Last updated: 2020-09-21
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A good review of RNA-Seq expression units from Harold Pimentel.
CPM: counts per million
RPKM: reads per kilobase per million
In general, need to consider 1. library size; 2. gene length. Because libraries sequenced at a greater depth will result in higher counts.
CPM and log-CPM transformations do not account for gene length differences as RPKM and FPKM values do. CPM and log-CPM values can be calculated using a counts matrix alone. Assuming that there are no differences in isoform usage between conditions, differential expression analyses look at gene expression changes between conditions rather than comparing expression across multiple genes or drawing conclusions on absolute levels of expression. In other words, gene lengths remain constant for comparisons of interest and any observed differences are a result of changes in condition rather than changes in gene length.
RPKM values are just as easily calculated as CPM values using the rpkm function in edgeR if gene lengths are available.
Observed counts as \(Y_{gk}\), gene length \(l_g\), true and unknown expression level (number of transcripts) \(\mu_{gk}\), total number of reads for library k \(N_k\), let \(s_k = \sum_g l_g \mu_{gk}\), then \[E(Y_{gk}) = \frac{\mu_{gk}l_g}{s_k}N_k\]
The ratio of expected value between two conditions is \[\frac{E(Y_{g1})}{E(Y_{g2})} = \frac{s_1}{s_2}\frac{N_2}{N_1}\frac{\mu_{g2}}{\mu_{g1}}\]
So \(s_k\) is the size of studied transcriptome in condition \(k\), \(\mu_{gk}/s_k\) is the relative expression of gene \(g\) in condition \(k\).
Let \(r\) denote the reference sample. Filter out 1. transcripts with null counts(The cases where \(Y_{gk} = 0\) or \(Y_{gr} = 0\) are trimmed); 2. the \(30\%\) more extreme \(M-value\): \(M_{gk}^r = \log\frac{y_{gk}/N_k}{y_{gr}/N_r}\); 3. the \(5\%\) more extreme \(A-value\): \(A_{gk}^r = 0.5*(\log\frac{y_{gk}}{N_k}+\log\frac{y_{gr}}{N_r})\).
After filtering out genes, we have a set of genes denoted as \(G^*\) whose neither value was trimmed. The scaling factor \(TMM_k\) is calculated as \[\log TMM_k = \frac{\sum_{g\in G^*}w_{gk}M_{gk}}{\sum_{g\in G^*}w_{gk}},\] where \(w_{gk} = \frac{N_k-Y_{gk}}{N_k Y_{gk}} - \frac{N_r-Y_{gr}}{N_r Y_{gr}}\). Usually the scaling factors are re-scaled such that they multiply to 1.
Define the reference as the geometric mean(less sensitive to extreme values than the standard mean) of samples \(Y_{gk}^r = (\Pi_k Y_{gk})^{1/n}\), then the scaling factor for sample \(k\) is calculated as \(median(\frac{Y_{gk}}{Y_{gk}^r})\).