J. Aitchison

Biplots of compositional data

Summary. The singular value decomposition and its interpretation as a linear biplot have proved to be a powerful tool for analysing many forms of multivariate data. Here we adapt biplot methodology to the specific case of compositional data consisting of positive vectors each of which is constrained to have unit sum. These relative variation biplots have properties relating to the special features of compositional data: the study of ratios, subcompositions and models of compositional relationships. The methodology is applied to a data set consisting of six-part colour compositions in 22 abstract paintings, showing how the singular value decomposition can achieve an accurate biplot of the colour ratios and how possible models interrelating the colours can be diagnosed.

On the Distribution of a Positive Random Variable Having a Discrete Probability Mass at the Origin

Abstract * This paper is a development of some of the estimation problems discussed by Utting and Cole [5]. The author wishes to express his indebtedness to J. A. C. Brown of the Department of Applied Economics for helpful criticism and for suggesting the application of the Poisson series distribution to the analysis of household composition. In a number of situations we are faced with the problem of determining efficient estimates of the mean and variance of a distribution specified by (i) a non-zero probability that the variable assumes a zero value, together with (ii) a conditional distribution for the positive values of the variable. This estimation problem is analyzed and its implications for the Pearson type III, exponential, lognormal and Poisson series conditional distributions are investigated. Two simple examples are given.

Logratio Analysis and Compositional Distance

The concept of distance between two compositions is important in the statistical analysis of compositional data, particularly in such activities as cluster analysis and multidimensional scaling. This paper exposes the fallacies in a recent criticism of logratio-based distance measures—in particular, the misstatements that logratio methods destroy distance structures and are denominator dependent. Emphasis is on ensuring that compositional data analysis involving distance concepts satisfies certain logically necessary invariance conditions. Logratio analysis and its associated distance measures satisfy these conditions.

On criteria for measures of compositional difference

Simple perceptions about the nature of compositions lead through logical necessity to certain forms of analysis of compositional data. In this paper the consequences of essential requirements of scale, perturbation and permutation invariance, together with that of subcompositional dominance, are applied to the problem of characterizing change and measures of difference between two compositions. It will be shown that one strongly advocated scalar measure of difference fails these tests of logical necessity, and that one particular form of scalar measure of difference (the sum of the squares of all possible logratio differences in the components of the two compositions), although not unique, emerges as the simplest and most tractable satisfying the criteria.

A new approach to null correlations of proportions

Much work on the statistical analysis of compositional data has concentrated on the difficulty of interpreting correlations between proportions with an assortment of tests for nullcorrelations, for independence except for the constraint, F-independence of bounded variables, neutrality in the mean and in the median. This paper questions the appropriateness of characterizing the dependence structure of proportions in terms of such concepts, suggests an alternative method of modeling, develops necessary distribution theory and tests, and illustrates the methodology in applications.

Logratios and Natural Laws in Compositional Data Analysis

The impossibility of interpreting correlations of raw compositional components and associated statistical methods has been clearly demonstrated over the last four decades and alternative statistical methodology developed. Despite this a return to the “traditional” use of raw components has been advocated recently and alternative methodology such as logratio analysis strongly criticized. This paper exposes the fallacies in this recent advocacy and demonstrates the constructive role that logratio analysis can play in geological compositional problems, in particular in the investigation of natural laws and in subcompositional investigations.

The statistical analysis of geochemical compositions

The analysis and interpretation of compositional data, such as major oxide compositions of rocks, has been traditionally plagued by the so-called constant-sum or closure problem. Particular difficulties have been the lack of a satisfactory, interpretable covariance structure and of rich, tractable, parametric classes of distributions on the simplex sample space. Consideration of logistic and logratio transformations between the simplex and Euclidan space has allowed the introduction of new concepts of covariance structure and of classes of logistic-normal distributions which have now opened up a substantial and meaningful array of statistical methodology for compositional data. From the motivation of a wide variety of practical geological problems we examine the range of possibilities with this new approach to the constant-sum problem.

Relative variation diagrams for describing patterns of compositional variability

In recognizing that a composition, such as a major oxide or sediment composition, provides information only about the relative, not the absolute, magnitudes of its components, this paper exposes the compositional variation array as the simplest and minimum way of summarizing the pattern of variability within a compositional data set. Such summaries are free of the notorious hazards of the constant-sum constraint and when depicted in relative variation diagrams can often provide substantial insights into the nature of the compositional variability. Concepts and practice are illustrated by reference to a number of real data sets.

Reducing the dimensionality of compositional data sets

The high-dimensionality of many compositional data sets has caused geologists to look for insights into the observed patterns of variability through two dimension-reducing procedures: (i)the selection of a few subcompositions for particular study, and (ii)principal component analysis. After a brief critical review of the unsatisfactory state of current statistical methodology for these two procedures, this paper takes as a starting point for the resolution of persisting difficulties a recent approach to principal component analysis through a new definition of the covariance structure of a composition. This approach is first applied for expository purposes to a small illustrative compositional data set and then to a number of larger published geochemical data sets. The new approach then leads naturally to a method of measuring the extent to which a subcomposition retains the pattern of variability of the whole composition and so provides a criterion for the selection of suitable subcompositions. Such a selection process is illustrated by application to geochemical data sets.

Bayesian Risk Ratio Analysis

Abstract The classical confidence interval approach has failed to find exact intervals, or even a consensus on the best approximate intervals, for the ratio of two binomial probabilities, the so-called risk ratio. The problem is reexamined from a Bayesian viewpoint, and a simple graphical presentation of the risk ratio assessment is given in such a way that sensitivity to the selected prior distribution can be readily examined.