Robust principal component analysis for compositional tables.

A data table arranged according to two factors can often be considered a compositional table. An example is the number of unemployed people, split according to gender and age classes. Analyzed as compositions, the relevant information consists of ratios between different cells of such a table. This is particularly useful when analyzing several compositional tables jointly, where the absolute numbers are in very different ranges, e.g. if unemployment data are considered from different countries. Within the framework of the logratio methodology, compositional tables can be decomposed into independent and interactive parts, and orthonormal coordinates can be assigned to these parts. However, these coordinates usually require some prior knowledge about the data, and they are not easy to handle for exploring the relationships between the given factors. Here we propose a special choice of coordinates with direct relation to centered logratio (clr) coefficients, which are particularly useful for an interpretation in terms of the original cells of the tables. With these coordinates, robust principal component analysis (rPCA) is performed for dimension reduction, allowing to investigate relationships between the factors. The link between orthonormal coordinates and clr coefficients enables to apply rPCA, which would otherwise suffer from the singularity of clr coefficients.

Cox regression survival analysis with compositional covariates: Application to modelling mortality risk from 24-h physical activity patterns.

Survival analysis is commonly conducted in medical and public health research to assess the association of an exposure or intervention with a hard end outcome such as mortality. The Cox (proportional hazards) regression model is probably the most popular statistical tool used in this context. However, when the exposure includes compositional covariables (that is, variables representing a relative makeup such as a nutritional or physical activity behaviour composition), some basic assumptions of the Cox regression model and associated significance tests are violated. Compositional variables involve an intrinsic interplay between one another which precludes results and conclusions based on considering them in isolation as is ordinarily done. In this work, we introduce a formulation of the Cox regression model in terms of log-ratio coordinates which suitably deals with the constraints of compositional covariates, facilitates the use of common statistical inference methods, and allows for scientifically meaningful interpretations. We illustrate its practical application to a public health problem: the estimation of the mortality hazard associated with the composition of daily activity behaviour (physical activity, sitting time and sleep) using data from the U.S. National Health and Nutrition Examination Survey (NHANES).

A new method for correlation analysis of compositional (environmental) data - a worked example.

Most data in environmental sciences and geochemistry are compositional. Already the unit used to report the data (e.g., µg/l, mg/kg, wt%) implies that the analytical results for each element are not free to vary independently of the other measured variables. This is often neglected in statistical analysis, where a simple log-transformation of the single variables is insufficient to put the data into an acceptable geometry. This is also important for bivariate data analysis and for correlation analysis, for which the data need to be appropriately log-ratio transformed. A new approach based on the isometric log-ratio (ilr) transformation, leading to so-called symmetric coordinates, is presented here. Summarizing the correlations in a heat-map gives a powerful tool for bivariate data analysis. Here an application of the new method using a data set from a regional geochemical mapping project based on soil O and C horizon samples is demonstrated. Differences to 'classical' correlation analysis based on log-transformed data are highlighted. The fact that some expected strong positive correlations appear and remain unchanged even following a log-ratio transformation has probably led to the misconception that the special nature of compositional data can be ignored when working with trace elements. The example dataset is employed to demonstrate that using 'classical' correlation analysis and plotting XY diagrams, scatterplots, based on the original or simply log-transformed data can easily lead to severe misinterpretations of the relationships between elements.

Statistical analysis of wines using a robust compositional biplot.

Eight phenolic acids (vanillic, gentisic, protocatechuic, syringic, gallic, coumaric, ferulic and caffeic) were quantitatively determined in 30 commercially available wines from South Moravia by gas chromatography-mass spectrometry. Raw (untransformed) and centered log-ratio transformed data were evaluated by classical and robust version of principal component analysis (PCA). A robust compositional biplot of the centered log-ratio transformed data gives the best resolution of particular categories of wines. Vanillic, syringic and gallic acids were identified as presumed markers occurring in relatively higher concentrations in red wines. Gentisic and caffeic acid were tentatively suggested as prospective technological markers, reflecting presumably some kinds of technological aspects of wine making.

Coffee aroma--statistical analysis of compositional data.

Solid-phase microextraction in headspace mode coupled with gas chromatography-mass spectrometry was applied to the determination of volatile compounds in 30 commercially available coffee samples. In order to differentiate and characterize Arabica and Robusta coffee, six major volatile compounds (acetic acid, 2-methylpyrazine, furfural, 2-furfuryl alcohol, 2,6-dimethylpyrazine, 5-methylfurfural) were chosen as the most relevant markers. Cluster analysis and principal component analysis (PCA) were applied to the raw chromatographic data and data processed by centred logratio transformation.