Composition Statistics (skbio.stats.composition)

This module provides functions for compositional data analysis.

Many omics datasets are inherently compositional – meaning that they are best interpreted as proportions or percentages rather than absolute counts.

Formally, sample \(x\) is a composition if \(\sum_{i=0}^D x_{i} = c\) and \(x_{i} > 0\), \(1 \leq i \leq D\) and \(c\) is a real-valued constant and there are \(D\) components (features) for this composition. In this module \(c=1\). Compositional data can be analyzed using Aitchison geometry [1].

However, in this framework, standard real Euclidean operations such as addition and multiplication no longer apply. Only operations such as perturbation and power can be used to manipulate this data.

This module allows two styles of manipulation of compositional data. Compositional data can be analyzed using perturbation and power operations, which can be useful for simulation studies. The alternative strategy is to transform compositional data into the real space. Right now, the centre log ratio transform (clr) and the isometric log ratio transform (ilr) [2] can be used to accomplish this. This transform can be useful for performing standard statistical methods such as parametric hypothesis testing, regression and more.

The major caveat of using this framework is dealing with zeros. In Aitchison geometry, only compositions with non-zero components can be considered. The multiplicative replacement technique [3] can be used to substitute these zeros with small pseudocounts without introducing major distortions to the data.

Differential abundance

Statistical tests for the differential abundance (DA) of components among groups of compositions.

ancom	Perform differential abundance test using ANCOM.
ancombc	Perform differential abundance test using ANCOM-BC.
dirmult_ttest	Perform t-test using Dirichlet-multinomial distribution.
dirmult_lme	Fit a Dirichlet-multinomial linear mixed effects model.
struc_zero	Identify features with structural zeros.

Note

Differential abundance tests will be moved to a separate module differential in the next release of scikit-bio. The current location will be kept as an alias.

Arithmetic operations

Manipulate compositional data within the Aitchison space.

centralize	Center data around its geometric average.
closure	Perform closure to ensure that all components of each composition sum to 1.
inner	Calculate the Aitchson inner product.
perturb	Perform the perturbation operation.
perturb_inv	Perform the inverse perturbation operation.
power	Perform the power operation.

Log-ratio transformation

Convert compositional data into log-ratio space to enable subsequent comparison and statistical analysis.

alr	Perform additive log ratio (ALR) transformation.
alr_inv	Perform inverse additive log ratio (ALR) transform.
clr	Perform centre log ratio (CLR) transformation.
clr_inv	Perform inverse centre log ratio (CLR) transformation.
ilr	Perform isometric log ratio (ILR) transformation.
ilr_inv	Perform inverse isometric log ratio (ILR) transformation.

Note

Arithmetic operations and log-ratio transformations support array formats compliant with the Python array API standard without transition through NumPy. For example, they can directly consume and return GPU-resident PyTorch tensors.

Correlation analysis

Measure the pairwise relationships of compositional data.

vlr	Calculate variance log ratio.
pairwise_vlr	Perform pairwise variance log ratio transformation.

Zero handling

Replace zero values in compositional data with positive values, which is necessary prior to logarithmic operations.

multi_replace

Replace all zeros with small non-zero values.

Basis construction

Generate basis vectors for compositional data via hierarchical partitioning, to allow for decomposition and transformation, such as ilr transform.

sbp_basis	Build an orthonormal basis from a sequential binary partition (SBP).
tree_basis	Calculate the sparse representation of an ilr basis from a tree.

References

[1]

V. Pawlowsky-Glahn, J. J. Egozcue, R. Tolosana-Delgado (2015), Modeling and Analysis of Compositional Data, Wiley, Chichester, UK

[2]

J. J. Egozcue., “Isometric Logratio Transformations for Compositional Data Analysis” Mathematical Geology, 35.3 (2003)

[3]

J. A. Martin-Fernandez, “Dealing With Zeros and Missing Values in Compositional Data Sets Using Nonparametric Imputation”, Mathematical Geology, 35.3 (2003)