and the latter accounts for the impact of varying read depth. It is easy to compute and satisfies our three primary criteria.

Empirical RNA-Seq data

RNA-Seq read data used in the present analysis was taken from a previous study[8] and is available through the sequence read archive (SRA id GSE20895). 14 RNA-Seq datasets were used, each corresponding to the sequencing reads from one “lane,” which is a physical compartment in the flow cell of the Illumina GA-II instrument. The 14 lanes corresponded to 4 RNA-Seq libraries, whereby 3 lanes of sequence data were available for each of two of the libraries and 4 lanes for each of the other two. Each library was synthesized from a different source of RNA. RNA sources corresponded to two experimental conditions, “SNL” (spinal nerve ligation) or “control.” Two independent RNA-Seq libraries, “biological replicates,” were available for each condition. The published analysis[8] found that biological replicates were similar and that the 2 experimental conditions were meaningfully different. Technical replicates were not compared in the previous report (instead reads originating from different lanes corresponding to the same RNA-Seq library were pooled) but are included in the comparisons made in the present study.

Additional file1: Table S1 lists the condition and replicate identification corresponding to each of the 14 lanes. The table illustrates how the available data allowed for three types of comparisons between RNA-Seq datasets: pairs of technical replicates (same library sequenced on different lanes); pairs of biological replicates (same experimental condition); and the two experimental conditions “SNL” versus “control”.

Mapping and annotation

RNA-Seq reads (50bp) were aligned to the rat reference genome (RGSC 3.4) by Bowtie[19]. We allowed for a maximum of two mismatches and considered only the uniquely mapped reads for the downstream analysis. This filtration step resulted in 108,636,496 million UMRs to the genome. Genome annotation ENSEMBL 65 was used including 22,921 protein coding genes. Overlapping exons of genes having multiple isoforms were combined resulting in a total of 222,097 exons (Additional file2). For the subsequent analysis exons served as bins, i.e., reads aligning to each exon were counted and the sum noted in the master read count file (Additional file3). The file has 222,097 rows (bins corresponding to exons) and 14 columns (corresponding to each of the lanes described above).

A universal pool of RNA-Seq reads for the simulation experiments

All uniquely mapped reads (from lane 1 to 3) from the first “control” RNA-Seq sample were combined resulting in 22.9 × 10⁶ reads in order to create a universal pool. The datasets for the in silico duplicates and replicates described below were generated from this pool. The in silico replicates created from the universal pool of RNA-Seq reads by random drawing are by definition only different due to stochastic (Poisson) variation of the sampling process (see Results). Similarly all 3 lanes from “control 2” were combined to create a second pool used as “contaminant” in the contamination experiments.

In silico replicates: “Perfect” replicates

A set of RNA-Seq datasets faithfully representing Poisson variation only (perfect non-identical replicates) was generated by randomly choosing sets of 5 x10⁶ reads from the universal pool by using the “sample” function in R (Additional file4) and a Java script (Additional file5). This process was repeated 200 times and used as reference of perfect replicates in the subsequent benchmarking of the different test statistics. The same procedure was further repeated with different set sizes (0.5 to 10 million reads per sample) in order to test for the influence of total read counts on the statistics.

In silico contamination

To test whether the statistical measures were sensitive to actual differences, we contaminated one in silico replicate out of a pair with 0;5;10;25;50;75;100% of a biological replicate (“control 2”) via computer simulation. In detail, the first sample was created by randomly drawing 5 million reads of the universal pool of “control 1” and the second by drawing x% of reads from “control 1” and y% of reads from file “control 2”, whereby x+y=100, corresponding to 5 million reads. The procedure was repeated 200x.

Processing of the empirical data

For Pearson’s correlation coefficient and Kappa, the 3 lanes of each of the two “control” and the 4 lanes for each of the two “SNL” condition were compared in a pairwise fashion, resulting in a total of 18 technical replicate comparisons (see Figure4). To compare the performance of the statistics on biological replicates we compared the first lane of “control 1” to the first lane of “control 2”, and lane 1 from “SNL 1” to lane 1 from “SNL 2”. To test whether Pearson’s r and Kappa were sensitive to different experimental conditions, we selected the first lane of each “control” and compared it to the first lane of each “SNL” sample. SERE is not restricted to pairwise comparisons, but allows to compare multiple samples or lanes simultaneously. Therefore, SERE yielded 4 values for the technical replicate comparisons, 2 for biological and 4 for different experimental conditions.

Simple Error Ratio Estimate (SERE)

The denominator is (M − 1) due to the constraint that${\displaystyle {\sum }_j\left(y_{\mathit{ij}}-{\widehat{y}}_{\mathit{ij}}\right)=0}$ for each exon i.

The SERE estimate is$s=\sqrt{\left(s^2\right)}$.

Simple algebra shows that for a singleton count, i.e., an exon that appears only once in only one of the lanes, SERE equals exactly 1. That is, singletons shrink the overall SERE estimate towards 1, whether or not the samples are actually replicates. Therefore, we modify the average in equation 2 to sum over only the non-singleton counts. The R code to calculate SERE is provided in Additional file6.

The unmodified measure of equation 2 is the measure of Poisson over-dispersion most often used in generalized linear models, see for instance the classic textbook of McCullagh and Nelder[20]. An alternative measure of overdispersion is based on the deviance statistic. Brown and Zhao[21] consider this measure along with two others in the context of random arrival data, e.g. calls to a support center, and show that it is inferior to s _i ² (equation 2) whenever there are small values for${\widehat{y}}_{\mathit{ij}}$. Their work corresponds to the case where all lanes have the same total count L _j ; extending their method shows that (M − 1)s _i ² will be distributed as a chi-square random variable with (M − 1) degrees of freedom and non-centrality parameter${\displaystyle \sum L_j^2}{\left(p_{\mathit{ij}}-{\overline{p}}_i\right)}^2$. The parameter p _ij is the true fraction of exon i within sample j and${\overline{p}}_i$ the fraction of exons in the (hypothetical) pooled sample. Under the null hypothesis of sample equality$p_{\mathit{ij}}={\overline{p}}_i$ the non-centrality parameter is zero. N(M − 1)s² will follow a chi-squared distribution with N(M − 1) degrees of freedom. This can be used to set 99 % confidence intervals on the non-centrality parameter, and through that on the SERE estimate. We provide the corresponding R code in Additional file7. Bins containing a read count of 0 in both samples of a pair contain no information and were therefore excluded from the comparative analyses of SERE and the two alternative parameters described below.

Pearson’s correlation coefficient

For a pair of lanes, we calculated the RPKM[1] for each lane. In the RPKM calculation we added a pseudo count of one read to each of the exons. The RPKM values were then log transformed and the Pearson’s correlation was calculated. The process was repeated using the raw counts to verify any influence of the RPKM transform itself, and using both the log and the square root of the raw counts. The latter is the variance stabilizing transform for Poisson data. None of these had any substantive impact on the results (data not shown).

Cohen’s simple Kappa statistic

The read counts were normalized to RPKM and divided into 9 bins of size: 0, 1-10, 11-20, 21-40, 41-80, 80-160, 161-320, 321-1000 and greater than 1000 RPKM as it was suggested by McIntyre et al. In order to compare a pair of replicates, a 9 × 9 table of counts was constructed, whereby each exon pair added to a cell of the table (see Additional file8). In this way, exons that were in agreement added to the diagonal of the table, whereas the total fraction of off-diagonal entries contributed to a measure of non-agreement.