'Equivalent numbers' for species, phylogenetic or functional diversity in a nested hierarchy of multiple scales

1. Many recent studies have searched to integrate species' functions and phylogenies in the measurement of biodiversity. To obtain easily interpretable measures, some researchers recommended diversity indices expressed in terms of equivalent numbers of species: the number of equally likely and maximally dissimilar species needed to produce the given value of diversity. Then, biodiversity is often calculated at three scales: within communities (alpha diversity), among communities (beta diversity) and in a region (gamma diversity). These three scales are, however, insufficient to tackle the organization of biodiversity in space because, for most organisms, there is a nested hierarchy of multiple scales characterized by different patterns and processes, from the small neighbourhood to the biosphere. 2. We developed methodologies for analysing species, functional, taxonomic or phylogenetic diversity in a hierarchy of multiple scales using equivalent numbers of species. As an example, we analysed the taxonomic and functional diversity of macroinvertebrate assemblages in the Loire River, France, at four levels: within sites (alpha diversity), among sites within geological regions (beta(1) diversity), among geological regions (beta(2) diversity) and at the river scale (gamma diversity). The new hierarchical approaches of biodiversity revealed very low differences among sites within regions and among regions in terms of taxonomy and functional traits (size and diet), despite moderate, significant species turnover among geological regions. 3. We compare our framework with those other authors have developed. We argue that different definitions of alpha, beta, gamma diversities are used in the literature reflecting different points of view on biodiversity. We make recommendations on how to normalize functional (or phylogenetic) dissimilarities among species to render sites and regions comparable, and discuss the pros and cons of our approach. 4. The hierarchical approaches of biodiversity in terms of 'equivalent numbers' respond to current demands to obtain intuitive, easily interpretable components of biodiversity. The approaches we propose go beyond current developments by considering a hierarchy of spatial scales and unbalanced sampling design. They will provide powerful tools to detect the ecological and evolutionary processes that act differently at different scales.