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Introduction

A good review of RNA-Seq expression units from Harold Pimentel.

CPM: counts per million

RPKM: reads per kilobase per million

So there must have something to do with the 0’s.

Let’s directly work with \(\hat{\sigma}^2\),

\[\begin{equation} \hat{\sigma}^2 = \frac{1}{n-2}(\sum_{i\in Group1}(y_i-\bar{y}_1)^2+\sum_{i\in Group2}(y_i-\bar{y}_2)^2), \end{equation}\]

where \(\bar{y}_1\) is the sample mean of group 1. Let \(n_1\) be the number of samples in group, \(p_1^0\) be the proportion of zeros or \(\log(s)\) in group 1, and \(y_0\) denote zero or \(\log(s)\), then

\[\begin{equation} \begin{split} \sum_{i\in Group1}(y_i-\bar{y}_1)^2 &= \sum_{\substack{i\in Group1 \\ y_i\neq y_0}}(y_i-\bar{y}_1)^2 + n_1p_1^0(y_0-\bar{y}_1)^2 \\&= \sum_{\substack{i\in Group1 \\ y_i\neq y_0}} y_i ^2 - 2p_1^0y_0\sum_{\substack{i\in Group1 \\ y_i\neq y_0}} y_i - \frac{1}{n_1}(\sum_{\substack{i\in Group1 \\ y_i\neq y_0}} y_i)^2 + n_1 p_1^0(1-p_1^0)y_0^2 \\&= n_1(1-p_1^0)\text{Var}\{y_i,\in Group1, y_i\neq y_0\} + \frac{(\sum_{\substack{i\in Group1 \\ y_i\neq y_0}} y_i)^2}{n_1/p_1^0-n_1} + n_1 p_1^0(1-p_1^0)y_0(y_0-2\bar{y}_{1,y_i\neq y_0}) \end{split} \end{equation}\] 

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Scientific Linux 7.4 (Nitrogen)

Matrix products: default
BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.so

locale:
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 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
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[13] promises_1.0.1  whisker_0.3-2   rmarkdown_1.10  tools_3.5.1    
[17] stringr_1.3.1   glue_1.3.0      httpuv_1.4.5    yaml_2.2.0     
[21] compiler_3.5.1  htmltools_0.3.6 knitr_1.20