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Several papers have recently raised the occurrence of some problems with between-group Principal Component Analysis (bgPCA). This method inflates the differences between the groups, and can even display completely artificial differences when none exist, for example when applied to random numbers tables with many variables (columns) and few individuals (rows). Lately, cross-validation has been proposed as a way to circumvent this problem. Here we present some tools and several functions of the ade4 package for the R statistical software to compute a bgPCA, test the presence of statistically significant groups, perform a cross-validation of this analysis and compute associated statistics. We also describe how to use these functions to avoid running into the spurious groups problem. Several examples, including a real data set and random numbers tables, are used to validate this approach in various experimental and numerical conditions. The integrated framework of the duality diagram, as implemented in ade4, allows to extend this approach to other multivariate analysis methods beyond principal component analysis, which could prove useful in the case of other types of variables. The R code and the real data table used to make the computations and graphs of this paper are available as supplementary material. [ABSTRACT FROM AUTHOR]

Microbial communities, which drive major ecosystem functions, consist of a wide range of interacting species. Understanding how microbial communities are structured and the processes underlying this is crucial to interpreting ecosystem responses to global change but is challenging as microbial interactions cannot usually be directly observed. Multiple efforts are currently focused to combine next‐generation sequencing (NGS) techniques with refined statistical analysis (e.g., network analysis, multivariate analysis) to characterize the structures of microbial communities. However, most of these approaches consider a single table of sequencing data measured for several samples. Technological advances now make it possible to collect NGS data on different taxonomic groups simultaneously for the same samples, allowing us to analyse a pair of tables. Here, an analytical framework based on co‐correspondence analysis (CoCA) is proposed to study the distributions, assemblages and interactions between two microbial communities. We show the ability of this approach to highlight the relationships between two microbial communities, using two data sets exhibiting various types of interactions. CoCA identified strong association patterns between autotrophic and heterotrophic microbial eukaryote assemblages, on the one hand, and between microalgae and viruses, on the other. We demonstrate also how CoCA can be used, complementary to network analysis, to reorder co‐occurrence networks and thus investigate the presence of patterns in ecological networks. [ABSTRACT FROM AUTHOR]

Correspondence analysis with linear external constraints on both the rows and the columns has been mentioned in the ecological literature, but lacks full mathematical treatment and easily available algorithms and software. This paper fills this gap by defining the method as maximizing the fourth-corner correlation between linear combinations, by providing novel algorithms, which demonstrate relationships with related methods, and by making a detailed study of possible biplots and associated approximations. The method is illustrated using ecological data on the abundances of species in sites and where the species are characterized by traits and sites by environmental variables. The trait data and environment data form the external constraints and the question is which traits and environmental variables are associated, how these associations drive species abundances and how they can be displayed in biplots. With microbiome data becoming widely available, these and related multivariate methods deserve more study as they might be routinely used in the future. [ABSTRACT FROM AUTHOR]

The ade4 package provides tools for multivariate analyses. Whereas new statistical methods have been added regularly in the package since its first release in 2002, the graphical functions, that are used to display the main outputs of an analysis, have not benefited from such enhancements. In this context, the adegraphics package, available on CRAN since 2015, is a complete reimplementation of the ade4 graphical functionalities but with large improvements. The package uses the S4 object system (each graph is an object) and is based on the graphical framework provided by lattice and grid. We give a brief description of the package and illustrate some important functionalities to build elegant graphs. [ABSTRACT FROM AUTHOR]

The spatial structuring of populations or communities is an important driver of their functioning and their influence on ecosystems. Identifying the (in)stability of the spatial structure of populations is a first step towards understanding the underlying causes of these structures. Here we studied the relative importance of spatial vs. interannual variability in explaining the patterns of abundance of a large herbivore community (8 species) at waterholes in Hwange National Park (Zimbabwe). We analyzed census data collected over 13 years using multivariate methods. Our results showed that variability in the census data was mostly explained by the spatial structure of the community, as some waterholes had consistently greater herbivore abundance than others. Some temporal variability probably linked to Park-scale migration dependent on annual rainfall was noticeable, however. Once this was accounted for, little temporal variability remained to be explained, suggesting that other factors affecting herbivore abundance over time had a negligible effect at the scale of the study. The extent of spatial and temporal variability in census data was also measured for each species. This study could help in projecting the consequences of surface water management, and more generally presents a methodological framework to simultaneously address the relative importance of spatial vs. temporal effects in driving the distribution of organisms across landscapes. [ABSTRACT FROM AUTHOR]

Assessing trait responses to environmental gradients requires the simultaneous analysis of the information contained in three tables: L (species distribution across samples), R (environmental characteristics of samples), and Q (species traits). Among the available methods, the so-called fourth-corner and RLQ methods are two appealing alternatives that provide a direct way to test and estimate trait-environment relationships. Both methods are based on the analysis of the fourth-corner matrix, which crosses traits and environmental variables weighted by species abundances. However, they differ greatly in their outputs: RLQ is a multivariate technique that provides ordination scores to summarize the joint structure among the three tables, whereas the fourth-corner method mainly tests for individual traitenvironment relationships (i.e., one trait and one environmental variable at a time). Here, we illustrate how the complementarity between these two methods can be exploited to promote new ecological knowledge and to improve the study of trait-environment relationships. After a short description of each method, we apply them to real ecological data to present their different outputs and provide hints about the gain resulting from their combined use. [ABSTRACT FROM AUTHOR]

Question How do multivariate methods perform in relating species- and community-level trait responses to the environment? Location (1) Field data from grazed semi-natural grasslands, NE Germany; (2) artificial data. Methods Research questions associated with trait-environment relationships were briefly reviewed and seven available methods evaluated. The main distinction between research questions is whether trait-environment relationships should be addressed at community or species level. A redundancy analysis ( RDA) of mean trait values of species in a plot weighted by their abundances ( CWM- RDA) is exclusively suitable for the community level. The other six methods address the species level. A double inertia analysis of two arrays ( RLQ) and double canonical correspondence analysis (double CCA) use combinations of ordinations to simultaneously analyse species and trait responses to the environment. A combination of the outlying mean index with generalized additive models ( OMI- GAM) predicts the response of species to environmental variables on trait gradients. RDA-RegTree first analyses species responses to the environment with RDA and then uses a regression tree to classify trait expressions according to scores of species responses on the ordination axes. Cluster regression uses cluster analyses and logistic regression to search for trait combinations with the best response to the environmental variables. This method models the distribution of functional groups on environmental gradients. All methods and data are available as R scripts. Results All methods consistently revealed the main trait responses to environment in the field data set, namely that life history was associated with available phosphorus while grazing intensity was related to leaf C: N ratio and canopy height. At community level, CWM- RDA gave a good overview of trait-environment relationships, as also provided by the species-based methods RLQ and double CCA. OMI- GAM revealed non-linear relationships in the field data set. Field and artificial data gave that the number and stability of functional groups produced by Cluster regression and RDA-RegTree varied more strongly than RLQ, double CCA and OMI- GAM. Conclusions Each method addresses particular ecological concepts and research questions. If a user asks for the response of average trait expressions of communities to environmental gradients, CWM- RDA may be the first choice. However, species-based methods should be applied to address questions regarding co-existence of different life histories or to assess how groups of species respond to environmental changes. The artificial data set revealed that the methods differed in sensitivity to gradient lengths and random data. [ABSTRACT FROM AUTHOR]

The identification of spatial structures is a key step in understanding the ecological processes structuring the distribution of organisms. Spatial patterns in species distributions result from a combination of several processes occuring at different scales: identifying these scales is thus a crucial issue. Recent studies have proposed a new family of spatial predictors (PCNM: principal coordinates of neighbours matrices; MEMs: Moran's eigenvectors maps) that allow for modelling of spatial variation on different scales. To assess the multi-scale spatial patterns in multivariate data, these variables are often used as predictors in constrained ordination methods. However, the selection of the appropriate spatial predictors is still troublesome, and the identification of the main scales of spatial variation remains an open question. This paper presents a new statistical tool to tackle this issue: the multi-scale pattern analysis (MSPA). This ordination method uses MEMs to decompose ecological variability into several spatial scales and then summarizes this decomposition using graphical representations. A canonical form of MSPA can also be used to assess the spatial scales of the species-environment relationships. MSPA is compared to constrained ordination using simulated data, and illustrated using the famous oribatid mites dataset. The method is implemented in the free software R. [ABSTRACT FROM AUTHOR]

Question: Are there spatial structures in the composition of plant communities? Methods: Identification and measurement of spatial structures is a topic of great interest in plant ecology. Univariate measurements of spatial autocorrelation such as Moran's I and Geary's c are widely used, but extensions to the multivariate case (i.e. multi-species) are rare. Here, we propose a multivariate spatial analysis based on Moran's I (MULTISPATI) by introducing a row-sum standardized spatial weight matrix in the statistical triplet notation. This analysis, which is a generalization of Wartenberg's approach to multivariate spatial correlation, would imply a compromise between the relations among many variables (multivariate analysis) and their spatial structure (autocorrelation). MULTISPATI approach is very flexible and can handle various kinds of data (quantitative and/or qualitative data, contingency tables). A study is presented to illustrate the method using a spatial version of Correspondence Analysis. Location: Territoire d'Etude et d'Expérimentation de Trois-Fontaines (eastern France). Results: Ordination of vegetation plots by this spatial analysis is quite robust with reference to rare species and highlights spatial patterns related to soil properties. [ABSTRACT FROM AUTHOR]

Abstract Classical multivariate analyses are based on matrix algebra and enable the analysis of a table containing measurements of a set of variables for a set of sites. Incomplete mapped data consist of measurements of a set of variables recorded for the same geographical region but for different zonal systems and with only a partial sampling of this zone. This kind of data cannot be analysed with usual multivariate methods because there is no common system of sites for all variables. We propose a new approach using GIS technology and NIPALS, an iterative multivariate method, to analyse the spatial patterns of this kind of data. Moreover, an extension of our method is that it can be used for areal interpolation purposes. We illustrate the method in analysing data concerning the distribution of roe deer weights over several years in a reserve. [ABSTRACT FROM AUTHOR]