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

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

Compositional data analysis of vote shares in the 2001 Australian Federal election

First application of compositional data analysis techniques to Australian election data

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 case study of how precision could be conveyed if the multivariate nature of data has to be taken into account. In the official release of the Swiss earnings structure survey, the total salary is broken down into several wage components. We follow Aitchison's approach for the analysis of compositional data, which is based on logratios of components. We first present diferent multivariate analyses of the compositional data whereby the wage components are broken down by economic activity classes. Then we propose a number of ways to assess precision

Regression analysis of compositional data when both the dependent variable and independent variable are components

It is well known that regression analyses involving compositional data need special attention because the data are not of full rank. For a regression analysis where both the dependent and independent variable are components we propose a transformation of the components emphasizing their role as dependent and independent variables. A simple linear regression can be performed on the transformed components. The regression line can be depicted in a ternary diagram facilitating the interpretation of the analysis in terms of components. An exemple with time-budgets illustrates the method and the graphical features

The single principle of compositional data analysis, continuing fallacies, confusions and misunderstandings and some suggested remedies

In any discipline, where uncertainty and variability are present, it is important to have principles which are accepted as inviolate and which should therefore drive statistical modelling, statistical analysis of data and any inferences from such an analysis. Despite the fact that two such principles have existed over the last two decades and from these a sensible, meaningful methodology has been developed for the statistical analysis of compositional data, the application of inappropriate and/or meaningless methods persists in many areas of application. This paper identifies at least ten common fallacies and confusions in compositional data analysis with illustrative examples and provides readers with necessary, and hopefully sufficient, arguments to persuade the culprits why and how they should amend their ways

Inference of distributional parameters from compositional samples containing nondetects

Low concentrations of elements in geochemical analyses have the peculiarity of being compositional data and, for a given level of significance, are likely to be beyond the capabilities of laboratories to distinguish between minute concentrations and complete absence, thus preventing laboratories from reporting extremely low concentrations of the analyte. Instead, what is reported is the detection limit, which is the minimum concentration that conclusively differentiates between presence and absence of the element. A spatially distributed exhaustive sample is employed in this study to generate unbiased sub-samples, which are further censored to observe the effect that different detection limits and sample sizes have on the inference of population distributions starting from geochemical analyses having specimens below detection limit (nondetects). The isometric logratio transformation is used to convert the compositional data in the simplex to samples in real space, thus allowing the practitioner to properly borrow from the large source of statistical techniques valid only in real space. The bootstrap method is used to numerically investigate the reliability of inferring several distributional parameters employing different forms of imputation for the censored data. The case study illustrates that, in general, best results are obtained when imputations are made using the distribution best fitting the readings above detection limit and exposes the problems of other more widely used practices. When the sample is spatially correlated, it is necessary to combine the bootstrap with stochastic simulation

Bayesian tools for zero counts in compositional data

The log-ratio methodology makes available powerful tools for analyzing compositional data. Nevertheless, the use of this methodology is only possible for those data sets without null values. Consequently, in those data sets where the zeros are present, a previous treatment becomes necessary. Last advances in the treatment of compositional zeros have been centered especially in the zeros of structural nature and in the rounded zeros. These tools do not contemplate the particular case of count compositional data sets with null values. In this work we deal with \count zeros" and we introduce a treatment based on a mixed Bayesian-multiplicative estimation. We use the Dirichlet probability distribution as a prior and we estimate the posterior probabilities. Then we apply a multiplicative modi¯cation for the non-zero values. We present a case study where this new methodology is applied. Key words: count data, multiplicative replacement, composition, log-ratio analysis

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 a freeware package, named CoDaPack was created. This software implements most of the basic statistical methods suitable for compositional data. In this paper we describe the new version of the package that now is called CoDaPack3D. 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 knowledge of 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 be zoomed and, in 3D, rotated. Also a customization menu is included and outputs could be saved in jpeg format. Also this new version includes an interactive help and all dialog windows have been improved in order to facilitate its use. To use CoDaPack one has to access Excel© and introduce the data in a standard spreadsheet. These should be organized as a matrix where Excel© rows correspond to the observations and columns to the parts. The user executes macros that return numerical or graphical results. There are two kinds of numerical results: new variables and descriptive statistics, and both appear on the same sheet. Graphical output appears in independent windows. In the present version there are 8 menus, with a total of 38 submenus which, after some dialogue, directly call the corresponding macro. The dialogues ask the user to input variables and further parameters needed, as well as where to put these results. The web site http://ima.udg.es/CoDaPack contains this freeware package and only Microsoft Excel© under Microsoft Windows© is required to run the software. Kew words: Compositional data Analysis, Software

Multivariate ARIMA Compositional Time Series Analysis

A compositional time series is obtained when a compositional data vector is observed at different points in time. Inherently, then, a compositional time series is a multivariate time series with important constraints on the variables observed at any instance in time. Although this type of data frequently occurs in situations of real practical interest, a trawl through the statistical literature reveals that research in the field is very much in its infancy and that many theoretical and empirical issues still remain to be addressed. Any appropriate statistical methodology for the analysis of compositional time series must take into account the constraints which are not allowed for by the usual statistical techniques available for analysing multivariate time series. One general approach to analyzing compositional time series consists in the application of an initial transform to break the positive and unit sum constraints, followed by the analysis of the transformed time series using multivariate ARIMA models. In this paper we discuss the use of the additive log-ratio, centred log-ratio and isometric log-ratio transforms. We also present results from an empirical study designed to explore how the selection of the initial transform affects subsequent multivariate ARIMA modelling as well as the quality of the forecasts

Compositional analysis of bivariate discrete probabilities

A joint distribution of two discrete random variables with finite support can be displayed as a two way table of probabilities adding to one. Assume that this table has n rows and m columns and all probabilities are non-null. This kind of table can be seen as an element in the simplex of n · m parts. In this context, the marginals are identified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclidean elements of the Aitchison geometry of the simplex can also be translated into the table of probabilities: subspaces, orthogonal projections, distances. Two important questions are addressed: a) given a table of probabilities, which is the nearest independent table to the initial one? b) which is the largest orthogonal projection of a row onto a column? or, equivalently, which is the information in a row explained by a column, thus explaining the interaction? To answer these questions three orthogonal decompositions are presented: (1) by columns and a row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independent two-way tables and fully dependent tables representing row-column interaction. An important result is that the nearest independent table is the product of the two (row and column)-wise geometric marginal tables. A corollary is that, in an independent table, the geometric marginals conform with the traditional (arithmetic) marginals. These decompositions can be compared with standard log-linear models. Key words: balance, compositional data, simplex, Aitchison geometry, composition, orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure, contingency table

Compositional data and Simpson's paradox

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

Compositional Time Series: An Application

The composition of the labour force is an important economic factor for a country. Often the changes in proportions of different groups are of interest. I this paper we study a monthly compositional time series from the Swedish Labour Force Survey from 1994 to 2005. Three models are studied: the ILR-transformed series, the ILR-transformation of the compositional differenced series of order 1, and the ILRtransformation of the compositional differenced series of order 12. For each of the three models a VAR-model is fitted based on the data 1994-2003. We predict the time series 15 steps ahead and calculate 95 % prediction regions. The predictions of the three models are compared with actual values using MAD and MSE and the prediction regions are compared graphically in a ternary time series plot. We conclude that the first, and simplest, model possesses the best predictive power of the three models

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 density estimation techniques in the context of compositional data analysis. Indeed, they gave two options for the choice of the kernel to be used in the kernel estimator. One of these kernels is based on the use the alr transformation on the simplex SD jointly with the normal distribution on RD-1. However, these authors themselves recognized that this method has some deficiencies. A method for overcoming these dificulties based on recent developments for compositional data analysis and multivariate kernel estimation theory, combining the ilr transformation with the use of the normal density with a full bandwidth matrix, was recently proposed in Martín-Fernández, Chacón and Mateu- Figueras (2006). Here we present an extensive simulation study that compares both methods in practice, thus exploring the finite-sample behaviour of both estimators

A Log-linear model of grain size influence on the geochemistry of sediments

Sediment composition is mainly controlled by the nature of the source rock(s), and chemical (weathering) and physical processes (mechanical crushing, abrasion, hydrodynamic sorting) during alteration and transport. Although the factors controlling these processes are conceptually well understood, detailed quantification of compositional changes induced by a single process are rare, as are examples where the effects of several processes can be distinguished. The present study was designed to characterize the role of mechanical crushing and sorting in the absence of chemical weathering. Twenty sediment samples were taken from Alpine glaciers that erode almost pure granitoid lithologies. For each sample, 11 grain-size fractions from granules to clay (ø grades <-1 to >9) were separated, and each fraction was analysed for its chemical composition. The presence of clear steps in the box-plots of all parts (in adequate ilr and clr scales) against ø is assumed to be explained by typical crystal size ranges for the relevant mineral phases. These scatter plots and the biplot suggest a splitting of the full grain size range into three groups: coarser than ø=4 (comparatively rich in SiO2, Na2O, K2O, Al2O3, and dominated by “felsic” minerals like quartz and feldspar), finer than ø=8 (comparatively rich in TiO2, MnO, MgO, Fe2O3, mostly related to “mafic” sheet silicates like biotite and chlorite), and intermediate grains sizes (4≤ø <8; comparatively rich in P2O5 and CaO, related to apatite, some feldspar). To further test the absence of chemical weathering, the observed compositions were regressed against three explanatory variables: a trend on grain size in ø scale, a step function for ø≥4, and another for ø≥8. The original hypothesis was that the trend could be identified with weathering effects, whereas each step function would highlight those minerals with biggest characteristic size at its lower end. Results suggest that this assumption is reasonable for the step function, but that besides weathering some other factors (different mechanical behavior of minerals) have also an important contribution to the trend. Key words: sediment, geochemistry, grain size, regression, step function

Statistical evidences of cyclic changes in volcanic gas chemistry composition by inverse modelling

The identification of compositional changes in fumarolic gases of active and quiescent volcanoes is one of the most important targets in monitoring programs. From a general point of view, many systematic (often cyclic) and random processes control the chemistry of gas discharges, making difficult to produce a convincing mathematical-statistical modelling. Changes in the chemical composition of volcanic gases sampled at Vulcano Island (Aeolian Arc, Sicily, Italy) from eight different fumaroles located in the northern sector of the summit crater (La Fossa) have been analysed by considering their dependence from time in the period 2000-2007. Each intermediate chemical composition has been considered as potentially derived from the contribution of the two temporal extremes represented by the 2000 and 2007 samples, respectively, by using inverse modelling methodologies for compositional data. Data pertaining to fumaroles F5 and F27, located on the rim and in the inner part of La Fossa crater, respectively, have been used to achieve the proposed aim. The statistical approach has allowed us to highlight the presence of random and not random fluctuations, features useful to understand how the volcanic system works, opening new perspectives in sampling strategies and in the evaluation of the natural risk related to a quiescent volcano