Greenacre 2023 - Wide Data, Zeros and Quasi-Coherence

Summary

For wide genomics data (hundreds to thousands of parts, many zeros) the paper argues that exact coherence and isometry are unnecessary: measures of incoherence and anisometry show common transformations are “near enough”. Amalgamated (summed) logratios are interpretable and zero-friendly; zeros, especially structural ones, remain “probably the thorniest and least-resolved issue”.

Overview

Sec. 6 shows CoDA for wide matrices (genomic sequencing, single-cell RNA-seq); Sec. 7 asks whether exact properties are needed; Sec. 8 concludes. The authors cite critiques that insisting on ILR/orthonormal coordinates is dogmatic.

Main Content

6 Wide data

Sequencing counts are relative: their size reflects sequencing depth, not molecules in the input. Normalisation (CPM = closure; effective library size) either keeps counts relative (scale-free CoDA) or assumes sizes are true sizes. CoDA is scale-free: counts are viewed as conditioned on totals via a multinomial model, with count probabilities as the compositions estimated as parameters. The authors sketch scRNA-seq exploration comparing log-ratio analysis (LRA) on modelled compositions with CA on relative counts (“Gene-principal LRA biplot” figures; thymocytes, stromal, vascular endothelial cell types). Zeros are numerous in such data, and the number of zeros relates to the total count before normalisation, with consequences for downstream analysis.

7 Are exact coherence and isometry necessary?

Incoherence and anisometry measures

Study random subcompositions of different sizes and compare their geometry (or effect sizes and p-values in model fits) with the full composition, using multidimensional-scaling stress or the Procrustes correlation: low stress near 0% or Procrustes correlation near 1 means high coherence. The same device against the CLR logratio geometry measures departure from isometry (“anisometry”).

• LRs are coherent but numerous; ALRs are coherent with $d$ variables but not exactly isometric; CLRs isometric but incoherent (all parts involved).
• Pivot/isometric logratios and ILR are “black boxes”: ratios of geometric means that users often cannot interpret.
• The paper lists dogmatic quotes such as “Compositional data analysis requires selecting an orthonormal basis” and argues interpretability should win.

Summed logratio (SLR), Eq. 12

For disjoint part sets $D_1,D_2$: $\mathrm{SLR}(D_1,D_2)=\log \frac{{\sum }_{j\in D_1}{x}_j}{{\sum }_{j\in D_2}{x}_j}$. SLRs: simple, subject-matter driven (fatty acids), coherent with respect to adding components, can absorb zeros inside an amalgam (a geometric mean cannot), but do not decompose into LR log-contrasts. A data-driven hierarchical clustering by amalgamation (Fig. 3) loses minimal explained logratio variance at each merge, and can be used for variable selection or supervised learning.

8 Reappraisal and conclusion

• Take the logratio approach as a reference, not a gold standard; use incoherence/anisometry diagnostics.
• Zeros (“Achilles heel of CoDA”): in time-use data, a true zero for working time replaced by a small value “is creating a fiction for the sake of being able to use the chosen method”; the method should cope with the true zero. Structural zeros should not be replaced. Replacement algorithms are usually applied without attention to their effects, particularly with very many zeros.
• If compositions serve as predictors or responses, transformation choice is guided by interpretability, with pairwise/additive logratios most attractive.
• The final “last word” is given to Aitchison himself (that the logratio approach is simple).

Examples

Zero strategies in a marketing mix

A channel that was off in some weeks has a structural zero share. Options: (i) add it to an “other” amalgam (SLR/amalgamation); (ii) model the zero pattern separately as in Tsagris 2018 - Dirichlet Regression with Zeros - Overview; (iii) a power transform/CA approach with no replacement. Replacing zeros with $ϵ$ makes results depend on $ϵ$.

Connections

Greenacre 2023 - Compositional Outcome Models in PyMC, Tsagris 2018 - Dirichlet Regression with Zeros - Zero Adjusted Dirichlet Regression

See Also

Greenacre 2023 - Aitchison CoDA 40 Years On - Overview