Compositional Data Analysis (CoDA)
Also known as: CoDA, Simplex Analysis, Log-Ratio Analysis, Bileşim Veri Analizi
Compositional Data Analysis (CoDA) is a branch of multivariate statistics designed for data that represent parts of a whole — proportions, percentages, or concentrations that sum to a constant. Introduced by John Aitchison in his landmark 1982 paper, CoDA recognises that standard Euclidean methods fail on the simplex and instead operates through log-ratio transformations that respect the relative nature of compositional information.
Key highlights
- Operates in a geometrically coherent sample space (the simplex), avoiding spurious correlations induced by the constant-sum constraint
- Log-ratio transformations are scale-invariant, so results do not depend on whether data are expressed as proportions, percentages, or parts-per-million
- Seamlessly integrates with established multivariate tools (PCA, regression, clustering) once data are transformed
- Aitchison distance provides an interpretable, mathematically rigorous measure of compositional difference
Intuition
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How it works
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When to use it
Use CoDA whenever observations are vectors of non-negative parts that carry information only through their relative magnitudes — geochemical element concentrations, microbiome taxon abundances, dietary macronutrient proportions, time-use surveys, or budget shares. Assumptions include strictly positive parts (zeros must be handled before analysis) and that relative variation, not absolute variation, is scientifically meaningful. Avoid CoDA when absolute amounts matter independently. Alternatives include Dirichlet regression for proportional outcomes and normalisation-followed-by-standard methods, though the latter lacks rigorous geometric justification.
Strengths & limitations
- Operates in a geometrically coherent sample space (the simplex), avoiding spurious correlations induced by the constant-sum constraint
- Log-ratio transformations are scale-invariant, so results do not depend on whether data are expressed as proportions, percentages, or parts-per-million
- Seamlessly integrates with established multivariate tools (PCA, regression, clustering) once data are transformed
- Aitchison distance provides an interpretable, mathematically rigorous measure of compositional difference
- Cannot handle exact zeros directly; observed zeros require subjective replacement strategies that may influence results
- Log-ratio coordinates are harder to interpret than the original parts, especially for domain scientists unfamiliar with CoDA
- The isometric log-ratio (ILR) basis is not unique, and different basis choices can complicate comparison across studies
- Software implementations vary in which transformation they apply by default, increasing risk of methodological inconsistency
Common pitfalls
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Applications
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Frequently asked
Why can't I just use standard statistics on proportions?
Proportions that sum to a constant are linearly dependent by construction, which artificially induces negative correlations among parts. Standard methods assume unconstrained Euclidean data; applying them to the simplex can yield misleading correlations, biased principal components, and regression estimates that violate the sum constraint. Log-ratio transformation removes these artefacts by moving data to a geometrically appropriate space.
How should I handle zeros in compositional data?
True zeros (a part is genuinely absent) and rounded zeros (a part is below detection) require different strategies. Rounded zeros are commonly replaced using multiplicative or Bayesian-multiplicative methods that preserve the compositional structure. True structural zeros may require partitioning the sample or using zero-inflated models. Simply adding a small constant to all zeros is generally discouraged because it distorts relative relationships.
What is the difference between ALR, CLR, and ILR transformations?
ALR (additive log-ratio) uses a single reference part and is easy to compute but not isometric. CLR (centred log-ratio) uses the geometric mean as reference and is isometric but produces a singular covariance matrix. ILR (isometric log-ratio) uses an orthonormal basis in the simplex, is fully isometric and non-singular, but requires choosing a basis, which complicates interpretation and cross-study comparison.
Sources
- 1.Aitchison, J. (1982). The statistical analysis of compositional data. Journal of the Royal Statistical Society: Series B, 44(2), 139–177.
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Cite this page
ScholarGate. (2026, June 2). Compositional Data Analysis. ScholarGate. https://scholargate.app/statistics/compositional-data-analysis