Subcompositional Coherence Research Articles

Aitchison’s Compositional Data Analysis 40 Years on: A Reappraisal

The development of John Aitchison’s approach to compositional data analysis is followed since his paper read to the Royal Statistical Society in 1982. Aitchison’s logratio approach, which was proposed to solve the problematic aspects of working with data with a fixed-sum constraint, is summarized and reappraised. It is maintained that the properties on which this approach was originally built, the main one being subcompositional coherence, are not required to be satisfied exactly—quasi-coherence is sufficient, that is near enough to being coherent for all practical purposes. This opens up the field to using simpler data transformations, such as power transformations, that permit zero values in the data. The additional property of exact isometry, which was subsequently introduced and not in Aitchison’s original conception, imposed the use of isometric logratio transformations, but these are complicated and problematic to interpret, involving ratios of geometric means. If this property is regarded as important in certain analytical contexts, for example, unsupervised learning, it can be relaxed by showing that regular pairwise logratios, as well as the alternative quasi-coherent transformations, can also be quasi-isometric, meaning they are close enough to exact isometry for all practical purposes. It is concluded that the isometric and related logratio transformations such as pivot logratios are not a prerequisite for good practice, although many authors insist on their obligatory use. This conclusion is fully supported here by case studies in geochemistry and in genomics, where the good performance is demonstrated of pairwise logratios, as originally proposed by Aitchison, or Box–Cox power transforms of the original compositions where no zero replacements are necessary.

Taxicab Correspondence Analysis and Taxicab Logratio Analysis: A Comparison on Contingency Tables and Compositional Data

In this paper, we attempt to see further by relating theory with practice: First, we review the principles on which three interrelated well developed methods for the analysis and visualization of contingency tables and compositional data are erected: Correspondence analysis based on Benzécri’s principle of distributional equivalence, Goodman’s RC association model based on Yule’s principle of scale invariance, and compositional data analysis based on Aitchison’s principle of subcompositional coherence. Second, we introduce a novel index named intrinsic measure of the quality of the signs of the residuals for the choice of the method. The criterion is based on taxicab singular value decomposition, on which the package TaxicabCA in R is developed. We present a minimal R script thatcan be executed to obtain the numerical results and the maps in this paper. Third, we introduce a flexible method based on the novel index for the choice of the constant to be added to contingency tables with zero counts so that logratio methods can be applied.

Pairwise ratio-based differential abundance analysis of infant microbiome 16S sequencing data.

Differential abundance analysis of infant 16S microbial sequencing data is complicated by challenging data properties, including high sparsity, extreme dispersion and the relative nature of the information contained within the data. In this study, we propose a pairwise ratio analysis that uses the compositional data analysis principle of subcompositional coherence and merges it with a beta-binomial regression model. The resulting method provides a flexible and easily interpretable approach to infant 16S sequencing data differential abundance analysis that does not require zero imputation. We evaluate the proposed method using infant 16S data from clinical trials and demonstrate that the proposed method has the power to detect differences, and demonstrate how its results can be used to gain insights. We further evaluate the method using data-inspired simulations and compare its power against related methods. Our results indicate that power is high for pairwise differential abundance analysis of taxon pairs that have a large abundance. In contrast, results for sparse taxon pairs show a decrease in power and substantial variability in method performance. While our method shows promising performance on well-measured subcompositions, we advise strong filtering steps in order to avoid excessive numbers of underpowered comparisons in practical applications.

Multivariate analysis of log-ratio transformed data and its priority in mining science, porphyry and polymetallic vein deposits case studies

Each mineralization style is characterized by typical signature associations between elements due to elemental interactions, therefore the coherence and closure effects problem must be overcome in geochemical processing. The coherence indicates the ratios between two components (rows or columns) remains the same whether they are considered in a subcomposition or in the full composition. The log-ratio transformation (LRT) has recognized as a standard procedure to support subcompositional coherence. The log-transformed data is applicable for geochemical data to unveil such associations, prior to applying the multivariate analysis like correspondence analysis (CA) and principal component analysis (PCA). At the present study, subcompositional coherence is overcome by inverse iso-metric log-ratio transformation for geochemical compositional data at two polymetallic and porphyry deposits. Based on Ilr-transformed data, Ag, Au, As, Pb, Te, Mo and rather S, W, Cu are enriched as polymetallic elements at Glojeh, while Au-Cu-(Mo) compositions indicate a porphyry deposit occurred in Dalli deposit. The ability to handle zero values in the data matrix and determining an elemental eccentricity from the center of each axis based on Euclidean distances are the advantages of CA method, with compression to LRT. Whereas, loading factors which spread in every direction and providing subcompositional coherence are the competitive advantages of PCA based on LRT, for both case studies. Results with these techniques show significant ability to draw an inference in such geochemical data, and in improving the performance of multivariate techniques using LRT.

Generalized linear models with linear constraints for microbiome compositional data.

Motivated by regression analysis for microbiome compositional data, this article considers generalized linear regression analysis with compositional covariates, where a group of linear constraints on regression coefficients are imposed to account for the compositional nature of the data and to achieve subcompositional coherence. A penalized likelihood estimation procedure using a generalized accelerated proximal gradient method is developed to efficiently estimate the regression coefficients. A de-biased procedure is developed to obtain asymptotically unbiased and normally distributed estimates, which leads to valid confidence intervals of the regression coefficients. Simulations results show the correctness of the coverage probability of the confidence intervals and smaller variances of the estimates when the appropriate linear constraints are imposed. The methods are illustrated by a microbiome study in order to identify bacterial species that are associated with inflammatory bowel disease (IBD) and to predict IBD using fecal microbiome.

Evaluation of elemental mineralization rank using fractal and multivariate techniques and improving the performance by log-ratio transformation

Background separation from anomalous populations in the polymetallic mineralization is a problematic issue in geochemical exploration. This study focuses on the applications of fractal modeling, correspondence analysis (CA), classical principal component analysis (PCA), and PCA method based on centred log-ratio (clr) transformation for the multi-elemental evaluation of geochemical data in Glojeh polymetallic mineralization, NW Iran. A Concentration-Number (C-N) multifractal modeling was applied to delineate individual self-similar patterns for elemental distributions simultaneously in terms of Au, Ag, Cu, Pb, and Zn. The differences between fractal dimensions and clockwise angle between fitted lines in the C-N Log-log plots could lead to evaluate mineralized rank corresponding to Au≥Ag≥Pb≫Cu≈Zn. A more comprehensive evaluation via CA and PCA of clr-transformed data could facilitate the vein structure identification in high-dimensional data. The PCA results of clr-transformed data revealed the potential intensification trend of (Pb, Ag, As, Te, Au)>(Mo, Zn, Be, Cu)>W>(S, Cd), that is more consistent with fractal and CA approaches. The closure effect problem was overcome by the clr transformation, and the total variance explained by the first factor increased from 28.2% in classical PCA to 43.7% in clr transformation. Accordingly, As, Sb and Cd were considered as potential pathfinder elements for Au in Glojeh deposit. The ability to handle zeros in the data matrix and determining an elemental eccentricity as a criterion are the advantages of CA method, while loading factors spread in a full circle and providing subcompositional coherence are the competitive advantages of PCA (calculated according to log-ratio transformation). However, the CA and PCA based on Log-Ratio transformation techniques showed significant potential to draw an inference in such deposits.

The Mathematics of Compositional Analysis

The term compositional data analysis is historically associated to the approach based on the logratio transformations introduced in the eighties. Two main principles of this methodology are scale invariance and subcompositional coherence. New developments and concepts emerged in the last decade revealed the need to clarify the concepts of compositions, compositional sample space and subcomposition. In this work the mathematics of compositional analysis based on equivalence relation is presented. The two principles are essential attributes of the corresponding quotient space. A logarithmic isomorphism between quotient spaces induces a metric space structure for compositions. Using this structure, the statistical analysis of compositions consists of analysing logratio coordinates.

An approach to measure distance between compositional diet estimates containing essential zeros

ABSTRACTFor many applications involving compositional data, it is necessary to establish a valid measure of distance, yet when essential zeros are present traditional distance measures are problematic. In quantitative fatty acid signature analysis (QFASA), compositional diet estimates are produced that often contain many zeros. In order to test for a difference in diet between two populations of predators using the QFASA diet estimates, a legitimate measure of distance for use in the test statistic is necessary. Since ecologists using QFASA must first select the potential species of prey in the predator's diet, the chosen measure of distance should be such that the distance between samples does not decrease as the number of species considered increases, a property known in general as subcompositional coherence. In this paper we compare three measures of distance for compositional data capable of handling zeros, but not satisfying some of the well-accepted principles of compositional data analysis. For compositional diet estimates, the most relevant of these is the property of subcompositionally coherence and we show that this property may be approximately satisfied. Based on the results of a simulation study and an application to real-life QFASA diet estimates of grey seals, we recommend the chi-square measure of distance.

Regression analysis for microbiome compositional data

One important problem in microbiome analysis is to identify the bacterial taxa that are associated with a response, where the microbiome data are summarized as the composition of the bacterial taxa at different taxonomic levels. This paper considers regression analysis with such compositional data as covariates. In order to satisfy the subcompositional coherence of the results, linear models with a set of linear constraints on the regression coefficients are introduced. Such models allow regression analysis for subcompositions and include the log-contrast model for compositional covariates as a special case. A penalized estimation procedure for estimating the regression coefficients and for selecting variables under the linear constraints is developed. A method is also proposed to obtain de-biased estimates of the regression coefficients that are asymptotically unbiased and have a joint asymptotic multivariate normal distribution. This provides valid confidence intervals of the regression coefficients and can be used to obtain the $p$-values. Simulation results show the validity of the confidence intervals and smaller variances of the de-biased estimates when the linear constraints are imposed. The proposed methods are applied to a gut microbiome data set and identify four bacterial genera that are associated with the body mass index after adjusting for the total fat and caloric intakes.

Analysis of soil bacteria susceptibility to manufactured nanoparticles via data visualization.

The impact of ZnO and TiO2 manufactured nanoparticles (MNPs) on soil bacterial communities for different exposure periods and MNP doses was explored via data visualization techniques. Interrelationships between MNP treatments and responses of bacterial taxa were illustrated by bipartite graphs, allowing fast identification of important soil bacterial taxa that are susceptible to MNPs. Contribution biplots with subcompositional coherence property were generated via log-ratio analysis (LRA), which jointly display the treatment distribution and the variance (contribution) of bacterial taxa. The LRA contribution biplots and nonmetric multi-dimensional scaling (NMDS) of the dataset, along with hierarchical clustering, demonstrated that high doses of ZnO and TiO2 MNPs caused significant compositional changes in soil bacterial communities. The suitability of family level for MNP taxonomic impact assessment was demonstrated by both the LRA biplots and simplified NMDSs with quantification provided by the distance correlation between MNP impacts summarized at different taxonomic levels. The present study demonstrates that visual exploration could potentially assist in knowledge discovery and interpretation of data on soil bacterial communities exposed to MNPs and thus evaluate the potential for environmental impacts.

Compositional statistical analysis of soil 31P-NMR forms

Phosphorus (P) forms determined by 31P nuclear magnetic resonance spectroscopy (31P-NMR) are compositional data (i.e., parts of some whole bounded between 0 and the unit of measurement). Because compositional data are intrinsically related to each other within a closed pre-defined compositional space, a simple log transformation, variable by variable, or any other transformation of the compositional variables may produce statistically erroneous results. However, most studies analyze the P forms as single components rather than parts of some whole such as total P (TP) or soil dry mass, leading systematically to methodological biases and conflicting interpretations. Compositional data analysis using centered log-ratio (clr) or isometric log-ratio (ilr) coordinates avoids such difficulties and preserves sub-compositional coherence in the analysis. The objective of this study was to compare classical and compositional methods for the statistical analysis of 31P-NMR P data expressed as proportions of TP or concentrations relative to soil dry mass. Two published datasets were used. Analyses of variance and regression analysis with soil pH were conducted on P species percentages scaled on TP or as untransformed concentrations scaled on a soil dry-weight basis as well as their ordinary log, centered log-ratios (clr) and isometric log-ratios (ilr). Contradictory F-statistics values and coefficients of correlation with soil pH were obtained for the untransformed and ordinary log transformed 31P-NMR P data expressed as proportions or concentrations. In contrast, statistical results were the same regardless of the measurement unit when P compound percentages were clr-transformed. Using orthogonal ilr coordinates, 31P-NMR P data were correlated to soil properties and to each other and synthesized into a multivariate distance without methodological bias. We conclude that the variance and regression analyses of molecular P species are scale-dependent and that the clr- and the ilr-transformations should be used to unbiasedly analyze the P fractions and avoid conflicting interpretations.

ZCompositions — R package for multivariate imputation of left-censored data under a compositional approach

zCompositions is an R package for the imputation of left-censored data under a compositional approach. It is pertinent when the analyst assumes that the relevant information is contained on the relative variation structure of the data. For instance, in cases where the experimental data are simultaneously measured in amounts related to a same total weight or volume. The approach is used in fields like geochemistry of waters or sedimentary rocks, environmental studies related to air pollution, physicochemical analysis of glass fragments in forensic science, and among many others. In these fields, rounded zeros and nondetects are usually regarded as left-censored data that hamper any subsequent data analysis. The implemented methods consider aspects of relevance for a compositional approach such as scale invariance, subcompositional coherence or preserving the multivariate relative structure of the data. Based on solid statistical frameworks, it comprises the ability to deal with single and varying censoring thresholds, consistent treatment of closed and non-closed data, exploratory tools, multiple imputation, MCMC, robust and non-parametric alternatives, and recent proposals for count data. Key methodological aspects, new contributions, computational implementation and the practical application of the approach are discussed.

Compositional Data Analysis (CoDA) as a tool to study the (paleo)ecology of coccolithophores from coastal-neritic settings off central Portugal

Whereas using the species percentages is the standard analytical procedure used to infer species ecological preferences, independently of taphonomical effects, the closure problem associated with closed number systems and subsequent inconsistency of determining percentages may lead to spurious correlations, biased statistical analysis and misleading interpretations. To avoid these problems, we applied Compositional Data Analysis (CoDA) to investigate the (paleo)ecological preferences and spatial distribution of coccolith assemblages preserved in seafloor sediments, using as a case-study the central Portuguese submarine canyons and adjacent shelf-slope areas. Results from using the isometric log-ratio (ilr) approach from CoDA are compared with results from using classical analytical methods, and further discussed.While providing scale invariance and subcompositional coherence, CoDA is revealed to be a consistent statistical tool to infer the (paleo)ecological preferences of coccolithophores, corroborating earlier work based on from percentage determinations. Results of this study clearly confirmed the coastal-neritic distribution of coccoliths from Gephyrocapsa oceanica, Helicosphaera carteri and Coronosphaera mediterranea, whereas coccoliths from Calcidiscus leptoporus, Umbilicosphaera sibogae, Umbellosphaera irregularis and Rhabdosphaera spp. more typically occur offshore. Differences between canyons and adjacent shelf and slope areas were also confirmed, namely the (a) greater importance of the coastal-neritic assemblage possibly resulting from local and persistent nutrient pumping in these areas, and (b) stronger mixing of coccoliths from coastal and oceanic species in upper canyon reaches, resulting from the focused coastward advection of more oceanic water masses along their axes.Unlike the results from both ilr-coordinates and the percentage approaches, both coccolith concentrations and fluxes showed that spatial trends in which the species ecological inter-relationships appear to be masked by taphonomical phenomena, especially towards the coast and in the canyons, suggesting the two latter approaches are not suitable to perform (paleo)ecological inferences in more dynamic coastal-neritic settings.Our study suggests that the (paleo)ecological signal preserved in the studied sediment samples is persistent enough to be revealed by both CoDA and percentages. Yet, given that CoDA provides the only statistical solution to coherently draw (paleo)ecological interpretations from compositional data, our recommendation is that CoDA should always be used to test and validate any ecological signals obtained from percentage distributions.

Colours and Cocktails: Compositional Data Analysis 2013 Lancaster Lecture

SummaryThe different constituents of physical mixtures such as coloured paint, cocktails, geological and other samples can be represented byd‐dimensional vectors called compositions with non‐negative components that sum to one. Data in which the observations are compositions are called compositional data. There are a number of different ways of thinking about and consequently analysing compositional data. The log‐ratio methods proposed by Aitchison in the 1980s have become the dominant methods in the field. One reason for this is the development of normative arguments converting the properties of log‐ratio methods to ‘essential requirements’ or Principles for any method of analysis to satisfy. We discuss different ways of thinking about compositional data and interpret the development of the Principles in terms of these different viewpoints. We illustrate the properties on which the Principles are based, focussing particularly on the key subcompositional coherence property. We show that this Principle is based on implicit assumptions and beliefs that do not always hold. Moreover, it is applied selectively because it is not actually satisfied by the log‐ratio methods it is intended to justify. This implies that a more open statistical approach to compositional data analysis should be adopted.

Ipsative measurement and the analysis of organizational values: an alternative approach for data analysis

In this paper, the analysis and test of ipsative data will be discussed, and some alternative methods will be suggested. Following a review of the literature about ipsative measurement, the Competing Values Framework will be presented as a major application in the field of organizational culture and values. An alternative approach for the intra-individual analysis and test of ipsative data will be suggested, which consists of: (i) a method that uses closed part-wise geometric means as a descriptive statistic; (ii) a nonparametric bootstrap test to create confidence intervals; and (iii) a permutation test to evaluate equivalence between ipsative scores. All suggested methods satisfy the three basic statistical requirements for the analysis of ipsative data, that is: scale invariance, permutation invariance, and subcompositional coherence. Our suggested approach can correctly compute and compare organizational culture profiles within the same organization, as will be demonstrated with an example. However, the problem of drawing inter-organizational contrasts in ipsative measurement still remains unsolved. Also, our alternative approach only allows for a relative interpretation of the results.

Dealing with Distances and Transformations for Fuzzy C-Means Clustering of Compositional Data

Clustering techniques are based upon a dissimilarity or distance measure between objects and clusters. This paper focuses on the simplex space, whose elements—compositions—are subject to non-negativity and constant-sum constraints. Any data analysis involving compositions should fulfill two main principles: scale invariance and subcompositional coherence. Among fuzzy clustering methods, the FCM algorithm is broadly applied in a variety of fields, but it is not well-behaved when dealing with compositions. Here, the adequacy of different dissimilarities in the simplex, together with the behavior of the common log-ratio transformations, is discussed in the basis of compositional principles. As a result, a well-founded strategy for FCM clustering of compositions is suggested. Theoretical findings are accompanied by numerical evidence, and a detailed account of our proposal is provided. Finally, a case study is illustrated using a nutritional data set known in the clustering literature.

Measuring Subcompositional Incoherence

Subcompositional coherence is a fundamental property of Aitchison’s approach to compositional data analysis and is the principal justification for using ratios of components. We maintain, however, that lack of subcompositional coherence (i.e., incoherence) can be measured in an attempt to evaluate whether any given technique is close enough, for all practical purposes, to being subcompositionally coherent. This opens up the field to alternative methods that might be better suited to cope with problems such as data zeros and outliers while being only slightly incoherent. The measure that we propose is based on the distance measure between components. We show that the two-part subcompositions, which appear to be the most sensitive to subcompositional incoherence, can be used to establish a distance matrix that can be directly compared with the pairwise distances in the full composition. The closeness of these two matrices can be quantified using a stress measure that is common in multidimensional scaling, providing a measure of subcompositional incoherence. The approach is illustrated using power-transformed correspondence analysis, which has already been shown to converge to log-ratio analysis as the power transform tends to zero.

Measuring Subcompositional Incoherence

Subcompositional coherence is a fundamental property of Aitchison's approach to compositional data analysis, and is the principal justification for using ratios of components. We maintain, however, that lack of subcompositional coherence, that is incoherence, can be measured in an attempt to evaluate whether any given technique is close enough, for all practical purposes, to being subcompositionally coherent. This opens up the field to alternative methods, which might be better suited to cope with problems such as data zeros and outliers, while being only slightly incoherent. The measure that we propose is based on the distance measure between components. We show that the two-part subcompositions, which are the most sensitive to subcompositional incoherence, can be used to establish a distance matrix which can be directly compared with the pairwise distances in the full composition. The closeness of these two matrices can be quantified using a stress measure that is common in multidimensional scaling, providing a measure of subcompositional incoherence. Furthermore, we strongly advocate introducing weights into this measure, where rarer components are weighted proportionally less than more abundant components. The approach is illustrated using power-transformed correspondence analysis, which has already been shown to converge to logratio analysis as the power transform tends to zero.

Distributional Equivalence and Subcompositional Coherence in the Analysis of Compositional Data, Contingency Tables and Ratio-Scale Measurements

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyze the ratios of the data values. A common approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property and can be applied to a wider class of methods. This weighted log-ratio analysis is theoretically equivalent to "spectral mapping", a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modeling. The weighted log-ratio methodology is used here to visualize frequency data in linguistics and chemical compositional data in archeology.

Log-Ratio Analysis Is a Limiting Case of Correspondence Analysis

It is common practice in compositional data analysis to perform the log-ratio transformation in order to preserve sub-compositional coherence in the analysis. Correspondence analysis is an alternative approach to analyzing ratio-scale data and is often contrasted with log-ratio analysis. It turns out that if one introduces a power transformation into the correspondence analysis algorithm, then the limit of the power-transformed correspondence analysis, as the power parameter tends to zero, is exactly the log-ratio analysis. Depending on how the power transformation is applied, we can obtain as limiting cases either Aitchison’s unweighted log-ratio analysis or the weighted form called “spectral mapping”. The upshot of this is that one can come as close as one likes to the log-ratio analysis, weighted or unweighted, using correspondence analysis.