Log-Ratio Transforms for Compositional Omics Data

Why Compositional Measurements Need Relative Coordinates

Many omics measurements are recorded as parts of a whole: for example, the proportions of identified molecules in a sample or feature abundances reported after normalization to a fixed total. In such data, the total is constrained, so an increase in one part's proportion can occur because that part increased, because other parts decreased, or both. A proportion alone does not establish an absolute change in the amount of that feature.

Log-ratio transforms express the information in these data as comparisons between parts. They are useful when the scientifically meaningful quantities are relative abundances, rather than the total amount measured. They do not make every omics data set compositional: the interpretation depends on how the measurements were produced and what the scientific question concerns.

The centered log-ratio (CLR) transform compares each part with the geometric mean of all parts. The additive log-ratio (ALR) transform compares each non-reference part with one chosen reference. Both require strictly positive entries, and both use natural logarithms here. A log-ratio is dimensionless: the units cancel in the ratio.

What the Coordinates Mean

A CLR coordinate is positive when its part is larger than the geometric mean of the parts in that sample, and negative when it is smaller. It is not a log of an absolute abundance. An ALR coordinate is positive when its part exceeds the selected reference and negative when it is smaller. For either transform, a difference of coordinates has a direct ratio interpretation.

Thus the difference between two CLR coordinates is the log of the ratio between those parts; the geometric mean cancels. For ALR coordinates with the same reference \(r\), subtracting two coordinates also cancels that reference: \(\operatorname{alr}_r(x)_i-\operatorname{alr}_r(x)_j=\ln(x_i/x_j)\). If the same feature and reference are compared across samples, the difference in that ALR coordinate is the log of the change in the feature-to-reference ratio.

Both transforms ignore a common multiplicative scale. If every part is multiplied by a positive constant \(c\), then \(g(cx)=c\,g(x)\), so \(\operatorname{clr}(cx)_i=\ln(cx_i/(c\,g(x)))=\operatorname{clr}(x)_i\). Also, \(\operatorname{alr}_r(cx)_i=\ln(cx_i/(cx_r))=\operatorname{alr}_r(x)_i\). This scale invariance is appropriate when the total is not part of the intended analysis, but it also means the transformed coordinates cannot recover that total.

Additive Log-Ratios and the Reference Choice

An ALR representation uses \(D-1\) coordinates because the reference part is compared with itself only implicitly: its ratio to itself is 1, with log equal to 0. Choosing a different reference changes the values and labels of the coordinates. It does not change the underlying pairwise ratios, which can be recovered from coordinate differences or from the original composition.

A chosen reference should be positive and measured comparably across samples. It should also be scientifically interpretable as a useful denominator. A low or technically unstable reference can make many ALR coordinates noisy, since every one divides by it. A CLR avoids selecting one reference part, but its denominator depends on all included parts; changing the feature set can therefore change CLR coordinates.

Constraints, Reconstruction, and a Short Derivation

The CLR coordinates always sum to zero. This follows directly from the definition:

The final equality holds because \(g(x)^D=\prod_{i=1}^{D}x_i\). Consequently, CLR coordinates lie in a \(D-1\)-dimensional subspace; they are linearly dependent, even though there are \(D\) coordinates. This is a property of the representation, not evidence of a calculation error. Treating all CLR coordinates as independent can lead to singular covariance matrices in methods that require an invertible covariance matrix.

ALR has exactly \(D-1\) coordinates and does not have this same zero-sum constraint. It is invertible as a representation of relative composition once the reference is specified. Set the reference value to 1 and each other value to the exponential of its ALR coordinate, then divide all values by their sum if proportions summing to 1 are needed. For example, if the ALR coordinates relative to part \(r\) are \(y_i\), set \(q_r=1\), \(q_i=e^{y_i}\), and \(p_i=q_i/\sum_{j=1}^{D}q_j\). Then \(p_i/p_r=e^{y_i}=x_i/x_r\), so the reconstructed composition has the same ratios as \(x\).

CLR can also be converted back to relative proportions: exponentiate each coordinate and close the resulting positive values to sum to 1. The zero-sum property implies that the geometric mean of the exponentiated CLR coordinates is 1; closure supplies the desired total. Neither inverse recovers the original absolute total, because that information was removed by scale invariance.

Assumptions, Diagnostics, and Alternatives

Before applying either transform, verify that the coordinates answer the scientific question and that the parts define a comparable composition for each sample. In particular, the selected parts should have consistent meanings across samples, be recorded on a comparable scale, and be strictly positive. The transform is mathematically defined for positive measurements, but mathematical eligibility alone does not establish that relative interpretation is scientifically appropriate.

• Check positivity. A zero or negative entry cannot be used in a logarithmic ratio as defined here. Do not silently replace it with a small positive value; handling zeros and detection limits requires a deliberate method.
• Check the parts and their measurement scale. Inconsistent feature definitions, units that differ across parts, or different sets of included parts can undermine comparisons. A common multiplicative factor applied to every part cancels, but part-specific calibration differences do not generally cancel.
• For ALR, inspect the reference. If the reference is near zero, unstable, or affected by detection limits, ratios involving it can become extreme or highly variable. Assess whether conclusions are sensitive to plausible reference choices.
• For CLR, record the feature set. Because the geometric mean includes every included part, adding or removing features can shift coordinates. Use a consistent feature set when comparing samples and document it.

The main alternatives answer different questions. An ALR is compact and can give a direct, interpretable comparison to a chosen reference, but conclusions expressed as individual coordinates depend on that choice. A CLR treats parts symmetrically and avoids selecting one denominator, but its coordinates are constrained to sum to zero and depend on the included parts. A direct pairwise log ratio, \(\ln(x_i/x_j)\), focuses on a particular comparison without defining a full transformed vector. By contrast, applying a log transform to each raw measurement separately does not itself remove a sample-wide scale: relative interpretation requires comparing logged parts or using an appropriate normalization strategy. The earlier material on normalizing proteomic and metabolomic intensities addresses intensity normalization; a log-ratio transform is not a substitute for deciding whether the measurements should be interpreted compositionally.

Common Mistakes and Practical Use

• Reading a log-ratio as an absolute effect. A higher CLR coordinate means the part is larger relative to the sample's geometric mean, not necessarily that its absolute abundance increased.
• Assuming the total is preserved. Multiplying every part by the same positive factor leaves both transforms unchanged. Keep separate total information when it matters.
• Calling the CLR coordinates independent. They sum to zero, so there is an exact linear dependence among the \(D\) coordinates.
• Ignoring an ALR reference change. Coordinate values from different references are not directly interchangeable. State the reference and interpret every coordinate as a comparison with it.
• Applying logarithms to zeros without a plan. The formulas require positive entries; an undefined value is not repaired by ignoring the issue.

A practical choice is to use CLR when a symmetric representation of the specified set of parts is useful and downstream methods can accommodate its linear dependence. Use ALR when a meaningful, stable reference supports clear comparisons and reference dependence is acceptable. In either case, report the parts included, the reference if applicable, and whether the measurements are treated as relative composition. These details determine what the resulting coordinates mean.