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# Table 1 Scaling normalization approaches derive their technical bias estimates from ratio of proportions

From: Analysis and correction of compositional bias in sparse sequencing count data

Technique | Proposed abundance measure, scale factor | Signal for compositional scale in |
---|---|---|

Total sum | \( \begin {array}{c} \frac {y_{gji}}{\tau _{gj} \cdot \Lambda _{gj}^{-1}},\\ \Lambda _{gj}^{-1}=1 \end {array} \) | |

TMM | \( \begin {array}{c} \frac {y_{gji}}{\tau _{gj} \cdot \Lambda _{gj}^{-1} }, \\ \Lambda _{gj}^{-1} = e^{\left [ \sum _{i: y_{ij}>0 ~\cap ~i\in \mathbf {trimmed\ set\ for\ j}} w_{ij} \log \left (\frac {q_{gji}}{q_{1ji}}\right) \right ]} \end {array} \) | \(\frac {q_{gji}}{q_{1ji}}\), ratio of proportions |

DESeq | \( \begin {array}{c} \frac {y_{gji}}{ C \cdot \tau _{gj} \cdot \Lambda _{gj}^{-1}} \propto \frac {y_{gji}}{\tau _{gj} \cdot \Lambda _{gj}^{-1}}, \\ \Lambda _{gj}^{-1} = \text {median}_{i} ~ \frac {q_{gji}}{\left [ \prod _{k} q_{ik} \right ]^{\frac {1}{n}}} \end {array} \) | \(\frac {q_{gji}}{\left [ \prod _{k} q_{ik} \right ]^{\frac {1}{n}}}\), ratio of proportions |

Median | \(\begin {array}{c} \frac {y_{gji}}{\tau _{gj} \cdot \Lambda _{gj}^{-1}}, \\ \Lambda _{gj}^{-1}=\text {median}_{i} ~q_{gji} \propto \text {median}_{i} ~\frac {q_{gji}}{1/p} \end {array}\) | \(\frac {q_{gji}}{1/p}\), ratio of proportions |

Upper quartile | \(\begin {array}{c} \frac {y_{gji}}{\tau _{gj} \cdot \Lambda _{gj}^{-1}}, \\ \Lambda _{gj}^{-1}=\text {upper quartile}_{i} ~q_{gji} \propto \text {upper quartile}_{i} ~\frac {q_{gji}}{1/p} \end {array}\) | \(\frac {q_{gji}}{1/p}\), ratio of proportions |

CLR Transformation | \(\begin {array}{c} \log \left (\frac {y_{gji}}{ \left [\prod _{i} y_{gji}\right ]^{\frac {1}{p}}} \right)\ \equiv \log \left (\frac {q_{gji}}{ \left [\prod _{i} q_{gji}\right ]^{\frac {1}{p}}} \right) \equiv \log \left (\frac {y_{gji}}{ \tau _{gj} \cdot \Lambda _{gj}^{-1}} \right),\\~\text {with} ~\Lambda _{gj}^{-1}=\left [\prod _{i} q_{gji}\right ]^{\frac {1}{p}} \propto \left [\prod _{i} \frac {q_{gji}}{1/p}\right ]^{\frac {1}{p}} \end {array}\) | \(\frac {q_{gji}}{1/p}\), closely tracks Median factors above; ratio of proportions |

Scran | \(\begin {array}{c}\frac {y_{gji}}{ \tau _{gj} \cdot \Lambda _{gj}^{-1} }, \\ \Lambda _{gj}^{-1}= \text {fit linear models to} ~\left \{ \frac {q_{1ji}}{\overline {q_{++i}} }, \dots, \frac {q_{nji}}{\overline {q_{++i}}} \right \}_{i=1}^{p} \end {array}\) | \(\frac {q_{gji}}{\overline {q_{++i}}}\), ratio of proportions |

Wrench | \(\begin {array}{c}\frac {y_{gji}}{ \tau _{gj} \cdot \Lambda _{gj}^{-1} }, \\ \Lambda _{gj}^{-1} = \frac {1}{p}\sum _{i} w_{ij} \frac {q_{gji}}{\overline {q_{++i}}}\end {array}\) | \(\frac {q_{gji}}{\overline {q_{++i}}}\), ratio of proportions |