New Features of CoDaPack. An Userfriendly Compositional Data Package

The statistical analysis of compositional data is commonly used in geological studies.As is well-known, compositions should be treated using logratios of parts, which aredifficult to use correctly in standard statistical packages. In this paper we describe thenew features of our freeware package, named CoDaPack, which implements most of thebasic statistical methods suitable for compositional data. An example using real data ispresented to illustrate the use of the package

The allometry of non-employment. What can compositional data analysis tell us about labour market performance across the UK's regions?

The low levels of unemployment recorded in the UK in recent years are widely cited asevidence of the country’s improved economic performance, and the apparent convergence of unemployment rates across the country’s regions used to suggest that the longstanding divide in living standards between the relatively prosperous ‘south’ and the more depressed ‘north’ has been substantially narrowed. Dissenters from theseconclusions have drawn attention to the greatly increased extent of non-employment(around a quarter of the UK’s working age population are not in employment) and themarked regional dimension in its distribution across the country. Amongst these dissenters it is generally agreed that non-employment is concentrated amongst oldermales previously employed in the now very much smaller ‘heavy’ industries (e.g. coal,steel, shipbuilding).This paper uses the tools of compositiona l data analysis to provide a much richer picture of non-employment and one which challenges the conventional analysis wisdom about UK labour market performance as well as the dissenters view of the nature of theproblem. It is shown that, associated with the striking ‘north/south’ divide in nonemployment rates, there is a statistically significant relationship between the size of the non-employment rate and the composition of non-employment. Specifically, it is shown that the share of unemployment in non-employment is negatively correlated with the overall non-employment rate: in regions where the non-employment rate is high the share of unemployment is relatively low. So the unemployment rate is not a very reliable indicator of regional disparities in labour market performance. Even more importantly from a policy viewpoint, a significant positive relationship is found between the size ofthe non-employment rate and the share of those not employed through reason of sicknessor disability and it seems (contrary to the dissenters) that this connection is just as strong for women as it is for men

The agreement between ipsative and normative questionnaires using compositional data analysis techniques

The main instrument used in psychological measurement is the self-report questionnaire. One of its majordrawbacks however is its susceptibility to response biases. A known strategy to control these biases hasbeen the use of so-called ipsative items. Ipsative items are items that require the respondent to makebetween-scale comparisons within each item. The selected option determines to which scale the weight ofthe answer is attributed. Consequently in questionnaires only consisting of ipsative items everyrespondent is allotted an equal amount, i.e. the total score, that each can distribute differently over thescales. Therefore this type of response format yields data that can be considered compositional from itsinception.Methodological oriented psychologists have heavily criticized this type of item format, since the resultingdata is also marked by the associated unfavourable statistical properties. Nevertheless, clinicians havekept using these questionnaires to their satisfaction. This investigation therefore aims to evaluate bothpositions and addresses the similarities and differences between the two data collection methods. Theultimate objective is to formulate a guideline when to use which type of item format.The comparison is based on data obtained with both an ipsative and normative version of threepsychological questionnaires, which were administered to 502 first-year students in psychology accordingto a balanced within-subjects design. Previous research only compared the direct ipsative scale scoreswith the derived ipsative scale scores. The use of compositional data analysis techniques also enables oneto compare derived normative score ratios with direct normative score ratios. The addition of the secondcomparison not only offers the advantage of a better-balanced research strategy. In principle it also allowsfor parametric testing in the evaluation

Validation of order rank scales based on compositional data analysis: a proposal

Usually, psychometricians apply classical factorial analysis to evaluate construct validity of order rankscales. Nevertheless, these scales have particular characteristics that must be taken into account: totalscores and rank are highly relevant

Assessing the Precision of Compositional Data in a Stratified Double Stage Cluster Sample: Application to the Swiss Earnings Structure Survey

Precision of released figures is not only an important quality feature of official statistics,it is also essential for a good understanding of the data. In this paper we show a casestudy of how precision could be conveyed if the multivariate nature of data has to betaken into account. In the official release of the Swiss earnings structure survey, the totalsalary is broken down into several wage components. We follow Aitchison's approachfor the analysis of compositional data, which is based on logratios of components. Wefirst present diferent multivariate analyses of the compositional data whereby the wagecomponents are broken down by economic activity classes. Then we propose a numberof ways to assess precision

CODAPACK3D. A new version of Compositional Data Package

The statistical analysis of compositional data should be treated using logratios of parts,which are difficult to use correctly in standard statistical packages. For this reason afreeware package, named CoDaPack was created. This software implements most of thebasic statistical methods suitable for compositional data.In this paper we describe the new version of the package that now is calledCoDaPack3D. It is developed in Visual Basic for applications (associated with Excel©),Visual Basic and Open GL, and it is oriented towards users with a minimum knowledgeof computers with the aim at being simple and easy to use.This new version includes new graphical output in 2D and 3D. These outputs could bezoomed and, in 3D, rotated. Also a customization menu is included and outputs couldbe saved in jpeg format. Also this new version includes an interactive help and alldialog windows have been improved in order to facilitate its use.To use CoDaPack one has to access Excel© and introduce the data in a standardspreadsheet. These should be organized as a matrix where Excel© rows correspond tothe observations and columns to the parts. The user executes macros that returnnumerical or graphical results. There are two kinds of numerical results: new variablesand descriptive statistics, and both appear on the same sheet. Graphical output appearsin independent windows. In the present version there are 8 menus, with a total of 38submenus which, after some dialogue, directly call the corresponding macro. Thedialogues ask the user to input variables and further parameters needed, as well aswhere to put these results. The web site http://ima.udg.es/CoDaPack contains thisfreeware package and only Microsoft Excel© under Microsoft Windows© is required torun the software.Kew words: Compositional data Analysis, Software

Compositional data and Simpson's paradox

Simpson's paradox, also known as amalgamation or aggregation paradox, appears whendealing with proportions. Proportions are by construction parts of a whole, which canbe interpreted as compositions assuming they only carry relative information. TheAitchison inner product space structure of the simplex, the sample space of compositions, explains the appearance of the paradox, given that amalgamation is a nonlinearoperation within that structure. Here we propose to use balances, which are specificelements of this structure, to analyse situations where the paradox might appear. Withthe proposed approach we obtain that the centre of the tables analysed is a naturalway to compare them, which avoids by construction the possibility of a paradox.Key words: Aitchison geometry, geometric mean, orthogonal projection

Markov chain montecarlo method applied to rounding zeros of compositional data: first approach

As stated in Aitchison (1986), a proper study of relative variation in a compositional data set should be based on logratios, and dealing with logratios excludes dealing with zeros. Nevertheless, it is clear that zero observations might be present in real data sets, either because the corresponding part is completelyabsent –essential zeros– or because it is below detection limit –rounded zeros. Because the second kind of zeros is usually understood as “a trace too small to measure”, it seems reasonable to replace them by a suitable small value, and this has been the traditional approach. As stated, e.g. by Tauber (1999) and byMartín-Fernández, Barceló-Vidal, and Pawlowsky-Glahn (2000), the principal problem in compositional data analysis is related to rounded zeros. One should be careful to use a replacement strategy that does not seriously distort the general structure of the data. In particular, the covariance structure of the involvedparts –and thus the metric properties– should be preserved, as otherwise further analysis on subpopulations could be misleading. Following this point of view, a non-parametric imputation method isintroduced in Martín-Fernández, Barceló-Vidal, and Pawlowsky-Glahn (2000). This method is analyzed in depth by Martín-Fernández, Barceló-Vidal, and Pawlowsky-Glahn (2003) where it is shown that thetheoretical drawbacks of the additive zero replacement method proposed in Aitchison (1986) can be overcome using a new multiplicative approach on the non-zero parts of a composition. The new approachhas reasonable properties from a compositional point of view. In particular, it is “natural” in the sense thatit recovers the “true” composition if replacement values are identical to the missing values, and it is coherent with the basic operations on the simplex. This coherence implies that the covariance structure of subcompositions with no zeros is preserved. As a generalization of the multiplicative replacement, in thesame paper a substitution method for missing values on compositional data sets is introduced

CoDaPack. An Excel and Visual Basic based software of compositional data analysis. Current version and discussion form upcoming versions

In the eighties, John Aitchison (1986) developed a new methodological approach for the statistical analysis of compositional data. This new methodology was implemented in Basic routines grouped under the name CODA and later NEWCODA inMatlab (Aitchison, 1997). After that, several other authors have published extensions to this methodology: Marín-Fernández and others (2000), Barceló-Vidal and others (2001), Pawlowsky-Glahn and Egozcue (2001, 2002) and Egozcue and others (2003). (...)

Bayesian tools for zero counts in compositional data

The log-ratio methodology makes available powerful tools for analyzing compositionaldata. Nevertheless, the use of this methodology is only possible for those data setswithout null values. Consequently, in those data sets where the zeros are present, aprevious treatment becomes necessary. Last advances in the treatment of compositionalzeros have been centered especially in the zeros of structural nature and in the roundedzeros. These tools do not contemplate the particular case of count compositional datasets with null values. In this work we deal with \count zeros" and we introduce atreatment based on a mixed Bayesian-multiplicative estimation. We use the Dirichletprobability distribution as a prior and we estimate the posterior probabilities. Then weapply a multiplicative modi¯cation for the non-zero values. We present a case studywhere this new methodology is applied.Key words: count data, multiplicative replacement, composition, log-ratio analysis

A comparison of the alr and ilr transformations for kernel density estimation of compositional data

In a seminal paper, Aitchison and Lauder (1985) introduced classical kernel densityestimation techniques in the context of compositional data analysis. Indeed, they gavetwo options for the choice of the kernel to be used in the kernel estimator. One ofthese kernels is based on the use the alr transformation on the simplex SD jointly withthe normal distribution on RD-1. However, these authors themselves recognized thatthis method has some deficiencies. A method for overcoming these dificulties based onrecent developments for compositional data analysis and multivariate kernel estimationtheory, combining the ilr transformation with the use of the normal density with a fullbandwidth matrix, was recently proposed in Martín-Fernández, Chacón and Mateu-Figueras (2006). Here we present an extensive simulation study that compares bothmethods in practice, thus exploring the finite-sample behaviour of both estimators

CODAMAT: a modern analogue techinque for compositional data

The quantitative estimation of Sea Surface Temperatures from fossils assemblages is afundamental issue in palaeoclimatic and paleooceanographic investigations. TheModern Analogue Technique, a widely adopted method based on direct comparison offossil assemblages with modern coretop samples, was revised with the aim ofconforming it to compositional data analysis. The new CODAMAT method wasdeveloped by adopting the Aitchison metric as distance measure. Modern coretopdatasets are characterised by a large amount of zeros. The zero replacement was carriedout by adopting a Bayesian approach to the zero replacement, based on a posteriorestimation of the parameter of the multinomial distribution. The number of modernanalogues from which reconstructing the SST was determined by means of a multipleapproach by considering the Proxies correlation matrix, Standardized Residual Sum ofSquares and Mean Squared Distance. This new CODAMAT method was applied to theplanktonic foraminiferal assemblages of a core recovered in the Tyrrhenian Sea.Kew words: Modern analogues, Aitchison distance, Proxies correlation matrix,Standardized Residual Sum of Squares

Studies in stoichiometry with compositional data

Geochemical data that is derived from the whole or partial analysis of various geologic materialsrepresent a composition of mineralogies or solute species. Minerals are composed of structuredrelationships between cations and anions which, through atomic and molecular forces, keep the elementsbound in specific configurations. The chemical compositions of minerals have specific relationships thatare governed by these molecular controls. In the case of olivine, there is a well-defined relationshipbetween Mn-Fe-Mg with Si. Balances between the principal elements defining olivine composition andother significant constituents in the composition (Al, Ti) have been defined, resulting in a near-linearrelationship between the logarithmic relative proportion of Si versus (MgMnFe) and Mg versus (MnFe),which is typically described but poorly illustrated in the simplex.The present contribution corresponds to ongoing research, which attempts to relate stoichiometry andgeochemical data using compositional geometry. We describe here the approach by which stoichiometricrelationships based on mineralogical constraints can be accounted for in the space of simplicialcoordinates using olivines as an example. Further examples for other mineral types (plagioclases andmore complex minerals such as clays) are needed. Issues that remain to be dealt with include thereduction of a bulk chemical composition of a rock comprised of several minerals from which appropriatebalances can be used to describe the composition in a realistic mineralogical framework. The overallobjective of our research is to answer the question: In the cases where the mineralogy is unknown, arethere suitable proxies that can be substituted?Kew words: Aitchison geometry, balances, mineral composition, oxides

Clustering compositional data trajectories

Our essay aims at studying suitable statistical methods for the clustering ofcompositional data in situations where observations are constituted by trajectories ofcompositional data, that is, by sequences of composition measurements along a domain.Observed trajectories are known as “functional data” and several methods have beenproposed for their analysis.In particular, methods for clustering functional data, known as Functional ClusterAnalysis (FCA), have been applied by practitioners and scientists in many fields. To ourknowledge, FCA techniques have not been extended to cope with the problem ofclustering compositional data trajectories. In order to extend FCA techniques to theanalysis of compositional data, FCA clustering techniques have to be adapted by using asuitable compositional algebra.The present work centres on the following question: given a sample of compositionaldata trajectories, how can we formulate a segmentation procedure giving homogeneousclasses? To address this problem we follow the steps described below.First of all we adapt the well-known spline smoothing techniques in order to cope withthe smoothing of compositional data trajectories. In fact, an observed curve can bethought of as the sum of a smooth part plus some noise due to measurement errors.Spline smoothing techniques are used to isolate the smooth part of the trajectory:clustering algorithms are then applied to these smooth curves.The second step consists in building suitable metrics for measuring the dissimilaritybetween trajectories: we propose a metric that accounts for difference in both shape andlevel, and a metric accounting for differences in shape only.A simulation study is performed in order to evaluate the proposed methodologies, usingboth hierarchical and partitional clustering algorithm. The quality of the obtained resultsis assessed by means of several indices

Possible solution of some essential zero problems in compositional data analysis

One of the tantalising remaining problems in compositional data analysis lies in how to deal with data sets in which there are components which are essential zeros. By anessential zero we mean a component which is truly zero, not something recorded as zero simply because the experimental design or the measuring instrument has not been sufficiently sensitive to detect a trace of the part. Such essential zeros occur inmany compositional situations, such as household budget patterns, time budgets,palaeontological zonation studies, ecological abundance studies. Devices such as nonzero replacement and amalgamation are almost invariably ad hoc and unsuccessful insuch situations. From consideration of such examples it seems sensible to build up amodel in two stages, the first determining where the zeros will occur and the secondhow the unit available is distributed among the non-zero parts. In this paper we suggest two such models, an independent binomial conditional logistic normal model and a hierarchical dependent binomial conditional logistic normal model. The compositional data in such modelling consist of an incidence matrix and a conditional compositional matrix. Interesting statistical problems arise, such as the question of estimability of parameters, the nature of the computational process for the estimation of both the incidence and compositional parameters caused by the complexity of the subcompositional structure, the formation of meaningful hypotheses, and the devising of suitable testing methodology within a lattice of such essential zero-compositional hypotheses. The methodology is illustrated by application to both simulated and real compositional data