Conceptual Issues with Compositional Data in Model Interaction Terms
In Statistical Modelling, Compositional Data (CoDa) consists of vectors where the components represent proportions of a total. Because these components are constrained to a constant sum (e.g., 100% or 1.0), they exist in a restricted space known as the Simplex. When "Super Users" attempt to include these variables in interaction terms ($X_1 X_2$) within a standard Euclidean framework, several conceptual failures occur.
1. The Closure Problem and Perfect Multicollinearity
The most immediate issue is that compositional data is inherently collinear. If you know $n-1$ components of a composition, the $n$-th component is automatically determined. Including all components plus an interaction term in a linear model leads to Singular Matrices.
- Spurious Correlation: Because the variables must sum to a constant, an increase in one component forces a decrease in others. This "built-in" correlation contaminates the interaction term, making it impossible to tell if the interaction is a real physical effect or a mathematical artifact of the closure.
- Interpretation Failure: In a standard interaction model, we ask: "How does the effect of $X_1$ change as $X_2$ increases?" In CoDa, $X_1$ cannot increase without $X_2$ (or some other part) decreasing. The ceteris paribus (all else held constant) assumption is physically impossible.
2. Non-Linearity of the Simplex Space
Interaction terms in OLS assume a Euclidean geometry where the distance between 1% and 2% is the same as the distance between 50% and 51%. In compositional data, this is not true. Changes near the boundaries (0% or 100%) carry much more information than changes in the middle.
- Relative vs. Absolute: Compositions are about relative differences. A standard interaction term ($X_i \times X_j$) operates on absolute values, which violates the scale invariance principle of compositional data.
- Curvature: The relationship between parts is often non-linear. Multiplying two proportions together in a "raw" format usually fails to capture the logarithmic nature of the ratios between parts.
3. Comparison: Raw Proportions vs. Log-Ratio Transformations
| Approach | Interaction Logic | Resulting Bias |
|---|---|---|
| Raw Proportions | $Y = \beta(X_1 \times X_2)$ | High (Spurious correlations, closure bias) |
| Additive Log-Ratio (ALR) | $\log(X_1/X_n) \times \log(X_2/X_n)$ | Low (Respects relative nature, choice of $X_n$ matters) |
| Isometric Log-Ratio (ILR) | Coordinates in Euclidean space | Minimum (Theoretically robust for interactions) |
4. The Solution: Interactions in Coordinates
To correctly model interactions with compositional data in 2026, experts on Cross Validated recommend transforming the data into Isometric Log-Ratio (ILR) coordinates. This maps the Simplex into real Euclidean space ($\mathbb{R}^{n-1}$).
Once in ILR coordinates, the components are no longer constrained by a constant sum, and standard interaction terms can be applied. However, the interpretation must be mapped back to the original proportions, which often requires calculating "Compositional Effects" or using visual tools like Compositional Heatmaps.
5. Checkpoint: Is your Interaction Meaningful?
- Sub-compositional Coherence: Does the interaction hold true if you only look at a subset of the parts? If not, the interaction is likely an artifact of the total sum.
- Permutation Invariance: Does the model's conclusion change if you swap which variable is the "base" or "denominator"? Robust models should remain stable.
Conclusion
Using compositional data in model interaction terms without transformation is a recipe for spurious results. The "Closed" nature of the data violates the fundamental assumptions of independence required for interaction analysis. By 2026, the use of Log-Ratio Transformations (CLR, ILR) has become the mandatory standard for any rigorous analysis involving parts of a whole. Only by moving the data out of the Simplex and into a coordinate system can we truly understand how the relative abundance of one part moderates the effect of another.
Keywords
compositional data interaction terms, closure problem statistics, Aitchison geometry CoDa, spurious correlation compositional data, ILR transformation interaction model, Cross Validated compositional data, simplex space modeling, log-ratio analysis 2026.
