Compositional ideas in the bayesian analysis of categorical data with application to dose finding clinical trials

Compositional random vectors are fundamental tools in the Bayesian analysis of categorical data.Many of the issues that are discussed with reference to the statistical analysis of compositionaldata have a natural counterpart in the construction of a Bayesian statistical model for categoricaldata.This note builds on the idea of cross-fertilization of the two areas recommended by Aitchison (1986)in his seminal book on compositional data. Particular emphasis is put on the problem of whatparameterization to use

Comparing correspondence analysis and the log-ratio alternative for representing categorical data

We compare correspondance análisis to the logratio approach based on compositional data. We also compare correspondance análisis and an alternative approach using Hellinger distance, for representing categorical data in a contingency table. We propose a coefficient which globally measures the similarity between these approaches. This coefficient can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions to the difference between the two representations. These three methods of representation can produce quite similar results. One illustrative example is given

Compositional analysis of archaeological glasses

At CoDaWork'03 we presented work on the analysis of archaeological glass composi-tional data. Such data typically consist of geochemical compositions involving 10-12variables and approximates completely compositional data if the main component, sil-ica, is included. We suggested that what has been termed `crude' principal componentanalysis (PCA) of standardized data often identi ed interpretable pattern in the datamore readily than analyses based on log-ratio transformed data (LRA). The funda-mental problem is that, in LRA, minor oxides with high relative variation, that maynot be structure carrying, can dominate an analysis and obscure pattern associatedwith variables present at higher absolute levels. We investigate this further using sub-compositional data relating to archaeological glasses found on Israeli sites. A simplemodel for glass-making is that it is based on a `recipe' consisting of two `ingredients',sand and a source of soda. Our analysis focuses on the sub-composition of componentsassociated with the sand source. A `crude' PCA of standardized data shows two clearcompositional groups that can be interpreted in terms of di erent recipes being used atdi erent periods, reected in absolute di erences in the composition. LRA analysis canbe undertaken either by normalizing the data or de ning a `residual'. In either case,after some `tuning', these groups are recovered. The results from the normalized LRAare di erently interpreted as showing that the source of sand used to make the glassdi ered. These results are complementary. One relates to the recipe used. The otherrelates to the composition (and presumed sources) of one of the ingredients. It seemsto be axiomatic in some expositions of LRA that statistical analysis of compositionaldata should focus on relative variation via the use of ratios. Our analysis suggests thatabsolute di erences can also be informative

Multivariate ARIMA Compositional Time Series Analysis

A compositional time series is obtained when a compositional data vector is observed atdifferent points in time. Inherently, then, a compositional time series is a multivariatetime 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, atrawl through the statistical literature reveals that research in the field is very much in itsinfancy and that many theoretical and empirical issues still remain to be addressed. Anyappropriate statistical methodology for the analysis of compositional time series musttake into account the constraints which are not allowed for by the usual statisticaltechniques available for analysing multivariate time series. One general approach toanalyzing compositional time series consists in the application of an initial transform tobreak the positive and unit sum constraints, followed by the analysis of the transformedtime series using multivariate ARIMA models. In this paper we discuss the use of theadditive log-ratio, centred log-ratio and isometric log-ratio transforms. We also presentresults from an empirical study designed to explore how the selection of the initialtransform affects subsequent multivariate ARIMA modelling as well as the quality ofthe 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 hasn rows and m columns and all probabilities are non-null. This kind of table can beseen as an element in the simplex of n · m parts. In this context, the marginals areidentified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclideanelements of the Aitchison geometry of the simplex can also be translated into the tableof probabilities: subspaces, orthogonal projections, distances.Two important questions are addressed: a) given a table of probabilities, which isthe nearest independent table to the initial one? b) which is the largest orthogonalprojection of a row onto a column? or, equivalently, which is the information in arow explained by a column, thus explaining the interaction? To answer these questionsthree orthogonal decompositions are presented: (1) by columns and a row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independenttwo-way tables and fully dependent tables representing row-column interaction. Animportant result is that the nearest independent table is the product of the two (rowand column)-wise geometric marginal tables. A corollary is that, in an independenttable, 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

Comparing methods for dimensionality reduction when data are density functions

Functional Data Analysis (FDA) deals with samples where a whole function is observedfor each individual. A particular case of FDA is when the observed functions are densityfunctions, that are also an example of infinite dimensional compositional data. In thiswork we compare several methods for dimensionality reduction for this particular typeof data: functional principal components analysis (PCA) with or without a previousdata transformation and multidimensional scaling (MDS) for diferent inter-densitiesdistances, one of them taking into account the compositional nature of density functions. The difeerent methods are applied to both artificial and real data (householdsincome distributions)

When a data set can be considered compositional?

Traditionally, compositional data has been identified with closed data, and the simplex has been considered as the natural sample space of this kind of data. In our opinion, the emphasis on the constrained nature ofcompositional data has contributed to mask its real nature. More crucial than the constraining property of compositional data is the scale-invariant property of this kind of data. Indeed, when we are considering only few parts of a full composition we are not working with constrained data but our data are still compositional. We believe that it is necessary to give a more precisedefinition of composition. This is the aim of this oral contribution

Conditional compositional biplots: theory and application

The biplot has proved to be a powerful descriptive and analytical tool in many areasof applications of statistics. For compositional data the necessary theoreticaladaptation has been provided, with illustrative applications, by Aitchison (1990) andAitchison and Greenacre (2002). These papers were restricted to the interpretation ofsimple compositional data sets. In many situations the problem has to be described insome form of conditional modelling. For example, in a clinical trial where interest isin how patients’ steroid metabolite compositions may change as a result of differenttreatment regimes, interest is in relating the compositions after treatment to thecompositions before treatment and the nature of the treatments applied. To study thisthrough a biplot technique requires the development of some form of conditionalcompositional biplot. This is the purpose of this paper. We choose as a motivatingapplication an analysis of the 1992 US President ial Election, where interest may be inhow the three-part composition, the percentage division among the three candidates -Bush, Clinton and Perot - of the presidential vote in each state, depends on the ethniccomposition and on the urban-rural composition of the state. The methodology ofconditional compositional biplots is first developed and a detailed interpretation of the1992 US Presidential Election provided. We use a second application involving theconditional variability of tektite mineral compositions with respect to major oxidecompositions to demonstrate some hazards of simplistic interpretation of biplots.Finally we conjecture on further possible applications of conditional compositionalbiplots

Weighted Logratio Biplots, Correspondence Analysis and Spectral Maps

Starting with logratio biplots for compositional data, which are based on the principle of subcompositional coherence, and then adding weights, as in correspondence analysis, we rediscover Lewi's spectral map and many connections to analyses of two-way tables of non-negative data. Thanks to the weighting, the method also achieves the property of distributional equivalence

On the use of principal components in contemporany population genetics: a case study

In human Population Genetics, routine applications of principal component techniques are oftenrequired. Population biologists make widespread use of certain discrete classifications of humansamples into haplotypes, the monophyletic units of phylogenetic trees constructed from severalsingle nucleotide bimorphisms hierarchically ordered. Compositional frequencies of the haplotypesare recorded within the different samples. Principal component techniques are then required as adimension-reducing strategy to bring the dimension of the problem to a manageable level, say two,to allow for graphical analysis.Population biologists at large are not aware of the special features of compositional data and normally make use of the crude covariance of compositional relative frequencies to construct principalcomponents. In this short note we present our experience with using traditional linear principalcomponents or compositional principal components based on logratios, with reference to a specificdataset

A unified approach for representing rows and columns in contingency tables

By using suitable parameters, we present a uni¯ed aproach for describing four methods for representing categorical data in a contingency table. These methods include:correspondence analysis (CA), the alternative approach using Hellinger distance (HD),the log-ratio (LR) alternative, which is appropriate for compositional data, and theso-called non-symmetrical correspondence analysis (NSCA). We then make an appropriate comparison among these four methods and some illustrative examples are given.Some approaches based on cumulative frequencies are also linked and studied usingmatrices.Key words: Correspondence analysis, Hellinger distance, Non-symmetrical correspondence analysis, log-ratio analysis, Taguchi inertia

Compositions in life science data

The aim of this talk is to convince the reader that there are a lot of interesting statisticalproblems in presentday life science data analysis which seem ultimately connected withcompositional statistics.Key words: SAGE, cDNA microarrays, (1D-)NMR, virus quasispecies

Ratio maps and correspondence analysis

We compare two methods for visualising contingency tables and developa method called the ratio map which combines the good properties of both.The first is a biplot based on the logratio approach to compositional dataanalysis. This approach is founded on the principle of subcompositionalcoherence, which assures that results are invariant to considering subsetsof the composition. The second approach, correspondence analysis, isbased on the chi-square approach to contingency table analysis. Acornerstone of correspondence analysis is the principle of distributionalequivalence, which assures invariance in the results when rows or columnswith identical conditional proportions are merged. Both methods may bedescribed as singular value decompositions of appropriately transformedmatrices. Correspondence analysis includes a weighting of the rows andcolumns proportional to the margins of the table. If this idea of row andcolumn weights is introduced into the logratio biplot, we obtain a methodwhich obeys both principles of subcompositional coherence and distributionalequivalence.

Country effects in ISSP-1993 environmental data: Comparison of SEM approaches

Structural equation models (SEM) are commonly used to analyze the relationship between variables some of which may be latent, such as individual ``attitude'' to and ``behavior'' concerning specific issues. A number of difficulties arise when we want to compare a large number of groups, each with large sample size, and the manifest variables are distinctly non-normally distributed. Using an specific data set, we evaluate the appropriateness of the following alternative SEM approaches: multiple group versus MIMIC models, continuous versus ordinal variables estimation methods, and normal theory versus non-normal estimation methods. The approaches are applied to the ISSP-1993 Environmental data set, with the purpose of exploring variation in the mean level of variables of ``attitude'' to and ``behavior''concerning environmental issues and their mutual relationship across countries. Issues of both theoretical and practical relevance arise in the course of this application.

What trees tell us: dendrochronological and statistical analysis of the data

Trees are a great bank of data, named sometimes for this reason as the "silentwitnesses" of the past. Due to annual formation of rings, which is normally influenced directly by of climate parameters (generally changes in temperature and moisture or precipitation) and other environmental factors; these changes, occurred in the past, are"written" in the tree "archives" and can be "decoded" in order to interpret what hadhappened before, mainly applied for the past climate reconstruction.Using dendrochronological methods for obtaining samples of Pinus nigra fromthe Catalonian PrePirineous region, the cores of 15 trees with total time spine of about 100 - 250 years were analyzed for the tree ring width (TRW) patterns and had quite high correlation between them (0.71 ¿ 0.84), corresponding to a common behaviour for the environmental changes in their annual growth.After different trials with raw TRW data for standardization in order to take outthe negative exponential growth curve dependency, the best method of doubledetrending (power transformation and smoothing line of 32 years) were selected for obtaining the indexes for further analysis.Analyzing the cross-correlations between obtained tree ring width indexes andclimate data, significant correlations (p<0.05) were observed in some lags, as forexample, annual precipitation in lag -1 (previous year) had negative correlation with TRW growth in the Pallars region. Significant correlation coefficients are between 0.27- 0.51 (with positive or negative signs) for many cases; as for recent (but very short period) climate data of Seu d¿Urgell meteorological station, some significant correlation coefficients were observed, of the order of 0.9.These results confirm the hypothesis of using dendrochronological data as aclimate signal for further analysis, such as reconstruction of climate in the past orprediction in the future for the same locality.