Compositional Data Analysis Methods Research Articles

A review of common methods used in the analysis of human microbiome sequencing data

The past two decades have seen a rapid rise in the development and use of sequencing technologies. The advent of high-throughput sequencing (HTS) has allowed scientists sequence the genomic content of entire microbial communities rapidly and affordably. Methods for data management, visualization, and analysis of this microbiome sequencing data are now widespread in open and closed source software tools. In this review, we identify common analysis methods used in microbiome sequencing studies, including methods for normalisation of abundance table data, visualization of high-dimensional microbiome data and the analysis of longitudinal microbiome sequencing data. In particular we find that a very small proportion of metagenomic studies made use of compositional data analysis methods and none that made use of compositional methods for longitudinal microbiome data.

Microbiome compositional data analysis for survival studies.

The growing interest in studying the relationship between the human microbiome and our health has also extended to time-to-event studies where researchers explore the connection between the microbiome and the occurrence of a specific event of interest. The analysis of microbiome obtained through high throughput sequencing techniques requires the use of specialized Compositional Data Analysis (CoDA) methods designed to accommodate its compositional nature. There is a limited availability of statistical tools for microbiome analysis that incorporate CoDA, and this is even more pronounced in the context of survival analysis. To fill this methodological gap, we present coda4microbiome for survival studies, a new methodology for the identification of microbial signatures in time-to-event studies. The algorithm implements an elastic-net penalized Cox regression model adapted to compositional covariates. We illustrate coda4microbiome algorithm for survival studies with a case study about the time to develop type 1 diabetes for non-obese diabetic mice. Our algorithm identified a bacterial signature composed of 21 genera associated with diabetes development. coda4microbiome for survival studies is integrated in the R package coda4microbiome as an extension of the existing functions for cross-sectional and longitudinal studies.

Comparison of the effectiveness of different normalization methods for metagenomic cross-study phenotype prediction under heterogeneity

The human microbiome, comprising microorganisms residing within and on the human body, plays a crucial role in various physiological processes and has been linked to numerous diseases. To analyze microbiome data, it is essential to account for inherent heterogeneity and variability across samples. Normalization methods have been proposed to mitigate these variations and enhance comparability. However, the performance of these methods in predicting binary phenotypes remains understudied. This study systematically evaluates different normalization methods in microbiome data analysis and their impact on disease prediction. Our findings highlight the strengths and limitations of scaling, compositional data analysis, transformation, and batch correction methods. Scaling methods like TMM show consistent performance, while compositional data analysis methods exhibit mixed results. Transformation methods, such as Blom and NPN, demonstrate promise in capturing complex associations. Batch correction methods, including BMC and Limma, consistently outperform other approaches. However, the influence of normalization methods is constrained by population effects, disease effects, and batch effects. These results provide insights for selecting appropriate normalization approaches in microbiome research, improving predictive models, and advancing personalized medicine. Future research should explore larger and more diverse datasets and develop tailored normalization strategies for microbiome data analysis.

Impacts of physical activity, sedentary behaviour, and sleep on depression symptoms in Canadian older adults 65 years of age and above: a compositional data analysis of the Canadian Longitudinal Study on Aging

BackgroundDaily time spent in moderate-to-vigorous physical activity (MVPA), light-intensity physical activity (LIPA), sedentary behaviour (SB) and sleep (collectively known as ‘movement behaviours’) are compositional, co-dependent variables. However, most studies examining effects of these behaviours on depression outcomes in older adults do not account for this. Study objectives were to use compositional data analysis methods to (1) examine the relationship between movement behaviour composition (daily time spent in MVPA, LIPA, SB, sleep) and depression symptoms, (2) estimate the extent to which changing time spent in any given movement behaviour within the movement behaviour composition was associated with changes in depression symptoms.Methods5643 older Canadian adults ≥ 65 years of age from the Canadian Longitudinal Study on Aging were studied using a quasi-longitudinal study design. Exposure was baseline daily movement behaviours; time spent in MVPA, LIPA and SB were derived from self-reported Physical Activity Scale for the Elderly responses. Night-time sleep was self-reported separately. Outcome was depression symptoms at follow-up obtained using the ten item Center for Epidemiologic Studies Depression Scale. Compositional data analysis was used to investigate associations between movement behaviours and depression symptoms.ResultsMovement behaviour composition was significantly associated with depression symptoms. Time spent in MVPA [exp(B) = 0.97 (95% CI: 0.94, 0.99)] and sleep [exp(B) = 0.91 (95% CI: 0.85, 0.97)] relative to the remaining movement behaviours were associated with lower depression symptoms scores. Relative time spent in SB was associated with higher depression symptoms scores [exp(B) = 1.09 (95% CI: 1.04, 1.15)]. Time displacement estimates revealed that the greatest change in depression symptoms scores occurred when time spent in MVPA was decreased and replaced with LIPA, SB, sleep or combination of these behaviours (+ 0.22 to 0.26 points increase on depression symptoms scores for 30 min/day displacements).ConclusionsDaily movement behaviour composition was associated with depression symptoms in older Canadians. Replacing time in MVPA with equivalent time from any other behaviour was associated with increased depression symptoms. Preserving time spent in MVPA may play a key role in mitigating and improving mental health in this demographic.

Auto encoder generative adversarial networks - based mineral prospectivity mapping in Lhasa area, Tibet

When conducting Mineral Potential Mapping (MPM) using multiple sources of data such as geology, geochemistry, and geophysics, it often encounters the challenges of complex and diverse data distributions within these datasets. Enhancing the capability to extract nonlinear data features and further uncover metallogenic information is a crucial research objective. This study utilizes the C-A multifractal approach to extract anomalous information related to metallogenic elements, employs compositional data analysis methods for quantitatively extracting geochemical associations of mineralization, and utilizes GIS spatial analysis techniques to quantitatively extract predictive indicators from various data sources, including geology, geophysics, and remote sensing, to construct an MPM prediction dataset. Building upon the foundation of AutoEncoder (AE), this study introduces a discriminator and employs the AEGAN (Auto Encoder Generative Adversarial Network) algorithm, which combines AutoEncoder and Generative Adversarial Network (GAN), for metallogenic prospectivity prediction in the Lhasa region. Compared to AE algorithms, AEGAN combines the strengths of AE and GAN, significantly improving the model's ability to reconstruct input data through the interaction between the generator and discriminator. Additionally, this study designs comparative experiments with AE, and the results demonstrate that the AEGAN model can more accurately identify the correlation between high anomaly areas and polymetallic deposits, providing a more precise delineation of anomalous extents. The Area Under the Receiver Operating Characteristic Curve (AUC) further validates the superior performance of the AEGAN model. These findings indicate that the AEGAN model exhibits outstanding capabilities in learning the internal connections and features among multiple data sources, holding significant potential for practical applications in mineral exploration.

Bayesian compositional regression with microbiome features via variational inference

The microbiome plays a key role in the health of the human body. Interest often lies in finding features of the microbiome, alongside other covariates, which are associated with a phenotype of interest. One important property of microbiome data, which is often overlooked, is its compositionality as it can only provide information about the relative abundance of its constituting components. Typically, these proportions vary by several orders of magnitude in datasets of high dimensions. To address these challenges we develop a Bayesian hierarchical linear log-contrast model which is estimated by mean field Monte-Carlo co-ordinate ascent variational inference (CAVI-MC) and easily scales to high dimensional data. We use novel priors which account for the large differences in scale and constrained parameter space associated with the compositional covariates. A reversible jump Monte Carlo Markov chain guided by the data through univariate approximations of the variational posterior probability of inclusion, with proposal parameters informed by approximating variational densities via auxiliary parameters, is used to estimate intractable marginal expectations. We demonstrate that our proposed Bayesian method performs favourably against existing frequentist state of the art compositional data analysis methods. We then apply the CAVI-MC to the analysis of real data exploring the relationship of the gut microbiome to body mass index.

Local Factors Impact Accuracy of Garlic Tissue Test Diagnosis

The low productivity of garlic in Brazil requires more efficient nutritional management. For this, environmental and fertilization-related factors must be adjusted to a set of local conditions. Our objective was to provide an accurate diagnosis of the nutrient status of garlic crops in southern Brazil. The dataset comprised 1024 observations, 962 as field tests conducted during the 2015–2017 period to train the model, and 61 field observations collected during the 2018–2019 period to validate the model. Machine learning models (MLM) related garlic yield to managerial, edaphic, plant, and climatic features. Compositional data analysis (CoDa) methods allowed classification of nutrients in the order of limitation to yield where MLM detected nutrient imbalance. Tissue analysis alone returned an accuracy of 0.750 in regression and 0.891 in classification about the yield cutoff of 11 ton ha−1. Adding all features documented in the dataset, accuracy reached 0.855 in regression and 0.912 in classification. Local diagnosis based on MLM and CoDa and accounting for local features differed from regional diagnosis across features. Local nutrient diagnosis may differ from regional diagnosis because several yield-impacting factors are taken into account and benchmark compositions are representative of local conditions.

Machine learning-based prediction of trace element concentrations using data from the Karoo large igneous province and its application in prospectivity mapping

In this study, we present a machine learning-based method to predict trace element concentrations from major and minor element concentration data using a legacy lithogeochemical database of magmatic rocks from the Karoo large igneous province (Gondwana Supercontinent). Wedemonstrate that a variety of trace elements, including most of the lanthanides, chalcophile, lithophile, and siderophile elements, can be predicted with excellent accuracy. This finding reveals that there are reliable, high-dimensional elemental associations that can be used to predict trace elements in a range of plutonic and volcanic rocks. Since the major and minor elements are used as predictors, prediction performance can be used as a direct proxy for geochemical anomalies. As such, our proposed method is suitable for prospective exploration by identifying anomalous trace element concentrations. Compared to multivariate compositional data analysis methods, the new method does not rely on assumptions of stoichiometric combinations of elements in the data to discover geochemical anomalies. Because we do not use multivariate compositional data analysis techniques (e.g. principal component analysis and combined use of major, minor and trace elements data), we also show that log-ratio transforms do not increase the performance of the proposed approach and are unnecessary for algorithms that are not spatially aware in the feature space. Therefore, we demonstrate that high-dimensional elemental associations can be modelled in an automated manner through a data-driven approach and without assumptions of stoichiometry within the data. The approach proposed in this study can be used as a replacement method to the multivariate compositional data analysis technique that is used for prospectivity mapping, or be used as a pre-processor to reduce the detection of false geochemical anomalies, particularly where the data is of variable quality. • Machine learning can predict trace element concentrations from major and minor element geochemical data. • Various machine learning algorithms can accurately predict trace element concentrations. • Log-ratio transformation of geochemical data is not always necessary when using machine learning. • Prediction residuals (difference between predicted and actual quantities) can be used to map geochemical anomalies.

Compositional Data Analysis of Periodontal Disease Microbial Communities

Periodontal disease (PD) is a chronic, progressive polymicrobial disease that induces a strong host immune response. Culture-independent methods, such as next-generation sequencing (NGS) of bacteria 16S amplicon and shotgun metagenomic libraries, have greatly expanded our understanding of PD biodiversity, identified novel PD microbial associations, and shown that PD biodiversity increases with pocket depth. NGS studies have also found PD communities to be highly host-specific in terms of both biodiversity and the response of microbial communities to periodontal treatment. As with most microbiome work, the majority of PD microbiome studies use standard data normalization procedures that do not account for the compositional nature of NGS microbiome data. Here, we apply recently developed compositional data analysis (CoDA) approaches and software tools to reanalyze multiomics (16S, metagenomics, and metabolomics) data generated from previously published periodontal disease studies. CoDA methods, such as centered log-ratio (clr) transformation, compensate for the compositional nature of these data, which can not only remove spurious correlations but also allows for the identification of novel associations between microbial features and disease conditions. We validated many of the studies’ original findings, but also identified new features associated with periodontal disease, including the genera Schwartzia and Aerococcus and the cytokine C-reactive protein (CRP). Furthermore, our network analysis revealed a lower connectivity among taxa in deeper periodontal pockets, potentially indicative of a more “random” microbiome. Our findings illustrate the utility of CoDA techniques in multiomics compositional data analysis of the oral microbiome.

Utilising CoDA methods for the spatio-temporal geochemical characterisation of groundwater; a case study from Lisheen Mine, south central Ireland

Lisheen Mine in County Tipperary, Ireland exploited an underground Pb/Zn massive sulphide deposit hosted in Carboniferous (Mississippian) carbonates. During the extraction phase, the mine workings (located at an average depth of 170 m below ground level), were continuously pumped to lower the groundwater level. Following mine closure in 2015, pumping ceased and eight groundwater wells in the surrounding area were sampled monthly over an 11-month period to monitor the effects of groundwater rebound. These wells draw water from the upper 30 m of a limestone/dolostone aquifer and the monthly samples were analysed for the concentration of 31 elements and compounds (SO4, Cl, NO3, F, NH4, NO2, P, Ca, Na, K, Mg, Fe, Mn, Cu, Zn, Pb, Al, Ni, Ba, As, Hg, B, Cr, Cd, Mo, Ag, Co, Sr, Be, Sb and U). All of the water can be described as Ca–HCO3 type as expected. Standard methods for analysing groundwater geochemistry data (e.g. piper diagrams etc.) are useful, differentiating groundwaters with first-order contrasting chemical signatures, for example, distinguishing Ca–HCO3-type from Na–HCO3-type water. Samples from the 8 monitoring wells appear to be broadly similar, using this approach. However, these major ion methods fail to further distinguish between different groundwaters. The use of multivariate statistical analytical techniques has become more common in groundwater studies in recent years, allowing the interaction of all the elements and compounds to be considered simultaneously. Compositional Data Analysis (CoDA) was used on the Lisheen dataset to gain a better understanding of the spatial and temporal variation in groundwater geochemistry. Ilr-ion plotting, Hierarchical Cluster Analysis (HCA) and Principal Component Analysis (PCA) through CoDA highlights the elements and compounds that account for the majority of the variance and at Lisheen these are nitrate, manganese, ammonium, sulphate and potassium. By displaying these data visually in a CoDA bi-plot, each location can be reliably ‘geochemically fingerprinted’ despite similar concentrations of major ions within a relatively small geographical area (<30 km2). Relabeling the bi-plot observations by date of recovery reveals how one particular groundwater well (PH) subtly varies over time, most likely as a result of seasonal land-use changes (input of compounds associated with fertiliser). This type of statistical analysis has broad applications in hydrology and hydrogeology including contaminant tracing and interaction, environmental studies, land-use planning and mineral exploration.

Editorial: Compositional data analysis and related methods applied to genomics—a first special issue fromNAR Genomics and Bioinformatics

Editorial: Compositional data analysis and related methods applied to genomics—a first special issue from<i>NAR Genomics and Bioinformatics</i>

Nutrient Diagnosis of Fertigated "Prata" and "Cavendish" Banana (Musa spp.) at Plot-Scale.

Fertigation management of banana plantations at a plot scale is expanding rapidly in Brazil. To guide nutrient management at such a small scale, genetic, environmental and managerial features should be well understood. Machine learning and compositional data analysis (CoDa) methods can measure the effects of feature combinations on banana yield and rank nutrients in the order of their limitation. Our objectives are to review ML and CoDa models for application at regional and local scales, and to customize nutrient diagnoses of fertigated banana at the plot scale. We documented 940 “Prata” and “Cavendish” plot units for tissue and soil tests, environmental and managerial features, and fruit yield. A Neural Network informed by soil tests, tissue tests and other features was the most proficient learner (AUC up to 0.827). Tissue nutrients were shown to have the greatest impact on model accuracy. Regional nutrient standards were elaborated as centered log ratio means and standard deviations of high-yield and nutritionally balanced specimens. Plot-scale diagnosis was customized using the closest successful factor-specific tissue compositions identified by the smallest Euclidean distance from the diagnosed composition using centered or isometric log ratios. Nutrient imbalance differed between regional and plot-scale diagnoses, indicating the profound influence of local factors on plant nutrition. However, plot-scale diagnoses require large, reliable datasets to customize nutrient management using ML and CoDa models.

Deep in the Bowel: Highly Interpretable Neural Encoder-Decoder Networks Predict Gut Metabolites from Gut Microbiome

BackgroundTechnological advances in next-generation sequencing (NGS) and chromatographic assays [e.g., liquid chromatography mass spectrometry (LC-MS)] have made it possible to identify thousands of microbe and metabolite species, and to measure their relative abundance. In this paper, we propose a sparse neural encoder-decoder network to predict metabolite abundances from microbe abundances.ResultsUsing paired data from a cohort of inflammatory bowel disease (IBD) patients, we show that our neural encoder-decoder model outperforms linear univariate and multivariate methods in terms of accuracy, sparsity, and stability. Importantly, we show that our neural encoder-decoder model is not simply a black box designed to maximize predictive accuracy. Rather, the network’s hidden layer (i.e., the latent space, comprised only of sparsely weighted microbe counts) actually captures key microbe-metabolite relationships that are themselves clinically meaningful. Although this hidden layer is learned without any knowledge of the patient’s diagnosis, we show that the learned latent features are structured in a way that predicts IBD and treatment status with high accuracy.ConclusionsBy imposing a non-negative weights constraint, the network becomes a directed graph where each downstream node is interpretable as the additive combination of the upstream nodes. Here, the middle layer comprises distinct microbe-metabolite axes that relate key microbial biomarkers with metabolite biomarkers. By pre-processing the microbiome and metabolome data using compositional data analysis methods, we ensure that our proposed multi-omics workflow will generalize to any pair of -omics data. To the best of our knowledge, this work is the first application of neural encoder-decoders for the interpretable integration of multi-omics biological data.

Counts: an outstanding challenge for log-ratio analysis of compositional data in the molecular biosciences.

Thanks to sequencing technology, modern molecular bioscience datasets are often compositions of counts, e.g. counts of amplicons, mRNAs, etc. While there is growing appreciation that compositional data need special analysis and interpretation, less well understood is the discrete nature of these count compositions (or, as we call them, lattice compositions) and the impact this has on statistical analysis, particularly log-ratio analysis (LRA) of pairwise association. While LRA methods are scale-invariant, count compositional data are not; consequently, the conclusions we draw from LRA of lattice compositions depend on the scale of counts involved. We know that additive variation affects the relative abundance of small counts more than large counts; here we show that additive (quantization) variation comes from the discrete nature of count data itself, as well as (biological) variation in the system under study and (technical) variation from measurement and analysis processes. Variation due to quantization is inevitable, but its impact on conclusions depends on the underlying scale and distribution of counts. We illustrate the different distributions of real molecular bioscience data from different experimental settings to show why it is vital to understand the distributional characteristics of count data before applying and drawing conclusions from compositional data analysis methods.

Variable selection in microbiome compositional data analysis.

Though variable selection is one of the most relevant tasks in microbiome analysis, e.g. for the identification of microbial signatures, many studies still rely on methods that ignore the compositional nature of microbiome data. The applicability of compositional data analysis methods has been hampered by the availability of software and the difficulty in interpreting their results. This work is focused on three methods for variable selection that acknowledge the compositional structure of microbiome data: selbal, a forward selection approach for the identification of compositional balances, and clr-lasso and coda-lasso, two penalized regression models for compositional data analysis. This study highlights the link between these methods and brings out some limitations of the centered log-ratio transformation for variable selection. In particular, the fact that it is not subcompositionally consistent makes the microbial signatures obtained from clr-lasso not readily transferable. Coda-lasso is computationally efficient and suitable when the focus is the identification of the most associated microbial taxa. Selbal stands out when the goal is to obtain a parsimonious model with optimal prediction performance, but it is computationally greedy. We provide a reproducible vignette for the application of these methods that will enable researchers to fully leverage their potential in microbiome studies.

The Quality of Urban Air in Barcelona: A New Approach Applying Compositional Data Analysis Methods

The main goal of this paper is to go some steps further to improve the understanding and manageability of air quality. Quality of atmospheric air in large cities is a matter of great importance because of its impact on the environment and on the health of the population. Recently, measures restricting access of private vehicles to the centre of large cities and other measures to prevent atmospheric air pollution are currently topical. The knowledge of air quality acquires special relevance to be able to evaluate the impact of those great social and economic measures. There are many indices to express air quality. In fact, quite every country has its own, depending on the main pollutants. In general, all indices ignore the compositional nature of the concentrations of air pollutants and do not apply methods of Compositional Data Analysis and have some other weak points such as leak of standardized scale. Therefore, the methodology used is founded on Compositional Data Analysis. The air quality index has an adequate correlation between input (concentrations) and output (air quality index), it distinguishes between air pollution and air quality and it has a 0-100 reference scale which makes easier interpretation and management of air quality expression. To illustrate the proposed method, an application is made to a series of air pollution data (Barcelona, 2001-2015). The results show the effectiveness of the 2008 European directive on ambient air quality.

Abstract P343: Associations of Adipose Tissue Distribution With Hypertension and Diabetes in African Ancestry Men: A Novel Approach Using Compositional Data Analysis

Background: Adipose tissue (AT) distribution, which differs by race/ethnicity, can differentially impact risk for cardiometabolic disease. Traditional analyses using AT volumes tend to place all ATs in a regression model, estimating an increase in one AT volume while “holding other volumes constant”; however, the compositional nature of the data suggests the other ATs cannot remain constant. Compositional data analysis (CODA) methods eliminate this concern by comparing logratios of components. CODA was designed for analyses of data like those of body composition , but has yet to be used in this field. We demonstrate the use of CODA for the comparison of visceral AT (VAT) and subcutaneous AT of the abdomen (ASAT) and thigh (TSAT) with prevalence of cardiometabolic diseases. Methods: This cross-sectional analysis used 615 African Caribbean men from Tobago aged 50-91 years (mean age 63.6 years, mean BMI 27.7 kg/m 2 ). AT volumes were obtained from CT scans of the abdomen (VAT and ASAT) and of the mid-thigh (TSAT). An additive log2 ratio transformation was applied to the ATs to generate the VAT, ASAT, and TSAT component effects relative to the other tissues captured in the respective CT scan. Ordinal logistic regression analyses were performed for increasing hypertension or glucose impairment categories (defined in Table) after adjustment for age, disease status, and lifestyle factors. Results: After full adjustment (Table), a 2-fold higher ASAT volume (relative to other abdominal tissues) was associated with higher odds, and a 2-fold higher TSAT volume (relative to other thigh tissues) was associated with lower odds, of being in a higher hypertension category. After BMI adjustment, no AT was associated with glucose impairment category. VAT was not a significant predictor for either outcome in African Ancestry men. Conclusion: CODA methods, which avoid problematic assumptions made in traditional analyses, found significant and opposing associations of TSAT and ASAT with hypertension categories in African Ancestry men.

Different and diverse anaerobic microbiota were seen in women living with HIV with unsuppressed HIV viral load and in women with recurrent bacterial vaginosis: a cohort study.

To compare the vaginal microbiota of women living with HIV (WLWH) with the vaginal microbiota of women with recurrent bacterial vaginosis (BV) and healthy women without HIV to determine if there are differences in the vaginal microbiome, what factors influence these differences, and to characterise HIV clinical parameters including viral load and CD4 count in relation to the vaginal microbiome. Observational cohort study. Canada. Women aged 18-49years who were premenopausal and not pregnant were recruited into three cohorts: healthy women, WLWH and women with recurrent BV. Demographic and clinical data were collected via interviews and medical chart reviews. Vaginal swabs were collected for Gram-stain assessment and microbiome profiling using the cpn60 barcode sequence. To compare overall community composition differences, we used compositional data analysis methods, hierarchical clustering and Kruskal-Wallis tests where appropriate. Clinical markers such as odour and abnormal discharge, but not irritation, were associated with higher microbial diversity. WLWH with unsuppressed HIV viral loads were more likely than other groups to have non-Gardnerella-dominated microbiomes. HIV was associated with higher vaginal microbial diversity and this was related to HIV viral load, with unsuppressed women demonstrating significantly higher relative abundance of Megasphaera genomosp. 1, Atopobium vaginae and Clostridiales sp. (all P<0.05) compared with all other groups. In WLWH, unsuppressed HIV viral loads were associated with a distinct dysbiotic profile consisting of very low levels of Lactobacillus and high levels of anaerobes. Vaginal microbiomes in WLWH with viral load >50copies/ml have distinct dysbiotic profiles with high levels of anaerobes.

Comparative geochemical study between the tap waters and the bottled mineral waters in Calabria (Southern Italy) by compositional data analysis (CoDA) developments

This work focuses on comparing the inorganic content of tap and bottled mineral waters in the Calabria region (Southern Italy) performed by using robust Compositional Data Analysis (CoDA) methods and related new developments associated with a classical graphical-numerical approach.Thirty-one samples of tap waters scattered throughout the Calabrian territory and twenty-one bottled natural mineral waters of various Calabrian brands were analyzed for major and trace inorganic components. In addition, a very sensitive method based on cold vapour atomic fluorescence spectrometry (CV-AFS) has been employed to determine total mercury (HgT) concentrations.Compositional data analysis was used to identify the main interrelationships among physico-chemical parameters, highlighting the differences between the tap waters and the bottled mineral waters. Some elements, such as Al, Mn, Fe, Zn, and Pb resulted particularly enriched in the tap waters. This group of enriched elements was probably controlled by corrosion processes from metal pipes, asking for more monitoring on the state of the public distribution water network. Li, Ca, Cr, Ni, and Cd, on the other hand, appear to be mostly contributed by natural sources affecting the overall variability of the bottled mineral waters. In this context, Cr, Cu, Hg, and Pb are mainly enriched in the sparkling bottled mineral waters.Innovative compositional indices did not highlight great differences from the Maximum Allowed Concentrations (MAC) in the two types of water, so that for the health of the consumer it is indifferent to drink tap or bottled mineral water. The results represent a fundamental base to develop monitoring plans in time to check for the maintenance of quality standards.

A comparison of generalised linear models and compositional models for ordered categorical data

Ordered categorical data occur in many applied fields, such as geochemistry, econometrics, sociology and demography or even transportation research, for example, in the form of results from various questionnaires. There are different possibilities for modelling proportions of individual categories. Generalised linear models (GLMs) are traditionally used for this purpose, but also methods of compositional data analysis (CoDa) can be considered. Here, both approaches are compared in depth. Particularly, different assumptions of the models on variability are highlighted. Advantages and disadvantages of individual models are pointed out. While the CoDa model may be inappropriate when the variability of the compositional coordinates depends on the regressors, for example, due to different total counts on which the coordinates are based, the GLM may underestimate the uncertainty of the predictions considerably in case of large-scale data.