The Statistical Analysis of Compositional Data

As long ago as 1897 Karl Pearson, in a now classic paper on spurious correlation, first pointed out dangers that may befall the analyst who attempts to interpret correlations between ratios whose numerators and denominators contain common parts. He thus implied that the analysis of compositional data, with its concentration on relation- ships between proportions of some whole, is likely to be fraught with difficulty. History has proved him correct: over the succeeding years and indeed right up to the present day, there has been no other form of data analysis where more confusion has reigned and where more improper and inadequate statistical methods have been applied. The special and intrinsic feature of compositional data is that the proportions of a composition are naturally subject to a unit-sum constraint. For other forms of constrained data, in particular for directional data where there is a unit-length constraint on each direction vector, scientist and statistician alike have readily appre- ciated that new statistical methods, appropriate to the special nature of the data, are required; and there now exists an extensive literature on the successful statistical analysis of directional data. It is paradox- ical that for compositional data, subject to an apparently simpler constraint, such an appreciation and development have been slower to emerge. In applications the unit-sum constraint has been widely ignored or wished away and inappropriate 'standard' statistical methods, devised for and successfully applied to unconstrained data, have been used with disastrous consequences.

Table of Contents:
1 Compositional data: some challenging problems.- 1.1 Introduction.- 1.2 Geochemical compositions of rocks.- 1.3 Sediments at different depths.- 1.4 Ternary diagrams.- 1.5 Partial analyses and subcompositions.- 1.6 Supervisory behaviour.- 1.7 Household budget surveys.- 1.8 Steroid metabolite patterns in adults and children.- 1.9 Activity patterns of a statistician.- 1.10 Calibration of white-cell compositions.- 1.11 Fruit evaluation.- 1.12 Firework mixtures.- 1.13 Clam ecology.- 1.14 Bibliographic notes.- Problems.- 2 The simplex as sample space.- 2.1 Choice of sample space.- 2.2 Compositions and simplexes.- 2.3 Spaces, vectors, matrices.- 2.4 Bases and compositions.- 2.5 Subcompositions.- 2.6 Amalgamations.- 2.7 Partitions.- 2.8 Perturbations.- 2.9 Geometrical representations of compositional data.- 2.10 Bibliographic notes.- Problems.- 3 The special difficulties of compositional data analysis.- 3.1 Introduction.- 3.2 High dimensionality.- 3.3 Absence of an interpretable covariance structure.- 3.4 Difficulty of parametric modelling.- 3.5 The mixture variation difficulty.- 3.6 Bibliographic notes.- Problems.- 4 Covariance structure.- 4.1 Fundamentals.- 4.2 Specification of the covariance structure.- 4.3 The compositional variation array.- 4.4 Recovery of the compositional variation array from the crude mean vector and covariance matrix.- 4.5 Subcompositional analysis.- 4.6 Matrix specifications of covariance structures.- 4.7 Some important elementary matrices.- 4.8 Relationships between the matrix specifications.- 4.9 Estimated matrices for hongite compositions.- 4.10 Logratios and logcontrasts.- 4.11 Covariance structure of a basis.- 4.12 Commentary.- 4.13 Bibliographic notes.- Problems.- 5 Properties of matrix covariance specifications.- 5.1 Logratio notation.- 5.2 Logcontrast variances and covariances.- 5.3 Permutations.- 5.4 Properties of P and QP matrices.- 5.5 Permutation invariants involving ?.- 5.6 Covariance matrix inverses.- 5.7 Subcompositions.- 5.8 Equivalence of characteristics of ?, ?, ?.- 5.9 Logratio-uncorrelated compositions.- 5.10 Isotropic covariance structures.- 5.11 Bibliographic notes.- Problems.- 6 Logistic normal distributions on the simplex.- 6.1 Introduction.- 6.2 The additive logistic normal class.- 6.3 Density function.- 6.4 Moment properties.- 6.5 Composition of a lognormal basis.- 6.6 Class-preserving properties.- 6.7 Conditional subcompositional properties.- 6.8 Perturbation properties.- 6.9 A central limit theorem.- 6.10 A characterization by logcontrasts.- 6.11 Relationships with the Dirichlet class.- 6.12 Potential for statistical analysis.- 6.13 The multiplicative logistic normal class.- 6.14 Partitioned logistic normal classes.- 6.15 Some notation.- 6.16 Bibliographic notes.- Problems.- 7 Logratio analysis of compositions.- 7.1 Introduction.- 7.2 Estimation of ? and ?.- 7.3 Validation: tests of logistic normality.- 7.4 Hypothesis testing strategy and techniques.- 7.5 Testing hypotheses about ? and ?.- 7.6 Logratio linear modelling.- 7.7 Testing logratio linear hypotheses.- 7.8 Further aspects of logratio linear modelling.- 7.9 An application of logratio linear modelling.- 7.10 Predictive distributions, atypicality indices and outliers.- 7.11 Statistical discrimination.- 7.12 Conditional compositional modelling.- 7.13 Bibliographic notes.- Problems.- 8 Dimension-reducing techniques.- 8.1 Introduction.- 8.2 Crude principal component analysis.- 8.3 Logcontrast principal component analysis.- 8.4 Applications of logcontrast principal component analysis.- 8.5 Subcompositional analysis.- 8.6 Applications of subcompositional analysis.- 8.7 Canonical component analysis.- 8.8 Bibliographic notes.- Problems.- 9 Bases and compositions.- 9.1 Fundamentals.- 9.2 Covariance relationships.- 9.3 Principal and canonical component comparisons.- 9.4 Distributional relationships.- 9.5 Compositional invariance.- 9.6 An application to household budget analysis.- 9.7 An application to clinical biochemistry.- 9.8 Reappraisal of an early shape and size analysis.- 9.9 Bibliographic notes.- Problems.- 10 Subcompositions and partitions.- 10.1 Introduction.- 10.2 Complete subcompositional independence.- 10.3 Partitions of order 1.- 10.4 Ordered sequences of partitions.- 10.5 Caveat.- 10.6 Partitions of higher order.- 10.7 Bibliographic notes.- Problems.- 11 Irregular compositional data.- 11.1 Introduction.- 11.2 Modelling imprecision in compositions.- 11.3 Analysis of sources of imprecision.- 11.4 Imprecision and tests of independence.- 11.5 Rounded or trace zeros.- 11.6 Essential zeros.- 11.7 Missing components.- 11.8 Bibliographic notes.- Problems.- 12 Compositions in a covariate role.- 12.1 Introduction.- 12.2 Calibration.- 12.3 A before-and-after treatment problem.- 12.4 Experiments with mixtures.- 12.5 An application to firework mixtures.- 12.6 Classification from compositions.- 12.7 An application to geological classification.- 12.8 Bibliographic notes.- Problems.- 13 Further distributions on the simplex.- 13.1 Some generalizations of the Dirichlet class.- 13.2 Some generalizations of the logistic normal classes.- 13.3 Recapitulation.- 13.4 The Ad(?,B) class.- 13.5 Maximum likelihood estimation.- 13.6 Neutrality and partition independence.- 13.7 Subcompositional independence.- 13.8 A generalized lognormal gamma distribution with compositional in variance.- 13.9 Discussion.- 13.10 Bibliographic notes.- Problems.- 14 Miscellaneous problems.- 14.1 Introduction.- 14.2 Multi-way compositions.- 14.3 Multi-stage compositions.- 14.4 Multiple compositions.- 14.5 Kernel density estimation for compositional data.- 14.6 Compositional stochastic processes.- 14.7 Relation to Bayesian statistical analysis.- 14.8 Compositional and directional data.- Problems.- Appendices.- A Algebraic properties of elementary matrices.- B Bibliography.- C Computer software for compositional data analysis.- D Data sets.- Author index.