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I am reading through the paper "A scaling normalization method for differential expression analysis of RNA-seq data" by Mark D Robinson, Alicia Oshlack, available here.

In this paper they introduce a normalization method called "trimmed mean of M-values" (TMM), which is apparently a normalization used in the edgeR package.

What is confusing to me is that on page 3 they define the value

$M_g = \log_2\left(\frac{Y_{gk}/N_k}{Y_{gk'}/N_k'}\right)$

where $k$ is an experiment to be normalized and $k'$ is a reference experiment. A few pages down on page 7, however, they define

$M_{gk}^r = \frac{\log_2\left(Y_{gk}/N_k\right)}{\log_2\left(Y_{gr}/N_r' \right)}$

where $k$ is an experiment to be normalized and $r$ is a reference experiment.

Are these two formulae intended to represent the same quantity? If they are supposed to be the same, which one has the correct order of logarithm/division? If they are not supposed to be the same, can anyone explain the difference? Thank you.

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