Transformation

Microbiome data

• High variability: feature abundances vary widely across samples
• Zero-inflation: many taxa absent in most samples
• Compositionality: only relative abundances observed; total counts per sample are arbitrary

Heteroscedasticity

Zero-inflation

Compositionality

• Sequencing measures relative abundances, not true microbial counts
• Values are parts of a whole
• Parts are linked by a fixed total (library size)
  • If one increases, at least one other must decrease

Absolute versus relative

Relative vs. quantitative profiles

RMP vs. QMP: conclusions can flip

Compositionality is not optional

“… microbiome datasets generated by HTS are compositional because they have an arbitrary total imposed by the instrument.”

“… HTS datasets derived from microbiome studies can and should be treated as compositions at all stages of analysis.”

Goals of transformation

• Mitigate biases in abundances
• Make features comparable across samples
• Meet assumptions of downstream statistical analyses

Relative abundance

Library sizes are arbitrary

\[ \text{X}_{ij} = \frac{\text{Count}_{ij}}{\sum_{k=1}^m \text{Count}_{ik}} \]

where

• \(i\) indexes the sample,
• \(j\) indexes the taxon,
• \(m\) is the total number of taxa.

Log-transformed relative abundance

Library sizes are arbitrary and data zero-inflated

\[ \text{X}_{ij} = \log \left( \text{Relative abundance}_{ij} + \epsilon \right) \]

where

• \(\epsilon\) is a small pseudocount to avoid \(\log(0)\).

Counts, proportions, logs

Centered log-ratio (CLR)

Library sizes are arbitrary and values are dependent

\[ \text{CLR}(x_i) = \log\left(\frac{x_i}{g(x)}\right),\quad g(x) = \left(\prod_{j=1}^{D} x_j \right)^{1/D} \]

• \(x_i\): abundance of bacteria \(i\)
• \(g(x)\): geometric mean across all bacteria in the sample

Intuition

1. Values dependent, only ratios matter

Intuition

1. Values dependent, only ratios matter
2. Standard statistics rely on subtraction and distance

Intuition

1. Values dependent, only ratios matter
2. Standard statistics rely on subtraction and distance

\[ \frac{x}{y} \;\longrightarrow\; a - b \;? \]

Intuition

1. Values dependent, only ratios matter
2. Standard statistics rely on subtraction and distance
   • Use log-scale

\[ \log\left(\frac{x}{y}\right) = \log(x) - \log(y) \]

Intuition

1. Values dependent, only ratios matter
2. Standard statistics rely on subtraction and distance
   • Use log-scale
3. Library sizes are arbitrary

Intuition

1. Values dependent, only ratios matter
2. Standard statistics rely on subtraction and distance
   • Use log-scale
3. Library sizes are arbitrary
   • Center the data

Centering

Centering the logs

\[ \begin{aligned} \text{CLR}(x_i) &= \log\left(\frac{x_i}{g(x)}\right) \\[2mm] &= \log(x_i) - \log(g(x)) \end{aligned} \]

Why the geometric mean?

\[ \text{Arithmetic mean} = \frac{2 + 8 + 32}{3} = \frac{42}{3} = 14 \]

\[ \begin{aligned} \text{Geometric mean} &= \sqrt[3]{2 \cdot 8 \cdot 32} \\[2mm] &= \exp\left( \frac{\log 2 + \log 8 + \log 32}{3} \right) \\[2mm] &= 8 \end{aligned} \]

Intuition: summary

1. Values dependent, only ratios matter
2. Standard statistics rely on subtraction and distance
   • Use log-scale
3. Library sizes are arbitrary
   • Center the data

Centered log-ratio (CLR)

\[ \begin{aligned} \text{CLR}(x_i) &= \log\left(\frac{x_i}{g(x)}\right) \\[2mm] &= \log(x_i) - \log(g(x)) \\[2mm] &= \log(x_i) - \frac{1}{D}\sum_{j=1}^{D}\log(x_j) \end{aligned} \]

CLR in practice

• Removes compositional constraints (constant sum)
• Values centered around zero, symmetric
• Allows standard statistical tools (PCA, correlations)
• Debate on whether it is necessary in DA

Counts versus CLR

Correct transformation depends on

1. Your research question
2. The analysis method

Use the simplest justified approach that works

Demonstration

library(mia)
data("GlobalPatterns", package = "mia")
tse <- GlobalPatterns

tse <- transformAssay(tse, method = "relabundance")
tse <- transformAssay(tse, method = "clr", pseudocount = TRUE)
assayNames(tse)

[1] "counts"       "relabundance" "clr"         

Exercises