Research Radar — 2026-11-01

Methodology Review: Compositional log-ratios: what relative measurements can and cannot tell you

English · About 10 minutes

R1 and R4 publisher abstracts/metadata only; R2 publisher full text; R3 indexed PMC primary full text and PubMed metadata, with direct PMC open showing a browser-check page. Toy examples and transfers are original, not empirical validation. Original ANCOM results are not presented as universal guarantees.

The intuition

Imagine an assay that reports how a fixed basket is divided, but not how large the basket was. A component can take a smaller share even when its amount has not fallen. Before choosing a statistical test, ask what the instrument has actually identified: an amount, a fraction, or a ratio. These are different scientific targets. The key question in this review is not how to recover a prettier embedding. It is which conclusions survive when the sample-specific total is unknown.

Aitchison made the simplex, the space of compositions, a central statistical object and developed transformed-normal models and notions of independence for it. Log-ratio thinking addresses a related practical problem: describe parts relative to other parts rather than silently treating a proportion as an absolute measurement. Changing coordinates does not restore missing totals; it makes the information that remains explicit. This is a foundational modeling perspective, not a promise that every count assay needs the same preprocessing. [R1]

A small worked example

Consider three fictional molecular species A, B and C. Their absolute amounts in sample one are \((10,20,30)\); in sample two they are \((10,20,90)\). The totals are 60 and 120. The reported fractions therefore change from \((1/6,1/3,1/2)\) to \((1/12,1/6,3/4)\). A and B both appear to halve as fractions, although their absolute amounts are unchanged. C increases threefold in amount but only one-and-a-half-fold in fraction. These numbers are an original arithmetic illustration, not experimental observations.

Now compare A with B. Their ratio is \(1/2\) in both samples, so its natural logarithm remains approximately \(-0.693147\). Compare C with B instead: the ratio moves from \(3/2\) to \(9/2\). The change in its natural logarithm is \(\ln(3)\approx1.098612\). Thus one contrast stays fixed and another changes, without requiring the totals. This is not a significance test: there is one constructed observation per condition, no estimated sampling variability, and no biological replication.

For a sharper identifiability check, replace the second absolute vector by \((5,10,45)\). It produces exactly the same second composition as \((10,20,90)\), but A and B now truly decrease. Replace it instead by \((20,40,180)\) and A and B increase. All three worlds give the same recorded fractions. An algorithm using only those fractions cannot choose between them. Collecting more decimal places does not resolve this ambiguity; a measurement of total amount, or a justified additional assumption, would change the information available.

Finally, remove C from the reported panel and renormalize the remaining parts. Both samples become \((1/3,2/3)\). The A-to-B ratio survives unchanged, whereas the individual fractions change. This provides a useful unit test for an implementation: compute the same pairwise log-ratio before and after restricting the panel. It should agree to numerical precision when both retained values are positive. Do not demand that every multivariate summary remain unchanged, because a summary involving all measured parts answers a different question after a part is removed.

The necessary mathematics

Let \(a_i\gt0\) be the amount of part \(i\), let \(D\) be the number of parts, and define the total \(T=\sum_{j=1}^{D}a_j\) and fraction \(x_i=a_i/T\). Here \(j\) also indexes parts, and \(\ln\) denotes the natural logarithm. Direct cancellation gives the elementary identity underlying the example:

A sample-specific positive multiplier disappears. The composition therefore identifies relative amounts, not its common scale. This algebra explains the scale-invariance principle discussed by Lovell and colleagues. [R2]

One can choose a positive reference part \(r\) and use additive log-ratio coordinates \(z_i=\ln(x_i/x_r)\) for every \(i\neq r\). Alternatively, define the geometric mean \(g(x)=(\prod_{j=1}^{D}x_j)^{1/D}\) and centered coordinates \(c_i=\ln(x_i/g(x))\). The product symbol multiplies all parts. Summing the centered coordinates gives zero because \(\ln g(x)\) is the average log component. Consequently they have only \(D-1\) independent dimensions; their full covariance matrix is singular. These identities follow directly from the definitions, not from a fitted distribution. [R1] [R4]

Egozcue and colleagues construct isometric log-ratio coordinates using orthonormal bases, preserving the relevant metric while removing coordinate redundancy. A basis must still be specified for interpretation. In our three-part example, the change in centered coordinates is \((-\ln(3)/3,-\ln(3)/3,2\ln(3)/3)\), approximately \((-0.366204,-0.366204,0.732408)\). Their sum is zero. A negative centered change for A means a decrease relative to the geometric mean, not proof of absolute loss. This calculation is a direct check of the definition, rather than a numerical result from the cited paper. [R4]

Assumptions and failure modes

Positive values are essential to these formulas. A zero could mean an absent component, an undetected component, or inadequate sampling; those possibilities are not interchangeable. The original ANCOM paper explicitly notes that adding a small constant before logging is arbitrary and warrants sensitivity investigation. Its discussion also excludes recovering a common absolute shift of every component from compositions alone. Do not read its reported simulation performance as universal false-discovery control for every sparse assay, sampling design, or zero-handling choice. [R3]

There are two immediate stress tests worth performing on our artificial example. First multiply all amounts within either sample by any positive number: every pairwise log-ratio should stay fixed. Then multiply only B by two: A-to-B and C-to-B should each drop by the natural logarithm of two. The second operation is not a library-size change; it changes the reference itself. This is why a convenient denominator is not automatically a stable biological control. Also keep the experimental unit visible: transforming measurements does not make repeated observations from the same donor independent.

Verified applications

Lovell and colleagues analyzed fission-yeast expression measurements and showed that relative-abundance correlations could disagree with absolute-abundance correlations. They used proportionality to construct association summaries and networks. Their worked analysis retained positive records: 3,031 mRNAs across 16 time points. This verifies a transcriptomics application, not a claim that the same preprocessing or association statistic is validated for sparse single-cell counts. [R2]

Mandal and colleagues introduced ANCOM and illustrated it with two human-gut microbiome datasets. Its log-ratio framework addresses group comparisons rather than simply displaying proportions. This establishes a concrete microbiome application of relative comparisons. The cited work does not establish that a transformed coordinate is a direct estimate of absolute microbial load, nor that its original implementation is the best choice for every contemporary study. [R3]

Possible transfers — speculative

A speculative transfer would be an audit of a targeted metabolite panel whose output is normalized within each sample. Before modeling an outcome, predefine two scientifically interpretable metabolite ratios and document why their denominators were chosen. Preserve any available absolute calibration rather than discarding it. Repeat the analysis with and without one abundant, unrelated panel member. The pairwise ratios should not move merely because that third member was removed; if they do, inspect the processing steps. This would test an implementation property, not demonstrate a disease biomarker.

A second proposed exercise is to simulate a perturbation that doubles every molecule in each sample while leaving all proportions fixed. Ask which parts of the workflow notice it. A composition-only pipeline should not: lack of a signal is correct for its input, even though the biological amount changed. Then add an independently measured total and compare the conclusions. Treat this as an identifiability demonstration, not as evidence that an external total is measured without error. A real study would need a separate measurement model and appropriate biological replicates.

What to try and what to read next

Try the three-part calculation before loading a large matrix. Write each claimed change in a full sentence: relative to B, relative to the geometric mean of this panel, or in absolute amount. If the denominator is missing from the sentence, the interpretation is unfinished. Save the feature list, transformations and zero-handling choices alongside the result. These small records make later changes of panel or preprocessing auditable.

For reading, start with Aitchison for the statistical framing, then Lovell for an omics example in which relative and absolute interpretations diverge. Use Mandal to see log-ratio comparisons embedded in an application, and Egozcue for the next mathematical step into nonredundant coordinates. The practical stopping rule is simple: you should be able to name an alternative absolute world that yields your observed composition. If you cannot rule it out with additional evidence, do not let a confident downstream model rule it out for you.

Reading list

• [R1] Primary reading: The Statistical Analysis of Compositional Data. J. Aitchison (1982). Read the foundational framing: proportions require their own sample space. This is the 1982 article, not the similarly titled 1986 book or reviews of that book.

• [R2] Related reading: Proportionality: A Valid Alternative to Correlation for Relative Data. Lovell, Pawlowsky-Glahn, Egozcue, Marguerat and Bähler (2015). Read the application and the distinction between a relative association and an absolute-abundance claim; the proposed proportionality statistic is not a universal replacement for every network model.

• [R3] Related reading: Analysis of composition of microbiomes: a novel method for studying microbial composition. Mandal, Van Treuren, White, Eggesbø, Knight and Peddada (2015). Read as a concrete log-ratio testing application, keeping its model assumptions and original simulation setting separate from any blanket error-control claim.

• [R4] Related reading: Isometric Logratio Transformations for Compositional Data Analysis. Egozcue, Pawlowsky-Glahn, Mateu-Figueras and Barceló-Vidal (2003). Read next for nonredundant coordinates and the geometry behind log-ratio analysis; this column does not derive the full isometric transformation.