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I would like to ask questions regarding Delta rule for log-cpm of this article, and this section has been asked before and may help for you to answer: this one.

(Q1) I would like to know the intuition behind the statement: "suppose that: $\text{var}(r)=\lambda+\phi\lambda^2$."

Here, $r$ and $\lambda$ are the number of a read count of a gene and its expected value. $\phi$ is a dispersion parameter.

(Q2) I believe that above is not based on the standard definition of variance. If so, how come the authors be able to apply the standard definition of variance to: $\text{var}(\log_2{r})(\log{2})^2\approx\dfrac{\text{var}(r)}{\lambda^2}$?

Thank you for your time.

ComfusedGuy
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  • The first expression is the negative binomial variance function, this distribution is commonly used to fit (overdispersed) counts as done by e.g. edgeR. limma rather treats the log cpm $y$ (composed of $\text{log}_2(r)$ and constants) as a log-normal variable. The assumption is that as long as $r$ is large then $\text{var}(y)\approx \text{var}(\text{log}_2(r))$. – PBulls Mar 16 '24 at 15:41
  • Thank you for your helpful answers. Yes, I was able to find the literature of edgeR, describing what you explained for Q1 (https://www.biorxiv.org/content/10.1101/2024.01.21.576131v1.full.pdf). I need to take a look into the mathematics for this. And from the above, I also need to think about Q2. My original concern was that to get the equation in Q2, one must use the properties from the standard definition of variance, but the literature sounded to me as if the authors redefined the variance in a new way. And, I was wondering if this is legit or wanted to know if the properties still hold. – ComfusedGuy Mar 16 '24 at 16:36
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    No, they haven't defined it differently; they are just approximating it and then working with the approximation. – jbowman Mar 16 '24 at 17:35
  • Hey all, I learned and understood the concept now. One of the traps was that I couldn't infer that they assumed the negative binomial distribution. Thanks a lot. – ComfusedGuy Mar 16 '24 at 21:02

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