Inverse analysis in non-parametric multivariate analyses: distinguishing groups of associated species which covary coherently across samples

For decades multivariate analysis has been recognised as being appropriate for the analysis and description of complex ecological datasets, such as are routinely generated in studies of biota along gradients in time or space. The main focus of analyses tends to be the description and analysis of patterns among samples and groups of samples. Early applications of multivariate analyses to ecological data also recognised the importance of, and gave equal weight to, understanding how variables (species or taxa, in biotic datasets) varied among samples and groups of samples, but such analyses have inherent difficulties. Among these are the facts that species do not vary independently of each other, that responses of species to gradients may not be monotonic, that there are generally many more species than samples, that abundances vary widely within and among species, and that some species are rare. Although some methods are routinely applied to explore species responses across and among samples to environmental gradients, few explicitly recognise that species do not vary independently. Within a very widely-used framework for the nonparametric multivariate analysis of ecological data we demonstrate how Similarity Profiles (SIMPROF) analysis and other approaches may be combined to analyse associations among species and to visualise those relationships. Type 2 SIMPROF determines whether observed associations could have arisen by chance. Type 3 SIMPROF detects statistically distinct subsets of species which respond to gradients in a coherent manner. How different groups respond is visualised using component line plots (coherent curves). We illustrate the method using a range of datasets. We show how the method discriminates groups of species which respond differently to a single gradient, or respond coherently to different environmental or anthropogenic pressure gradients. We demonstrate how these approaches extend naturally to analyses of other types of multivariate data where the identification of coherent groups of variables is of interest. (C) 2013 Elsevier B.V. All rights reserved.