Statistical Analysis Of Compositional Data Research Articles

Gmcoda: Graphical model for multiple compositional vectors in microbiome studies.

Microbes are essential components in the ecosystem and participate in most biological procedures in environments. The high-throughput sequencing technologies help researchers directly quantify the abundance of microbes in a natural environment. Microbiome studies explore the construction, stability, and function of microbial communities with the aid of sequencing technology. However, sequencing technologies only provide relative abundances of microbes, and this kind of data is called compositional data in statistics. The constraint of the constant-sum requires flexible statistical methods for analyzing microbiome data. Current statistical analysis of compositional data mainly focuses on one compositional vector such as bacterial communities. The fungi are also an important component in microbial communities and are always measured by sequencing internal transcribed spacer instead of 16S rRNA genes for bacteria. The different sequencing methods between fungi and bacteria bring two compositional vectors in microbiome studies. We propose a novel statistical method, called gmcoda, based on an additive logistic normal distribution for estimating the partial correlation matrix for cross-domain interactions. A majorization-minimization algorithm is proposed to solve the optimization problem involved in gmcoda. Through simulation studies, gmcoda is demonstrated to work well in estimating partial correlations between two compositional vectors. Gmcoda is also applied to infer cross-domain interactions in a real microbiome dataset and finds potential interactions between bacteria and fungi. Gmcoda is open source and freely available from https://github.com/huayingfang/gmcoda under GNU LGPL v3.

A guideline for the statistical analysis of compositional data in immunology

A guideline for the statistical analysis of compositional data in immunology

“The Statistical Analysis of Compositional Data” by John Aitchison (1986): A Bibliometric Overview

This paper presents a complete bibliometric analysis of Aitchison’s 1986 seminal book “The Statistical Analysis of Compositional Data.” We have set three objectives. The first is to analyze the academic structure of Aitchison’s 1986 book. Results reveals that although the work has received citations uninterruptedly since its publication, the number of these has increased very significantly over the past 4 years. This is due to the significant increase in the number of publications on the theme of “Compositional Data Analysis” in fields related to “geoscience” over the last few years. The second objective is to determine which main journals Aitchison’s book has been cited in. The results highlight that the main journals are indexed under the following WoS category: “Geosciences, Multidisciplinary” and “Ecology.” Of these, “Mathematical Geosciences” and “Computers, Geosciences” stand out. The third objective is to determine the main topics analyzed in the principal papers published by authors citing Aitchison’s book. Our results show that the keywords in the main papers to have cited Aitchison’s 1986 book originate from the geoscience field, since many of them are related to concepts directly linked to this field and refer to terms related to “biodiversity,” “geodiversity,” “geoheritage,” and “georesources.” Lastly, the analysis shows how the CoDA methodology is now in a phase of exponential growth, expanding to other fields. This implies that geoscience is becoming consolidated in the scientific literature as one of the branches of modern science that has given rise to a new mathematical theory of great impact.

A Bibliometric Analysis of the 35th anniversary of the paper "The Statistical Analysis of Compositional Data" by John Aitchison (1982)

This study presents a comprehensive bibliometric analysis of the paper published by John Aitchison in the Journal of the Royal Statistical Society. Series B (Methodological) in 1982. Having recently reached the milestone of 35 years since its publication, this pioneering paper was the first to illustrate the use of the methodology "Compositional Data Analysis" or "CoDA". By October 2019, this paper had received over 780 citations, making it the most widely cited and influential article among those using said methodology. The bibliometric approach used in this study encompasses a wide range of techniques, including a specific analysis of the main authors and institutions to have cited Aitchison' paper. The VOSviewer software was also used for the purpose of developing network maps for said publication. Specifically, the techniques used were co-citations and bibliographic coupling. The results clearly show the significant impact the paper has had on scientific research, having been cited by authors and institutions that publish all around the world.

Ternary plots of public interest in cultural human-nature interactions: Google Trends-based culturomics of the mediterranean olive triptych

Abstract The Mediterranean olive triptych –i.e. the conceptual integration of an ecosystem (the olive-grove), a product (olive-oil) and a nutrition/diet scheme (Mediterranean olive-based diet)– is a discernable cultural model in Europe. In this paper, we attempt to explore the potential and limitations of statistical analysis of compositional data –especially ternary plotting techniques– vs. typical representations of Google Trends-based Conservation Culturomics. In that perspective, typical representations of GT-data in published Culturomics literature, e.g. reference term/country/language/time, are considered as descriptors of variation of public interest (2D trend lines, 3D graphics and triangular projections). Ternary plots, i.e. a typical representation of compositional data, conventionally differ significantly from the previous representations, since the primary GT-data are weighted so that ternary sum equals to 1 (or 100%). We present the technique for such a calculation method. The comparison of these representation techniques allows for strongly diverging interpretations. Lower level interpretations show significant differences between (Mediterranean) countries and/or groups-of-countries (Mediterranean vs. non-Mediterranean) where higher ternary compilations do suggest that cultural pattern across Europe actually diverge. The overall conclusion of this research is that GT-based Culturomics should overcome earliest simplistic approaches, such as linear regressions; and, it has to progressively construct a new epistemology where the significance of accumulation of empirical evidence should be strictly connected to scientific paradigm shift.

Compositional data analysis as an alternative paradigm for nutritional studies

Although the compositional nature of dietary data is well recognized, little attention has been given to the methods specifically developed for the statistical analysis of compositional data. The use of standard statistical procedures that ignore the relative nature of compositional elements can lead to spurious results when applied to crude data. This note proposes using a compositional data approach for the statistical analysis of nutritional data. The analyses of data from an Italian population study of 673 non-diabetic women aged 40-74 years were based on isometric log-ratio (ilr) transformation applied to three dietary compositions: seven-part macronutrients, nine-part vitamins and six-part minerals. The ilr transformation produces new variables that represent specific contrasts (balances) between the compositional parts. Different sequential binary partitions have been described and used as a means of flexibly defining balances on the basis of a researcher's interest, and the new variables (that are suitable for undergoing standard statistical procedures) have been included as covariates in linear regression models in order to examine the isocaloric associations between specific dietary balances and waist circumference (WC). Regardless of the dietary compositions, total energy was positively and fibre intake was negatively associated with WC. Net of these effects and keeping constant the proportional relationships between the other dietary components, WC was negatively associated with the relative increase in starches, and positively associated with the relative niacin content. Vitamin C was negatively associated with WC, whereas folate content was positively related. The proposed approach allows a clear interpretation of the relative roles of different dietary components within a holistic overview of a diet. The analyses involving WC provide some useful insights.

Analysing Activities in a Classroom - Remembrances of Professor John Aitchison in Hong Kong with Applications to a Service Provider

Compositional data analysis formed a main focus of statistical activities at Hong Kong University when Professor John Aitchison was head of the Statistics department. It was part of a new Master's degree in Statistics that he set up, and as this was the first such post graduate degree to be offered in Hong Kong, it attracted many gifted statisticians from the Government Statistical Service and other employments making it a very lively program.
 Acknowledging the constrained nature of many types of data led to a new way of looking at proportions and percentages of components making up data items. Professor Aitchison's seminal book The statistical analysis of compositional data contains background theory and many examples of data arising from a wide variety of applications such as geology, economics and human behaviour. One example was an analysis of the daily activities of a statistician. This prompted an analysis of classroom activities in a range of classes and schools encountered during teacher training within the Professional Educational Studies department of Hong Kong University. It was found that the nature of the target class and the school level affected the pattern of lesson activities with more listening carried out in the higher target classes and higher level schools. More time was spent dealing with educational equipment in lower level schools.
 Data analytics is increasingly popular in all walks of life and many small and medium enterprises are realising the benefits. Compositional data forms a large part of internal company operational data and its analysis can provide useful insights. For example, the change in proportions of different activities undertaken over time is important information for a service provider. In addition to biplots and coordinate representations of isometric log ratios, using ternary diagrams to illustrate proportions is an informative way to share findings with company staff.

Spatial analysis of compositional data: A historical review

Like the statistical analysis of compositional data in general, spatial analysis of compositional data requires specific tools. A historical overview of their development is presented in three steps: (a) the recognition of the problem, known as spurious spatial covariance, (b) first attempts to use the logratio approach, and (c) the application of the principle of working in coordinates using isometric logratio representations. Also mentioned are the use of matrix-valued variation-variograms as a tool to model crossvariograms, and the simplicial approach to indicator kriging, that solves inconsistencies in the standard approach to indicator kriging.

Metasomatism and graphite formation at a lithological interface in Malaspina (Alpine Corsica, France)

Multiple pieces of geologic evidence suggest that interfaces between contrasted lithologies exert a strong control on the fate of volatiles in subduction zones. Here we present results from a contact between serpentinites and sediments, located in Corsica and metamorphosed in the blueschist facies during the alpine orogeny. It was shown previously that carbonates in the sediments have been reduced to graphitic carbonaceous material within a 5–10-cm-thick reaction zone at the contact with serpentinites. In an effort to investigate the mechanisms governing this unusual process, bulk rock geochemical analyses incorporating a statistical analysis of compositional data are presented. Observations show that the fate of C was decoupled from that of other elements such as O, H, and large-ion lithophile elements—e.g. K, Sr, Ba...,As—that were extensively leached from the reaction zone. Notably, Na is strongly enriched in the reaction zone and structurally linked to pectolite. Reducing conditions, manifested by the depletion of O in the reaction zone compared to the bulk metasediment, were likely maintained by the presence of Fe(II) in the serpentinite. Moreover, thermodynamic calculations show that the low solubility of carbon in COH fluids at high-pressure and low-temperature conditions was the main driver for graphite precipitation synchronously with carbonate destabilization. This may have been kinetically favored by the presence of already existing graphitized carbonaceous material and phengite in the metasediment. Limited lateral flow might have contributed as well to the geochemical and petrological patterns observed in these rocks.

Model-based replacement of rounded zeros in compositional data: Classical and robust approaches

The log-ratio methodology represents a powerful set of methods and techniques for statistical analysis of compositional data. These techniques may be used for the estimation of rounded zeros or values below the detection limit in cases when the underlying data are compositional in nature. An algorithm based on iterative log-ratio regressions is developed by combining a particular family of isometric log-ratio transformations with censored regression. In the context of classical regression methods, the equivalence of the method based on additive and isometric log-ratio transformations is proved. This equivalence does not hold for robust regression. Based on Monte Carlo methods, simulations are performed to assess the performance of classical and robust methods. To illustrate the method, a case study involving geochemical data is conducted.

Statistical comparisons of heavy metal pollutants between seven regions of the Polish exclusive economic zone

This paper addresses three intractable difficulties associated with the statistical analysis of compositional data, such as percentages or ppm. These are: (1) that such data do not follow multivariate normal distributions thus rendering inappropriate, standard parametric statistical tests and estimation procedures, (2) the covariance/correlation coefficients between specific pairs of components are determined in whole or in part by the presence or absence of other components, and, (3) the negative bias property. That is, at least one covariance and therefore at least one correlation, must be negative, hence the remaining correlations are prevented from ranging freely between −1 and +1. It follows that correlation coefficients formed from compositional data are not only not absolute, but also frequently spurious. Standard multivariate procedures based on them are unreliable, and intrinsic associations between components inferred from strong positive correlations in particular, are potentially false. In a recent 2009 paper, it was reported that 59 surface sediment samples from 7 regions in the Polish exclusive economic zone had been chemically analyzed for 16 elements. Enrichment factors together with crude correlation coefficients between selected elements were presented. All these quantities were computed from the initial raw compositional data resulting from the chemical analyses In this paper, a statistical procedure is presented which is distinctly different to the enrichment factor computations based on the same raw compositional data. The procedure generates a log-ratio measure of the abundance of each element in each of the seven regions, thus enabling comparisons of relative levels of pollution between the regions. Although the two techniques are quite unrelated, it is shown that in general, extremely high or low measures of the relative abundances in the regions are associated with correspondingly high or low values of the enrichment factors in the same regions that were reported in the 2009 paper. That is, the statistical analysis confirms the results of the enrichment factor data in the identification of the most to the least polluted regions. In an additional analysis, the residue term was excluded from each sediment sample by rescaling the 16 element concentrations to sum to 100%, thus forming 59 residue-free sub-compositions. Crude correlation coefficients were computed for pairs of elements of this sub-compositional data. These revealed that certain correlations based on the initial raw data that were reported in the 2009 paper for the same pairs of elements, were not only inconsistent, but sometimes also contradictory. Such contradictions imply that intrinsic geochemical element associations inferred in that paper from such correlations were false.

Balance-dendrogram. A new routine of CoDaPack

A statistical analysis of compositional data based on the Aitchison geometry of the simplex requires an appropriate basis for representing the data. A simple and intuitive way of building such a basis employs a sequential binary partition of the compositional vector. The partition, together with some statistical summaries of the coordinates, or balances, can be represented in a dendrogram-type graph. In this paper we introduce an implementation of this methodology inside CoDaPack, which is freeware. An example with real data illustrates the use of the Balance-Dendrogram routine.

A modified EM alr-algorithm for replacing rounded zeros in compositional data sets

The presence of rounded zeros results in an important drawback for the statistical analysis of compositional data. Data analysis methodology based on log-ratios cannot be applied under these conditions. In this paper rounded zeros are considered as a special kind of missing data. Thus, an EM-type computational algorithm for replacing them is provided. The procedure is based on the additive logistic transformation and assumes an additive logistic normal model for the data. First, the alr transformation moves data from the constrained simplex space to the unconstrained real space. Next, missing transformed data are imputed by using modified EM steps. Last, imputed data are transformed back into the simplex space to obtain a compositional data set free of rounded zeros. Additionally, a sequential strategy is proposed for the case of rounded zeros in all the components of a composition. This work focuses on the algorithm's properties and on computational implementation details. Also, its effectiveness on simulated data sets with a range of detection limits is analyzed. Special attention is paid on the effects on the covariance structure of a compositional data set. Results confirm the good behavior of our proposal. Finally, MATLAB routines implementing the algorithm are made available to the reader.

A. Buccianti, G. Mateu-Figueras and V. Pawlowsky-Glahn (eds): Compositional data analysis in the geosciences: from theory to practice

It is already 25 years since the publication of John Aitchison’s seminal paper on the statistical analysis of compositional data, which finally provided a solution to the problem identified by Karl Pearson in the nineteenth century of spurious correlation when there is a constant sum constraint. Log-ratio transformations provide a way to ‘‘escape’’ the constraint of a constant sum in a coherent way, which enables the consistent analysis of sub-compositions. It is surprising that such a fundamental advance has still not been fully embraced by scientific disciplines that deal frequently with compositional data, such as geology. Given this lack of adoption of such a fundamental advance, a book like this one is especially invaluable as it provides an excellent balance between the theoretical advances of the 20 years since John Aitchison’s book and detailed analysis of practical problems that shows that log-ratio methods provide more meaningful solutions in practice. Overall, there are four theoretical chapters, eight chapters that apply the methods to real datasets and three chapters, which cover the software available for compositional data analysis. Although the book was initially based on conference contributions, the chapters are quite consistent in notation and level, suggesting hard work by the editors! The introductory chapter by Pawlowsky-Glahn and Egozcue explains how log-ratio transformations solve the spurious correlations of Pearson and covers the advances since the Aitchison book, including isometric-log-ratio (ilr) transformations, which provide an alternative to the additive-log-ratio (alr) and centred-log-ratio (clr) transformations. However, arguably, the major benefits of the ilr transformations are mathematical (an orthogonal basis) and geometric (neat link with the implied geometry on the simplex) rather than statistical as the results transformed back to the simplex are not affected by the choice of transformation and the ilr transformations can be harder to interpret. Centering the data at the geometric mean, as a special case of the perturbation operation, is shown to help in visualization, together with biplots, which provide a useful graphical summary of the variation in the simplex. The second theoretical chapter by Egozcue and Pawlowsky-Glahn provides a detailed description of the geometry on the simplex including the distance metric, which is simple for the clr and ilr representation/transformations. While this chapter is interesting and important from a mathematical and geometrical perspective it is not clear that it provides additional statistical or geological insight. The third theoretical chapter by Martin-Fernandez and Thio-Henestrosa discusses possible solutions to the problem of zero components, treating the zeros as though the true value is missing due to detection limits rather than a complete absence of the component. They examine the possibility of using non-parametric imputation following the methods of Rubin and Little for missing data and conclude that multiplicative adjustments are superior to additive or simple adjustments, although they emphasize the necessity of sensitivity analysis to examine the robustness in the context of the analysis selected. They also claim that artificial correlation is caused when there is more than one component with zeros and non-parametric methods of imputation are used, but it seems that this outcome reflects the use of a specific non-parametric method, rather than a weakness of non-parametric methods J. Bacon-Shone (&) Social Sciences Research Centre, The University of Hong Kong, Pokfulam Road, Hong Kong e-mail: johnbs@hku.hk

Multivariate Statistical Analysis of Compositional Data within the Simplex

Many multivariate statistical tools assume the data is drawn from a continuous and unbounded sample space, such as the d-dimensional set of real numbers R. In these spaces, variables can take on values ranging from −∞ to +∞, including zero. In contrast, many of the data that appear in quantitative microanalysis experiments are drawn from more restricted spaces. The counts in an XEDS detector channel, for example, are constrained to be non-negative. It does not make sense to speak of a 10 eV wide X-ray channel with negative counts, and when commonly-applied algorithms such as Principal Components Analysis (PCA) return results with negative values, many analysts are confused. Compositional data are even more constrained than count data. Consider the microanalysis of a Ni-Al-Fe alloy in an SEM using XEDS. The measured composition at each pixel on the sample is expressed in terms of the mass fraction of each component of the mixture, such as Ni0.25Al0.5Fe0.25. The mass fraction data for each component is constrained to vary from 0 to 1, an extremely narrow range. To complicate matters further, the three component values are jointly constrained to sum to one, i.e. xNi + xAl + xFe = 1. Mathematically, the components are said to fall within a simplex, a severely restricted subspace of R.

A hyperspherical transformation forecasting model for compositional data

Although Aitchison’s [Aitchison, J., 1986. The Statistical Analysis of Compositional Data, Chapman and Hall, London] method of logratio transformation of compositional data is widely used in various domains, it is limited by the assumption of a strict non-negativity of the components and the requirement of special treatments in practice of the zero components. We propose a dimension-reduction approach through a hyperspherical transformation that is capable of resolving the difficulty in maintaining non-negativity and unit-sum in forecasting compositional data over time. Applying the proposed model to a numerical simulation with a 4D compositional data embedded with zero components and forecasting the three production sectors in the Chinese economy both demonstrate the usefulness and validity of the new approach.

About this title

Since Karl Pearson wrote his paper on spurious correlation in 1897, a lot has been said about the statistical analysis of compositional data, mainly by geologists such as Felix Chayes. The solution appeared in the 1980s, when John Aitchison proposed to use Iogratios. Since then, the approach has seen a great expansion, mainly building on the idea of the ‘natural geometry’ of the sample space. Statistics is expected to give sense to our perception of the natural scale of the data, and this is made possible for compositional data using Iogratios. This publication will be a milestone in this process.This book will be of interest to geologists using statistical methods. It includes the intuitive justification of the methodology, convincing through case studies and presenting user-friendly software, which includes a section for those who need to see the proof of the mathematical consistency of the methods used.

Compositional Data Analysis of Geological Variability and Process: A Case Study

Developments in the statistical analysis of compositional data over the last two decades have made possible a much deeper exploration of the nature of variability and the possible processes associated with compositional data sets from many disciplines. In this paper, we concentrate on geochemical data. First, we explain how hypotheses of compositional variability may be formulated within the natural sample space, the unit simplex, including useful hypotheses of sub-compositional discrimination and specific perturbational change. Then we develop through standard methodology, such as generalised likelihood ratio tests, statistical tools to allow the systematic investigation of a lattice of such hypotheses. Some of these tests are simple adaptations of existing multivariate tests but others require special construction. We comment on the use of graphical methods in compositional data analysis and on the ordination of specimens. The recent development of the concept of compositional processes is then explained, together with the necessary tools for a staying-in-the-simplex approach, such as the singular value decomposition of a compositional data set. All these statistical techniques are illustrated for a substantial compositional data set, consisting of 209 major oxide and trace element compositions of metamorphosed limestones from the Grampian Highlands of Scotland. Finally, we discuss some unresolved problems in the statistical analysis of compositional processes.

Dealing with Compositional Data: The Freeware CoDaPack

The statistical analysis of compositional data based on logratios of parts is commonly used in geological studies. In particular, descriptive methods of reduction of dimensionality as the biplot or Principal Component Analysis are very useful tools in a multivariate context. These methods are difficult to use correctly with compositional data in standard packages. In this paper we describe a freeware package, named CoDaPack, which implements most of the basic statistical methods suitable for compositional data. An example using real data is presented to illustrate the use of the package.

Dealing with Zeros and Missing Values in Compositional Data Sets Using Nonparametric Imputation

The statistical analysis of compositional data based on logratios of parts is not suitable when zeros are present in a data set. Nevertheless, if there is interest in using this modeling approach, several strategies have been published in the specialized literature which can be used. In particular, substitution or imputation strategies are available for rounded zeros. In this paper, existing nonparametric imputation methods—both for the additive and the multiplicative approach—are revised and essential properties of the last method are given. For missing values a generalization of the multiplicative approach is proposed.