{ "id": "epdx7dqj6y", "slug": "compositional-data-analysis", "title": "Compositional data analysis", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis" }, { "id": "physical.math.statistics.multivariate-analysis.alpha-a-to-me", "label": "Titles A to Me", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.multivariate-analysis.alpha-a-to-me" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math.statistics.multivariate-analysis", "label": "Non-geographic · 1800 to 1945: Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.multivariate-analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics" }, { "id": "geo.nongeo.t1800.physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.multivariate-analysis" } ] }, { "id": "geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "label": "Non-geographic · 1946 to 2000: Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.multivariate-analysis", "label": "Multivariate association and dimension reduction", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.multivariate-analysis" } ] } ], "excerpt": "Compositional data analysis (CoDA) is the branch of statistics for data carrying only relative information, such as percentages and proportions, analyzing logratios within the simplex since Aitchison's 1986 monograph.", "snippet": "Compositional data analysis (CoDA) is the branch of statistics for data carrying only relative information, such as percentages and proportions, analyzing logratios within the simplex since Aitchison's 1986 monograph.", "node": "physical.math.statistics.multivariate-analysis.alpha-a-to-me", "markdown": "# Compositional data analysis\n\nCompositional data analysis (CoDA) is the branch of statistics that handles data carrying only relative information, such as percentages, proportions, or parts-per-million values that sum to a fixed total. Because the components of each observation are constrained to a constant sum, ordinary multivariate methods applied directly to the raw values produce spurious correlations and distorted covariance structure; CoDA instead analyzes logarithms of ratios between components, within a geometry of the simplex designed for constrained data.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436) • [2](https://ww2.amstat.org/meetings/proceedings/2014/data/assets/pdf/312239_89024.pdf) The field rests on the logratio approach established by J. Aitchison's 1986 monograph *The Statistical Analysis of Compositional Data*.[3](https://doi.org/10.1007/978-94-009-4109-0)\n\n| Key fact | Detail |\n|---|---|\n| Definition | Nonnegative data carrying relative, not absolute, information, often under a constant-sum constraint (proportions summing to 1, percentages to 100)[1](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436) |\n| Core problem | Closure induces spurious correlations among the components of fixed-sum compositional data, and correlations between ratios sharing a common component can also arise from that shared denominator, independently of closure[2](https://ww2.amstat.org/meetings/proceedings/2014/data/assets/pdf/312239_89024.pdf) • [25](https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1004075) |\n| Solution | Transform to logarithms of component ratios, after which standard univariate and multivariate methods apply[1](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436) |\n| Main transformations | Additive logratio (alr), centered logratio (clr), and isometric logratio (ilr), each with distinct trade-offs[4](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf) |\n| Central practical obstacle | Zeros, since logratios are undefined at zero; sparse sequencing tables can reach about 90% zero cells[1](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436) • [5](https://digitalcollection.zhaw.ch/items/a8e26557-b5a4-4c85-ac66-00b14d2d2c2b/full) |\n| Founding work | Aitchison's 1986 monograph, preceded by his 1982 read paper to the Royal Statistical Society[3](https://doi.org/10.1007/978-94-009-4109-0) • [6](https://arxiv.org/html/2201.05197v3) |\n| Software | R packages including compositions, zCompositions, robCompositions, ALDEx2, philr, and CoDaLoMic[7](https://f1000research.com/articles/7-1278) |\n\n## How it works\n\nThe sample space of a D-part composition is the simplex, the set of vectors with nonnegative components and a constant sum κ, where κ is 1 for proportions, 100 for percentages, \\( 10^{6} \\) for ppm, or \\( 10^{9} \\) for ppb.[4](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf) Within this space, ordinary addition and scalar multiplication are replaced by two operations: perturbation, defined as \\( z = x \\oplus y = C[x_{1} \\cdot y_{1}; \\ldots; x_{D} \\cdot y_{D}] \\), where C denotes closure (rescaling to sum κ), and the power operation, \\( z = \\lambda \\odot x = C[x_{1}^{\\lambda}; \\ldots; x_{D}^{\\lambda}] \\).[4](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf) These operations, introduced in Aitchison's 1986 monograph, make the simplex an [Abelian group](https://www.edgepedia.org/abelian-group) and underpin what is now called Aitchison geometry.[3](https://doi.org/10.1007/978-94-009-4109-0) • [8](https://www.ajs.or.at/index.php/ajs/article/download/vol45-4-4/525) Distances are measured with the Aitchison distance, built from sums of squared logratio differences, and dependence between parts is summarized by the variation matrix \\( T = [\\tau_{ij}] \\) with \\( \\tau_{ij} = \\mathrm{Var}\\,\\ln(x_{i}/x_{j}) \\); values of \\( \\tau_{ij} \\) near zero indicate that parts \\( x_{i} \\) and \\( x_{j} \\) are proportional.[4](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf)\n\nAitchison showed that inferences are identical whether a transformation technique or a staying-in-the-simplex approach is adopted.[9](http://leg.ufpr.br/lib/exe/fetch.php/pessoais:abtmartins:a_concise_guide_to_compositional_data_analysis.pdf) The guiding principle is scale invariance: compositions provide information only about relative magnitudes of the parts, so analysis should be in terms of component ratios, and logarithms make those ratios mathematically tractable.[9](http://leg.ufpr.br/lib/exe/fetch.php/pessoais:abtmartins:a_concise_guide_to_compositional_data_analysis.pdf)\n\n## How it is done\n\nA typical workflow deals with zeros, chooses a logratio transformation, and then applies standard multivariate methods to the transformed coordinates.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436) Once data are transformed to logratios, regular methods such as dimension reduction, clustering, and modeling can be used.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436) The compositional mean is recovered as the closed geometric mean, \\( \\mathrm{Mean}_{A}[X] = \\mathrm{clr}^{-1}(\\mathrm{Mean}[\\ln X]) \\), so estimates return to the simplex through the inverse clr.[4](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf)\n\nZeros demand a decision before any logratio is computed. They are classified as below-detection-limit zeros, structural zeros, and missing-at-random values, and even careful imputation can yield very different ratios depending on the rest of the data set.[10](http://stat.boogaart.de/Publications/iamg06_s07_01.pdf) Alternatives to replacement include projecting clr-transformed compositions with missing parts onto the orthogonal complement of the missing parts' clr directions, which gives unbiased mean and variance estimators without imputation,[10](http://stat.boogaart.de/Publications/iamg06_s07_01.pdf) amalgamating components to remove zeros,[1](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436) and arcsine conversion, which is well defined at zero and needs no replacement.[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC11609487/)\n\nThe main R implementations are compositions, zCompositions (multivariate imputation of left-censored compositional data), robCompositions (robust analysis), ALDEx2, philr, and CoDaLoMic.[7](https://f1000research.com/articles/7-1278) • [1](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436) ALDEx2, introduced by Andrew D Fernandes and colleagues in 2014 in [Microbiome](https://www.edgepedia.org/microbiome), models reads as proportions using [Monte Carlo](https://www.edgepedia.org/monte-carlo) instances sampled from a [Dirichlet distribution](https://www.edgepedia.org/dirichlet-distribution) and handles zeros in a Bayesian context.[12](https://pmc.ncbi.nlm.nih.gov/articles/PMC4030730/)\n\n## Origin\n\n[Karl Pearson](https://www.edgepedia.org/karl-pearson) warned in 1897 in the Proceedings of the Royal Society of London against interpreting correlations between ratios whose numerators and denominators contain common parts, showing that if X, Y, and Z are uncorrelated, X/Z and Y/Z will not be uncorrelated.[13](https://doi.org/10.1098/rspl.1896.0076) • [2](https://ww2.amstat.org/meetings/proceedings/2014/data/assets/pdf/312239_89024.pdf) The warning went largely unheeded until the 1960s, when geologists including Chayes, Krumbein, Sarmanov, and Vistelius, and the biologist Mosimann, raised the problem again; Chayes's 1960 paper in the [Journal of Geophysical Research](https://www.edgepedia.org/journal-of-geophysical-research) connected spurious correlation to constant-sum data and showed that some correlations between components must be negative because of the unit-sum constraint.[9](http://leg.ufpr.br/lib/exe/fetch.php/pessoais:abtmartins:a_concise_guide_to_compositional_data_analysis.pdf) • [14](https://doi.org/10.1029/jz065i012p04185) • [2](https://ww2.amstat.org/meetings/proceedings/2014/data/assets/pdf/312239_89024.pdf)\n\nA proper methodology emerged in the 1980s: the logistic-normal distribution on the simplex was presented by J. Aitchison and S. M. Shen in 1980 in [Biometrika](https://www.edgepedia.org/biometrika),[15](https://doi.org/10.2307/2335470) • [6](https://arxiv.org/html/2201.05197v3) • [16](http://wiki.leg.ufpr.br/lib/exe/fetch.php/pessoais:abtmartins:thestatisticalanalysisofcompositionaldata.pdf) and the 1986 monograph established the field.[3](https://doi.org/10.1007/978-94-009-4109-0) The work was a reaction against unconstrained analysis of raw percentages, which ignores the confinement of data points to a simplex.[16](http://wiki.leg.ufpr.br/lib/exe/fetch.php/pessoais:abtmartins:thestatisticalanalysisofcompositionaldata.pdf)\n\n## Variants\n\nThe three classical transformations differ in reference and geometry. The additive logratio maps a D-part composition to \\( \\mathbb{R}^{D-1} \\) as \\( \\mathrm{alr}(x) = [\\ln(x_{1}/x_{D}); \\ldots; \\ln(x_{D-1}/x_{D})] \\), with inverse \\( \\mathrm{alr}^{-1}(y) = C[\\exp([y; 0])] \\); it enables unconstrained analysis but uses an arbitrary reference part and defines coordinates in an oblique basis, so it is not an isometry.[4](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf) • [7](https://f1000research.com/articles/7-1278) The centered logratio treats parts symmetrically, \\( \\mathrm{clr}(x) = \\ln(x/g(x)) \\) with \\( g(x) = (x_{1} x_{2} \\cdots x_{D})^{1/D} \\) and inverse \\( \\mathrm{clr}^{-1}(z) = C[\\exp(z)] \\), but the resulting D coordinates are singular (they sum to zero) and clr is not subcompositionally coherent.[4](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf) • [6](https://arxiv.org/html/2201.05197v3) • [7](https://f1000research.com/articles/7-1278) The isometric logratio, \\( \\mathrm{ilr}_{V}(x) = \\mathrm{clr}(x) \\cdot V \\) with \\( V \\cdot V^{t} = I_{D-1} \\), yields \\( D-1 \\) linearly independent, isometric coordinates; its disadvantage is that interpretation as ratios of geometric means is complicated.[4](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf) • [6](https://arxiv.org/html/2201.05197v3) Balances provide interpretable ilr coordinates: a sequential binary partition of the parts defines \\( D-1 \\) balances whose variances sum to the total sample-wise variance, an approach introduced by J. J. Egozcue and V. Pawlowsky-Glahn in 2005 in Mathematical Geology.[17](https://doi.org/10.1007/s11004-005-7381-9) • [7](https://f1000research.com/articles/7-1278)\n\nDomain-specific variants include PhILR, a phylogenetic ilr transform for microbiota data described by Justin D Silverman and colleagues in 2017 in eLife,[18](https://doi.org/10.7554/elife.21887) and ANCOM, described in Microbial Ecology in Health and Disease, together with its bias-corrected successor ANCOM-BC in [Nature Communications](https://www.edgepedia.org/nature-communications).[19](https://doi.org/10.3402/mehd.v26.27663) • [20](https://doi.org/10.1038/s41467-020-17041-7) Recent additions include the chiPower transformation, a power-based alternative to logratios described by Michael Greenacre in 2024 in Advances in Data Analysis and [Classification](https://www.edgepedia.org/classification) that permits zero values.[21](https://doi.org/10.1007/s11634-024-00600-x) • [6](https://arxiv.org/html/2201.05197v3)\n\n## Applications\n\nCoDA originated in geochemistry, where data recorded as percent or ppm are subject to the constant-sum constraint that precludes much ordinary statistical analysis.[22](https://www.jstage.jst.go.jp/article/geosoc/112/3/112_3_173/_article/-char/en) [High-throughput sequencing](https://www.edgepedia.org/high-throughput-sequencing) is the other major application area: RNA-seq and 16S rRNA datasets are compositional because total reads per sample are not informative, and they map to the Aitchison simplex rather than [Euclidean space](https://www.edgepedia.org/euclidean-space), whereas standard tools assume Euclidean count data.[12](https://pmc.ncbi.nlm.nih.gov/articles/PMC4030730/)\n\n## Limitations and alternatives\n\nThe named failure modes follow from the theory. Subcompositional coherence, the requirement that results on a subset of parts match those on the full composition, is a defining principle, and clr violates it; linear constraints on the simplex also become non-linear in logratio terms.[2](https://ww2.amstat.org/meetings/proceedings/2014/data/assets/pdf/312239_89024.pdf) • [7](https://f1000research.com/articles/7-1278) Simulations show that both the proportion of zeros and the choice of zero-replacement value substantially affect power and false discovery rate of tests on ALR/CLR-transformed data, and that ILR preserves geometric properties but may have lower statistical power than ALR/CLR in high-dimensional or small-sample settings.[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC11609487/)\n\nAgainst alternatives, ANCOM-BC corrects unequal sampling fractions through a sample-specific offset in a linear regression model, and in simulations ANCOM and ANCOM-BC control FDR at the nominal level except when sample sizes are below 10, while other methods inflate FDR increasingly with sample size; RNA-seq methods such as DESeq2 and edgeR perform poorly on microbiome data because their normalization assumes very few taxa are differentially abundant.[23](https://www.nature.com/articles/s41522-020-00160-w) A 38-dataset benchmark found wide variation among differential abundance tools, with ALDEx2 and ANCOM-II the most conservative and consistent, and recommended avoiding edgeR and uncorrected LEfSe for 16S data and using a consensus of methods.[24](https://www.nature.com/articles/s41467-022-28034-z)\n\n## References\n\n1. [Compositional Data Analysis (Annual Review of Statistics and Its Application)](https://www.annualreviews.org/content/journals/10.1146/annurev-statistics-042720-124436)\n2. [Compositional Data: An Overview (Bacon-Shone, JSM 2014 proceedings)](https://ww2.amstat.org/meetings/proceedings/2014/data/assets/pdf/312239_89024.pdf)\n3. [J. Aitchison (1986). The Statistical Analysis of Compositional Data. .](https://doi.org/10.1007/978-94-009-4109-0)\n4. [Compositional Data Analysis in a Nutshell (Tolosana-Delgado, 2008)](http://www.sediment.uni-goettingen.de/staff/tolosana/extra/CoDaNutshell.pdf)\n5. [Comparison of zero replacement strategies for compositional data with large numbers of zeros (Lubbe, Filzmoser & Templ, 2021, Chemometrics and Intelligent Laboratory Systems 210:104248)](https://digitalcollection.zhaw.ch/items/a8e26557-b5a4-4c85-ac66-00b14d2d2c2b/full)\n6. [Aitchison's Compositional Data Analysis 40 Years On: A Reappraisal (Greenacre)](https://arxiv.org/html/2201.05197v3)\n7. [Visualizing balances of compositional data (F1000Research, 2018)](https://f1000research.com/articles/7-1278)\n8. [The Mathematics of Compositional Analysis (Austrian Journal of Statistics)](https://www.ajs.or.at/index.php/ajs/article/download/vol45-4-4/525)\n9. [A Concise Guide to Compositional Data Analysis (Aitchison)](http://leg.ufpr.br/lib/exe/fetch.php/pessoais:abtmartins:a_concise_guide_to_compositional_data_analysis.pdf)\n10. [Concepts for handling of zeros and missing values in compositional data (van den Boogaart & Tolosana-Delgado, IAMG 2006)](http://stat.boogaart.de/Publications/iamg06_s07_01.pdf)\n11. [Review and revamp of compositional data transformation: A new framework combining proportion conversion and contrast transformation](https://pmc.ncbi.nlm.nih.gov/articles/PMC11609487/)\n12. [Unifying the analysis of high-throughput sequencing datasets: characterizing RNA-seq, 16S rRNA gene sequencing and selective growth experiments by compositional data analysis (Gloor et al., Microbiome 2014; ALDEx2)](https://pmc.ncbi.nlm.nih.gov/articles/PMC4030730/)\n13. [Karl Pearson (1897). Mathematical contributions to the theory of evolution., On a form of spurious correlation which may arise when indices are used in the measurement of organs. Proceedings of the Royal Society of London.](https://doi.org/10.1098/rspl.1896.0076)\n14. [F. Chayes (1960). On correlation between variables of constant sum. Journal of Geophysical Research Atmospheres.](https://doi.org/10.1029/jz065i012p04185)\n15. [J. Aitchison, S. M. Shen (1980). Logistic-Normal Distributions: Some Properties and Uses. Biometrika.](https://doi.org/10.2307/2335470)\n16. [Aitchison (1982) read paper full text, Royal Statistical Society](http://wiki.leg.ufpr.br/lib/exe/fetch.php/pessoais:abtmartins:thestatisticalanalysisofcompositionaldata.pdf)\n17. [J. J. Egozcue, V. Pawlowsky-Glahn (2005). Groups of Parts and Their Balances in Compositional Data Analysis. Mathematical Geology.](https://doi.org/10.1007/s11004-005-7381-9)\n18. [Justin D Silverman and colleagues (2017). A phylogenetic transform enhances analysis of compositional microbiota data. eLife.](https://doi.org/10.7554/elife.21887)\n19. [Siddhartha Mandal and colleagues (2015). Analysis of composition of microbiomes: a novel method for studying microbial composition. Microbial Ecology in Health and Disease.](https://doi.org/10.3402/mehd.v26.27663)\n20. [Huang Lin, Shyamal Das Peddada (2020). Analysis of compositions of microbiomes with bias correction. Nature Communications.](https://doi.org/10.1038/s41467-020-17041-7)\n21. [Michael Greenacre (2024). The chiPower transformation: a valid alternative to logratio transformations in compositional data analysis. Advances in Data Analysis and Classification.](https://doi.org/10.1007/s11634-024-00600-x)\n22. [Problems in compositional data analysis and their solutions (Ohta & Arai, J. Geol. Soc. Japan, 2006)](https://www.jstage.jst.go.jp/article/geosoc/112/3/112_3_173/_article/-char/en)\n23. [Analysis of microbial compositions: a review of normalization and differential abundance analysis (npj Biofilms and Microbiomes, 2020)](https://www.nature.com/articles/s41522-020-00160-w)\n24. [Microbiome differential abundance methods produce different results across 38 datasets (Nature Communications)](https://www.nature.com/articles/s41467-022-28034-z)\n25. [Proportionality: A Valid Alternative to Correlation for Relative Data](https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1004075)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Multivariate association and dimension reduction › Titles A to Me*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/compositional-data-analysis", "markdown_url": "https://www.edgepedia.org/compositional-data-analysis.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. 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Edgepedia Community License 1.0.", "speakable": "Compositional data analysis is the branch of statistics for data carrying only relative information, such as percentages and proportions, analyzing logratios within the simplex since Aitchison's 1986 monograph." }