Greenacre 2023 - Aitchison CoDA 40 Years On - Overview

Summary

Greenacre, Grunsky, Bacon-Shone, Erb and Quinn (2023) review Aitchison’s logratio approach to compositional data (data on the simplex, sum-to-one) 40 years after his 1982 read paper. Their thesis: exact subcompositional coherence and exact isometry are not needed; simple pairwise/additive logratios (ALR) or even Box-Cox power transforms are “quasi-coherent” and “quasi-isometric” enough, and zeros remain the “Achilles heel”.

Overview

Question. What properties must a transformation of compositional data satisfy, and are the complicated isometric logratios (ILR/pivot) necessary?

Contribution. A guided tour of CoDA properties (scale invariance, coherence, dominance, distributional equivalence, isometry), a worked 6,799-sample geochemical example (Tellus), wide-data genomics, and measures of incoherence/anisometry to quantify departures rather than demand exactness.

Key result. On Tellus, the ALR with reference Al reaches Procrustes correlation 0.990 with the full logratio geometry; correspondence analysis of Box-Cox ($\alpha =0.5$) power-transformed data needs no zero replacement and gives similar structure.

Motivating PyMC examples

E70 “Compositional data: parts of a whole (Dirichlet, logistic-normal, Dirichlet-multinomial)” (Aitchison’s Arctic lake sediments; 2024 UK general election, 542 English constituencies) — closure and spurious correlation, ALR/CLR/ILR, Dirichlet regression, logistic-normal with LKJ and covariate-dependent scale — at https://redam94.github.io/pymc-challenges/examples.html. See Greenacre 2023 - Compositional Outcome Models in PyMC and Tsagris 2018 - Dirichlet Regression with Zeros - Overview.

Main Content — Source Map

Paper section	Covered in
Abstract, 1 Introduction	this note; Greenacre 2023 - Simplex, Closure and Core CoDA Properties
2 Tellus geochemical data set (N=6799, D=52, zeros)	Greenacre 2023 - Tellus Application, Variable Selection and Correspondence Analysis
3.1 Scale invariance; 3.2 Coherence; 3.5 Dominance; 3.6 Distributional equivalence	Greenacre 2023 - Simplex, Closure and Core CoDA Properties
3.3 Pairwise/additive logratios; 3.4 CLR; 3.7 Log-contrasts; 3.8 Isometry/ILR; 3.9 Dispersion measures	Greenacre 2023 - Logratios ALR CLR ILR and Log-Contrasts
4 Application (weighting, ALR selection, CA alternative, quasi-coherence)	Greenacre 2023 - Tellus Application, Variable Selection and Correspondence Analysis
5 Variable selection	Greenacre 2023 - Tellus Application, Variable Selection and Correspondence Analysis
6 Wide data (genomics, scRNA-seq)	Greenacre 2023 - Wide Data, Zeros and Quasi-Coherence
7 Are exact coherence and isometry necessary? (7.1-7.5 incl. amalgamation/SLR)	Greenacre 2023 - Wide Data, Zeros and Quasi-Coherence
8 Reappraisal and conclusion (zeros, “last word”)	Greenacre 2023 - Wide Data, Zeros and Quasi-Coherence
(concept translation)	Greenacre 2023 - Compositional Outcome Models in PyMC

Relevance to marketing measurement

Channel-share, budget-allocation and market-share outcomes are compositions: modelling shares with unconstrained regressions produces spurious negative correlations and predictions outside $[0,1]$. A log-ratio response model respects the constraint, and conclusions about “share of spend” should be checked for subcompositional coherence when channels are aggregated or dropped. See Generalized Linear Models, Hierarchical Models.

Connections

Zero handling continues in Tsagris 2018 - Dirichlet Regression with Zeros - Overview; directional (square-root) view of compositions appears in Pewsey 2021 - Directional Statistics - Spherical, Toroidal and Other Topics.

See Also

Source index: _Index.md in this folder.